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M. Ajayan, L. S. Schadler, P. V. Braun Nanocomposite Science and Technology + +# Related Titles from Wiley-VCH + +# Colloids and Colloid Assemblies + +2003, ISBN 3-527-30660-9 + +Decher, G., Schlenoff, J.B. + +Multilayer Thin Films Sequential Assembly of Nanocomposite Materials + +2003, ISBN 3-527-30440-1 + +Komiyama, M., Takeuchi, T., Mukawa, T., Asanuma, H. + +Molecular Imprinting From Fundamentals to Applications + +2003, ISBN 3-527-30569-6 + +Krenkel, W. + +High Temperature Ceramic Matrix Composites + +2001, ISBN 3-527-30320-0 + +Ko ¨hler, M., Fritzsche, W. + +2003, ISBN 3-527-30484-3 Nanotechnology 2004, ISBN 3-527-30750-8 + +P. M. Ajayan, L. S. Schadler, P. V. Braun + +# Nanocomposite Science and Technology + +# + +Dept. of Materials Science and Engineering Rensselaer Polytechnic Institute Troy, NY 12180-3590 USA + +Linda S. Schadler Dept. of Materials Science and Engineering Rensselaer Polytechnic Institute Troy, NY 12180-3590 USA + +# + +Dept. of Materials Science and Engineering University of Illinois at Urbana-Champaign Urbana, IL 61801 USA + +This book was carefully produced. Nevertheless, authors and publisher do not warrant the infor- mation contained therein to be free of errors. Readers are advised to keep in mind that state- ments, data, illustrations, procedural details or other items may inadvertently be inaccurate. + +Library of Congress Card No.: Applied for. + +British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. + +Die Deutsche Bibliothek – CIP Cataloguing-in-Publication-Data Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de + +$\copyright$ 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim + +All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. + +Printed in the Federal Republic of Germany + +Printed on acid-free paper + +Composition Mitterweger & Partner, Plankstadt Printing Strauss Offsetdruck GmbH, Mo ¨rlenbach Bookbinding Litges & Dopf Buchbinderei GmbH, Heppenheim Cover Design: Gunter Schulz, Fußgo ¨nheim + +ISBN 3-527-30359-6 + +# Contents + +1 Bulk Metal and Ceramics Nanocomposites 1 + +Pulickel M. Ajayan + + 1.1 Introduction 1 + + 1.2 Ceramic/Metal Nanocomposites 3 + + 1.2.1 Nanocomposites by Mechanical Alloying 6 + + 1.2.2 Nanocomposites from SolGel Synthesis 8 + + 1.2.3 Nanocomposites by Thermal Spray Synthesis 11 + + 1.3 Metal Matrix Nanocomposites 14 + + 1.4 Bulk Ceramic Nanocomposites for Desired Mechanical Properties 18 + + 1.5 Thin-Film Nanocomposites: Multilayer and Granular Films 23 + + 1.6 Nanocomposites for Hard Coatings 24 + + 1.7 Carbon Nanotube-Based Nanocomposites 31 + + 1.8 Functional Low-Dimensional Nanocomposites 35 + + 1.8.1 Encapsulated Composite Nanosystems 36 + + 1.8.2 Applications of Nanocomposite Wires 44 + + 1.8.3 Applications of Nanocomposite Particles 45 + + 1.9 Inorganic Nanocomposites for Optical Applications 46 + + 1.10 Inorganic Nanocomposites for Electrical Applications 49 + + 1.11 Nanoporous Structures and Membranes: Other Nanocomposites 53 + + 1.12 Nanocomposites for Magnetic Applications 57 + + 1.12.1 Particle-Dispersed Magnetic Nanocomposites 57 + + 1.12.2 Magnetic Multilayer Nanocomposites 59 + + 1.12.2.1 Microstructure and Thermal Stability of Layered Magnetic Nanocomposites 59 + + 1.12.2.2 Media Materials 61 + + 1.13 Nanocomposite Structures having Miscellaneous Properties 64 + + 1.14 Concluding Remarks on Metal/Ceramic Nanocomposites 69 + +2 Polymer-based and Polymer-filled Nanocomposites 77 Linda S. Schadler + + 2.1 Introduction 77 + + 2.2 Nanoscale Fillers 80 + + 2.2.1 Nanofiber or Nanotube Fillers 80 + + 2.2.1.1 Carbon Nanotubes 80 + + 2.2.1.2 Nanotube Processing 85 + + 2.2.1.3 Purity 88 + + 2.2.1.4 Other Nanotubes 89 + + 2.2.2 Plate-like Nanofillers 90 + + 2.2.3 Equi-axed Nanoparticle Fillers 93 + + 2.3 Inorganic FillerPolymer Interfaces 96 + + 2.4 Processing of Polymer Nanocomposites 100 + + 2.4.1 Nanotube/Polymer Composites 100 + + 2.4.2 Layered FillerPolymer Composite Processing 103 + + 2.4.2.1 Polyamide Matrices 107 + + 2.4.2.2 Polyimide Matrices 107 + + 2.4.2.3 Polypropylene and Polyethylene Matrices 108 + + 2.4.2.4 Liquid-Crystal Matrices 108 + + 2.4.2.5 Poly methyl meth a cry late/Polystyrene Matrices 108 + + 2.4.2.6 Epoxy and Polyurethane Matrices 109 + + 2.4.2.7 Polyelectrolyte Matrices 110 + + 2.4.2.8 Rubber Matrices 110 + + 2.4.2.9 Others 111 + + 2.4.3 Nanoparticle/Polymer Composite Processing 111 + + 2.4.3.1 Direct Mixing 111 + + 2.4.3.2 Solution Mixing 112 + + 2.4.3.3 In-Situ Polymerization 112 + + 2.4.3.4 In-Situ Particle Processing Ceramic/Polymer Composites 112 + + 2.4.3.5 In-Situ Particle Processing Metal/Polymer Nanocomposites 114 + + 2.4.4 Modification of Interfaces 117 + + 2.4.4.1 Modification of Nanotubes 117 + + 2.4.4.2 Modification of Equi-axed Nanoparticles 118 + + 2.4.4.3 Small-Molecule Attachment 118 + + 2.4.4.4 Polymer Coatings 119 + + 2.4.4.5 Inorganic Coatings 121 + + 2.5 Properties of Composites 122 + + 2.5.1 Mechanical Properties 122 + + 2.5.1.1 Modulus and the Load-Carrying Capability of Nanofillers 122 + + 2.5.1.2 Failure Stress and Strain Toughness 127 + + 2.5.1.3 Glass Transition and Relaxation Behavior 131 + + 2.5.1.4 Abrasion and Wear Resistance 132 + + 2.5.2 Permeability 133 + + 2.5.3 Dimensional Stability 135 + +2.5.4 Thermal Stability and Flammability 136 + + 2.5.5 Electrical and Optical Properties 138 + + 2.5.5.1 Resistivity, Permittivity, and Breakdown Strength 138 + + 2.5.5.2 Optical Clarity 140 + + 2.5.5.3 Refractive Index Control 141 + + 2.5.5.4 Light-Emitting Devices 141 + + 2.5.5.5 Other Optical Activity 142 + + 2.6 Summary 144 + +3 Natural Nano bio composites, Biomimetic Nanocomposites, and Biologically Inspired Nanocomposites 155 Paul V. Braun + +3.1 Introduction 155 + + 3.2 Natural Nanocomposite Materials 157 + + 3.2.1 Biologically Synthesized Nanoparticles 159 + + 3.2.2 Biologically Synthesized Nanostructures 160 + + 3.3 Biologically Derived Synthetic Nanocomposites 165 + + 3.3.1 Protein-Based Nanostructure Formation 165 + + 3.3.2 DNA-Templated Nanostructure Formation 167 + + 3.3.3 Protein Assembly 169 + + 3.4 Biologically Inspired Nanocomposites 171 + + 3.4.1 Lyotropic Liquid-Crystal Templating 178 + + 3.4.2 Liquid-Crystal Templating of Thin Films 194 + + 3.4.3 Block-Copolymer Templating 195 + + 3.4.4 Colloidal Templating 197 + + 3.5 Summary 207 + +4 Modeling of Nanocomposites 215 + +4.1 Introduction The Need For Modeling 215 + + 4.2 Current Conceptual Frameworks 216 + + 4.3 Multiscale Modeling 217 + + 4.4 Multiphysics Aspects 220 + + 4.5 Validation 221 + +Index 223 + +# Preface + +The field of nanocomposites involves the study of multiphase material where at least one of the constituent phases has one dimension less than $100\;\mathrm{nm}$ . The promise of nanocomposites lies in their multi functionality, the possibility of realizing unique combinations of properties unachievable with traditional materials. The challenges in reaching this promise are tremendous. They include control over the distribution in size and dispersion of the nanosize constituents, tailoring and understanding the role of interfaces between structurally or chemically dissimilar phases on bulk proper- ties. Large scale and controlled processing of many nanomaterials has yet to be achieved. Our mentor as we make progress down this road is mother Nature and her quintessential nanocomposite structures, for example, bone. + +We realize that a book on a subject of such wide scope is a challenging endeavour. The recent explosion of research in this area introduces another practical limitation. What is written here should be read from the perspective of a dynamic and emerging field of science and technology. Rather than covering the entire spectrum of nanocom- posite science and technology, we have picked three areas that provide the basic con- cepts and generic examples that define the overall nature of the field. In the first chap- ter we discuss nanocomposites based on inorganic materials and their applications. In the second chapter polymer based nanoparticle filled composites are detailed with an emphasis on interface engineering to obtain nanocomposites with optimum perform- ance. The third chapter is about naturally occurring systems of nanocomposites and current steps towards naturally inspired synthetic nanocomposites. Finally a short chapter contributed by our colleagues highlights the possibility of using theoretical models and simulations for understanding nanocomposite properties. We hope our readers will find the book of value to further their research interests in this fas- cinating and fast evolving area of nanocomposites. + +Troy, July 2003 P. M. Ajayan, L. S. Schadler and P. V. Braun + +1 + +# Bulk Metal and Ceramics Nanocomposites + +Pulickel Ajayan + +# 1.1 Introduction + +The field of nanocomposite materials has had the attention, imagination, and close scrutiny of scientists and engineers in recent years. This scrutiny results from the simple premise that using building blocks with dimensions in the nanosize range makes it possible to design and create new materials with unprecedented flexibility and improvements in their physical properties. This ability to tailor composites by using nanosize building blocks of heterogeneous chemical species has been demon- strated in several interdisciplinary fields. The most convincing examples of such de- signs are naturally occurring structures such as bone, which is a hierarchical nano- composite built from ceramic tablets and organic binders. Because the constituents of a nanocomposite have different structures and compositions and hence properties, they serve various functions. Thus, the materials built from them can be multifunc- tional. Taking some clues from nature and based on the demands that emerging tech- nologies put on building new materials that can satisfy several functions at the same time for many applications, scientists have been devising synthetic strategies for pro- ducing nanocomposites. These strategies have clear advantages over those used to produce homogeneous large-grained materials. Behind the push for nanocomposites is the fact that they offer useful new properties compared to conventional materials. + +The concept of enhancing properties and improving characteristics of materials through the creation of multiple-phase nanocomposites is not recent. The idea has been practiced ever since civilization started and humanity began producing more efficient materials for functional purposes. In addition to the large variety of nanocom- posites found in nature and in living beings (such as bone), which is the focus of chapter 3 of this book, an excellent example of the use of synthetic nanocomposites in antiquity is the recent discovery of the constitution of Mayan paintings developed in the Mesoamericas. State-of-the-art characterization of these painting samples reveals that the structure of the paints consisted of a matrix of clay mixed with organic colorant (indigo) molecules. They also contained inclusions of metal nanoparticles encapsu- lated in an amorphous silicate substrate, with oxide nanoparticles on the substrate [1]. The nanoparticles were formed during heat treatment from impurities (Fe, Mn, Cr) present in the raw materials such as clays, but their content and size influenced the optical properties of the final paint. The combination of intercalated clay forming a superlattice in conjunction with metallic and oxide nanoparticles supported on the amorphous substrate made this paint one of the earliest synthetic materials resem- bling modern functional nanocomposites. + +Nanocomposites can be considered solid structures with nanometer-scale dimen- sional repeat distances between the different phases that constitute the structure. These materials typically consist of an inorganic (host) solid containing an organic component or vice versa. Or they can consist of two or more inorganic/organic phases in some combinatorial form with the constraint that at least one of the phases or fea- tures be in the nanosize. Extreme examples of nanocomposites can be porous media, colloids, gels, and copolymers. In this book, however, we focus on the core concept of nanocomposite materials, i.e., a combination of nano-dimensional phases with dis- tinct differences in structure, chemistry, and properties. One could think of the na- nostructured phases present in nanocomposites as zero-dimensional (e.g., embedded clusters), 1D (one-dimensional; e.g., nanotubes), 2D (nanoscale coatings), and 3D (embedded networks). In general, nanocomposite materials can demonstrate different mechanical, electrical, optical, electrochemical, catalytic, and structural prop- erties than those of each individual component. The multifunctional behavior for any specific property of the material is often more than the sum of the individual compo- nents. + +Both simple and complex approaches to creating nanocomposite structures exist. A practical dual-phase nanocomposite system, such as supported catalysts used in het- erogeneous catalysis (metal nanoparticles placed on ceramic supports), can be pre- pared simply by evaporation of metal onto chosen substrates or dispersal through solvent chemistry. On the other hand, material such as bone, which has a complex hierarchical structure with coexisting ceramic and polymeric phases, is difficult to duplicate entirely by existing synthesis techniques. The methods used in the prepara- tion of nanocomposites range from chemical means to vapor phase deposition. + +Apart from the properties of individual components in a nanocomposite, interfaces play an important role in enhancing or limiting the overall properties of the system. Due to the high surface area of nanostructures, nanocomposites present many inter- faces between the constituent intermixed phases. Special properties of nanocomposite materials often arise from interaction of its phases at the interfaces. An excellent ex- ample of this phenomenon is the mechanical behavior of nanotube-filled polymer composites. Although adding nanotubes could conceivably improve the strength of polymers (due to the superior mechanical properties of the nanotubes), a noninteract- ing interface serves only to create weak regions in the composite, resulting in no en- hancement of its mechanical properties (detailed in chapter 2). In contrast to nano- composite materials, the interfaces in conventional composites constitute a much smaller volume fraction of the bulk material. + +In the following sections of this chapter, we describe some examples of metal/cera- mic nanocomposite systems that have become subjects of intense study in recent years. The various physical properties that can be tailored in these systems for specific applications is also considered, along with different approaches to synthesizing these nanocomposites. + +# 1.2 Ceramic/Metal Nanocomposites + +Many efforts are under way to develop high-performance ceramics that have promise for engineering applications such as highly efficient gas turbines, aerospace materials, automobiles, etc. Even the best processed ceramic materials used in applications pose many unsolved problems; among them, relatively low fracture toughness and strength, degradation of mechanical properties at high temperatures, and poor resis- tance to creep, fatigue, and thermal shock. Attempts to solve these problems have involved incorporating second phases such as particulates, platelets, whiskers, and fibers in the micron-size range at the matrix grain boundaries. However, results have been generally disappointing when micron-size fillers are used to achieve these goals. Recently the concept of nanocomposites has been considered, which is based on passive control of the microstructures by incorporating nanometer-size second-phase dispersions into ceramic matrices [2]. The dispersions can be characterized as either + +![](images/07f2edacf4b2a2f630c18af3f31982bb834f916b5ac2c0cf1608387f224f582e.jpg) +Fig. 1.1 New concept of ceramic metal nano- into ceramic materials. This is a completely new composites with inter- and intra-granular designs: method to fabricate materials with excellent me- properties of ceramic materials can be improved by chanical properties (such as high strength and nanocomposite technology. This technique is based toughness), due to the desirable microstructure of on passive control of the microstructures by in- ceramics (Source:[228] Reprinted with permission) corporating nanometer-sized second dispersions + +intragranular or intergranular (Figure 1.1). These materials can be produced by incor- porating a very small amount of additive into a ceramic matrix. The additive segregates at the grain boundary with a gradient concentration or precipitates as molecular or cluster sized particles within the grains or at the grain boundaries. Optimized proces- sing can lead to excellent structural control at the molecular level in most nanocom- posite materials. Intragranular dispersions aim to generate and fix dislocations during the processing, annealing, cooling, and/or the in-situ control of size and shape of matrix grains. This role of dispersoids, especially on the nano scale, is important in oxide ceramics, some of which become ductile at high temperatures. The intergra- nular nanodispersoids must play important roles in control of the grain boundary structure of oxide $(\mathrm{Al}_{2}\mathrm{O}_{3}$ , $\mathrm{MgO}$ ) and nonoxide $(\mathrm{Si}_{3}\mathrm{N}_{4}$ , SiC) ceramics, which improves their high-temperature mechanical properties [3–6]. The design concept of nanocom- posites can be applied to ceramic/metal, metal/ceramic, and polymer/ceramic com- posite systems. + +Dispersing metallic second-phase particles into ceramics improves their mechanical properties (e.g., fracture toughness). A wide variety of properties, including magnetic, electric, and optical properties, can also be, tailored in the composites due to the size effect of nanosized metal dispersions, as described later in the chapter. Conventional powder metallurgical methods and solution chemical processes like sol–gel and co- precipitation methods have been used to prepare composite powders for ceramic/me- tal nanocomposites such as $\mathrm{Al}_{2}\mathrm{O}_{3}/\mathrm{W}$ , Mo, Ni, Cu, Co, Fe; $\mathrm{{ZrO_{2}/N i}}$ , Mo; MgO/Fe, Co, Ni; and so on. The powders are sintered in a reductive atmosphere to give homoge- neous dispersions of metallic particles within the ceramic matrices. Fracture strength, toughness, and/or hardness are enhanced due to microstructural refinement by the nanodispersions and their plasticity. For transition metal particle dispersed oxide cera- mic composites, ferromagnetism is a value-added supplement to the excellent me- chanical properties of the composites [7,8]. In addition, good magnetic response to applied stress was found in these ceramic/ferromagnetic-metal nanocomposites, al- lowing the possibility of remote sensing of initiation of fractures or deformations in ceramic materials. + +Nanocomposite technology is also applicable to functional ceramics such as ferro- electric, piezoelectric, varistor, and ion-conducting materials. Incorporating a small amount of ceramic or metallic nanoparticles into $\mathrm{BaTiO}_{3}$ , $\mathrm{ZnO}$ , or cubic $\mathrm{ZrO}_{2}$ can significantly improve their mechanical strength, hardness, and toughness, which are very important in creating highly reliable electric devices operating in severe en- vironmental conditions [9]. In addition, dispersing conducting metallic nanoparticles or nanowires can enhance the electrical properties, as described later. Dispersion of soft materials into a hard ceramic generally decreases its mechanical properties (e.g., hardness). However, in nanocomposites, soft materials added to several kinds of cera- mics can improve their mechanical properties. For example, adding hexagonal boron nitride to silicon nitride ceramic can enhance its fracture strength not only at room temperature but also at very high temperatures up to $1500\,^{\circ}\mathrm{C}$ . In addition, some of these nanocomposite materials exhibit superior thermal shock resistance and machin- ability because of the characteristic plasticity of one of the phases and the interface regions between that phase and the hard ceramic matrices. + +Advanced bulk ceramic materials that can withstand high temperatures $(>\!1500\,^{\circ}\mathrm{C})$ without degradation or oxidation are needed for applications such as structural parts of motor engines, gas turbines, catalytic heat exchangers, and combustion systems. Such hard, high-temperature stable, oxidation-resistant ceramic composites and coatings are also in demand for aircraft and spacecraft applications. Silicon nitride $\mathrm{(Si_{3}N_{4})}$ and silicon carbide/silicon nitride $\mathrm{(SiC/Si}_{3}\mathrm{N}_{4}\mathrm{)}$ composites perform best in adverse high-temperature oxidizing conditions. Commercial $\mathrm{Si}_{3}\mathrm{N}_{4}$ can be used up to ${\sim}1200\,^{\circ}\mathrm{C}$ , but the composites can withstand much higher temperatures. Such Nano- composites are optimally produced from amorphous silicon carbonitride (obtained by the pyrolysis of compacted poly hy dr ido methyl sila zane $[\mathrm{CH}_{3}\mathrm{SiH-NH}]_{m}[(\mathrm{CH}_{3})_{2}\mathrm{Si-NH}]_{n}$ at about $1000\,^{\circ}\mathrm{C})$ ), which produces crystallites of microcrystals of $\mathrm{Si}_{3}\mathrm{N}_{4}$ and nanocrys- tals of SiC [10] (Figure 1.2). The oxidation resistance, determined by TGA analysis, arises from the formation of a thin (few microns) silicon oxide layer. + +Processing is key to the fabrication of nanocomposites with optimized properties. Some examples of commonly used processes for creating nanocomposites are dis- cussed below. + +![](images/f33de2a0f211d3058f0f47c35e0262fc859a378d17f6a5c3b2213c6046891363.jpg) +Fig. 1.2 Calculated phase diagrams of the system Si/B/C/N allows for the creation of high-temperature ceramic nanocomposites. The system Si/B/C/N is being investigated with respect to processing new covalent materials. Based on this system, several nanocomposites $\mathsf{(S i C/S i_{3}N_{4})}$ have been developed that can, for example, withstand high temperatures $\left(\sim\right]500\,^{\circ}\mathsf{C}\mathrm{)}$ without degradation or oxidation [10]. (Source [229] used with permission) alternative web site: http://aldix.mpi-stuttgart.mpg.de/E_head.html, used with permission + +# 1.2.1 Nanocomposites by Mechanical Alloying + +Mechanical alloying was originally invented to form small-particle (oxide, carbide, etc.) dispersion-strengthened metallic alloys (Figure 1.3) [11]. In this high-energy ball milling process, alloying occurs as a result of repeated breaking up and joining (weld- ing) of the component particles. The process can prepare highly metastable structures such as amorphous alloys and nanocomposite structures with high flexibility. Scaling up of synthesized materials to industrial quantities is easily achieved in this process, but purity and homogeneity of the structures produced remains a challenge. In addi- tion to erosion and agglomeration, high-energy milling can provoke chemical reac- tions that are induced by the transfer of mechanical energy, which can influence the milling process and the properties of the product. This idea is used to prepare magnetic oxide-metal nanocomposites via mechanically induced displacement reac- tions between a metal oxide and a more reactive metal [12,13]. High-energy ball milling can also induce chemical changes in non metallurgical systems, including si- licates, minerals, ferrites, ceramics, and organic compounds. The interest in mechan- ical alloying as a method to produce nanocrystalline materials is due to the simplicity of the method and the possibility for scaled-up manufacturing. + +Displacement reactions between a metal oxide and a more reactive metal can be induced by ball milling [14]. The reaction may progress gradually, producing a nano- composite powder. In some cases, the reaction progresses gradually, and a metal/me- tal-oxide nanocomposite is formed. Milling may also initiate a self-propagating com- + +![](images/7f3d28df4ad9c0f654c9eca4474c0c759236991a945c3a1f071f3e52a2d4c555.jpg) +Fig. 1.3 Schematic of the for- mation process of typical nano- composite microstructures by the mechanical alloying method. (Source [230, 11] used with permission) + +bustive reaction [15]. The nature of such reactions depends on thermodynamic para- meters, the microstructure of the reaction mixture, and the way the microstructure develops during the milling process. The mechanical stresses developed during high impact hits can also initiate combustion in highly exothermic systems, melting the reaction mixture and destroying the ultrafine (nanocrystalline) microstructure. Milling mixtures of ceramic and metal powders can induce mechanochemical reac- tions, and this process is an efficient way of producing nanocermets [16]. Depending on the thermodynamics of the metal/metal-oxide systems and the kinetics of the ex- change (displacement) reactions during processing, various nanocomposite systems could evolve. As an example, the reduction of metal oxides with aluminum during reactive ball milling can result in nanocomposites of $\mathrm{Al}_{2}\mathrm{O}_{3}$ and metallic alloys (Fe, Ni, Cr; particularly binary alloy systems), and such ceramics with ductile metal inclu- sions produce toughened materials with superior mechanical properties [17]. These nanocomposite materials also have better thermo mechanical properties, such as high- er thermal shock resistance, due to better metal–ceramic interfacial strength. + +Ball milling by direct milling of a mixture of iron and alumina powders has been used to prepare nanocomposites with magnetic phases, such as nanoparticles of iron embedded in an insulating alumina matrix [18]. The average particle size can be re- duced to the $10\mathrm{-nm}$ range, as indicated by x-ray diffraction linewidths and electron microscopy. The magnetic properties of this system (e.g., saturation magnetization and coercivity) can be tailored by changing the phase composition, particle size, and the internal stresses accumulated during milling. In this system, the iron nano- particles were formed with lattice strains of about 0.005; coercivities up to $400\,\mathrm{De}$ were achieved. The magnetization of the iron particles is $25\,\%\!-\!40\,\%$ less than that expected for bulk iron. Systems of smaller magnetic particles embedded in a nonmagnetic ma- trix can be prepared by high-energy ball milling [19]. For example, nanocomposites of ${\mathrm{Fe}}_{3}\mathrm{O}_{4}$ particles dispersed in Cu have been prepared by ball milling a mixture of ${\mathrm{Fe}}_{3}\mathrm{O}_{4}$ and Cu powders directly, as well as by enhanced ball milling-induced reaction between CuO and metallic iron [20]. Both processes result in magnetic semi-hard nanocom- posites with a significant super para magnetic fraction, due to the very small particle sizes of the dispersed magnetic phase. In situ chemical reactions provide a means to control the ball milling process and to influence the microstructure and magnetic properties of the product. Nanocomposite magnets (such as hard magnetic SmCoFe phases in soft magnetic Fe/Co systems), discussed in detail later in this chapter, are routinely prepared by mechanical milling and heat treatment. The metastable nano- crystalline/amorphous structures inherently obtained in mechanically alloyed pow- ders result from repeated deformation and fracture events during collisions of pow- ders with the balls. Plastic deformation in powders initially occurs through the forma- tion of shear bands, and when high dislocation densities are reached, the shear bands degenerate into randomly oriented subgrains. The large surface area of the nanocrys- talline grains often helps in the transformation of crystalline into amorphous struc- tures [21]. Deformation-induced defect density and the local changes in temperature due to impacts affect the diffusion coefficients of the several species involved during the milling process. In fact, the final microstructure and stoichiometry of mechanically milled samples often reflects the competing processes of milling-induced disorder and diffusion-limited recovery, rather than being solely dependent on the starting material (e.g., depending on whether the starting mixtures are pre-alloyed or in their elemental forms). + +# 1.2.2 Nanocomposites from Sol–Gel Synthesis + +Aerogels, due to their high-porosity structure, are clearly an ideal starting material for use in nanocomposites. Aerogels are made by sol–gel [22,23] polymerization of se- lected silica, alumina, or resorcinol-formaldehyde monomers in solution and are ex- tremely light (densities ${\sim}0.5\!-\!0.001\,\mathrm{g\cc^{-1}})$ but highly porous, having nanosize pores. In nanocomposites derived from aerogels, the product consists of a ‘substrate’ (e.g., silica aerogel) and one or more additional phases (of any composition or scale). In the composites, there is always at least one phase whose physical structures have dimen- sions on the order of nanometers (the particles and pores of the aerogel). The addi- tional phases may also have nanoscale dimensions or may be larger. The systems most commonly made are silica-based nanocomposite systems [24], but this approach can be extended to other aerogel (alumina, etc.) precursors. + +Aerogel nanocomposites can be fabricated in various ways, depending on when the second phase is introduced into the aerogel material. The second component can be added during the sol–gel processing of the material (before supercritical drying). It can also be added through the vapor phase (after supercritical drying), or chemical modification of the aerogel backbone may be effected through reactive gas treat- ment. These general approaches can produce many varieties of composites. A non- silica material is added to the silica sol before gelation. The added material may be a soluble organic or inorganic compound, insoluble powder, polymer, biomaterial, etc. The additional components must withstand the subsequent processing steps used to form the aerogel (alcohol soaking and supercritical drying). The conditions encountered in the $\mathrm{CO}_{2}$ drying process are milder than in the alcohol drying process and are more amenable to forming composites. If the added components are bulk insoluble materials, steps must be taken to prevent its settling before gelation. The addition of soluble inorganic or organic compounds to the sol provides a virtually unlimited number of possible composites. Two criteria must be met to prepare a com- posite by this route. First, the added component must not interfere with the gelation chemistry of the aerogel precursor. Possible interference is difficult to predict in ad- vance, but it is rarely a problem if the added component is reasonably inert. The second problem is the leaching out of the added phases during the alcohol soak or supercri- tical drying steps. This problem can be a significant impediment if a high loading of the second phase is desired in the final composite. When the added component is a metal complex, it is often useful to use a chemical binding agent that can bind to the silica backbone and chelate the metal complex. Many use this method to prepare na- nocomposites of silica aerogels or xerogels. After the gel is dried, the resulting nano- composite consists of an aerogel with metal atoms or ions uniformly (atomically) dis- persed throughout the material. Thermal post-processing creates nanosize metal par- + +![](images/2576750faf0930de45921a25e6cf96c6bee05017cb55ea487fe9f5a5fc5a4055.jpg) +Fig. 1.4 (a) Microstructure of aerogel-encapsulated phase nanocomposite. (b) Left image of three pieces of nanocomposites shows silica aerogel samples that have been coated with silicon nanoparticles by chemical vapor methods. The composites emit red light when excited with ultraviolet light. Right image of six pieces of nanocomposites prepared by adding metal salts or other compounds to a sol before gelation; they show different colors depending on the metal species present. The deep blue aerogel contains nickel; the pale green, copper; the black, carbon and iron; the orange, iron oxide. (Source, the silica aerogel photo gallery [231] used with permission) + +ticles within the aerogel matrix. Such composites can have many applications. An example is their use as catalysts for gas-phase reactions or for catalyzed growth of nanostructures. + +Vapor phase infiltration through the open pore network of aerogels provides another route [25] to creating various forms of aerogel-based nanocomposites; almost any com- pound can be deposited uniformly throughout an aerogel. In fact, adsorbed materials in silica aerogels can be modified into solid phases by thermal or chemical decom- position. The same is true for materials that have a porous interior structure, such as zeolites. The nanosize pores within these porous hosts can be utilized for depositing a second phase by chemical or vapor phase infiltration and thermal decomposition. Recently, single-walled carbon nanotubes have been deposited within pores of zeolites to create nanocomposite materials that have unique properties, such as superconduc- tivity [26]. + +Some examples of nanocomposites (Figure 1.4) that have been created out of silica- based aerogel matrices are the following: + +Silica aerogel/carbon composites [27]: These can be made by the decomposition of hydrocarbon gases at high temperatures. The fine structure of aerogels allows the decomposition to take place at a low temperatures $(200\!-\!450\,^{\circ}\mathrm{C})$ ). Carbon loadings of $1\%\!-\!800\,\%$ have been observed. The carbon deposition is uniform throughout the substrate at lower loadings, but at higher loadings, the carbon begins to localize at the exterior surface of the composite. These nanocomposites have interesting prop- erties, such as electrical conductivity (above certain loadings) and higher mechanical strength relative to the aerogel. + +Silica aerogel/silicon composites [28]: Thermal decomposition of various organosilanes on a silica aerogel forms deposits of elemental silicon. In this case, rapid decomposi- tion of the silane precursor leads to deposits localized near the exterior surface of the aerogel substrate. The nanocomposite, with 20–30-nm diameter silicon particles, ex- hibits strong visible photo luminescence at $600\ \mathrm{nm}$ . + +Silica aerogel/transition-metal composites [29]: Organo/transition-metal complexes can be used to deposit metal compounds uniformly through the aerogel volumes. The compounds can be thermally decomposed to their base metals. These intermedi- ate composites, due to the disperse nature of the metallic phase and hence their high reactivity, can be converted to metal oxides, sulfides, or halides. The loading of the metallic phase can be changed by repeated deposition steps. The nanocomposites contain crystals of the desired metal species with sizes in the range of $5\,{-}\,100\;\mathrm{nm}$ in diameter. + +![](images/12724e6842ec5e3f3dc577b416d9d511291074fd4e8ea0e50c98b709c4486c22.jpg) +Fig. 1.5 Photo luminescence intensity (irradiance) vs. oxygen pressure (concentration gives a similar plot) at two temperatures measured with a prototype sen- sor made of silica aerogels. The photo luminescence intensity is indirectly proportional to the amount of gaseous oxygen within the aerogel. The quench- ing of photoluminescence by oxygen is observed in many luminescent materials. Source [232] used with permission) + +The chemical structure of the silica (or other oxide) backbone of an aerogel can also be easily modified. For example, silica aerogel surfaces can be partially reduced by hydrogen. The resulting composite consists of thin interior surface layers of oxy- gen-deficient silica $(\mathrm{SiO_{x}})$ . This material exhibits strong visible photo luminescence at $490\!-\!500\:\mathrm{nm}$ when excited by ultraviolet $(330\ \mathrm{nm})$ light. The chemical process used to change the surface characteristics of the aerogel does not alter the physical shape or optical transparency of the original structure. This composite is the founda- tion for the aerogel optical oxygen sensor [30] (Figure 1.5), which is based on the fact that the intensity of photo luminescence is indirectly related to the oxygen concentra- tion in the nanocomposite. + +# 1.2.3 Nanocomposites by Thermal Spray Synthesis + +Thermal spray processing is a commercially relevant, proven technique for processing nanostructured coatings [31]. Thermal spray techniques are effective because agglom- erated nanocrystalline powders are melted, accelerated against a substrate, and quenched very rapidly in a single step. This rapid melting and solidification promotes the retention of a nanocrystalline phase and even amorphous structure. Retention of the nanocrystalline structure leads to enhanced wear behavior, greater hardness, and sometimes a reduced coefficient of friction compared to conventional coatings. + +Figure 1.6 shows a generalized thermal spray process [32]. To form the starting powders, conventional powders can be cryomilled to achieve a nanocrystalline struc- ture [33–35]. Under the right conditions, for example, Fe alloyed with Al, precipitates form, and these precipitates stabilize the nanoscale grain structure to $75\,\%$ of the melt- ing temperature of the pure metal. Pure metals (except for aluminum) require some alloying before the nanocrystalline structure is stable at elevated temperatures [36]. For $\mathbb{W}\mathrm{C}/\mathrm{Co}$ and $\mathrm{Cr}_{3}\mathrm{Cr}_{2}/\mathrm{NiCr}$ , the hard particles are broken into nanometer-size grains, and they are embedded in the binder [37, 38]. Other systems have also been milled for thermal spraying, such as steel [39] and $\mathrm{NiCr/Cr}_{3}\mathrm{C}_{2}$ . In all cases, there appears to be some nitrogen or oxygen contamination. + +The nanoscale powders, prepared by various techniques, must be agglomerated so that grains on the order of $50\,\mathrm{nm}$ can be introduced into the thermal spray gun. Unlike sintering of ceramics, this agglomeration does not prevent full densification. A reason- ably narrow particle size distribution ensures uniform heating. Nanocrystalline feed- stock is generally injected internally (inside the torch), but powders can be injected externally. The type of flame or jet produced depends on the thermal spray techni- que, and within each technique, gas heating and gas flow parameters can control the velocity and temperature profile. The temperature and velocity profile, combined with the spray distance (the distance from the end of the nozzle to the substrate), control the temperature that the powders reach. Successive impact of particles in a molten or viscous state on the substrate or on previously deposited layers of material forms a coating. + +The ability to maintain the nanocrystalline structure during processing and upon consolidation is critical to improving its properties because it is the nanoscale micro- + +![](images/e1918e51a540988b86b09b517c529c73d4ed766d8fec3d88fee2c34f97cabf37.jpg) +Fig. 1.6 Schematic for a generalized thermal spray microstructure of the coating, hardness, wear re- process, showing the different variables used. The sistance, etc.) are affected by a multidimensional qualities of the coatings (bonding to the substrate, parameter space + +structure that leads to the unique properties. Several parameters are critical: (a) The thermal stability of the agglomerated powders: nanocrystalline materials can experi- ence grain growth at temperatures well below the temperatures observed for conven- tional materials. The high surface area drives this growth. (b) The degree of melting that occurs in flight: this can be controlled by the spray distance, the temperature of the jet, and the velocity of the jet, and optimal parameters are determined primarily by experiment. (c) The cooling rate: a high cooling rate leads to high nucleation and slow grain growth, which promotes the formation of nanocrystalline grains. The sys- tems that tend to maintain their nanocrystalline structure even at elevated temperature are apt to have impurities or a second phase that stabilize the grain structure [40]. For example, cryomilling often results in nanoscale particles (oxides, nitrides, or oxyni- trides) [41] that fix the grain boundaries. In addition, significant impurities or excess solute atoms at the grain boundaries also limit grain growth [42, 43]. + +Plasma spraying and high velocity oxy fuel (HVOF) processes are the most widely used thermal spray methods for producing nanocrystalline and nanocomposite coat- ings. In plasma spraying, an electric arc is used to ionize an inert gas to produce a highly energetic thermal plasma jet with gas temperatures and velocities of approxi- mately $11\,000\;\mathrm{K}$ and $2000\,\mathrm{ms^{\cdot1}}$ . Vacuum plasma spraying and low-pressure plasma spraying have been used to effectively process WC/Co nanocomposite coatings. Use of HVOF involves an internal combustion chamber in which fuel (hydrogen, propylene, acetylene, propane) is burned in the presence of oxygen or air (HVAF). This results in a hypersonic gas velocity. The particle velocities are higher than the $800\,\mathrm{ms^{\cdot1}}$ achieved with plasma spray, and the thermal energy is lower (it may reach 3000 K), which reduces superheating and particle vaporization. The high speed and low temperatures result in more strongly adhering and more homogeneous coatings with lower oxide content. + +WC/Co coatings are of great interest because they already exhibit excellent wear properties. Nanostructuring further increases the wear resistance and decreases the coefficient of friction. Thermally sprayed WC/Co coatings, however, do not always exhibit improved properties. WC/Co coatings sprayed via HVOF exhibited decreased wear resistance due to decomposition of the carbide phase during spraying [44]. Na- nostructured powders reach temperatures almost $500~^{\circ}$ higher than their conventional counterparts. Vacuum plasma spraying, however, resulted in coatings with signifi- cantly improved wear resistance and lower coefficient of friction, presumably because the Ar atmosphere prevented oxidation of the carbide phase [45]. + +$\mathrm{Cr}_{3}\mathrm{C}_{2}/\mathrm{NiCr}$ composites are also used in applications where wear resistance is re- quired, but they have an added advantage over WC/Co, which has excellent corrosion resistance. Nanostructuring of these coatings has also resulted in improved hardness and scratch resistance, as well as reduced coefficient of friction. The improved homo- + +![](images/6875eb9fa5faf413039a133f8c3f31f0bf29225f80da8007adf7d28ac0058b4d.jpg) +Fig. 1.7 Microstructures of thermal-sprayed $\mathsf{C r}_{3}\mathsf{C}_{2}/\mathsf{N i C r}$ coatings. Top micrograph shows conventional coating and bot- tom micrograph shows nano- composite microstructure. A uniform, dense microstructure is observed in the nanostruc- tured coatings, compared to an inhomogeneous microstructure in the conventional coating. (Source [37] used with permis- sion) + +geneity of these structures, as well as a high density of $\mathrm{Cr}_{2}\mathrm{O}_{3}$ nanoparticles (formed by oxidation during the thermal spray process), compared to conventional materials, cause the improved properties. Figure 1.7 shows an example of the improved homo- geneity of nanostructured coatings. Ceramics such as alumina/titania and zirconia have also been thermally sprayed, and the nanostructured powders have lead to sub- micron final grain sizes in the coatings. Key to achieving excellent properties is mini- mizing the degree of melting [46] so as to maintain the nanostructure in the final coating. On the other hand, significant deformation or splatting of the particles is required upon impact, to assure a large surface contact between the particles [47]. Thus, some melted particles lend well to continuous, good-quality coatings. + +# 1.3 Metal Matrix Nanocomposites + +During the past decade, considerable research effort has been directed towards the development of in situ metal-matrix composites (MMCs), in which the reinforce- ments are formed by exothermal reactions between elements or between elements and compounds [48]. With this approach, MMCs with a wide range of matrix materi- als (including aluminum, titanium, copper, nickel, and iron), and second-phase par- ticles (including borides, carbides, nitrides, oxides, and their mixtures) have been pro- duced. Because of the formation of stable nanosized ceramic reinforcements, in situ MMCs exhibit excellent mechanical properties. + +MMCs are a kind of material in which rigid ceramic reinforcements are embedded in a ductile metal or alloy matrix. MMCs combine metallic properties (ductility and toughness) with ceramic characteristics (high strength and modulus), leading to great- er strength to shear and compression and to higher service temperature capabilities. The attractive physical and mechanical properties that can be obtained with MMCs, such as high specific modulus, strength, and thermal stability, have been documented extensively. Interest in MMCs for use in the aerospace and automotive industries and other structural applications has increased over the past 20 years. This increase results from the availability of relatively inexpensive reinforcements and the development of various processing routes that result in reproducible microstructure and properties. + +The family of discontinuously reinforced MMCs includes both particulates and whiskers or short fibers. More recently, this class of MMCs has attracted considerable attention as a result of (a) availability of various types of reinforcement at competitive costs, (b) the successful development of manufacturing processes to produce MMCs with reproducible structure and properties, and (c) the availability of standard or near- standard metal working methods, which can be utilized to fabricate these composites. The particulate-reinforced MMCs are of particular interest due to their ease of fabrica- tion, lower costs, and isotropic properties. Traditionally, discontinuously reinforced MMCs have been produced by several processing routes such as powder metal- lurgy, spray deposition, mechanical alloying (MA), and various casting techniques. All these techniques are based on the addition of ceramic reinforcements to the matrix materials, which may be in molten or powder form. For conventional MMCs, the reinforcing phases are prepared separately prior to the composite fabrication. Thus, conventional MMCs can be viewed as ex situ MMCs. In this case, the scale of the reinforcing phase is limited by the starting powder size, which is typically on the order of micrometers to tens of micrometers and rarely below $1\,\mu\mathrm{m}$ . Other main drawbacks that must be overcome are interfacial reactions between the reinforcements and the matrix and poor wettability between the reinforcements and the matrix due to the surface structure of the reinforcements as well as to contamination. + +The properties of MMCs are widely recognized to be controlled by the size and volume fraction of the reinforcements as well as by the nature of the matrix/reinforce- ment interfaces. An optimum set of mechanical properties can be obtained when fine, thermally stable ceramic particulates are dispersed uniformly in the metal matrix. Efforts have been made to meet such requirements and have led to the development of novel composites – in situ MMCs in which the reinforcements are synthesized in a metallic matrix by chemical reactions between elements or between element and com- pound during fabrication of the composite. Compared with conventional MMCs pro- duced by ex situ methods, in situ MMCs exhibit the following advantages: (a) forma- tion of reinforcements that are thermodynamic ally stable in the matrix, leading to less degradation in elevated-temperature services; (b) reinforcement/matrix interfaces that are clean, resulting in strong interfacial bonding; (c) the formation of reinforcing par- ticles of a finer size with a more uniform distribution in the matrix, which yields better mechanical properties. + +The great potential that in situ metal matrix nanocomposites offer for widespread applications has resulted in the development of a variety of processing techniques for production during the past decade. Using these routes, in situ composites with a wide range of matrix materials (including aluminum, titanium, copper, nickel, and iron) and second-phase particles (including borides, carbides, nitrides, oxides, and their mixtures) have been produced. Particularly attractive among the several techniques available for synthesizing MMCs are the solidification processes in which the reinfor- cing particles are formed in situ in the molten metallic phase prior to its solidification. What makes them so attractive is their simplicity, economy, and flexibility. The judi- cious selection of solidification processing techniques, matrix alloy compositions, and dispersoids can produce new structures and affect a unique set of useful engineering properties that are difficult to reach in conventional monolithic materials. Specifically, the solidification conditions that are present during processing play an important role in dictating the microstructure and the mechanical and physical characteristics of these structures. Microstructure refinement arising from rapid solidification proces- sing (RSP) offers a potential avenue for alleviating solute segregation and enhancing dispersion hardening by substantially reducing the size of the reinforcing phases and modifying their distribution in the matrix. As example, RSP of Ti/B or Ti/Si alloys accompanied by large undercoolings and high cooling rates is very effective in produ- cing in situ Ti-based nanocomposites containing large volume fractions of reinforcing particles [49]. These particles are formed in situ in Ti/B or Ti/Si alloys either upon solidification or, subsequently, by controlled decomposition of the resulting supersa- turated solid solutions. + +More recently, several workers have used the RSP route [50] to fabricate in situ TiC particulate-reinforced Al-based composites. In their work, master material ingots were prepared by melting a mixture of Al, Ti, and graphite powder in a graphite-lined in- duction furnace under an argon atmosphere, followed by direct chill cast. Chill block melt spinning was used to prepare rapidly solidified samples in ribbon form. These ribbons were further milled into powders $(100\pm250\,\mu\mathrm{m})$ , which were subsequently canned and degassed and then extruded into rods. In situ formed TiC particles of $40\pm$ $80\;\mathrm{nm}$ were reported to be distributed uniformly in the aluminum matrix with a grain size of $0.3\,\pm\,0.85\,\,\mu\mathrm{m}$ . The authors reported that RSP can significantly refine the microstructure of the composites. For TiC/Al composites, the microstructure is often characterized by the presence of agglomerated TiC particles. These particles, with a size of $0.2\pm1.0\,\mu\mathrm{m}$ , accumulate at the Al subgrain or the grain boundaries. The larger particles have polyhedral morphology, and the smaller ones are round or globular. In comparison, typical rapidly solidified microstructures consist of a uniform, fine-scale dispersion of TiC particles with a size of $40\pm80\;\mathrm{nm}$ in an Al supersaturated matrix of $0.30\pm0.85\,\mu\mathrm{m}$ grain size. One main advantage of RSP is its ability to produce alloy compositions not obtainable by conventional processing methods. Furthermore, RSP materials have excellent compositional homogeneity, small grain sizes, and homoge- neously distributed fine precipitates or dispersoids. + +The homogeneity of composite materials is crucially important to high-performance engineering applications such as in the automotive and aircraft industries. A uniform reinforcement distribution in MMCs is essential to achieving effective load-bearing capacity of the reinforcement. Nonuniform distribution of reinforcement can lead to lower ductility, strength, and toughness of the composites. Nanoscale ceramic par- ticles synthesized in situ are dispersed more uniformly in the matrices of MMCs, leading to significant improvements in the yield strength, stiffness, and resistance to creep and wear of the materials. For example, in situ fabrication of TiC-reinforced Al, Al/Si, and $\mathrm{{Al/Fe/V/Si}}$ matrix composites by the RSP route is far more effective in + +![](images/90e814c11144ec6eee25af587477f09b59bf0f203188bb97a932412b8a4c927b.jpg) + $10^{8}\,\mathrm{m}$ + +![Tab. 1.1 Typical magnetic properties of nanocrystal/amorphous composites and amorphous alloys. Alloy $\mathsf{l}\colon\mathsf{F e}_{73.5}\circ\mathsf{C u}_{1}\;\mathsf{N b}_{3}\;\mathsf{S i}_{13.5}\;\mathsf{B}_{9}$ (at. $\%)$ ). Alloy 2 $\mathsf{;F e_{73.5}\ C u_{1}\ N b_{3}\ S i_{16.5}\ B_{6}}$ (at. $\%)$ . Core loss: $100~{\sf k H z},\,0.2~{\sf T}$ [52]. ](images/aec1f349d07506e9ae9e81a91599c31dbb6e3018c124dcff449cc537d930c75e.jpg) + +improving the tensile properties of these composites, due to the formation of a refined microstructure. The in situ composites exhibit excellent strength at room temperature and elevated temperatures. The values of strength ( r ) increased with increasing vol- ume fractions or decreasing diameters of dispersed-phase (TiC) particles (Figure 1.8). When the volume fraction of dispersed particles is about $15\,{-}\,30\,\mathrm{vol}\,\%$ and the particle diameters $40\,{-}\,80\ \mathrm{nm}$ , the values of $\upsigma$ are $120\,{-}\,270~\mathrm{MPa}$ and $200\,{-}\,350~\mathrm{MPa}$ , respec- tively (Figure 1.8). + +Nanocrystalline materials in general are single- or multi-phase polycrystals with grain sizes in the nanometer range. Owing to the extremely small dimensions, many properties of nanocrystalline samples are fundamentally different from, and often superior to, those of conventional polycrystals and amorphous solids. For exam- ple, nanocrystalline materials exhibit increased strength or hardness, improved duc- tility or toughness, reduced elastic modulus, enhanced diffusivity, higher specific heat, enhanced thermal expansion coefficient, and superior soft magnetic properties in comparison with conventional polycrystalline materials [51]. Crystallizing completely amorphous solids under proper heat treatment conditions can result in formation of nanocrystalline materials. However, controlled crystallization of amorphous alloys can + +![](images/21d45caab2fa7d41dc053365967bba99245eb037d4915c34f45ab8be0acd0d01.jpg) +Fig. 1.9 Compressive stress– strain curves of amorphous and partly crystallized $\mathsf{Z r}_{57}\mathsf{A l}_{10}\mathsf{C u}_{20}\mathsf{N i}_{8}\mathsf{T i}_{5}$ alloy nano- composite (a) as-cast, (b) 40 vol. $\%$ nanocrystals, (c) 45 vol. $\%$ nanocrystals and (d) 68 vol. $\%$ nanocrystals. The sample con- taining a volume fraction of $40\,\%$ nanocrystals (b) seems to pro- vide the best compromise be- tween strength and ductility. (Source [53] used with permis- sion) + +be used to obtain partially crystallized materials with nanosized crystallites embedded in the residual amorphous matrix. This special nanocrystal/amorphous nanocompo- site structure allows the materials to exhibit excellent mechanical or magnetic proper- ties. The $\mathrm{Fe/Cu/Nb/Si/B}$ alloys are a good example of this type of material. The Fe/Si/ $\mathsf{B}/M$ ( M : additives) alloys properly prepared by annealing amorphous alloys and hav- ing bcc Fe solid solution and 10-nm-diameter nanostructures embedded in the resi- dual amorphous matrix, show excellent soft magnetic properties (Table 1.1). The na- nocrystal/amorphous composite shows high saturation flux density, low magnetostric- tion, and excellent soft magnetic properties. This nanocomposite, therefore, is ex- pected to find use in many kinds of magnetic devices such as choke coils and trans- formers. + +Production of bulk nanostructured composites with amorphous matrices has been carried out by die casting and mechanical alloying and subsequent consolidation at elevated temperatures in $\mathrm{Zr}$ -based alloys [53]. The distribution of finely dispersed na- nocrystals increases the flow stress significantly. For example, a $\mathrm{Zr_{57}A l_{10}C u_{20}N i_{8}T i_{5}}$ sample containing a volume fraction of $40\,\%$ nanocrystals (Figure 1.9, curve b) seems to provide the best compromise between strength and ductility. + +# 1.4 Bulk Ceramic Nanocomposites for Desired Mechanical Properties + +Over the past half century, ceramics have received significant attention as candidate materials for use as structural materials under conditions of high temperature, wear, and chemical attack that are too severe for metals. However, one characteristic of these ceramics that has prevented them from being widely used is their inherent brittleness. Thus, significant scientific effort has been directed towards making ceramics more flaw-tolerant through design of their microstructure. An important example of high-toughness structural ceramics is self-reinforced silicon nitrides, which were first developed during the 1970s [54]. These materials have high toughness and room-tem- perature strength, along with good resistance to corrosion and oxidation. However, their high-temperature $({>}1000\,^{\circ}\mathrm{C})$ strength is compromised by low creep resistance and the occurrence of subcritical crack growth. These phenomena are caused by the softening of a glassy phase that is located at the grain boundaries as the temperature is increased. A possible way to overcome this problem is the fabrication of $\mathrm{Si}_{3}\mathrm{N}_{4}/\mathrm{Si}\mathrm{C}$ nanocomposites [55]. Also, several approaches have been used to improve the proces- sability (the sinterability of these materials is rather poor) and high-temperature prop- erties of monolithic silicon nitrides, with limited success. + +Considerable attention has been devoted to ‘functionally graded nanocomposite ma- terials’, for which gradually varying the dispersion (nanoparticles) to matrix ratio in chosen directions continuously changes the material. An example of such a material is SiC dispersions in a C (pyrolytic graphite) matrix, which has served well as thermal barriers on the space shuttle due to its excellent resistance to oxidation and thermal shock. One route to preparing these composite systems is by chemical vapor deposi- tion using multicomponent gas reactions. For example, 10–100-nm sized SiC disper- sions can be prepared with pyrographite using precursors of $\mathrm{SiCl_{4}}/\mathrm{C}_{3}\mathrm{H}_{8}/\mathrm{H}_{2}$ or $\mathrm{SiCl_{4}}/$ $\mathrm{CH}_{4}$ in CVD. The entire range of compositions from carbon to SiC has been prepared by this method [56]. Changing the deposition conditions can control the morphologies of the second phase and hence the microstructure of the composites; this control in- fluences the mechanical properties of the material. Nanocrystalline carbide-embedded composites, particularly those with amorphous or diamond-like carbon matrices, can be considered for tribological applications. The challenges here are to minimize for- mation of the $\mathrm{sp}^{2}$ soft graphite-like phase during synthesis and to retain a high fraction of the hard a-C matrix. Pulsed laser deposition (in which laser-ablated plumes from graphite are intercepted by a low-energy metal plasma produced by magnetron sput- tering) [57] at near room temperature has been used successfully to create ${\sim}10\ \mathrm{nm}$ TiC/TiCN nanocrystals embedded uniformly in a-C with diamond-like characteris- tics. Hardness values as high as 60 GPa, coefficient of friction values as low as 0.1, and high toughness values have been achieved in these films, which have high tribological value and possible applications in surface-protection coating technol- ogies. The two-phase heterogeneous structure in the nanocomposites provides crack deflection mechanisms, reducing the tendency toward easy brittle failure in these hard composites. + +Significant interest was generated in 1991 when Niihara [58] reported large improve- ments in both the fracture toughness and the strength of materials with a unique microstructure: ceramics with nanometer-range particles ( $.20\,{-}\,300\,\mathrm{\nm})$ ) embedded within a matrix of larger grains and at their grain boundaries. These ceramics report- edly showed up to $200\,\%$ improvement in both strength and fracture toughness, better retention of strength at high temperatures, and better creep properties. Materials like SiC, $\mathrm{Al}_{2}\mathrm{O}_{3}$ , $\mathrm{ZrO}_{2}$ and $\mathrm{Si}_{3}\mathrm{N}_{4}$ are excellent candidates for demanding structural applica- tions due to their mechanical and thermo mechanical properties (Figure 1.10) [59]. The incorporation of fine SiC and $\mathrm{Si}_{3}\mathrm{N}_{4}$ particles (smaller than $300~\mathrm{{nm}}$ ) in an alumina + +![](images/bfcd90b899bf364ca0d98f334526f79747f11728332258348687e9f6f8ecba56.jpg) +Fig. 1.10 Mechanical properties of a SiC/zirconia-toughened mullite nano- composite as a function of nanosized SiC content. (Source [59] used with permission) + +$(\mathrm{Al}_{2}\mathrm{O}_{3}.$ , a structural ceramic material) matrix first demonstrated the concept of struc- tural nanocomposites. The dispersion of these particles improved the fracture tough- ness from 3 to $4.8\ \mathrm{MPa}\:\mathrm{m}^{1/2}$ and the strength from 350 to $1050~\mathrm{MPa}$ at only 5 vol. $\%$ additions of SiC [60]. The bending strength of such composite was also measured to be $\sim\,1$ GPa. Further improvements in strength to about $1500\ \mathrm{MPa}$ were achieved by annealing the samples at $1300\,^{\circ}\mathrm{C}$ . The high strengths were maintained to about $1000\,^{\circ}\mathrm{C}$ . For $\mathrm{Si}_{3}\mathrm{N}_{4}/\mathrm{Si}C$ nanocomposites, several processing techniques have been used, including conventional powder processing, sol–gel processing, and polymer precursor processing; the SiC particles can originate from admixed commercial SiC powders, SiCN powders produced by plasma synthesis, in situ reaction pyrolysis of carbon-coated $\mathrm{Si}_{3}\mathrm{N}_{4}$ particles, and pyrolysis of a polycarbosilane based on SiCN precursor. An example of conventional processing is co-milling of the solid precursor powder materials followed by hot pressing, which produces nanocomposites posses- sing both inter- and intra- granular SiC particles [61]. Intragranular particles are most effective in toughening the material, because they are mainly responsible for crack deflection and crack impediment. Intergranular particles are often detrimental (in- itiating cracks), but they could provide some advantages by grain refinement during processing. The toughness of the composite depends on the volume fractions of these two types of dispersoids, and controlling these fractions precisely is challenging. In polymer-based processing, a mixture of $\mathrm{Si}_{3}\mathrm{N}_{4}$ powder, sintering additives, and poly- methyl phenyl si lane is pyrolyzed at ${\sim}1000\,^{\circ}\mathrm{C}$ in Ar and sintered in nitrogen. Another possibility is preparing an amorphous $\mathrm{Si/N/C}$ powder by crosslinking and pyrolysis of poly methyl sila zane [62]. Whereas the conventional process leads to a micronanostruc- ture with nano-sized SiC dispersions mainly inside $\mathrm{Si}_{3}\mathrm{N}_{4}$ grains, the polymer-proces- sing route results in atto (nano-nano) structures. Hybrid polymer/powder processing can also be applied to other composite microstructures such as $\mathrm{Al}_{2}\mathrm{O}_{3}/\mathrm{SiO}$ . Coating a silicon-containing polymer (polycarbosilane) onto alumina powder, followed by pyr- olysis, can result in a finer SiC nanophase [63]. Novel, superior processing routes can be used to prepare well dispersed ceramic nanocomposites, for example $\mathrm{Al}_{2}\mathrm{O}_{3}/\mathrm{SiO}$ . Colloidal consolidation and reaction sintering is one such process [64]: micro-size particles of the two phases are colloidally dispersed and consolidated to form uniform compacts. The SiC phase particles are then oxidized to reduce them to nanometer-size cores. The interfacial reaction between the oxidized SiC particles and the alumina matrix produces mullite. The advantage of this process is that particle sizes need not be brought down to nanoscale by milling and can be controlled well by oxidation. Also, due to the volume increase that occurs during reaction sintering, shrinkages often seen during sintering are small. + +Materials fabricated from polymer-derived powders made by hot pressing have yielded the best mechanical properties. These techniques are expensive and limit the shapes that can be fabricated. Gas-pressure sintering or pressure-less sintering are the most attractive processing techniques. However, to date, research on gas-pres- sure sintering has used mixed powders, which result in poor powder dispersion, ag- glomeration of SiC, and changes in the glass phase chemistry due to reaction with SiC. Thus, the focus for future processing rests on routes that produce commercially viable powders with uniform dispersion of SiC particles (with controlled size and volume fraction) that can be fabricated into dense components by gas-pressure or pressure-less sintering. The effects of reinforcement on the mechanical properties of silicon nitride nanocomposites is not well understood. As with the $\mathrm{Al}_{2}\mathrm{O}_{3}/\mathrm{SiO}$ nanocomposites, the presence of the SiC at the grain boundaries restricts grain growth, resulting in the formation of a matrix of finer $\mathrm{Si}_{3}\mathrm{N}_{4}$ grains. In fact, at higher volume percent of re- inforcements, grain growth is reported to be severely restricted, resulting in fine $a$ - $\mathrm{Si}_{3}\mathrm{N}_{4}$ grains and a composite showing superplastic (large strain to failure) beha- vior. The ‘nanosize’ strengthening and toughening effect due to thermal expansion mismatch proposed for $\mathrm{Al}_{2}\mathrm{O}_{3}/\mathrm{SiO}$ composites needs to be critically examined for $\mathrm{Si}_{3}\mathrm{N}_{4}/\mathrm{Si}\mathrm{C}$ composites: the difference between the thermal expansion coefficients of $\mathrm{Si}_{3}\mathrm{N}_{4}$ and SiC is small and in the opposite sense as in $\mathrm{Al}_{2}\mathrm{O}_{3}/\mathrm{SiO}$ . The presence of a liquid phase in $\mathrm{Si}_{3}\mathrm{N}_{4}$ and its composites further complicates matters: reactions between the reinforcements and the liquid phase could alter its composition and quan- tity, thus changing the sintering behavior and creep resistance. Thus, a more funda- mental understanding of the effect of nanosized SiC reinforcements on the behavior of $\mathrm{Si}_{3}\mathrm{N}_{4}$ matrix composites is required. In particular, systematic studies of the effects of reinforcement size and volume fraction on the microstructure, processing, and properties of these nanocomposites are needed. Such studies exist for composites in which the reinforcement size is in the several-micron range, and in these stu- dies, both the reinforcement size and volume fraction significantly affect the fracture toughness and strength [65]. + +For unreinforced $\mathrm{Si}_{3}\mathrm{N}_{4}$ ceramics, there is sufficient evidence that creep in the liquid phase-sintered materials is controlled by grain boundary transport and sliding. Thus, the incorporation of fine reinforcements at the grain boundaries may improve creep resistance [66]. The amount and viscosity of the intergranular phase due to chemical reactions and accelerated crystallization, the hindrance of grain-boundary sliding due to fixing by SiC particles, and the obstruction of easy diffusion paths by the SiC par- ticles explain the better creep resistance of the nanocomposites [67]. For $\mathrm{Al}_{2}\mathrm{O}_{3}/\mathrm{SiO}$ nanocomposites, the better creep resistance (5 vol. $\%$ SiC nanoparticles reduces the tensile creep of $\mathrm{Al}_{2}\mathrm{O}_{3}$ by 2–3 orders of magnitude) is explained by SiC nanopar- ticles fixing the grain boundaries, resulting in less grain-boundary sliding, a small viscoelastic contribution to creep, and enhanced grain-boundary strength, allowing plastic deformation of grains through dislocation motion [68]. The presence of non- bridged grain boundaries causes a rearrangement of the microstructure due to initial grain sliding. But this initial sliding process is stopped easily as boundaries fixed by the SiC nanoparticles are encountered; hence, only lattice mechanisms based on disloca- tion motion or, to a lesser extent, viscoelastic mechanism, contribute to creep. How- ever, in general, the studies on nanocomposites and the exact mechanisms of creep resistance have not been entirely conclusive, and systematically investigating the ef- fects of various microstructural parameters as a functions of particle size, oxidation, and volume of the dispersed phase is desirable. An additional factor that needs careful control is the volume fraction of reinforcements on the grain boundary as opposed to within the grain. The erosion properties of such composites also deserve mention. The residual surface stress induced by grinding and polishing nanocomposites (e.g., $\mathrm{Al}_{2}\mathrm{O}_{3}/\mathrm{SiC}$ ) and monolithic alumina are quite different, and studies show that the residual surface stress in the nanocomposite is more sensitive to surface treatment than that in the corresponding monolithic structure. Direct observations by transmis- sion electron microscopy suggest that deformation micromechanisms, from twinning in the alumina to dislocation generation in the nanocomposite, dominate the plastic deformation induced by the surface treatments. + +Crack-tip bridging by particles is a primary mechanism of strengthening ceramic nanocomposites. Small brittle particles (for example, silicon carbide particles in ${\mathrm{Al}}_{2}{\mathrm{O}}_{3}/$ SiC composites) cause crack tip bridging at small distances behind the cracks [69]. Residual stresses around the particles cause the strengthening mechanism to operate effectively even at small volume fraction loading of SiC. The increase in the nanocom- posite strength due to the reduction in the critical flaw size is achieved by the fine dispersion of nanoparticles [70]. Intergranular fracture, which is responsible for frac- ture in monolithic alumina, is suppressed in the nanocomposite, because the crack extension along grain boundaries is suppressed by particles that strongly bond the matrix–matrix interfaces. Hence, transgranular fracture is a common mode of frac- ture in nanocomposites. Tensile stresses in the intragranular particles induce the cracks to extend to the particles, leading to particle bridge toughening. + +Finally, dispersing metallic second-phase particles into ceramics is not only suitable for improving the mechanical properties of ceramics, but also presents a wide variety of advantages for addition of new functions such as magnetic, electric, and optical proper- ties, due to the size effect of nanosized metal dispersion. Granular films can be made with a ceramic phase embedded with nanosize metal granules [71] (e.g., $\mathrm{Fe/Al}_{2}\mathrm{O}_{3}$ , Fe/ $\mathrm{SiO}_{2}$ ). Such films display unusual and often enhanced transport, optical, and magnetic properties. An important parameter that affects the physical behavior of granular films is the percolation threshold of the embedded metal phase. Percolation effects in na- nocomposites are discussed later in this chapter. A dramatic manifestation of this percolation effect is the change, by many orders of magnitude, of the electrical resis- tivity (insulator-to-metal transition) as the percolation threshold is passed. The percola- tion threshold is generally found to be in the range of 0.5–0.6 metal volume fractions for nanocomposite films [72]. The mechanical properties can also undergo changes near the percolation threshold. The obvious explanation for this is that a change occurs in deformation behavior as the embedded phase forms connected networks that spread through the matrix phase. Conventional powder metallurgical methods and solution chemical processes like sol–gel, and coprecipitation methods can also be used to pre- pare composite powders for ceramic/metal nanocomposites such as $\mathrm{Al}_{2}\mathrm{O}_{3}/\mathrm{W}$ , Mo, Ni, Cu, Co, Fe; $\mathrm{{ZrO_{2}/N i}}$ , Mo; MgO/Fe, Co, Ni and so on. They are sintered in a reductive atmosphere that gives homogeneous dispersion of metallic particles within the ceramic matrices. The microstructural refinement by nanodispersion and the plasticity enhance the fracture strength, toughness, and/or hardness. + +In summary, the advantage of ceramic nanocomposites lies not only in the mechan- ical strength of the composite material but also in other mechanical properties such as fracture toughness, hardness, and creep resistance. The degree of improvement in these properties depends on the composite systems involved. Although there are some generalities in the strengthening and toughening mechanisms in such compo- sites, such as crack deflection and crack-tip bridging by the dispersed particles, the actual size, location, and volume fraction of particles strongly influence the final out- come with respect to mechanical behavior. + +# 1.5 Thin-Film Nanocomposites: Multilayer and Granular Films + +Thin-film nanocomposites are films consisting of more than one phase, in which the dimensions of at least one of the phases is in the nanometer range. These nanocom- posite films can be categorized as multilayer films, in which the phases are separated along the thickness of the film, or granular films, in which the different phases are distributed within each plane of the film (Figure 1.11). Multilayered thin-film nano- composites consist of alternating layers of different phases and have a characteristic thickness on the order of nanometers. These films are usually used for their enhanced hardness, elastic moduli, and wear properties. The elastic moduli is higher in multi- layered thin films than in homogeneous thin films of either component. The super- modulus effect [73] is observed in some metallic systems, by which, at certain char- acteristic thicknesses (typically ${\sim}2~\mathrm{nm}$ , corresponding to a bilayer consisting of one layer of each phase) of the film, the elastic modulus increases by more than $200\,\%$ . The most satisfactory explanation of this effect assumes an incoherent interface between the adjacent layers [74], suggesting that atoms are displaced from their equilibrium positions and that, during loading, all the layers undergo compression, which results in a higher resistance to deformation. The increase in hardness of the multilayer na- nocomposites has been explained [75] by considering that, when ultrathin films of materials with different dislocation line lengths are stacked, the strength approaches the theoretical limit. The dislocations cannot move from layer to layer, due to the difference in dislocation line lengths, and the films are thin enough that independent dislocation sources do not become operative. Conventional thin-film deposition tech- niques (sputtering, physical vapor deposition, CVD, electrochemical deposition, etc.) can produce multilayer nanocomposites, and the excellent flexibility of these techni- ques creates extremely thin films of uniform compositions. + +![](images/0334f9400c31282695561872bcd629e2c136d737f668cde00543ea838c38b5c4.jpg) +Fig. 1.11 Schematic of possible microstructures for nanocomposite and nanostructured coatings: isotropic dispersed multiphase microstructure (e.g., TiC/amorphous carbon), multilayered mi- crostructure (e.g., TiN/TiC) for nanocomposite coatings, and homogeneous alloyed microstructure as possible homogeneous nanostructured coatings (e.g., NiCoCrAlY alloy) + +Granular nanocomposite films are those that contain both phases (metal and cera- mic) in the same layer of the film and have no abrupt interfaces across the film thick- ness as in multilayer films. Here, one phase can be in the nanosize range (similar to dispersions), or both phases can have nanocrystalline grains distributed contiguously and laterally in the film. Granular nanocomposite coatings are also common; in these, the matrix phase is a polymer and the dispersed phase is inorganic. Granular films in which at least one distributed phase has electrical/magnetic properties are mainly used in electrical and magnetic applications [76]. In certain systems consisting of metal and ceramic particles (such as iron oxide/silver and alumina/nickel), changing the fraction of the phases present can alter the magnetic properties. At small volume fractions of the metal component, the material exhibits ferromagnetic behavior. Beyond a certain volume fraction of the metal phase, the ferromagnetic ordering gives way to superparamagnetism, because, once the metal particles attain percolation, the film behaves essentially like the metallic phase. An important parameter that critically affects the properties of granular films is the percolation threshold of the metal, which typically corresponds to a metal volume fraction of 0.5–0.6. A dramatic observation at percolation is the large (several orders of magnitude) change in electrical resistivity of such films, going from an insulating ceramic-like behavior to a conducting-metal behavior. Similar techniques (such as CVD or electrochemical methods) as used in multilayer film growth can also be used to prepare homogeneous nanostructured com- posite films. As an example, nanostructured AlN/TiN composite films were made by CVD using high-speed deposition of gaseous precursors $(\mathrm{AlCl}_{3}$ , $\mathrm{TiCl_{4}}$ , $\mathrm{NH}_{3}$ ) to form insoluble solid mixtures [77]. In the films so generated, the grain sizes corresponded to $8\,\mathrm{nm}$ (AlN) and $6\:\mathrm{nm}$ (TiN). The grain size range allows the films to have better duc- tility and greater toughness compared to bulk AlN/TiN composites. + +# 1.6 Nanocomposites for Hard Coatings + +Improved wear resistance, good high-temperature stability, and improved friction properties are important characteristics of good coatings for use in applications such as cutting tools. Most widely used coatings are made from TiN, TiC, TiAlN, $\mathrm{CrN}$ , diamond-like carbon (DLC), WC/C, $\mathrm{MoS}_{2}$ , $\mathrm{Al}_{2}\mathrm{O}_{3}$ , etc. For improved coatings in which lower friction, increased life time, increased toughness, higher thermal sta- bility, and in some cases, environmental (biomedical, for example) compatibility are needed, new types of materials are being considered, including nanocomposite ma- terials [78]. Nanocomposite structures such as multilayers or even isotropic coatings can be made from nanoscale entities with properties superior to single-phase materi- als. This approach of using nanocomposites is an alternative to using specific alloying elements in single-phase coating materials (to improve properties such as hardness) and provides far better flexibility in tailoring multifunctional coatings. + +Nanocomposite coatings usually consist of two or more phases combined as multi- ple layers [79] or as homogeneous isotropic multiphase mixtures. Classical multilayer (3 to many layers) coatings have total thicknesses of several micrometers, and the many layers are typically used to provide toughness (by crack deflection at the many interfaces) and properties such as oxidation resistance (due to, e.g., TiAlN layers). Building multilayer structures also provides better tribological properties [80]. Gradi- ent layers are also often introduced to counterbalance the vast differences in thermal expansion coefficients between multiple layers, which would cause internal stresses and delamination at the interfaces between layers. In nanoscale multilayer coatings, when the thickness of each layer is in the nanometer range, superlattice effects can increase the hardness and other properties of the coatings. The increased hardness of such coatings (e.g., TiN/VN), where the superlattice period is a few nanometers, can be orders of magnitude higher than that of the corresponding base materials [81, 82]. The hardness increase in these nanoscale multilayer films results mainly from hindering dislocation movements across the sharp interfaces between two materials having vastly different elastic (particularly shear modulus) properties and lattice mismatch (coherency strain) [83]. One interesting point to note is that, if the individual layers are very thin (less than $3\,{-}\,5~\mathrm{nm}$ typically), the hardness increase can disappear, be- cause the strain field around the dislocations falls mainly outside the particular layer. Also, the sharpness of the interface also affects the hardening mechanisms, because high-temperature interdiffusion between layers can decrease the sharp varia- tion in shear modulus. Another possible reason for increased hardness and improved chemical stability is chemical differences intentionally inserted at the interfaces so as to induce strains and electronic interactions; for example, TiAlN/CrN multilayer struc- tures are more efficient than TiAlN films [84]. The hardness effect resulting from lattice mismatch disappears at multilayer periods greater than $10~\mathrm{{nm}}$ , due to lattice relaxation (Figure 1.12). Hence, for practical applications, the thickness of the layers in these coatings is designed based on the above considerations so as to obtain the opti- mum hardness values and wear properties with appropriate temperature stability. In fact, for many coating applications (e.g., coatings for cutting tools), toughness at high operating temperatures and chemical stability are more crucial than hardness alone [85]. Commercial multilayer coatings with multilayer periods in the nanoscale range do exist; for example, WC/C coatings used in the cutting tool industry. When the multilayer nanoscale coatings are deposited by periodic variation of the deposition (e.g., sputtering) conditions, a templating effect is often observed. In building superlattice structures from different materials of different structures, the layer first deposited can force the next layer (of the different material) to adopt the crystallo- graphic structure of the first layer; this occurs, for example, in $\mathrm{TiN}/\mathrm{Cr}_{2}\mathrm{N}$ coatings in which $\mathrm{Cr_{2}N}$ is forced into the structure of the TiN (fcc) underlayer [86] and in TiN/AlN, in which the AlN (originally wurtzite) is forced into the NaCl structure of the TiN. + +In addition to nanoscale multilayer coatings, it is also possible to fabricate isotropic nanocomposite coatings consisting of crystallites embedded in an amorphous matrix, with grain sizes in the nanometer range. These coatings generally have one phase that is hard (load bearing; e.g., transition metal carbides and nitrides) and a second phase that acts as a binder and provides structural flexibility (amorphous silicon nitride, amorphous carbon). The formation of these composite structures involves phase se- paration between two materials (which show complete immiscibility in solid solution), + +![](images/c56e01642c4163d2630327f4ca429c137b2727c5d4b7d3c8da8c13ff254a148d.jpg) +Fig. 1.12 Hardness of several multilayered nanocomposite coatings as a function of multilayer period in the coatings. The individual layers in these coatings can differ in inherent elastic properties (modulus) as well as in structure and lattice spacing. (Source [78] used with permission) + +which are often codeposited, for example, by sputtering or plasma deposition. Unlike in multilayer composite systems, the possible material compositions and particle sizes in nanocomposite coatings are restricted by material properties and deposition con- ditions. Typically, these nanocomposite coatings are deposited by plasma-assisted che- mical vapor deposition (PACVD) or physical vapor deposition (PVD). Very hard $({\sim}50-$ $60\,\mathrm{\mathrm{GPa},}$ ) coatings of nanocomposites have been made from a $\mathrm{TiN}/\mathrm{a}{\cdot}\mathrm{Si}_{3}\mathrm{N}_{4}$ system, using PACVD from $\mathrm{TiCl_{4}}$ , $\mathrm{SiCl_{4}/S i H_{4}}$ , and $\mathrm{H}_{2}$ at about $600\,^{\circ}\mathrm{C}$ [87]. The nanocomposite con- tains nanocrystalline TiN $(4\!-\!7\ \mathrm{nm})$ in a matrix of amorphous $\mathrm{Si}_{3}\mathrm{N}_{4}$ . The gas phase nucleation (uncontrollable rates), chlorinated precursors (unreacted species remain in the process, contaminating the films), and high processing temperatures are disad- vantages in this process. PVD processing can be used to prepare the same coatings by sputtering Tian dSi targets in nitrogen gas at room temperatures.The disadvantage is that the films are of inferior quality, and the hardness is lower than the PACVD-de- posited nanocomposite coatings. In addition to improved hardness, nanocomposite coatings also show better oxidation resistance in comparison to TiN coatings. + +It is interesting to note and analyze the high hardness of these nanocrystalline par- ticles/amorphous matrix coatings. Typically in single-phase nanocrystalline materials, the major increase in hardness comes from the lack of plastic deformation because the dislocations face far more barriers to mobility. However, hindered dislocation alone cannot explain the superior hardness properties of these nanocomposite coatings. + +Since the nano crystal lites within the amorphous matrices are only a few nanometers large, dislocation formation simply does not occur, and hence, plastic deformation is largely quenched. Plastic deformation occurs through a pseudo-plastic deformation in which the nanocrystals move against each other. Since this process requires higher energy, resistance to plastic deformation in these materials is relatively high. Crystal- line carbides in amorphous carbon matrices are a good example of this category of nanocomposites and can provide hard, low-friction coatings [88]. Ti/C systems are most promising in this regard. Advances in laser-assisted deposition techniques (e.g., magnetron sputtering-assisted pulsed-laser deposition) have aided the fabrica- tion of hard composites with nanocrystalline and amorphous phases [89]. The volume fraction and particle size of the coatings can be adjusted to obtain optimum properties in terms of toughness and hardness. The design of appropriate particle size allows for the optimum generation of dislocations and micro- and nano-cracks, which result in a self-adjustment in composite deformation from hard elastic to plastic at loads exceed- ing the elastic limit. Thus, compliance of the coatings is improved and catastrophic brittle failure is avoided. Such load-adaptive nanocomposites based on optimal design of the composite microstructure are extremely useful in applications subject to wear. + +Hard coatings such as TiN and Ti-C-N/DLC have significant advantages in aero- space systems. Apart from their hardness, toughness, and low friction properties, these coatings also resist attack by corrosive fluids that are used (as engine oils or lubricants) in aircraft engines and parts. Magnetron sputter-assisted pulsed-laser de- position is a good candidate technique for depositing such coatings in different con- figurations, such as functional-gradient, multilayer, and granular nanocomposites [90]. The tailoring of the different architectures in creating appropriate coatings that blend the various tribological properties is key to creating high-performance hard coatings. Nanostructured coatings provide several pathways for reducing stress in hard coatings and terminating cross-sectional dislocations, with possibilities for controlling dislocation movement by dispersion strengthening or lattice mismatching between alternate layers, as discussed above. + +Other systems that create excellent nanocomposite-based, low friction, hard coatings are based on carbide particles (Ti, Ta, Nb, etc.) in DLC matrices [91–93] (Figure 1.13). Several techniques, such as PVD, pulsed-laser deposition, and reactive magnetron sputtering, are used to deposit these coatings. These typically contain larger crystalline particles $(10\,{-}\,50\ \mathrm{nm})$ surrounded by thick amorphous carbon coatings $(5\ \mathrm{nm})$ ). The particle sizes are large enough to allow dislocations but are too small for crack pro- pagation. The larger grain separation allows incoherency strains to develop and, under loading, cracks to originate between crystallites, which allow pseudoplastic deforma- tion. Thus, their hardness is also much higher $(\sim\!30\,\mathrm{GPa})$ ) than that of single-crystalline carbide materials, and these coatings have, in addition, much higher toughness. These types of coatings can also be modified by introducing other elements; for example, W or $\mathrm{Cr}$ for creating optically absorptive coatings for solar energy converters [94] and materials such as $\mathrm{MoS}_{2}$ $\mathrm{(TiN/MoS_{2}}$ , $\mathrm{TaB_{2}/M o S_{2}}$ ) for lubricating coatings [95]. + +Metal carbide/ductile metal systems are considered for cutting tools, because the carbide phase provides hardness and the metal provides toughness. For example, com- posites such as WC/Co, WC/TiC/Co are commonly used in cutting and forming tool + +![](images/8a70659db12ff604cc815e312adafb10b3b0f0a04368fd9140c3571635f23ffb.jpg) +Fig. 1.13 Plan-view transmis- sion electron microscope ex- aminations of W-DLC layers: (a) electron diffraction pattern of a W-DLC with 9.7 at. $\%$ W, (b) electron diffraction pattern of a W-DLC with 32.0 at. $\%$ W, (c) dark-field micrograph of the W- DLC layer with 9.7 at. $\%$ W, taken with WC rings. (Source [93] used with permission) + +applications, and these can be synthesized by various routes [96]. These nanocompo- sites, in which the particle or grain sizes of the component phases are in the nan- ometer range, have much better mechanical properties (strength, hardness, tough- ness). Typically, reductive decomposition of W- and Co-containing salts, followed by gas-phase carburization (with $\mathrm{CO/CO_{2}})$ are used to prepare nanocomposites such as WC/Co. These processes produce carbon-deficient metastable carbide phases with inferior mechanical properties. Alternative approaches, in which a polymer pre- cursor such as poly acrylonitrile is used as the carbon source during the chemical synthesis and subsequent heat treatment to obtain high-quality nanocomposites (par- ticle sizes ${\sim}50\,{-}\,80\;\mathrm{nm})$ ), have been developed [97]. $\mathrm{TaC/Ni}$ nanocomposites are inter- esting from two aspects: excellent thermal stability and outstanding mechanical prop- erties [98]. They are used as surface coatings for protection against wear and corrosion. These nanocomposites can be prepared by devitrification of sputtered amorphous films of $\mathrm{{Na/Ta}/C}$ of non stoic hio metric composition. The grain size of the matrix Ni is about $10\,{-}\,30\;\mathrm{nm}$ , and the TaC particles $({\sim}10\,{-}\,15\;\mathrm{nm})$ ) are uniformly distributed in the matrix. Even at $700\,^{\circ}\mathrm{C}$ , no grain growth is observed, suggesting excellent ther- mal stability for these nanocrystalline duplex-phase composites. The measured hard- ness of the composite (12 GPa) matches that of conventional WC/Co nanocomposites at a much reduced volume fraction $(35\,\%)$ of the composite. + +Chemical and physical vapor deposition (CVD and PVD, respectively), and laser ablation have been used to prepare a variety of superhard nanocomposites made of nitrides, borides, and carbides. In these systems, the hardness of the nanocomposite significantly exceeds that given by the rule of mixtures in bulk. For example, the hard- ness of nanocrystalline $\mathrm{{M}_{n}\mathrm{{N/a}\mathrm{{Si}_{3}\mathrm{{N}_{4}}}}}$ $\mathrm{(M=Ti}$ , W, V, etc.) nanocomposites with the optimum content of $\mathrm{Si}_{3}\mathrm{N}_{4}$ (close to the percolation threshold) reaches 50 GPa, although that of the individual nitrides does not exceed 21 GPa. For a binary solid solution, such as $\mathrm{TiN}_{1\cdot x}C_{x},$ the hardness increases monotonically with increasing $_x$ from that of TiN to the that of TiC, following the rule of mixtures. Recently, super- hardness has been achieved in coatings consisting of a hard transition-metal nitride and a soft metal that does not form thermodynamic ally stable nitrides, such as nano- crystalline $\mathrm{M}_{n}\mathrm{N}{\cdot}\mathrm{M}^{\prime}$ ( $\mathrm{TM}=\mathrm{Ti}$ , Cr, $\mathrm{Zr}$ , $\mathrm{M}^{\prime}=\mathrm{Cu}$ , Ni) [99]. However, these systems have low thermal stability and their hardness decreases upon annealing to $\geq400\,^{\circ}\mathrm{C}$ . + +The generic concept for the design of novel superhard nanocomposites that are stable up to high temperatures $({\sim}1000\,^{\circ}{\bf C},$ , which is very important in industrial applications) is based on thermodynamic ally driven segregation in binary (and ternary) systems. These systems display immiscibility and undergo spinodal decomposition even at such temperatures. In these systems, any small local fluctuation in the composition of the mixed phase decreases the free energy of the system, thus leading to spontaneous segregation and, as a result, a nanocomposite that remains stable against grain or particle coarsening develops. For many hard coatings (e.g., PVD deposited films, in which the compressive stress is typically 4 to 6 GPa or even greater than 10 GPa), the measured enhancement of the apparent hardness and elastic modulus is due to high biaxial compressive stress. It is important to perform annealing experi- ments in such systems to verify what the real, ‘intrinsic’ hardness of the film is. The observed high resistance of superhard nanocomposites to crack formation has been studied in terms of conventional fracture mechanics scaled down to dimensions of 1–2 nm. An analysis of indentation curves measured on superhard nanocomposites in terms of Hertzian elastic response shows that they are indeed strong materials. The stress concentration factor based on nanoscale flaws is low, and therefore, the stress needed to propagate such a small nanocrack is very high. The propagation of such nanocracks in 3D nanocomposites involves much deflection and branching of the plane of the cracks, which hinder growth of the nanocracks. However, the self- organization of these systems, due to thermodynamic ally driven spinodal segregation, results in a very low concentration of intrinsic flaws. Thus, the remarkably high re- sistance of these nanocomposites to crack formation can be understood in terms of a high threshold for the initiation of larger microcracks, which may lead to their cata- strophic growth. + +Carbon–carbon nanocomposites : Carbon–carbon composites have received much attention due to their tremendous potential for applications in the aerospace, automo- bile, and energy industries. This material’s attractive properties include low specific weight, low thermal expansion, high thermal shock resistance, and high strength over large temperature ranges. They are usually made with carbon fiber reinforcement, with the matrix being disordered or partially graphitized carbon infiltrated from var- ious resins, pitches, and gaseous (hydrocarbon) precursors. The main limitation of + +![](images/503c9a6d377bc3e51696bc914dd1d730679794cd28d9eb4d9ee9cfe047e22466.jpg) +Fig. 1.14 Comparison of the tensile strength (a), modulus (b), and electrical resistivity (c) of composite carbon fibers with 1 and 5 wt. $\%$ nanotube loadings with the corresponding values in unmodified isotropic pitch fibers. Each data point represents an average value for strength, modulus, or resistivity obtained from 12 composite fibers. Standard deviations of experimental values are strength, $15\,\%$ ; modulus, $15\,\%$ ; and resistivity, $20\%$ . (Source [100] used with permission) + +these composites is their susceptibility to oxidation, but surface modification or coat- ing of the fibers or addition of appropriate ceramic refractory fillers that shield against oxygen attack at high temperatures alleviates this problem. The advent of several novel forms of nanocarbons has led to attempts to create nanocomposites with nanocarbon + + (e.g., nanotubes) components. In a recent experiment, single-walled carbon nanotubes + + $\lnot5\,\%$ by weight) were uniformly dispersed in an isotropic petroleum pitch matrix to make nanotube-based carbon nanocomposite fibers (Figure 1.14), with better electri- cal and mechanical properties than isotropic pitch-based carbon fibers [100]. The ten- sile strength increased nearly $100\,\%$ , and the electrical conductivity increased four- fold. The advent of the new carbon nanostructures suggests that novel carbon–carbon composites can be prepared with properties far superior to those of conventional car- bon–carbon-fiber-based composites. The challenges lie in achieving proper disper- sion of these nanostructures and tailoring the interfacial strength between the nano- carbons and the matrix. + +# 1.7 Carbon Nanotube-Based Nanocomposites + +The mechanical behavior of carbon nanotubes is exciting, since nanotubes are seen as the ultimate carbon fiber ever made [101]. The most important application of nano- tubes, based on their mechanical properties, will be as reinforcements in composite materials. Chapter 2 includes a detailed discussion of nanotube-based polymer com- posites; here, we briefly discuss some developments regarding ceramic/metal-matrix- based nanotube composites. The nanotube reinforcements promise to increase the fracture toughness of the composites by absorbing energy through their highly flex- ible elastic behavior during deformation, which will be especially important for nano- tube-based ceramic matrix composites. Possible applications are in lightweight armor or conductive durable ceramic coatings. An increase in fracture toughness on the order of $10\,\%$ has been seen in nanotube/nanocrystalline SiC ceramic composite fab- ricated by a hot-pressing method at 2273 K (25 MPa in Ar for $^{1\textrm{h}}$ ) [102]. + +Nanoscale ceramic powders with carbon nanotubes provide another opportunity for creating dense ceramic-matrix composites with enhanced mechanical properties. The strength and fracture toughness of hot-pressed $a$ -alumina is typically much greater than that of conventional grain-size polycrystalline alumina. Addition of carbon nano- tubes to the alumina results in lightweight composites with even greater strength and fracture toughness (Figure 1.15). The mechanical properties of such composites de- pend strongly on the processing method and surface treatment of the carbon nano- tubes. Sintered alumina has high strength, hardness, and fracture toughness. An ex- citing possibility, as well as a processing challenge, is incorporating carbon nanotubes into an alumina-matrix composite to improve these properties. Alumina ( c -phase) matrix composites with 5–20 vol. $\%$ MWNT (multiwalled nanotubes) have been fab- ricated [103]. The $\gamma$ -phase alumina powder was transformed to $a$ -alumina (with mean particles sizes close to $60\;\mathrm{nm}$ ) during sintering at $1300\,^{\circ}\mathrm{C}.$ . The MWNT were lightly oxidized at $640\,^{\circ}\mathrm{C}$ in air, which removed the disordered carbonaceous material and + +![](images/3606b56d075809b0abbf184d5b25d9239253f1ff46ccbbb506db180560f54f08.jpg) +Fig. 1.15 (a) Fracture tough- ness of multiwalled nanotube (MWNT)/alumina nanocompo- site compared with that of sin- tered nanophase alumina. The inset in (a) shows an indent and resulting cracks. The micro- structure of the nanocomposite is shown in (b). (c) Diametral strengths of alumina-matrix na- notube composites hot-pressed in Ar at $\rceil300^{\circ}\mathsf{C}$ and $60\,\mathsf{M P a}$ for 1 h and containing various amounts of MWNT. (Source [103]) + +![](images/17ef750cc240ef0bfcfc3900d12ee425fafae564a157e12a65aea171298253e2.jpg) + +made it easier to disperse the nanotubes. The alumina- MWNT mixtures were dis- persed in organic solvents via ultrasound, dried, and then sintered by hot pressing in a graphite die at $1300\,^{\circ}\mathrm{C}$ and a pressure of $60\ \mathrm{\Mpa}$ . The structures of the MWNT remained the same before and after processing, without any visible degrada- tion. The density of the sintered composites was $>97\,\%$ of the theoretical density. + +The diametral strengths of alumina-MWNT composites with different MWNT con- tents indicate great improvements in fracture toughness with addition of nanotubes. The bulk alumina made from nanoparticles had a strength of $654~\mathrm{MPa}$ , compared to typical strengths of $200\,{-}\,350\;\mathrm{MPa}$ for sintered bulk alumina, and was even comparable to strengths reported for single-crystal alumina (sapphire) of $350\!-\!1000\:\mathrm{MPa}$ . With the addition of purified, well-dispersed nanotubes, the strength first increased at 5 and 10 vol. $\%$ MWNT and then decreased for higher vol. percents of MWNT. The fracture toughness of bulk alumina and alumina/MWNT composites shows that, for 5 vol. + + $\%$ MWNT composites, the toughness increased by about $25\,\%$ to about $5.1\,\mathrm{\MPa} + +$ $\sqrt\mathrm{m}$ . The toughness values were higher than those reported for single-crystal alumina + + (sapphire) and polycrystalline alumina. The hardness of the composite depends on how well the nanotubes are dispersed in the matrix and how the nanotube surface is modified (with functional groups via oxidation) to produce anchors to the ma- trix; for the best value, the Vickers hardness is $20.4~\mathrm{GPa}$ compared to $18.4~\mathrm{GPa}$ for bulk alumina. From the general trends in the results of a few scattered reports in the literature, the strength and fracture toughness of bulk structures hot-pressed from nanophase ceramic powders is much higher than the typical strength and tough- ness of conventional polycrystalline ceramics. The addition of small amounts of na- notubes to nanophase ceramics to form dense nanocomposite improves the diametral strength and fracture toughness. However, the real improvements occur when the nanotubes are purified and well dispersed in the matrix, so that large aggregates do not create macrodefects in the structures, leading to easy fracture and lower strength. Vacuum hot pressing also removes entrapped gases in the composite pow- der mixture, preventing formation of residual stresses and reduction of strength. These processing improvements show the great promise of nanophase alumina/ MWNT composites for light-weight, high-strength, high-toughness materials and in applications such as impact-resistant body armor. + +It is widely perceived that carbon nanotubes will allow the construction of compo- sites with extraordinary properties. Metal-matrix composites, as discussed before, are a new range of advanced materials and normally, the liquid-phase fabrication method is an efficient process because of its simple processing. If carbon nanotubes can remain stable within some suitable high-strength metal matrix, outstanding nanocomposites may be obtained. The rapid solidification technique allows the extension of alloying levels and more refined microstructures, which can lead to enhanced mechanical and physical properties. In addition, rapid solidification processing may help carbon na- notubes to survive the process because of the controllable melting step, which is the most important issue to be solved in fabricating nanotube-reinforced composites. Ra- pid solidification processing has achieved the successful synthesis of nanotube/ $\mathrm{Fe_{80}P_{20}}$ metallic glass nanocomposites [104]. Their properties show that the addition of nanotubes greatly enhances the thermal stability of the glass matrix and increases + +![](images/fa72ba00bc8a5a6b74f28b5d0dfac73f9d704d59d1f95d2e5c8ddd79a04ceddf.jpg) +Fig. 1.16 Magnetic susceptibility vs. temperature for various amorphous al- loys in nonisothermal heat treatments. The temperatures at the end of the first transition are indicated. The first peak in the plots corresponds to the first crystallization process. The onset temperature of crystallization for amorphous $\mathsf{F e}/\mathsf{P}/2\,\%$ CNTs is about $100~\mathsf{K}$ higher than that for amorphous Fe/P alloy at the same heating rate. (Source [105] used with permission) + +the low-temperature electrical resistivity by $70\,\%$ (Figure 1.16; Table 1.2). The activa- tion energy of crystallization for the nanotube composite is higher than those of both the $\mathrm{Fe_{80}P_{20}}$ metallic glass and the metallic glass containing $2\%$ carbon. The onset temperature of crystallization for the nanotube composite is almost $100~\mathrm{K}$ higher than for the original glass, according to magnetothermal analysis (MTA). + +![Tab. 1.2 The onset temperatures at a heating rate of $5\;\mathsf{K}\;\mathsf{m i n}^{\cdot1}$ and activation energy of crystallization for various amorphous metallic glass nanocomposite materials containing carbon (C) and Carbon Nanotubes (CNTs) [105]. ](images/c34b8420ac6bdc841eca833c133d220e3c0c123bd27e638d66b4b410a529b050.jpg) + +# 1.8 Functional Low-Dimensional Nanocomposites + +Recent years have seen a drive towards creating heterogeneous nanostructures, which could serve as multifunctional materials and essentially satisfy the definition of func- tional nanocomposites. These stand out from the bulk nanocomposites that are the main subject of this book, but will have important applications as nanoscale sen- sors, electronic and optical nanodevices, nanoprobes, nano electromechanical sys- tems, unique reinforcements, and drug-delivery media. These are particularly inter- esting for layered material systems in which atomic layers of different material com- positions (e.g., graphite, dichalcogenides, clays, etc.) can be assembled into different geometries on the nanoscale. Some excellent examples, which are described below, are endohedral fullerenes (metal/carbon systems), filled carbon nanotubes, boron/car- bon/nitrogen nanostructures, and nanosize coaxial cables from a wide range of layered materials. + +What are the various synthetic strategies for creating such composite nanoscale structures? One simple, elegant idea is to use infiltration of nanoporous struc- tures, commonly used in bulk composites for creating heterogeneous materials. At the nanoscale, this problem is not trivial, especially if the process occurs through the infiltration of a liquid or molten phase in which the viscosity of the infiltrating phase is high. A very useful example of such a nanocomposite structure is nanotubes and nanowires built into porous alumina templates. The alumina template can be prepared by anodization of Al metal in acidic solvents, and it is possible to create templates with straight, random pores that have diameters of $20\,{-}\,50\;\mathrm{nm}$ and are sev- eral micrometers long, running through the thickness of the alumina template [106]. Once the templates with pores are prepared, they can be filled (via electrochemical deposition) with nanowires (such as Au, Co, etc.; Figure 1.17) by changing the elec- trochemical conditions and introducing metal-containing solvents into the electrolyte [107]. The structure of the metal wires can be tailored (amorphous or crystalline) by changing the electrochemical deposition conditions. The distinct difference between nanowires created by other techniques and by template-based synthesis is that the latter technique offers the possibility of creating well-separated individual nanowires with adjustable diameters. Structures such as carbon nanotubes can also be deposited in the pores defined by such templates, by infiltration of gaseous hydrocarbon precur- sors (e.g., acetylene, benzene) and subsequent annealing [108]. Electrochemical tech- niques for materials processing (from highly anisotropic wires inside porous tem- plates to particle-dispersed thin-film nanocomposites) provide a powerful alternative technique to vapor phase deposition and solid state reaction for the synthesis of na- nomaterials and nanocomposites [109]. Their main advantage is control of the process so as to modulate dimensions and compositions. Their disadvantages are mainly poor crystallinity (being a low-temperature deposition process) and contamination with im- purities from the electrochemical baths. + +![](images/7abed47a41a0ef2f2f5578c20ceaf9e499d92355f3f889d29433ce5574a92b1e.jpg) + +![](images/1854f3018c4b83c4eb71b424071730c0fadf5ae369ec48ab7acfe9f0aa417af1.jpg) +Fig. 1.17 Nanowires grown in alumina templates. Top schematic shows how pores in a straight-pore anodic alumina template are filled with metal by electrode position. Scanning electron micrograph of Co nanowire-filled template is shown in lower image; an individual nanowire removed from the template is shown (right). (Courtesy Prof. Mamoun Mu- hammed, The Royal Institute of Technology Stockholm, Sweden) + +# 1.8.1 Encapsulated Composite Nanosystems + +Here, we describe the creation and characterization of composite carbon/metal-based nanostructures (systems consisting of combinations of a layered material and a solid phase, which have been studied by several groups in recent years). These are unique nanocomposite systems with promise for many interesting applications. + +Graphite-encapsulated metal/ceramic structures are interesting systems, and the generation of metal/carbon nanosystems requires knowledge of the interactions of metal atoms and the graphite lattice. The experimental study of the behavior of indi- vidual metal atoms in a graphitic environment is difficult and has been successful in only a few cases. In 1993, Ruoff et al. [110] performed carbon-arc discharge experi- ments using C anodes filled with metals and metal carbides and obtained carbon na- notubes and polyhedral carbon nanoparticles that contained both encapsulated metals and metal carbides. Polyhedral graphitic particles generally contain a hollow core, which is given shape by the outer faceting of the particle, and metal nanoparticles can easily fill these cores during formation (Figure 1.18). Metal surfaces are expected to act as nucleation sites for graphitic carbon to surround. The expected tendency of preexisting graphitic filaments is to wrap around metallic nano crystal lites and close by + +![](images/d81f16c1cf4008425bfc84956da8f273b16d7c5fef9ab649156d409bcd78431b.jpg) +Fig. 1.18 (a) Schematic of an endohedral fullerene. Structural model of a La atom in ${\mathsf{C}}_{82}$ fullerene (b) Encapsulated nanoparticle of lanthanum carbide inside a carbon polyhedral particle (approximately $50\:\mathsf{n m}$ in diameter) produced in an arc discharge with a composite carbon/La electrode. (c) Encapsulated nanoparticles of metals (Fe, Co, Au) in spherical graphitic layers (carbon onions). These structures were produced by electron irradiation of carbon soot containing the metal species + +saturating all carbon dangling bonds. The properties of encapsulated metal crystals have been the subject of many experimental studies [111]. Graphite-coated ferromag- netic metal crystals should show a reduced magnetic coupling that depends on the thickness of the graphite shell [112], with potential applications in magnetic recording media. Guerret-Piecourt et al. [113] sought general principles relating to the nature and structure of the filling materials, for efficient filling of nanotubes made via the electric- arc method (Figure 1.19). Several metals and/or their compounds have been encap- sulated in carbon cages and studied; e.g., Ti, Cr, Fe, Co, Ni, Cu, Zn, Mo, Pd, Sn, Ta, W, Gd, Dy, and Yb. In later experiments, the authors also managed to fill hollow carbon nanostructures with semiconducting structures such as Se, S, Sb, and Ge by similar processes [114]. S, present as impurity in the carbon, was later reported to play a major role in the effective formation of the filled carbon nanotube/nanoparticle structures. The above technique has made it possible to encapsulate a large number of carbides nanocystallites [115, 116] $(\mathrm{LaC}_{2}$ , $\mathrm{Y}\mathbf{C}_{2}.$ , $\mathrm{{CoC}_{2}}$ , ${\sf G d}_{2}{\sf C}_{3}$ , TiC, $\mathrm{V}_{4}\mathrm{C}_{3}$ , $\mathrm{ZrC}$ , TaC, MoC, NbC, HfC). With pure metals such as Fe, Ni, or Co, which strongly interact with carbon, encapsulating nanowires inside nanotubes is challenging. However, experiments have shown that onion-like nanoparticles of metal-containing carbon clusters can be pro- duced relatively easily with the arc method. Here, the metal or metal carbide encap- sulate is protected by the C layer coating, which effectively insulates the magnetic metal particles from oxidization upon exposure to the environment and retains its magnetic properties. It is noteworthy that high melting point refractory metal carbides such as TaC, NbC, and MoC, which exhibit superconducting transitions at low tem- peratures, have been inserted into nanotubes and graphitic nanoparticles [117]. Bulk magnetic susceptibility measurements of encapsulated TaC, MoC, and NbC revealed diamagnetic responses and superconducting transitions between $10~\mathrm{K}$ and $14~\mathrm{K}$ , re- spectively. The structure and state of these encapsulated nanostructures affects their physical properties; for example, a shrinkage in the lattice parameters for the (fcc) MoC encapsulated structure appears to decrease the Tc by $3.8~\mathrm{K}.$ . + +Magnetic alloy (e.g., FeCo) nanoparticles encapsulated in carbon layers can also be made in radio frequency (RF) torch reactors, and acetylene was used to generate a large number of carbon-coated particles [118]. The carbon lamination provides several func- tions for these nanoparticles that have low magneto crystalline anisotropy and large saturation magnetization (making them valuable as soft magnetic materials). These functions include oxidation protection, prevention of coarsening and particle coales- cence during growth, shielding interparticle magnetic interactions, and even reducing eddy current losses in high frequency environments. This technique can readily pro- duce particle sizes anywhere from 5 to $50\,\mathrm{nm}$ . Other techniques, such as the tungsten arc, blown arc, or electric arc generation, can be used to create encapsulated nanos- tructures of similar dimensions. + +The first experiments on filling of nanotubes date to 1993, where the filling was done via capillary forces, which pushed molten metals and compounds into the cav- ities of nanotubes during oxidative opening of their tips (Figure 1.20) [119]. Since then, the most efficient filling of hollow nanostructures to create heterogeneous nanoma- terial systems has been achieved by chemical treatment. Tsang et al. [120] filled larger + +![](images/fd15bb09d28b9b6d13c5bec8dafc7e2de3620b1069f2490b7acdaf7a45ad1a6d.jpg) +Fig. 1.19 (a) High-resolution transmission electron micrograph of a Mn metal-filled carbon nanotube made by electric arc discharge with electrodes of carbon filled with Mn (b) Encapsulated nanotubes with Fe carbides, formed during a chemical vapor deposition process involving hydrocarbon and metalorganic precursors + +![](images/5a708ba42bfee0dc70cede7b2f6ec5fc2e0fafd1881649cc71ae105c39a0c7e8.jpg) +Fig. 1.20 Top: (a) High-resolu- tion transmission electron mi- cro graph of aKI crystal in a 1.4-nm diameter SWNT, (b) its structural representation. Encapsulation of halides and other materials within SWNTs enables the study of 1D crystal structures formed on the smallest scale possible. Bottom: (c,d)Schematic of a filled MW NT and a corresponding transmis- sion electron micrograph of Pb oxide nanowire-filled nanotube, formed by capillary filling of the molten material inside nanotube cavities.In(d)the horizontal lines indicate graphite planes which have a separation of $0.34\ \mathsf{n m}$ . (Source, top image and sche- matic [233] used with permis- sion) + +multi-walled carbon nanotubes oxidatively by opening the ends in nitric acid and then filling with solutions containing metal oxides (e.g., oxides of Ni, Co, Fe, U, Mo, Sn, Nd, Sm, Eu, La, Ce, Pr, Y, Zr, Cd) using capillary force. They were then dried to form encapsulated solid materials inside the nanotube cavities. Several other material sys- tems such as Pd, Ag, Au, AuCl, proteins, and enzymes have been introduced into nanotubes with this method. Recently it was reported that single-walled nanotubes (SWNTs) can be opened selectively and filled by wet-chemistry techniques [121]. Treat- ment with concentrated HCl leads to the selective opening of SWNTs at their tips, and foreign materials may be drawn in, in a similar fashion as MWNTs; the much smaller $({\sim}1\;\mathrm{nm})$ cavities in SWNT make the filling more challenging. Usually, the nanowires formed inside the nanotubes are single crystals. Alternative strategies to fill nanotube cavities with materials have been explored. For example, exposing SWNT nanotubes filled with $\mathrm{AgCl/AgBr}$ to light or an electron beam can reduce the encapsulated com- pounds, resulting in extremely small Ag nanowires [122]. Similarly, constrained che- mical reactions within narrow channels can be used to create and characterize nano- wire structures of many materials. These promise to be ideal structures for studying 1D properties (electrical, thermal, optical) in materials for applications such as quan- tum wires and magnetic storage arrays. Characterization of these structures is often challenging, but recent progress in highly spatially resolved microscopy and spectro- scopy techniques have enabled researchers to carefully study the structure–property relationships in confined nanocomposite wires. SWNTs containing fullerenes in their inner hollow are another unique form of nanocomposite structure with interesting electronic properties. ${\bf C}_{60}$ insertion into SWNT could occur during their formation in the arc or laser process. Very recently, researchers have managed to introduce ${\sf C}_{60}$ inside SWNTs in high yield (Figure 1.21) by thermally annealing ${\sf C}_{60}$ powders over SWNTs above $600\,^{\circ}\mathrm{C}$ under vacuum [123]. This class of hybrid materials may form the basis of functional devices for applications in electronics, biomedicine, and micro electromechanical systems. Similarly, endohedral fullerenes [124] (or full- erenes containing single or multiple metal atoms, which in their own right can be considered novel nanocomposite systems; e.g., $\mathrm{La}_{2}@{\cal C}_{80}$ , $\mathrm{Gd@C_{82}}$ , etc.) have been introduced into SWNTs through the vapor phase. These structures are breaking new ground because creation of nanocomposite structures with discrete atomic spe- cies in their cores are resulting in the discovery of novel properties of 1D nanoscale materials. + +Recent theoretical studies suggest that the presence of a metal nanowire inside a nanotube greatly alters the mechanical properties of the nanotube, for example, sup- pressing tube-buckling instability. Increasing tube diameter increases the bending strength; however, in contrast to hollow tubes, there is no decrease in the maximum deflection before buckling. Analysis of the principal bending vibrational mode shows a lowering of the frequency, associated with increased tube inertia. Remarkably, metal- filled tubes exhibit strong damping, whereas unfilled single-walled and multi-walled tubes show no damping of oscillations (Figure 1.22). Simulations of multi-walled na- notubes revealed the lack of damping behavior; the dissipation of the mechanical en- ergy is intrinsically associated with tube filling by a dissimilar material [125]. These studies demonstrated the benefits of filling tubes with solids for modifying the bend- ing strength and flexibility, suggesting implications for nanotube-based elements in micromechanical devices or nanoprobes. Experimental studies on these unique 1D nanocomposite systems are eagerly awaited, since a variety of them have already been fabricated. + +Condensed-phase electrolytic processing provides an alternative way to fabricate nanotubes with or without encapsulated metals. The technique involves passage of an electric current through graphite electrodes immersed in molten mixtures of ionic salts (e.g., LiCl, $\mathrm{5nCl}_{2}$ ) under inert gas atmospheres [126]. The graphite crucible anode + +![](images/ba812ebfb2205cd35040aa0368414326d44a3262105fc9b75937dba35192def2.jpg) +Fig. 1.21 (a) Transmission electron micrograph of a $\mathsf{L a}_{2}@\mathsf{C}_{80}@\mathsf{S W N T}$ assembly. Each ${\mathsf{C}}_{80}$ molecule appears as a dark circle, and the SWNT $71.4\,\mathsf{n m}$ diameter) itself appears as two parallel lines. (b) Effect of inserted Gd metal lo fullerenes (GdMFs) on the topography and band structure of a single-walled nanotube (SWNT). The illustra- tion of a (11,9) GdMF-SWNT. Schematic representation of es- timated modulation of conduc- tion (Ec) and valence (Ev) band edges is also shown. (Source: top image [234], bottom part [235] used with permission) + +![](images/0923d8b5799ae69edf5a71c09be647baefa88c3b35084856fdda8ef9058c7826.jpg) +Fig. 1.22 Damping in pure vs. metal-filled nanotubes, simula- tion results. Top: bending of filled nanotubes; filling makes the nanotubes stiffer to bending and buckling. Oscillatory fre- quency $(\mathsf{f}_{\circ})$ of vibration, as a function of tube diameter for filled and hollow tubes as func- tion of tube diameter. The filled nanotube composite structures show high damping during vi- brations. (Source [125]) + +contains the electrolyte salt, which is heated by an external furnace. When the salt mixture melts, the cathode (graphite rod) is immersed in the electrolyte. During elec- trolysis, the graphite electrode interacts with the molten electrolyte, forming nanos- tructures. If pure LiCl is used as the electrolyte, the carbonaceous material formed consists of tangled carbon nanotubes, some of which contain the carbide of the salt (i.e., LiC). Adding small amounts of low-melting metals such as Sn, Pb, and Bi to the electrolyte forms nanowires of the respective metals. The formation of car- bide nanowires is based on the reduction of metal ions on the graphite cathode surface and intercalation of the Li between the graphite layers, followed by the formation of $\mathrm{Li}_{2}\mathrm{C}_{2}$ . Carbon then precipitates in the form of nanotubes. When Sn, Pb, and Bi are added to the electrolyte, the cations derived from these metals are discharged at the electrode surfaces by the electrons provided by oxidation of the carbide. This ultimately forms metallic nanowires surrounded by graphitic sheets in the nanotube form. This technique has also generated carbon-encapsulated $\mathrm{{Sn/Pb}}$ nanowires [127]. + +Pyrolysis of hydrocarbons over metal catalysts at elevated temperatures also pro- duces filamentous structures containing metal and carbon [128]. If the catalyst con- centrations are high $(>\!10\!-\!20\,\%)$ , encapsulation of the metal catalysts can produce nanowire composites. Pyrolysis of organic compounds or other carbon sources such as ${\sf C}_{60}$ in the presence of transition metal species (e.g., metallorganics such + +![](images/3c4d143522810752f6cbe76d37710a615e6cbf66fd05054abb9f15bd44a8f097.jpg) +Fig. 1.23 Scanning electron micrograph showing the fibrous structure of carbon nanotube $/\mathsf{V}_{2}\mathsf{O}_{5}$ nanocomposite. The individual nanotubes (seen as tiny filaments) are finely coated with subnanometer layers of the oxide, uniformly on the surface, modifying the nanotube surface properties. The structure was made by mixing nanotubes and oxide powder and annealing. + +as ferrocene) leads to nanotubes containing these metals or their alloys [129]. Metal nanowires over carbon structures can also be prepared via substitution. Recent work suggests that suspending SWNTs and exposing them to metal vapor can be used to fabricate nanowires of Pt, Au, Ti, etc. Hollow nanotubes of the metal are formed around nanotube templates [130]. Composite nanotube-based nanocomposites con- sisting of nanotubes and other materials can also be created by coating nanotube tem- plates by capillary effects. Nanoscale skins of layered oxides such as $\mathrm{V}_{2}\mathrm{O}_{5}$ can be uni- formly coated onto nanotube surfaces [131] (Figure 1.23). It is also possible to create composite nanowire systems by CVD techniques, in which individual layers of lami- nated materials such as $\mathrm{MoS}_{2}$ and ${\mathbb{W}}{\mathbb{S}}_{2}$ are coated onto multi-walled carbon nanotubes [132]. The electronic properties of these coated tubes are dominated by their outermost shells (oxides or sulfides, which are generally semiconducting or insulating). How- ever, the inner tube structure (carbon nanotubes) may dominate the mechanical prop- erties. In this way, it will be possible to tailor both the mechanical and electronic prop- erties of these 1D systems. Physical property characterization of these nanocomposite systems is challenging, but much excitement exists about uncovering a rich range of properties inherent to these nanoscale multiphase systems. + +Multiple-phase nanowires, constituting different structures along the wire length, may also become important in future nanotechnology. The advantage here will be the ability to form controlled junctions, which could have semiconducting properties (metal–semiconductor junctions), novel deformation behavior (whereby toughness may be increased when they are used as reinforcements), photo luminescence (if + +![](images/e0a4f3bb19ca3da764abc5fb7d4e0918f842c67d5fa945a4eef3519534b25ae1.jpg) +Fig. 1.24 High resolution transmission electron micro- graph of a heterojunction formed between an Fe nanowire and a multiwalled carbon nanotube. The structure was formed in the vapor phase during pyrolysis of hydrocarbon and ferrocene. Similar structures can also be made by step- wise electrode position inside alumina nanoporous mem- branes. + +phases are semiconducting with distinct n- or p- character), or thermoelectric-junction properties. Carbon nanotube–metal junctions have been made by using excess cat- alyst (Fe) in the chemical vapor deposition process or by electrochemical deposition in porous alumina templates (Figure 1.24) [133, 134], but the properties of these multiple- phase nanowires have not been evaluated. Controlled solid-state reactions have been used to form heteronanowires of single-walled carbon nanotubes and nanorods of silicon carbide and transition-metal carbides (TiC) [135]. The reactions are carried out via contact annealing with a metal (or Si) solid surface and nanostructures such as nanotubes. The structures created are some of the smallest heterojunctions that have ever been made and will play a large role in the fabrication of future hybrid nanodevices. Similarly, techniques such as laser ablation have been used to form coax- ial cables containing he t ero compositional layers. For example, high resolution trans- mission electron microscopy and spatially resolved electron energy loss spectroscopy have characterized wires that consist of $\beta$ -silicon carbide sheathed inside amorphous silicon oxide surrounded by graphitic shells made of C or h-BN [136] (Figure 1.25). These coaxial nanocables of semiconductor-insulator-metal/semiconductor structures could find applications in nanoscale electronics. The formation of such multiple-layer nanocable structures is challenging and has to be controlled through the self-assembly of various elemental and molecular species in the vapor phase during relatively short growth times, and the directed assembly of such complex structures is one of the most crucial steps in the creation of useful low-dimensional functional nanocomposite structures. + +Recent work has also demonstrated that semiconductor nanowires can be control- lably grown in cross-bar array architectures, and, if the crossing nanowires can be doped into n- or p- type, then light emission can result at the crossing junctions when a voltage is applied across the wires. High-temperature laser-assisted deposition + +![](images/7ebb21b7b32d40c514e308c8432ed749c930bf1f175f156dc771e16c4895427a.jpg) + +has been used to grow single-crystal indium phosphide (InP) nanowires using catalytic gold particles [137], either in p-doped or n-doped form. Superlattice nanowires can also be made with alternating blocks of different semiconducting materials (e.g., Si and SiGe) on a single wire [138]. A hybrid pulsed-laser ablation/chemical vapor deposition method can be used to custom-create various he t ero structures on individual nano- wires. These structures systematically contain junctions and interfaces between the multiple blocks present in the same wire, enabling them to be useful in miniaturized transistors, LEDs, and lasers. + +# 1.8.2 Applications of Nanocomposite Wires + +Of several applications proposed for composite nanowire structures, only one is dis- cussed here. Magnetic recording exploits the hysteresis behavior of ferromagnetic materials: the residual magnetization of a tape or magnetic disc is a record of the maximum field intensity the material experienced with respect to the magnitude and direction of the applied field. For long-term storage, usable magnetic materials should have a reasonably high coercivity, because external magnetic fields should not alter the magnetization. However, the coercive force should not be too large if the medium is to be reusable, since the magnetic fields used during recording should be capable of remagnetizing the material for new data recording. Recently, feature sizes on magnetic disks made using complicated e-beam lithography protocols have shrunk to the sub- $\cdot100\;\mathrm{nm}$ range [139]. Ferromagnetic Fe nanowires could pro- vide an ideal system for recording, but the tendency to oxidize is strong in nanoscale metallic systems. The encapsulation of Fe nanowires within carbon nanotubes will assure that the metal is maintained in the reduced state, due to the protective C coat- ing. These encapsulated nanowires exhibit magnetic coercivities larger than those associated with pure Ni and Co nanowires [140]. The coercive field decreases linearly with increasing temperature, revealing the presence of anisotropy. The small size, anisotropy, and the single (isolated) domain nature of the encapsulated Fe crystals appear to be responsible for this enhanced coercivity. The magnetic properties of en- capsulated Fe nanowires do not degrade with time, because they are sealed within the graphitic tubes. These nanowire arrays may have significant potential in magnetic data storage device applications (e.g., quantized magnetic disks) due to their size and ani- sotropic behavior, with the possibility of drastically increasing the attainable recording density. Film structures of this kind, based on single-domain elements, exhibit the best available storage densities (e.g., $65\;\mathrm{{Gb}\;i n^{2}})$ ) [141]. The type of magnetic nanocom- posite particles should also find applications in the fabrication of fine-particle magnets for use in magnetic inks and as toners in xerography. + +# 1.8.3 Applications of Nanocomposite Particles + +Several strategies are being pursued for creating nanoparticles containing multiple shell structures (for multi functionality), with each shell composed of different materi- als. Template-based approaches are effective in this situation, for which an inner re- movable template particle (silica, polymer beads) can be used to coat shells of other materials (e.g., a metal) via multi-step colloidal or vapor-phase assembly and can later be easily removed to create empty shells. These could even be filled with different materials to produce multiple-shell composites [142]. Creating uniform coatings on particle templates by colloidal self-assembly is based on the concept of self-assembled organic molecular species. The two ends of the molecules to be joined have specific functional groups (e.g.. thiols, amines, carboxylic groups) that can be targeted for spe- cific interactions with the template and the clusters that are used to make the coatings. Uniform, dense packing of the molecules around the templates leads to close packing + +![](images/2c39dacb082ffc6407194b46b2cc48bdeb19f5155a3fe2f03ac87544682ac27f.jpg) +Fig. 1.26 Fabrication of multilayer nanoparticle self-assembled monolayers of bifunctional organic structures by colloidal synthesis. Top: micrograph of molecules (e.g., mer cap to prop yl tri meth oxys i lane). silica spheres being continually coated with gold Multiple-layer structures can be fabricated by col- nanoparticles until a uniform coating is produced. loidal assembly techniques. (Courtesy Prof. Ma- The process of attachment of gold particles (shown moun Muhammed) in the schematic below) occurs via connectors of + +of the clusters that form a porous but space-filled shell around the template (Figure 1.26). Noninteracting metal-coated magnetic particles $(\mathrm{SiO}_{2}/\mathrm{Au}$ , $\mathrm{Fe}_{3}\mathrm{O}_{4}/\mathrm{Au}$ , $\mathrm{{SiO}/\mathrm{{Co}}}$ , etc.) or coated semiconducting particles (PbS/CdS) are examples of such composite particle structures. These structures can have applications in magnetic recording, as multilayered catalyst materials, or in drug delivery. In such applications of drug de- livery systems [143], for which a biocompatible outer layer and a drug-containing inner core are necessary, the multilayered particle approach will be crucial. As an alternative to colloidal templating, structures such as PbS-coated CdS nanocomposite particles (a few nanometers in diameter) may be synthesized by ion displacement in inverse mi- croemulsions and could be useful in nonlinear optical applications. The observed large refractive nonlinearity in these nanocomposite particles may be attributed to the op- tical Stark effect and to strong interfacial and inter nano particle interactions. + +# 1.9 Inorganic Nanocomposites for Optical Applications + +Nonlinear optical effects, such as nonlinear optical absorption and second- and third- order optical nonlinearities, can be used to make optical limiters, optical modulators, etc. Although many organic materials have high optical absorption and nonlinearity, their thermal and optical stabilities are poor. Often, it helps to create hybrids or com- posites (organic/inorganic, inorganic/inorganic) that have acceptable optical proper- ties and stability. Quantum-confined nanoparticles have been extensively used in the fabrication of such composites because of their novel optical and electronic properties. The organic-matrix nanocomposites for optical applications are described in chapter 2, which briefly discusses some of the organic materials that contain dispersed nanopar- ticles in various hosts. + +Recent advances in controlling the fabrication and dispersion of semiconductor nanoparticles in polymer and ceramic matrices have suggested possible uses for such nanocomposites in optical applications. A good example of an optically func- tional ceramic nanocomposite is GaAs nanocrystals embedded in $\mathrm{SiO}_{2}$ matrix. The interest in the novel optical properties of semiconductor nanocrystals [144] has re- sulted in strategies to package them as nanocomposites. A variety of techniques, such as colloidal synthesis, self-assembly, and electrochemistry, can be used to pro- duce the semiconductor nanoparticles; however, ion coimplantation $(\mathrm{Ga}+,\mathrm{As}+)$ is an efficient way of creating well-dispersed nanocomposite materials (e.g., GaAs/ $\mathrm{SiO}_{2}^{\prime}$ ) [145]. Typically, the ions are implanted at fluxes of ${\sim}10^{16}\,\mathrm{cm}^{2}$ into $100\,\mathrm{nm}$ silica films on Si substrates and annealed at appropriate temperatures to create nanocrystals of GaAs (several nanometers in size) in the matrix (Figure 1.27). Photo luminescence studies show an efficient, broad luminescence band in the visible and near-infrared spectral regions due to quantum confinement in GaAs nanocrystals and defect states in $\mathrm{SiO}_{2}$ . + +Nanoparticles in matrices have interesting photo luminescence properties, due to the effects of quantum confinement on their optical properties. Indirect band semi- conductors such as Si and Ge have very poor luminescence efficiency (hence, efficient + +![](images/94043ef2cbf273a0f411403726644c72946a39516001d5039cb709d88037b69a.jpg) +Fig. 1.27 Transmission electron micrograph, left, photo luminescence band observed in the red and GaAs nanocrystals embedded inside ${\sf S i O}_{2}$ glass near infrared spectral regions. The efficient lumi- matrix. Sequential ion implantation followed by nescence is attributed to both quantum confine- thermal annealing was used to form GaAs nano- ment states in GaAs nanocrystals and defects in crystals in ${\sf S i O}_{2}$ films. Right, efficient, broad ${\sf S i O}_{2}$ . (Source [145] used with permission) + +light emitters cannot be fabricated directly from them), because the band center tran- sition is optically forbidden. But, by making the particle sizes smaller than the exciton Bohr radius ( $4.9\,\mathrm{nm}$ for Si, $24.3\;\mathrm{nm}$ for Ge), the resulting confinement produces an increase in oscillator strength, shifting the luminescence to higher energies; this al- lows tailoring novel optical materials, which are otherwise impossible. To take full advantage of the particle size effect, the particles must be separated, and the best pos- sible way of accomplishing this is to disperse the particles in a matrix. Understanding and minimizing the interaction of the host matrix and the particles that create interface states are crucial to many of the optical applications. Embedding nanoparticles in dif- ferent matrices and studying the optical properties provides a way to decipher interface effects. Ge quantum dots have been studied in various hosts, and one way to minimize host interaction is to embed them in oxygen-free environments, such as AlN; AlN/Ge multilayer structures deposited by pulsed-laser deposition show a blue-shifted photo- luminescence peak [146]. Similarly, indium oxide (InO) nanoparticles dispersed (by solvent-phase impregnation) within the pores of mesoporous silica (prepared by sol– gel technique) show multiple photo luminescence peaks related to the size and struc- ture of the particles [147]. Amorphous InO particles $(<\!6\ \mathrm{nm})$ show a photolumines- cence peak that is down-shifted compared to slightly larger crystalline particles. For larger aggregates of InO particles, no photo luminescence is observed. Si nanoparticles themselves are interesting from the point of view of photo luminescence, and porous Si, which contains small particle domains, has been studied in great detail. The pro- blem with porous Si is the structural nonuniformity, lack of reproducibility in emis- sion and aging, and environmental degradation problems. Incorporating light-emit- ting Si structures in a matrix that is chemically inert and has a wide band gap suitable for quantum confinement can overcome this problem. Nanocomposites of Si nano- particles $(\sim\!5\ \mathrm{nm})$ ) embedded in polycrystalline diamond matrix have been prepared and studied. The room temperature photo luminescence behavior of such materials in the $1.6\!-\!2.5~\mathrm{eV}$ range shows a strong increase in emission efficiency from the Si na- nostructures. The studies conducted on this nanocomposite reveal that selecting the size of the embedded Si particles makes it possible to tune the luminescence frequency in the yellow–green spectral region. + +Nanocomposites of nanosized metal particles in transparent dielectrics can also be applied as nonlinear optical materials in photonic devices. These materials are char- acterized by large third-order optical nonlinearity $(\chi^{3})$ and fast response times, which are important for device applications such as optical computing, real time holography, and phase conjugators. To incorporate metal nanoparticles into dielectrics, several techniques such as ion implantation, sol–gel techniques, and sputtering can be em- ployed. It is even possible to prepare graded layers of metal nanoparticle distributions (with different particle sizes and interparticle separations) by implantation, and these structures can be carefully tailored to produce interacting and noninteracting nano- particle layers that produce different optical response in the plasma resonance fre- quencies. Several of these composite systems have been studied, for example, $\mathrm{Au}/$ $\mathrm{SiO}_{2}$ , $\mathrm{Ag/SiO_{2}}$ , $\mathrm{{Au}/\mathrm{Al}}_{2}\mathrm{{O}}_{3}$ , $\mathrm{{Au}/T i O}_{2}$ with metal concentrations varying from $15\,\%-$ $60\,\%$ , for nonlinearity and plasma resonance frequency shifts. High values of suscept- ibilities $(\chi^{3}\sim6\times10^{-7}$ esu, compared to low values of ${\sim}10^{\cdot12}\!-\!10^{\cdot11}$ esu for glass) have been reported for composites containing an optimized fraction of the metal compo- nent [148, 149]. Groups II–VI semiconducting nanocrystals (e.g., CdS) prepared in glass hosts have been studied in great detail, because of their large optical nonlinear response and small carrier lifetimes. Enhancement of carrier recombination rates is observed in nanocrystals with large surface-to-volume ratios, and the enhancement results from increases in the density of surface states and fast surface recombination and capture due to multiphoton emission. Surface recombination in nanocrystals em- bedded in glass matrices exhibits thermally activated nonradiative recombination, which is enhanced at reduced particle size. The semiconductor/glass interfaces in the composites may give rise to deep traps responsible for photoactive phenomena. + +In recent years, nanoparticles have been used to make transparent nanocomposite structures having high refractive indices. Polymers containing inorganic particles in a range of $1\!-\!100\:\mathrm{nm}$ (nanocomposites) are interesting in this regard. In contrast to composites having particles in the micron size range, nanocomposites do not scatter light and are interesting for optical applications. Preparation of nanocomposites with refractive indices over the entire range of ${<}1$ to ${>}3$ , which is by far the lowest and highest ever achieved for a polymer composite, has been possible. Transparent poly- meric materials can be coated with surface layers ${\sim}100\;\mathrm{nm}$ thick) of UV-absorbing nanocomposites to inhibit degradation of the polymer by UV light [150]. High-refrac- tive-index transparent materials are mainly used for improving the optical coupling efficiencies in photonic devices. The refractive indices of polymers vary between 1.3 and 1.7, those for inorganic semiconductors vary between 2 and 5, and for high bandgap semiconductors this value is ${<}3$ . The technological challenge in creat- ing near-transparent high-refractive-index organic/inorganic nanocomposites lies in the creation of nanoparticles in the size range $20\,{-}\,40\;\mathrm{nm}$ , to produce high-refrac- tive-index nanoparticles with low absorption coefficients in the visible range. Exam- ples of such composites are PbS, InP, GaP, Ge, and Si particles in a gelatin matrix. The particle sizes, as well as the loading fraction of the particles, affect the refractive index and absorption coefficient. Very small particles show reduced refractive index. But typically there is a critical size above which the value of refractive index reaches the bulk value (for $\mathrm{Pb}\,\mathrm{S}$ this is ${\sim}25\;\mathrm{nm}$ ). Hence, to obtain predictable values of refrac- tive index, it is important to have good dispersion of particles. The particles useful for creating these nanocomposites can be produced through a variety of techniques, ran- ging from mechanical attrition [151] (followed by centrifugation to obtain near-uni- form Si particles) to colloidal processing. Nanocomposites can then be prepared by mixing and spin coating (or casting) aqueous solutions of gelatin with suspension of nanoparticles in appropriate organic solvents. + +# 1.10 Inorganic Nanocomposites for Electrical Applications + +The incorporation of particles of a non piezoelectric phase into a piezo material is unlikely to improve properties such as the strain per unit field or the charge per unit force, but the dielectric loss of the nanocomposite material under high alternat- ing electric fields is an interesting issue. High-drive electrical applications are often restricted by reduced efficiency, due to heating caused by the large dielectric loss at high fields. A clear distinction has been shown between weak and strong field behavior in materials such as $\mathrm{BaTiO}_{3}$ . Above a critical field level, the dielectric constant and dielectric loss increase dramatically, due to hysteretic domain wall motion and the resultant reorientation by spontaneous polarization. It is possible that incorporating nanosized particles (SiC of small volume percentages, ${\sim}2.5\,\%)$ ) into an electroceramic matrix $(\mathrm{BaTiO}_{3})$ ) may hinder domain wall motion sufficiently to reduce dielectric losses at high fields [152]. Accordingly, the microstructure requires the particles to be situ- ated intragranularly. This concept is similar to that used in the development of hard ferromagnetic materials, for which heat treatment is used to produce a fine dispersion of particles within the microstructure, which increases the resistance to domain wall motion. In addition to the possible effect of the particles on piezoelectric properties, additional issues that require examination include the influence of any secondary phases produced by solid-state reactions between the matrix and nanoparticles; in a $\mathrm{BaTiO}_{3}/\mathrm{SiO}$ system, heat treatment and hot pressing can form new phases such as $\mathrm{Ba_{2}T i S i_{2}O_{8}}$ . When such reaction phases are present in the composite, the permit- tivity of the system is lowered. Addition of silicon carbide leads to a change in the microstructure of the material and a decrease in grain size. This outcome has been reported in structural systems, which show corresponding improvements in mechanical properties. In addition, the fracture mode was transformed from transgra- nular for monolithic $\mathrm{BaTiO}_{3}$ to intergranular for the nanocomposite. Many studies have been undertaken to fabricate piezoelectric particle-dispersed ceramic nanocom- posites from other materials: $\mathrm{Pb}(\mathrm{Zr},\mathrm{Ti})\mathrm{O}_{3}$ nanocomposites have been prepared from high-purity PZT powder and small amounts of oxides $(\mathrm{Al}_{2}\mathrm{O}_{3}$ , $\mathrm{MgO}$ , etc.) [153]. Small additions of a second phase improve the mechanical properties, such as hardness and fracture toughness, of the PZT. The nanocomposite’s reduced grain size during pro- cessing with the second phase is responsible for the improved mechanical properties. The piezoelectric properties of these composites remain essentially unchanged, although for the MgO composites, the electromechanical coupling factor (which is important for actuator applications) becomes larger. + +Nanocomposites can also be useful for such applications as electrical contact ma- terials, particularly to replace the toxic $\mathrm{Ag/C d O}$ contacts being used now. The reactive milling process has been used for manufacturing $\mathrm{Ag/S n O}_{2}$ nanocomposites. A high- energy ball mill controls the reaction to obtain nanosized $\mathrm{SnO}_{2}$ particles in an $\mathrm{Ag}$ matrix [154]. Then this powder is hot pressed into electrical contacts with superior erosion resistance and good thermal and electrical conductivity. Such cermet-like na- nocomposites (in different configurations) have applications in other areas, like sen- sors. Many reports have appeared on gas-sensing technologies based on the measure- ment of changes in the electrical resistances of metal oxides (such as $\mathrm{SnO}_{2}$ , $\mathrm{TiO}_{2},$ ) as the environment is changed. The sensitivity and selectivity of these metal oxide sensors can be affected by deposition of discontinuous metal films on their surface. Metals such as Pd, Pt, etc. have been used as surface activators. Similarly, incorporating metal clusters into bulk oxides to form nanocomposites also makes it possible to tailor the effective bulk electrical response of sensor materials. This incorporation allows the range of resistivity changes caused by changes in the carrier concentration at the sur- face or bulk of the metal oxide sensor material to be altered. + +Although ceramic/metal composites are exciting mostly due to improved mechan- ical properties, the temperature-dependent electrical behavior of metal in ceramic na- nocomposites is quite interesting [155]. As examples, Ag and Pb particles have been prepared in silica gels by electrochemical methods. DC electrical characterization of these materials shows surprising changes from semiconducting to metallic behavior as a function of temperature and depending on particle size. The electron conduction mechanism in these nanocomposites is complicated but, in general, occurs via two conduction mechanisms: electron tunneling between metal particles and metallic con- duction due to the presence of metal percolated paths. The particle size and micro- structure of the composite can be changed to adjust these properties [156]. The variable conductivity of these composites can be used in applications, such as electromagnetic induction shielding. + +# Percolation Effects and Transport Phenomena in Composite Systems + +To understand nanocomposites in terms of conduction processes, we need a basic understanding of the bulk materials as well as of the composite filler. This is where certain issues arise. When dealing with organic or inorganic insulators and semicon- ductors, we may rely on a form of charge transport through the medium in terms of variable range-hopping conductivity when we add fillers to the system [157]. However, the conductivity is determined by the dominant charge carrier, because the carrier’s mobility in hopping from one site to the next is the dominating feature of the con- duction mechanism. A detailed study of hopping and its effect on conductivity has been summarized earlier [158, 159]. + +For metal filler systems in insulating matrices, the dominating feature is the type of filler, with particular regard for its size and shape [160]. Changing the electrical transport properties of these insulators with nanosize fillers is necessary, because we want to retain the innate properties that make the insulators useful. We also want to induce electronic properties that allow us to make quantitative determinations without altering the chemical and/or physical nature of the dominant medium (in- sulator). For the simple percolative effect, we must look at the insulating/metallic mix. In general, for bulk transport properties, the conductivity follows a power law behavior: + +$$ +\upsigma=\upsigma_{0}\big(p-p_{c}\big)^{t}\ \ \ \ (\mathrm{for}\ t=2) +$$ + +where $\upsigma$ denotes the conductivity and $p$ represents the mass fraction [161]. In the past, this behavior has been shown to agree with 3D behavior, by Monte-Carlo simulations for thick films [162]. We do not deal with this behavior in detail here, except to note that the study of superconductivity in percolating films is relevant for the interpretation of critical temperature and heat capacity data [163]. A more detailed application of per- colation theory deals with the effect of varying the number of interconnections present in a random system [164]. Fourier et al. [165] proposed a model in bulk materials, based on the Fermi–Dirac distribution, that describes the critical insulator-to-conduc- tor transition: + +$$ +\log(\upsigma_{c})=\log(\upsigma_{n c})+\frac{\log(\upsigma_{f})-\log(\upsigma_{n c})}{1+\exp[b(p-p_{c})]} +$$ + +where $\upsigma_{c}$ is the composite conductivity, $\upsigma_{f}$ is the filler conductivity, $\upsigma_{n c}$ is the nanocry- stalline conductivity, $p$ is the mass fraction, and $b$ is an empirical parameter that leads to the change in conductivity at the percolation threshold $p_{c}$ . + +On a macro scale, we look at transport properties in terms of 3D processes. The dimensionality of the charge responds to its environment in the x, y, and z direc- tions, as well as to the field that affects it. The simple phrase ‘the path of least resis- tance’ is very true for 3D transport. For years, many organic and inorganic materials were examined for their conduction dimensionality on the nanoscale. + +In examining composites for their electrical (and electronic) properties, considera- tion of the volume or mass fraction addition (where we add a metallic component to our insulator) is crucial. However, where this volume addition occurs, the structural phenomena have a large bearing on the composite’s transport properties. Alterna- tively, the metal particles can be distributed randomly, but their proximity (path length) and size can be varied, depending on the technique for adding them. Some clusters will aggregate and therefore nucleate to form larger grain sizes, but other deposition techniques result in islands of metallic clusters. This heterogenity has a dramatic effect on the conduction process as well as on the dimensionality of the conduction pathway. Variable ranges for the critical mass fraction of added filler $(5\,\%\!-\!50\,\%)$ required to reach the percolative threshold have been reported. Why should we care about these values and how are they important? The reason is both complex and simple. To influence transport phenomena, both the shape and size of the filler are crucial. In addition, the manner of deposition and the resulting pathway and mean distance between islands of particles or colloids also directly influ- ence the dimensionality of the charge carriers. We should be aware of certain effects of various fillers. The electrical conductivity of very thin metal films can be much lower than in a bulk material. Closer examination of the thin films reveals a discontinuous structure, where the metal appears as clusters or islands. Detailed examinations have been made of the transition of materials from metallic ballistic transport to insulating, indicating that it is percolative in effect. + +As mentioned above, in recent years, the size-dependent properties of nanocrystal- line solids have attracted much attention. By gaining an in-depth knowledge of these materials, we can start to use nanocrystalline fillers as components in emerging com- plex composites. Small changes in the physical size of a simple nanocrystalline ma- terial can lead to large changes in its physical properties [166]. In general, to under- stand such fillers, we must examine many complexities before expanding the inves- tigations into macro- or even nanoscale composites. Work in this field is still inade- quate for a complete understanding of percolation effects and dimensionality when nanofillers are used in composites (Figure 1.28) [167]. However, recent studies of carbon nanotubes as a filler in a conjugated polymer have led to some interesting results. Carbon nanotubes were used as fillers in thin-film nanocomposites, and al- tered conductivity was measured. What makes this effect interesting is that a reason- ably well characterized material (the carbon nanotube) was used as a filler in a semi- conducting polymer – one whose low conductivity despite conjugation falls in the insulating range [168]. The effect at percolation was to change the conductivity of the composite by 10 orders of magnitude. + +![](images/5dd878218dc159f8b420c73dc60810632221f7add6ba8c5736f5ace55235f139.jpg) + +# 1.11 Nanoporous Structures and Membranes: Other Nanocomposites + +Nanoporous materials are materials with pores that have dimensions in the nanoscale. In many ways, these materials can be considered nanocomposites, because they have one phase (empty space) that is either randomly dispersed or ordered within a solid matrix. We have already discussed examples of some of these structures, such as hol- low carbon nanotubes with straight hollow channels or porous alumina templates made by electrolytic methods. Examples of random pore distribution are zeolites, me- soporous silica, and porous silicon, all of which have important applications. In this section, we discuss in a little more detail the importance of a few select structures in the realm of nanocomposites. An exhaustive treatise of porous materials and mem- branes is beyond the scope of this book but has been dealt with elsewhere [169]. + +The transformation of a chemically homogeneous material into a porous structure by some form of chemical reaction controlling pore size is challenging, because che- mical homogeneity and selectivity during such processes do not appear to go hand in hand. However, several strategies can be used (Figure 1.29). Examples include zeolites + + (pore diameter ${\sim}0.3\!-\!0.4\ \mathrm{nm})$ ), Al foils converted to alumina having straight pores + + (diameter ${\sim}7\mathrm{-}250~\mathrm{nm}$ ) via an electrochemical self-organization, compact polymeric membranes and also mica plates and glassy metals treated by irradiation or particle (e.g., ion) beams that also develop straight pores (‘chemical drilling’), porous carbon structures generated by carbonization of appropriate polymeric precursors and sub- sequently subjected to an activation reaction (diameter $0.3\!-\!4~\mathrm{nm})$ ), and cyclodextrin polymers [170] (diameter ${\sim}0.8~\mathrm{nm}$ ). + +The pore sizes in porous structures and membranes range from $0.3\;\mathrm{nm}$ to $10\,\mu\mathrm{m}$ . Pore dimensions below $0.3\!-\!0.4\:\mathrm{nm}$ may also be considered, but these are nearly the dimensions of ordered or disordered interstices in solids. However, the upper limit of $10\,\mu\mathrm{m}$ covers the pores of most membranes. Pore size uniformity (and distribution) is obviously important for applications such as separations on any size scale [171]. In addition, the physical and chemical character of the pore surfaces can play a signifi- cant role at smaller pore sizes, especially when gases or vapors are to be separated. In general, the fabrication of a defect-free porous membrane addresses two problems: one is the creation of pores of the desired local geometry (size, void fraction, and pore coordination number); the other is the achievement of mechanical robustness while repeating the local arrangement over macroscopic distances. + +Polymeric membranes useful at lower temperature $(20\!-\!100\,^{\circ}\mathrm{C})$ and for large-scale routine applications are an important class of porous materials [172]. Ceramic mem- branes, compared with their polymeric counterparts, are usually more fragile, more expensive, and harder to fabricate. On the other hand, ceramic membranes tend to withstand harsher separation conditions (e.g., high temperatures, solvents, and cor- rosive or fouling-favoring environments) and can often be cleaned easily. Carbon membranes are based on appropriate polymeric or pitch precursors, and they are in- termediate (in character and properties) between polymeric and ceramic membranes [173]. The properties of carbon membranes are closer to the properties of their ceramic counterparts: high temperature and chemical stability (except in oxidative environ- + +![](images/79b005cd075a96ea01989c12f3b4fa49f6be2a320e5524ba36fe59d3a3828e96.jpg) +Fig. 1.29 2D schematics of a selection of 3D bicontinuous structures com- monly encountered in nanoporous materials. Except for structure (e), the void part in indicated by black. (a) Fully ordered pore network, as in zeolites (pore size $0.4\!-\!1\ \mathsf{n m})$ ). (b) Generic nanoporous material with an imprecisely tailored porous network. (c) Structure generated by compression/mild sintering of solid, nearly round nanoparticles. Pores down to a few nanometers are achievable. (d) Structure typically generated by spinodal decomposition (a Corning silica-based product with $4\;\mathsf{n m}$ pores) (e) Nanoporous carbon with a reference nanodo- mained graphitic structure. Black lines represent graphitic planes and slit- shaped pores correspond to missing graphitic material. (f) Nanoporous ma- terial with tubular pores in a parallel semiordered arrangement, anodic alumina (with a minimum pore size in the $\mathsf{10\mathrm{-}n m}$ range), and various mesoporous zeolites (pore size in the range of several nanometers). (Courtesy Dr. Kon- stantinos Beltsios, Institute of Physical Chemistry, NCSR, Demokritos, Greece) + +ments at temperatures ${>}350{-}400\,^{\circ}\mathrm{C})$ ). Current interest in carbon membranes focuses mostly on those appropriate for gas separations; at least for air separation, these mem- branes exhibit a combination of permeability and selectivity superior to those known so far for polymeric materials. Examples of applications include separation of $\mathrm{CO}_{2}$ from $\mathrm{CH}_{4}$ in landfill gas, production of oxygen-enriched air, and separation of propy- lene from propane. As a consequence of some of the routes for their preparation and/ or their final structure, BN and SiC membranes constitute a bridge between carbon and generic ceramic membranes. Because BN is superior to C in terms of oxidative resistance, developing BN versions of C membranes is potentially interesting. In ana- logy to the fabrication of porous carbon structures from pitch and other precursors, appropriate organic precursors can be found for BN fibers. Deposition of a precursor organosilicon polymer or oligomer [174] (e.g., polycarbosilane) on porous ceramic substrates, followed by pyrolysis, can be used to fabricate silicon carbide or Si-O-C containing ceramic top layers of composite membranes. Pores from ${<}1$ nm to meso- pores can be generated, depending on the precursor and the processing. In very recent years, self-assembled single-walled carbon nanotube-based arrays have been proposed for membranes (Figure 1.30). The membrane action is based on quantized transport of water through nanotubes [175] ( ${\zeta}{<}1\;\mathrm{nm}$ pore size), mimicking water transport in phy- siological structures through tubules of similar dimensions. + +Several forms of well-defined nanostructured carbons (Figure 1.31) exist, and these forms have important practical applications, such as development of electronic, cat- alytic, and hydrogen storage systems. These materials can be synthesized by thermal chlorination of various carbides. It has so far been possible to synthesize nanoporous carbon, nanoparticles with multi-walled, band-like lamellar graphitic carbon, and tur- bostratic carbon. The nanoporous amorphous carbon consists of very high-surface- area carbon with a tunable, very narrow, pore size distribution and is suitable in ap- plications such as supercapacitors and specific adsorbents. Template-based (e.g., quartz, silicates) synthesis techniques can also be used to prepare such materials. Structures such as mesoporous aluminum silicate or nanoporous silica have well-or- dered arrays of open structures randomly connected by pores (pore size ${<}3.5\,\mathrm{nm}$ ). The pores can be filled with organic solvents, heated, then pyrolyzed at high temperature; finally, the template is etched off to produce nanoporous carbons [176]. + +Semiconductor and metal porous membranes have specialized applications [177]. The most important example of metal membranes is hydrogen-separating palladium- based membranes [178], which work on the basis of selective diffusion of gaseous species through the metal lattice. Composite membranes also exist, in which one or more top or buried layers are added to a bulk membrane material. At least one of the layers plays an active role in separation or provides defect healing. Composite materials, consisting of a matrix material (most often a polymer) and a dispersion of a + +![](images/8ed2661822e42cda98ca01bf3a5dcc41e655eb498bae3814804be3699ecf7209.jpg) +Fig. 1.30 Above: schematic of a possible multiple layer membrane consisting of self-assembled nanotubes (along the arrows), packed together densely. Between the membrane layers is a water-salt mixture. Simulation suggests that osmotic pressure can drive selective transport of species across the mem- branes, as shown by molecules moving through the nanotubes. Such porous architectures can find use in many applications ranging from bioseparations to drug delivery. (Courtesy Prof. Shekhar Garde, Department of Chemical Engi- neering, Rensseloer Polytechnic Institute, Troy, New York, USA) + +![](images/ebbc76f2e2a20d3eede78ea8aa5a2015e817dafe83e5069b5b566bab448f4d85.jpg) +Fig. 1.31 Atomic-level sche- matic of nanoporous carbon structure with loaded metal particles. High surface area of porous carbon materials are ideal for holding dispersed metal catalyst particles for use in het- erogeneous catalysis. (Source: [236] used with permission) + +standard nanoporous material (such as zeolites or microporous carbon), can also be useful in catalysis and separation applications. + +Straight-pore alumina membranes (which were briefly described above, in the sec- tion on preparation of templates for the growth of nanowires and nanotubes) are pre- pared by anodic oxidation of thin aluminum foils, in the electrolysis of a solution of oxalic, phosphoric, chromic, or sulfuric acid [179]. The electrolyte is chosen among those having some capacity for dissolving alumina. As electrolysis progresses, self- organization occurs; the aluminum oxide formed in the anode undergoes spatial re- distribution, resulting in a honeycomb-like structure, with straight pores distributed in a nearly hexagonal arrangement. Pore sizes in the $10\,{-}\,250\;\mathrm{nm}$ range, porosities on the order of 0.2–0.3, and film thickness in the $10\!-\!100\,\mu\mathrm{m}$ range are usually obtained. For separation applications requiring small pores, it is preferable to develop asymmetric versions of anodic alumina membranes. In addition to asymmetry achievable by de- positions, changing the conditions during growth, especially through voltage changes, makes asymmetry possible. In addition to the anodic porous alumina structures cre- ated, novel approaches being tested include, for example, the use of a master pattern (made of, e.g., SiC) with nanosize features to imprint well-defined surface textures that guide the formation of nearly defect-free pore channels with high aspect ratio over a wide area [180]. Long-range ordered channel arrays with dimensions on the order of millimeters and a channel density of $10^{10}\,\mathrm{cm}^{.2}$ can be obtained, with aspect ratios of over 150 for individual channels. + +Although the limited thickness achievable by the aluminum anodization approach makes the membranes produced unlikely candidates for extensive use as gas-separa- tion membranes, a substantial amount of research has been dedicated to generating model asymmetric ceramic structures with additional functional layers deposited for specific functions; oriented microcrystals of porous phosphate structures and zeolite- like solids (which have tailorable pore sizes of $2\!-\!20\:\mathrm{nm}$ ) can be grown on the surface of straight-pore alumina membranes to provide added functionality in separation ap- plications [181, 182]. Straight-pore alumina membranes have been used as templates for generating a porous metal membrane by electroless plating [183]. Dissolution can remove the original and intermediate templates. + +Polymeric nanoporous materials and nanocomposite materials have improved the technology of polymeric membranes drastically. In a complete reversal of roles for organic and ceramic matter in the building of organized gas- or vapor-selective ma- terials, it has been possible to infiltrate the channels of zeolite-like materials with di- mensionally constrained active organic (and metal-containing organic) phases. The architecture of the inorganic zeolite can be used to build other porous structures, including porous carbons. For example, a zeolite structure exposed to propylene gas at a high temperature, after carbonization and acid washing, yields a micropor- ous/nanoporous carbon material [184]. + +# 1.12 Nanocomposites for Magnetic Applications + +# 1.12.1 Particle-Dispersed Magnetic Nanocomposites + +Magnetic nanocomposites in which magnetic species are dispersed within nonmag- netic or magnetic matrices are practically useful. Magnetic recording, giant-magneto resistance, and magnetic refrigeration are some important areas in which magnetic nanocomposites have relevance. Nanocomposite magnets with both particle/matrix systems being magnetic have received much attention, because they may have high remanence (associated with exchange coupling at interfaces separating hard and soft magnetic phases) and large energy product, $(\mathrm{BH})_{\mathrm{max}},$ relative to conventional magnetic materials [185]. The remanence enhancement can be as high as $0.8~\mathrm{M_{s}};$ , the saturation magnetization, compared to the isotropic value of $0.5\,\mathrm{M_{s}}$ . Many reports have appeared on the enhanced magnetic properties of nanocomposite magnets made of hard magnetic phases such as $\mathrm{nd}_{2}\mathrm{Fe}_{14}\mathrm{B}$ and $\mathrm{S}\mathrm{m}_{2}\mathrm{Fe}_{17}\mathrm{N}_{3}$ within soft magnetic matrices (e.g., a –Fe) [186]. The effect of the strength of intergrain exchange interaction is im- portant in these materials and influences properties of nanocomposite magnets such as coercivity and maximum energy product [187]. Decreasing the grain size of the material and reducing the intergrain exchange interaction can increase the maximum energy product to as high as 200 kJ $\mathrm{m}^{3}$ [188]. In general, the coercivity also increases (and peaks at some value) with a decrease of the exchange constant compared to the intragrain exchange interaction. Nanosize magnetic particles typically have grain sizes smaller than the critical single domain particle size. Also typically in magnetic nano- composite films, the coercivity increases with grain size if the grains are isolated (no interaction), but when the grains start contacting and exchange interaction kicks in, the coercivity falls rapidly with grain size. The coercivity is highest at percolation, where the grains just start touching each other. In a very recent study, exchange- coupled nanocomposites were fabricated by nanoparticle self-assembly. Here, both FePt and ${\mathrm{Fe}}_{3}\mathrm{O}_{4}$ particles were incorporated as nanometer-scale building blocks into binary assemblies, which were subsequently annealed to obtain $\mathrm{FePt}/\mathrm{Fe}_{3}\mathrm{Pt}$ nanocom- posites, in which FePt is a magnetically hard phase and $\mathrm{Fe}_{3}\mathrm{Pt}$ a soft phase. An opti- mum exchange coupling, and therefore an optimum energy product, is obtained by independently tuning the size and composition of the individual building blocks [189]. + +The improvements in magnetic properties in nanocomposites can be explained as follows. A nanocomposite magnet can be considered as hard and soft magnetic grains connected by magnetically interacting springs. The intergrain interaction has a ten- dency to align the magnetization in each grain in the same direction. When a mag- netic field is applied in the opposite direction, reversal of magnetization occurs only in the soft grain (until a critical field value is reached). The magnetization in these grains is reversed when the field is removed, resulting in high values of reversible suscept- ibility. However, for high applied fields, which reverse the magnetization even in the hard grains, the magnetization in the soft grains does not reverse upon removal of the field. This effect is most notable in magnets (Nd/Fe/B) having grain sizes smaller than $50~\mathrm{nm}$ . The strength of interaction between the grains is crucial in determining the properties of the nanocomposite magnets; in addition, the magnetic properties also depend on the grain size, the fraction of soft and hard phases present, and the shape of the grains. + +Different synthesis routes can be used to produce magnetic composites. Melt spin- ning/splat cooling of alloys accompanied by heat treatments such as fast annealing have been used to prepare alloy ribbons of nanocrystalline and nanocomposite mag- nets (e.g., containing $\mathrm{nd}_{2}\mathrm{Fe}_{14}\mathrm{B}$ , $\mathrm{Fe}_{3}\mathrm{B}$ , and $a$ -Fe). The most successful technique has been mechanical alloying of two-phase mixtures (e.g., $a$ -F $\mathrm{{^\circe/S m}_{2}F e_{17}N_{3}}$ ), which typi- cally results in two-phase mixtures consisting of a nanocrystalline soft magnetic ma- trix and an amorphous hard magnetic phase. Annealing and crystallization at higher temperatures lead to nanocomposite magnets that show superior magnetic behavior. The evolution of the microstructure and grain morphology during mechanical milling and subsequent heat treatments affects the magnetic properties and has to be con- trolled precisely. Cosputtering from multiple targets is another method capable of fabricating uniform nanocomposite structures. Uniform thin-film nanocrystalline co- balt grains $(10\,{-}\,20\ \mathrm{nm})$ surrounded by graphitic layers have been prepared by this technique followed by post-annealing treatments [190–192]. The $\mathrm{Co}/\mathrm{C}$ system is im- miscible, with a negative interdiffusion coefficient, and because the carbides in this system are metastable, they decompose into the elemental forms. Hence, uniform films of segregated magnetic and encapsulant layers can be fabricated for controlling the magnetic interaction between the different Co grains present in the films. Such films may prove useful for high-density magnetic recording, because carbon provides good isolation, so as to reduce the intergrain exchange interaction. The thickness of the film deposited also needs to be controlled for preparing optimal recording media, and this thickness should be comparable to the grain size $({\sim}10\,{-}\,15~\mathrm{nm})$ ) within the films for best performance. + +# 1.12.2 Magnetic Multilayer Nanocomposites + +The increase in storage density of hard disk drives and the parallel decline in cost per megabyte (MB) of storage is happening rapidly [193]. Along with disk performance and price, sputter equipment is evolving in both design and efficiency. Current standard processing modules involve heating, dc or RF sputter deposition, and cooling. Further improvements and required modifications of these modules, as well as new proces- sing steps such as substrate cleaning, multilayer deposition, and plasma chemical vapor deposition of carbon are necessary in the future. Magnetic nanocomposites are the end result of many of those improvements. + +# 1.12.2.1 Microstructure and Thermal Stability of Layered Magnetic Nanocomposites + +Granular solids consist of small metal granules embedded in an immiscible medium, which may be insulating or, less frequently, metallic [194]. Like other artificially struc- tured solids, granular solids have intricate structure on the nanometer scale and extra degrees of freedom within which the physical properties can be manipulated to achieve tailored materials for applications and for explorations of physical phenom- ena. The relevant extra degree of freedom for granular solids is the granule size. In magnetic granular solids, all such granules are single domains exhibiting ‘hard’ magnetic properties. On the other hand, for samples with large metal volume frac- tions, the metal granules form an infinite network, exhibiting metallic conductivity and ‘soft’ magnetic properties, because the conducting paths and magnetic closure structure are readily facilitated. For most common metals (e.g., Fe, Au, Co, Cu), the ultrafine solid granules can be made as small as 10 to 100 nanometers. This is the size range in which finite-size effects, single-domain magnetic properties, and other phenomena are observed. Work on this type of nanostructured film began with the pioneering studies of Abeles et al. [195]. These films have attracted consider- able attention in recent years because of theoretical suggestions dealing with their physical properties and also because of practical applications, among which the use of ferromagnetic nanocomposites as data recording media stands out. For this application, superior magnetic performance can be tailored by manipulating the size and volume fraction of the particles. Other advantages also derive from the struc- ture of the film, in that the insulating portion of the film provides chemical and me- chanical protection to the metal particles. + +A variety of deposition methods can be used to produce granular, layered magnetic solids. Of these, sputtering is the most versatile. Sputtering is most often applied via a single homogenous target, or by cosputtering. Sequential deposition is also applicable if layer formation can be avoided. Generally, because the nanostructures of granular metal solids are strongly influenced by the processing conditions, deposition para- meters such as deposition rate, sputtering pressure, and substrate temperature must be tightly controlled. Multilayer thin films with an under layer and a cover layer (total film thicknesses are usually in the range of several micrometers) have been prepared with multiple-gun dc and RF sputtering systems. The multilayers were de- posited onto glass or silicon substrates. Flowing high-purity gas (Ar, $\mathrm{N}_{2}$ , etc.) was used in sputtering. The sputtering parameters, including gas pressure, the power rates of the sputtering guns, and the target-to-substrate distance, were chosen to produce films with maximum coercivity. Heat treating the as-deposited films in a furnace or rapid thermal annealing yielded nanocomposite films with random nanostructure. Gener- ally, high coercivity is obtained only after heat treatment. + +As an example, cobalt nitride films, CoN, in pure form and also as a nanocomposite in a boron nitride or silicon nitride matrix were generated by reactive sputtering of cobalt metal, cobalt boride, or cobalt silicide as targets, respectively, in nitrogen plas- ma [196]. Cobalt nitride decomposes into the elements when heated under vacuum at $500\,^{\circ}\mathrm{C}.$ The nanostructure of the composites was preserved in the heat treatment, thus creating a fine dispersion $(<\!10\:\mathrm{nm})$ ) of cobalt particles in a ceramic matrix. The mag- netic properties of the nanocomposites were studied in detail; the precursor cobalt nitride is paramagnetic but the cobalt dispersions, having dimensions smaller than single magnetic domain size, show characteristics typical of systems such as super- paramagnetism and, at temperatures lower than the blocking temperature, marked hysteresis. The coercive fields at $5\;\mathrm{K}$ for the BN and $\mathrm{Si}_{3}\mathrm{N}_{4}$ nanocomposites are 3250 and $850~\mathrm{De}$ , respectively. + +The problem of long-term stability of written bits in magnetic data storage devices places a fundamental limit on the possible increase in area density for conventional magnetic recording. Because decreased magnetic grain size is required for the im- proved media signal-to-noise ratio that is needed to support higher area densities, thermal energy becomes significant relative to the switching energy, and the playback signal amplitude decays over time. Thermal stability will ultimately limit the maxi- mum area density achievable with conventional longitudinal recording. The key as- pects of the media microstructure contributing to thermal stability are the grain size and grain size distribution, alloy composition, alloy segregation, lattice defects, and strain. Random nucleation processes occurring during media deposition create grain size distributions. Continued increases in area density will require tighter grain size distributions and improved microstructural control of very thin magnetic layers [197]. + +Controlling the media grain size and tightening the grain size distribution are prob- ably the most difficult, yet most important, aspects of media processing. Due to the exponential dependence of the decay on $K_{\mu}V/k T,$ ( $\mathit{K}_{\mu}V$ is the product of the uniaxial anisotropy energy density and the switching volume), small changes in $V$ determine the difference between stable and unstable bits, with the smallest grains in the dis- tribution decaying first. Thermal decay experiments have shown that the data cannot be well explained without taking the contribution of the grain size distribution into account. Slow lateral grain growth rates or high nucleation rates favor smaller average grain sizes. Materials or growth conditions that limit surface mobility reduce the grain size. Because a minimum number of atoms are needed to form a critical nucleus, a narrower grain size distribution is expected as the grain size approaches the size of the critical nucleus. To significantly improve the grain size distribution, one needs a reg- ular array of nucleation sites or a new type of media in which the magnetic grains do not grow together to form a fully dense film. Although research on so-called self-as- sembled quantum dot arrays for media is ongoing, these approaches are still far from being implemented for production media. + +# 1.12.2.2 Media Materials + +Area densities in longitudinal magnetic recording have reached 36 Gbits $\mathrm{{in}^{\cdot2}}$ in labora- tory demonstrations [198] and 10 Gbits $\mathrm{{in}^{\cdot2}}$ in m a designs already in use $12\pm4$ nm grains) [199]. Roadmaps toward 40–100 Gbits in $\mathrm{{in}^{\cdot2}}$ have been proposed [200]. At such densities, tight control of the media microstructure, especially grain size, grain size dispersion, and chemical isolation to break exchange are necessary, in order to keep the media noise within acceptable bounds [201]. Continued grain size scaling to dia- meters considerably below $11\!-\!12\:\mathrm{nm}$ should be possible, allowing for densities well beyond the currently perceived 40–100 Gbits $\mathrm{{in}^{\cdot2}}$ limit. This prospect has been the driving force behind industrial and academic research in the area of thin-film hard magnet materials, which promise minimal thermally stable grain sizes down to $2\!-\!3\;\mathrm{nm}$ and a more than 10-fold potential density gain. However, for extremely high density recording media, typically defined as ${>}100$ Gbits $\mathrm{{in}^{\cdot2}}$ , the critical values of relevant magnetic properties will be stringent for achieving low media noise: coer- civities of $4~\mathrm{keV}$ and weak exchange coupling in grains ${<}10\,\mathrm{nm}$ are estimates. + +High- $K_{\mu}$ (uniaxial magneto crystalline anisotropy) materials are attractive for ultra- high-density magnetic recording applications (e.g., audio tapes, video cassettes, mag- netic memory devices) because they allow smaller, thermally stable media grains (Fig- ure 1.32). Particle-dispersed recording systems are also possible targets for future magneto-optical data storage. For the particulates, high magnetization is crucial, be- cause it determines the strength of the field to be detected during read/write. Metals such as Fe and Co have high magnetization, but one drawback is that they oxidize easily when particle sizes are small. Alloys have advantages in this regard and encap- sulation in a medium (matrix) can solve several problems associated with chemical stability of the particulates. The search for higher magnetic recording densities pur- + +![](images/f542e85ea97a20d158cebae69884d631e8a8798566532ea3eaa942a1431b0ef4.jpg) +Fig. 1.32 Schematic of multilayer microstructures of typical nanocomposites used in magnetic re- cording media. The area bound by the thick jagged lines corresponds to the transition between bits (dimension of each bit ${\sim}80\times400\,\mathrm{\Omegam})$ ). Within each bit, the grains of the magnetic layer, CoCrPt, have uniaxial anisotropy due to the random orientation of magnetic moments. The total magnetic moment is the vector sum of all magnetic orientations (shown by arrows M). The orientation of the Cr layers can induce magnetization in the plane of the film by their epitaxial relationship. The thickness of the films correspond to $20\;\mathsf{n m}$ for the CoCrPt layer and ${\sim}40~\mathsf{n m}$ for the Cr layer. (Courtesy Prof. M. Shima, Department of Materials Science $\&$ Engi- neering, Rensselaer Polytechnik Institute, Troy, New York 12180, USA.) + +sues particle sizes that are ${<}10\,\mathrm{nm}$ . With such small particle sizes, high magnetocrys- talline anisotropy is needed to avoid thermal and field fluctuations that can destroy the magnetization in recorded locations. When particle sizes become that small, in many material systems the magnetization can become inherently unstable, due to effects such as superparamagnetism. Intrinsic magnetic properties of several potential alter- native media alloys can be found in reference [202]. Recent studies have emphasized rare earth intermetallic compounds with high magnetic anisotropy and small grain sizes in the search for the best magnetic properties for high-density recording media. + +Nanocomposites, e.g., metal/carbon, metal/oxide, or immiscible metal/metal mix- tures, are prospective routes toward generating ${<}10\cdot\mathrm{nm}$ granular structures, needed for ultrahigh density recording. Much work has been reported on nanocrystalline rare- earth transition-metal films, most prominently ${\mathsf{C}}{\mathsf{o}}_{5}{\mathsf{S}}{\mathrm{m}}.$ - and $\mathrm{Co}_{17}\mathrm{Sm}_{2}$ -based films. High-density recording studies have been reported on $\mathrm{CoSM/Cr}$ structures with coer- civities in the 2000–4000 Oe range. Another example is nanocrystalline high coerciv- ity $\mathrm{CoPr}/\mathrm{Cr}$ films, for which coercivities of about 8000 Oe and grain sizes of $10\,\mathrm{nm}$ have been obtained. + +Face-centered tetragonal (fct) structures of near-equiatomic composition, CoPt and FePt, have been studied extensively. Both compositions result in large coercivities, close to the theoretical maximum obtained for 12–15-nm-thick epitaxial Fe/Pt (001) films with perpendicular orientation grown on $\mathrm{MgO}$ substrates. ‘Nanometer- size’ CoPt particles $\mathrm{{'}100\,{-}\,300\ n m)}$ obtained by dc-magnetron sputtered, and post-an- nealed CoPt films exhibited coercivities of about $30\,{-}\,37~\mathrm{kOe}$ . Annealed nanocompo- site $\mathrm{CoPt}/\mathrm{C}$ films may be tailored for magnetic recording media applications. Some of the common problems with hard magnet nanomaterials are unfavorably high proces- sing temperatures during annealing or growth, which may adversely affect grain growth. As a consequence, the noise performance is compromised, as was recently found in perpendicular recording studies of polycrystalline ordered Fe/Pt (001) thin films. Codeposited nanophase materials, e.g., $\mathrm{CoPt/Ag}$ and $\mathrm{CoPt}/\mathrm{C}$ , may allevi- ate some of these problems. CoPt alloys undergo phase transition from fcc to the more anisotropic face-centered tetragonal (fct) phase at higher temperature $(835\,^{\circ}\mathrm{C})$ , which makes this material magnetically hard. RF magnetron sputtering from targets contain- ing CoPt alloys and Ag can be used to prepare nanocomposite $\left(\mathrm{CoPt/Ag}\right)$ , and post- annealing treatment embeds the hard magnetic phase. Large coercivities, which can be tailored over the whole range by changing the processing route and hence the micro- structure, were achieved for these multilayer films of CoPt fct phases in a fcc Ag ma- trix, with average grain sizes in the range of $7\!-\!100\:\mathrm{nm}$ [203]. Different matrix materi- als (other than Ag) can also be used, serving as the isolation material between the hard magnetic grains, for example, silica and carbon. Carbon should be an ideal isolation material, because it does not react with Co or $\mathrm{Pt}$ to form carbides. $\mathrm{CoPt}/\mathrm{C}$ nanocom- posite films have been prepared that show the high coercivity and chemical stability required for high density recording media. Exciting new developments in high density recording media may further derive from chemical synthesis routes, self-assembly of magnetic nanoparticles, and template-based growth of magnetic nanowire arrays; these options are currently being explored for generating monodisperse, ordered na- noparticle arrays. + +In addition to the common applications outlined above, magnetic nanocomposite materials also have applications in ferrofluids, imaging, and separation technologies and as magnetic carriers for drug delivery. The challenge here is to make the magnetic nanocomposite particles (e.g., silica-coated iron oxide) resistant to agglomeration and chemical attack. The magnetic particles are often coated with protective and functional materials such as silica, which can be done effectively and easily by sol–gel synthesis or liquid phase coating processes (Figure 1.33). The silica coating provides protection against chemical attack and also provides a shell that can be modified chemically to make functional nanocomposites [204]. The sol–gel process (performed in organic solvent via hydrolysis of tetra eth oxys i lane) is good for making thin inorganic films with tailored porosity; however, because the coating is porous, chemical protection is not necessarily achieved. Supersaturated monosilicic acid can be used to achieve homogeneous thick coatings on particles. Often, a combination of the two processes is used to obtain uniform coatings of silica on magnetic particles. The key in this technology is to provide a protective coating without any reduction in magnetic proper- ties such as saturation magnetization (important for magnetic carrier technology) and to provide an outer surface for additional easy modification. Thin silica films also ensure the stability of the magnetic nanocomposites against aggregation and metal (Fe) release to the environment, which may contaminate the systems during use. Iso- tropic ferrofluid dispersion can be obtained for these silica-coated magnetic particles (due to electrostatic repulsion and strong hydration forces between the outer silica surfaces) in the absence of a magnetic field, and the dispersion can be easily switched + +![](images/68b4b1ec6ed4f1b7bca2c7f70a97c6cbf798803bd6e12bc303ee7b4de1fae4e7.jpg) +Fig. 1.33 Schematic and micrograph showing the fabrication of core shell structures. Several possible routes can be used to make such structures, for example, vapor phase coating or multi-step precipitation via nanoemulsion processing. The micrograph shows cobalt (core)-nickel oxide shell magnetic nanocomposite particles in the $20\ \mathsf{n m}$ size range. Such structures could find applications in magnetic xerography and as ferrofluids. (Micrograph courtesy Prof. Mamoun Muhammed) + +to an anisotropic structure in external magnetic fields. These nanocomposites have applications as stable ferrofluids and in magnetic carrier applications. Magnetic par- ticles embedded in polymer matrices (e.g., iron oxides inside a conductive polyaniline matrix) have also been synthesized for the fabrication of conductive, superparamag- netic plastic films [205]. Deposition of conductive polymer chains and magnetic na- noparticles in a layer-by-layer fashion may lead to new properties such as giant mag- netoresistance (GMR) and novel organic-based nanostructured GMR materials. + +# 1.13 Nanocomposite Structures having Miscellaneous Properties + +Photo electrochemical properties are inherent to semiconducting nanostructures. Elec- trochemical deposition from baths loaded with colloidal nanoparticles of the semicon- ductor (e.g., Ni films incorporating $\mathrm{TiO}_{2}$ , CdS, etc.) can produce metal films contain- ing photoactive semiconducting particles. Much larger photocurrents are produced from these nanocomposite electrodes compared to conventional semiconductor elec- trodes and contiguous nanocrystalline semiconductor electrodes [206]. Separating the semiconductor nanostructures within a matrix in a nanocomposite system and creat- ing intimate contacts between the particles and the molecular redox media can im- prove quantum yields in the photoexcitation processes. Controlling the local environ- ments of the semiconductor species in the nanocomposite can modulate electron transfer and carrier mobility at the electrodes. Such photoelectrodes provide an interesting context for studying charge transfer processes at semiconductor–molecu- lar contacts. As noted here, electrochemical deposition to fabricate nanocomposites is a powerful synthetic approach that has been used in several other instances to create nanocomposites with enhanced properties. One example is the formation of $\mathrm{Ni/}$ $\mathrm{Al}_{2}\mathrm{O}_{3}$ films [207], where ${\sim}50\;\mathrm{nm}\;\mathrm{Al}_{2}\mathrm{O}_{3}$ particles are uniformly distributed in a Ni matrix. The only requirement for this synthesis process is a stable inclusion phase (ceramic or semiconductor) and its well-dispersed suspension in the electrochemical solution. The $\mathrm{Ni/Al}_{2}\mathrm{O}_{3}$ nanocomposite shows enhanced mechanical properties due to reinforcement of the ductile metal matrix by the hard ceramic nanoparticles (disper- sion hardening). These can be used as coatings that retain most of the properties of the metal. + +Applications of $\mathrm{TiO}_{2}$ colloids and thin films extend from photovoltaics to catalytic devices and sensors. Nanocrystalline $\mathrm{SiO}_{2}$ and ordered structures have been synthe- sized by the sol–gel method. Processing conditions can be controlled to produce var- ious interesting (isotropic to anisotropic) morphologies and structures in the micro- structure of the films that can be deposited. Rod-like or truncated pyramidal morphol- ogies are formed at the anatase phase as the temperature is varied during processing [208]. Controlling the base used in the colloidal synthesis can make aligned rod-like structures self-organize into regular arrays. The ordering occurs due to surface charges and forces that can be altered by controlling the basicity and the dielectric constant of the colloidal medium. Thin-film optoelectronic devices with semiconductor and metal nanoparticles are hot candidates for research these days [209]. Alternating layers of $\mathrm{SiO}_{2}$ thin films and modified nanoparticles (semiconductor–metal layers) have been prepared for nonlinear optical applications [210]. Layer-by-layer assembly and surface sol–gel processing are used to fabricate these nanocomposites. Surface-stabilized (with organic monolayers such as thiols) gold nanoparticles $(4\!-\!5\ \mathrm{nm})$ can, in addi- tion, be assembled into layers forming close-packed monoparticulate layers; multi- layer films of $\mathrm{SiO}_{2}$ /Au can be prepared this way. The surface hydroxyl groups present on the Au particles act as an active component in the surface sol–gel process [211], which can be used to make metal oxide thin films with nanometer-scale control of film thickness. + +The optical properties of small semiconductor particles are interesting, showing quantum confinement effects and shifts in their absorption spectra. Also, dielec- tric/nanocrystalline semiconductor interfaces enhance carrier recombination to pico- seconds, with the possibility of ultrafast optical applications. Applications of semicon- ductor polymer nanocomposites in optics are discussed briefly in the next chapter. Similarly, metal/semiconductor hybrid nanoparticles produce novel effects such as enhanced optical nonlinearity [212] and shifted plasmon resonances for the metal particles [213]. Such hybrid nanocomposite particles (CdS/Ag) have been prepared in glass matrices by a modified melt–quench process [214]. The hybrid CdS-coated Ag nanoparticles exhibit red-shifted plasmon resonance absorption spectra, which can be tuned by the heat treatment that controls the particle size and other interfacial effects (Figure 1.34). Arrays of nanometer-sized CdSe clusters have been fabricated inside the pores of porous silicon matrices. The purpose here is to increase photolu- minescence by enhancing the light-emitting centers by chemical infiltration [215] (using first an aqueous $\mathrm{{Cd}(C O O)_{2}}$ solution, resulting in the formation of CdO, and subsequently selenidation by passage of $\mathrm{H}_{2}\mathrm{Se}$ ). The nanosize semiconducting clus- ters $(3\!-\!5~\mathrm{nm})$ provide the high-density light-emitting centers in this nanocomposite on top of the intense photo luminescence spectra in the visible range from the porous silicon. Electro chemically depositing metal (Au) within pores of optically transparent nanoporous alumina templates can be used to prepare interesting optical nanocom- posites. The plasmon resonance absorption band of the metal nanoparticles as a func- tion of particle size and aspect ratio (nanoparticle vs. nanowires in the channels of the alumina) was studied in such composites [216]. As the particle size decreased, the $\lambda_{\mathrm{max}}$ (wavelength of maximum absorption intensity) approached theoretical values and reached a quasi-static lower limit for particles of ${\sim}16~\mathrm{nm}$ , irrespective of the aspect ratio. + +Functional nanocomposites in which nanoparticles are deposited inside appropriate substrates can be used as sensors that react by changing optical properties in response to gas exposure. CdO-doped $\mathrm{SiO}_{2}$ nanocomposites have been prepared by RF cosput- tering, and show gas-sensing capabilities [217]. This nanocomposite showed a change in optical transmittance during exposure to nitrogen oxide gas. For such nanocompo- sites to function, the embedded nanoparticles should change optical transmittance upon exposure to gases, the matrix should have porosity for the gas molecules to dif- fuse and interact with the particles, and the matrix should be transparent to allow detection of the changing optical signals. + +![](images/5a1aef98ea04d0fd27fe9da59b1e51e0c71f260f49e6b4da64ad317c1cb0add9.jpg) +Fig. 1.34 Optical (luminescence) property of CdS-coated $\mathsf{A g}$ particles embedded in a glass matrix nanocomposite. Plot shows the plasmon peak position as a function of the ratio of the CdS shell volume to the total particle volume in the nanocomposite particle. Semiconductor/metal hybrid nanoparticle composites were prepared in a glass host by a modified melt/quench process and examined with respect to their optical properties. The coated hybrid nanoparticles exhibit surface plasmon resonance absorption spectra that are red-shifted relative to that of a dispersion of homogeneous Ag nanoparticles in the same host. (Source [214] used with permission) + +Many nanocomposite systems derive their usefulness from the fact that value-added functions can be advantageous, in addition to the primary function of the nanostruc- tured phase. For example, we already discussed the dispersion-induced toughening and strengthening of ceramic nanocomposites $(\mathrm{Al}_{2}\mathrm{O}_{3}/\mathrm{SiO}$ , $\mathrm{Si}_{3}\mathrm{N}_{4}/\mathrm{Si}C$ , etc.), and, if the dispersoid phase acquire add new functions, such as piezo electricity and ferroe- lectricity, then new applications can result. Conventional powder metallurgy techni- ques (milling, mixing, and hot pressing) can be used to prepare barium titanate-dis- persed $\mathrm{MgO}$ nanocomposites [218]. The criteria for the formation of well-dispersed nanocomposites are that the dispersoids do not react with the matrix phase during sintering. In these perovskite-dispersed composites, the mechanical properties were not degraded, but new piezoelectric properties were added, enabling possible applications as sensors. Such materials have sensing ability for crack propagation by using electromotive forces or by introducing internal strains via electrical im- pulses, resulting in increased fracture toughness. The introduction of piezo- and fer- ro-electricity into structural ceramic nanocomposites will create new materials that exhibit ‘intelligent’ functions [219]. For transition-metal dispersed-oxide ceramic com- posites, ferromagnetism is also compatible with its excellent mechanical properties. In addition, good magnetic response to applied stress was found in these ceramic/fer- romagnetic-metal nanocomposites, showing the possibility of remote sensing of frac- ture or deformation of ceramic materials. + +Nanocomposites can also be effective as catalytically active membranes (e.g., cata- lytic converters in automobiles), in which a catalytically active phase is uniformly dis- tributed in a matrix of high porosity. The operating conditions for such materials are quite demanding, because they are used over a wide range of temperatures and che- mical environments. Nanostructured $\gamma\mathrm{-Al}_{2}\mathrm{O}_{3}$ catalyst supports with a dispersed cer- ium oxide phase and supporting noble-metal nanoparticle (Pt, Pd, etc.) heterogeneous catalysts are commonly used in automotive exhaust streams. The ceria phase serves as an oxygen-storage component, to buffer sudden changes in the fuel/air ratio; it also stabilizes the alumina structure and the noble metal particle dispersion. Calcination of sol–gels (prepared from Al and Ce chlorides and nitrides), followed by microwave or autoclave heating, have been used to prepare such nanocomposites [220]. The micro- wave-generated powder contains nanocrystalline ceria and alumina phases, suitable for catalytic applications. Note that, in nanocomposite structures, microstructure evo- lution and the final dispersion structure and its relation to the matrix decide the func- tionality of the composite. For example, if the final product in the above alumina/ceria composite is autoclaved instead of microwaved, a coarse microstructure that has no catalytic ability results. Silica/copper oxide nanocomposites have been prepared by a sol–gel process (controlled hydrolysis/polymerization of sodium metasilicate and cop- per nitrate via a microemulsion technique with anionic surfactants) and show promise in catalytic applications [221]. These nanocomposites have specific surface areas ran- ging from 200 to $400\,\mathrm{m}^{2}\,\mathrm{g}^{\ensuremath{-}1}$ and are mesoporous, with pore sizes in the range $3\!-\!6\:\mathrm{nm}$ . Nanocomposites are also being considered for use as photocatalysts to decompose environmental pollutants and harmful microorganisms. As an example of such ma- terials, transparent nanocomposite anatase films have been deposited at low tempera- tures on various substrates by a sol–gel method [222]. For the substrate, materials such as porous $\mathrm{SiO}_{2}$ work well because, for such applications, a high surface area, transparency to light (including UV), and durability for the photocatalytic activity are required. + +Low-dimensional catalytic nanocomposites can also be envisioned. For example, molybdenum carbide has been considered as a substitute catalyst for metals like Ru in several heterogeneous catalysis systems. Sonochemically prepared nanocompo- sites of molybdenum oxide and carbide dispersed as discontinuous surface coatings on microspheres of silica are being considered as nanocomposite catalysts and fit the profile of the optimal catalyst material, considering that the composites retain the nanoscale dimensions of the active phases, provide effective interaction between the catalyst and the solid support, and have a uniform distribution of the active phase on the support surface [223]. The formation of strong interfacial Si-O-Mo bonds is observed in this material. The use of sonochemical activation is yet another approach to breaking down and dispersing precursor materials into nanocrystalline forms and simultaneously activating surfaces onto which the dispersed phase can be uniformly coated. + +Oxidation-resistant coating systems have been developed, based on nanocomposite technology, for applications such as resistive heating elements for industrial melting. + +Materials such as Mo, which has several excellent electro mecha no thermal properties, are used for such applications, but suffer from severe oxidation problems. $\mathrm{MoSi_{2}}$ is a substitute material for such applications, due to its excellent resistance to oxidation at high temperatures. The high temperature strength of this material is a problem, and incorporation of SiC into the $\mathrm{MoSi_{2}}$ matrix provides a compromise. The best solution is to put $\mathrm{MoSi_{2}/S i C}$ coatings onto Mo substrates, by techniques such as RF magnetron sputtering of composite targets [224]. As deposited films, they are usually amorphous but can be converted into nanocrystalline form by annealing under vacuum at ${>}1000\,^{\circ}\mathrm{C}.$ . The high interdiffusion of Si and C in these films can be minimized if the sputtered films are made in a nitrogen atmosphere. Pyrolysis of $\mathrm{MoSi_{2}}$ particles coated with polycarbosilane and subsequent hot-pressing can also be used to prepare bulk $\mathrm{MoSi_{2}/S i C}$ nanocomposite materials for high-temperature applications [225]. Uniform dispersions of SiC particles are obtained in the $\mathrm{MoSi_{2}}$ matrix after densifica- tion. The room-temperature and high-temperature flexural strengths of this nanocom- posite are several times those of unreinforced matrix. + +DLC-based nanocomposites possess very interesting properties and hence can be targeted for several applications (a high-profile application being the DLC coating on the Gillette Mach 3 razor blade). Amorphous diamond-like/quartz-like $\mathrm{(C;H/\Omega}$ Si:O) composites and metals containing such materials constitute a class of DLC- based nanocomposites with unique bulk and surface properties. These nanocompo- site materials have excellent diffusion barrier properties, high wear resistance, and low coefficient of friction, in addition to good adhesion to most substrates, variable elec- trical resistivity, thermal shock resistance, and controllable optical properties. Films of diamond-like nanocomposites can be deposited by plasma-assisted vapor deposition of organosilane precursors and involves C- and Si-containing free radicals and, option- ally, transition-metal atoms [226]. These nanocomposites can find applications in tri- bology, especially in magnetic recording read/write heads. DLC/ $\mathrm{SiO}_{2}$ nanocomposites have been prepared with better optical clarity as thin films [227]. The films of the nanocomposite (Figure 1.35) were deposited at room temperature by RF plasma, from tetra eth oxys i lane (TEOS) as the precursor. The advantage of this precursor is that it serves as the source for both the DLC and the $\mathrm{SiO}_{2}$ during a simultaneous deposition process. By introducing $\mathrm{O}_{2}$ into the plasma, the surface of the deposited film can be made more hydrophilic, changing its wetting characteristics. + +![](images/4e4f21e9fd0f0245e3f5e1d8f3b0130bd1adbcf11337226898057c1ea15bd58b.jpg) +Fig. 1.35 Beer–Lambert plot of normalized ab- sorbance of $\mathsf{D L S/S i O}_{2}$ nanocomposites as a func- tion of volume fraction of DLC. With increasing RF power the normalized absorbance increases line- arly with the volume fraction of DLC in the nano- composite thin films. The slope is the optical ab- sorption, and the value obtained for the nano- composite is similar to typical polar absorptions found for organic compounds. (Source [227] used with permission) + +# 1.14 Concluding Remarks on Metal/Ceramic Nanocomposites + +This chapter includes examples of how ceramic and metal-based nanocomposite materials can be useful in several applications and how they can be superior to conventional composites and to the separate, independent constituent phases. Thischapter in no way provides an exhaustive list, and the field is still evolving, providing new directions for applications of nanocomposites. Some of the synthesis techniques were discussed, as were challenges involved in fabricating nanocomposites with optimized properties. Synthesis parameters, which determine the micro- structure, greatly affect the final properties of the nanocomposite, particularly with respect to tailoring the interface between the constituent phases. In the next chapter we discuss another important class of nanocomposites based on polymer matrices. + +# Acknowledgements + +I greatly appreciate the help of Dr. Bing Qing Wei in putting together parts of this chapter and several of the figures. I also thank Dr. Seamus Curran and Dr. Robert Vajtai for their help in preparing this chapter. I thank the National Science Founda- tion (in particular the Nanoscale Science and Engineering Center at RPI) for funding support. + +# References + +1 M. Jose-Yacaman, L. Rendon, J. Arenas, M. C. Serra Puche, Science, 273, 223 (1996). 2 K. Niihara , Metal Powder Report, 54(3), 37 (1999). 3 Y. H. Choa, A. Nakahira, K. Niihara, J. Mater. Sci., 35, 3143 (2000). 4 K. Niihara, Y. K. Jeong, T. Sekino, Y.-H. Choa, Ceram. Trans., Adv. Ceram. Matrix Composites IV, Am. Ceram. Soc., We- sterville, OH, USA, 96, 185 (1999), 5 J.-F. Yang, T. Sekino, Y.-H. Choa, K. Nii- hara, T. Ohji, J. Am. Ceram. Soc., 84, 406 (2001). 6 M. Yoshimura, T. Ohji, M. Sando, K. 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Materials with synergistic properties are chosen to create composites with tailored properties; for example, high-modulus but brittle carbon fibers are added to low-modulus polymers to create a stiff, light- weight composite with some degree of toughness. In recent years, however, we have reached the limits of optimizing composite properties of traditional micro- meter-scale composite fillers, because the properties achieved usually involve compro- mises. Stiffness is traded for toughness, or toughness is obtained at the cost of optical clarity. In addition, macroscopic defects due to regions of high or low volume fraction of filler often lead to breakdown or failure. + +Recently, a large window of opportunity has opened to overcome the limitations of traditional micrometer-scale polymer composites – nanoscale filled polymer compo- sites – in which the filler is ${<}100$ nm in at least one dimension (Figure 2.1). Although some nanofilled composites (carbon black [1] and fumed silica [2, 3] filled polymers) have been used for more than a century, research and development of nanofilled poly- + +![](images/78a7242887d6d36b285816c94afa16cd72f77477bb0c99b9bedc9a790dda9cfa.jpg) +Fig. 2.1 Schematic of nanoscale fillers + +Nanocomposite Science and Technology. Edited by P.M. Ajayan, L.S. Schadler, P.V. Braun Copyright $\copyright$ 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30359-6 + +mers has greatly increased in recent years, for several reasons. First, unprecedented combinations of properties have been observed in some polymer nanocomposites [4]. For example, the inclusion of equi-axed nanoparticles in thermoplastics, and particu- larly in semicrystalline thermoplastics, increases the yield stress, the tensile strength, and Young’s modulus [5] compared to pure polymer. A volume fraction of only 0.04 mica-type silicates (MTS) in epoxy increases the modulus below the glass transition temperature by $58\,\%$ and the modulus in the rubbery region by $450\,\%$ [6]. In addition [7], the permeability of water in poly(e-caprolactone) decreases by an order of magni- tude with the addition of $4.8\,\%$ silicate by volume. Yano et al. [8] showed a $50\,\%$ decrease in the permeability of polyimides at a $2\%$ loading of MTS. Many of these nanocomposites are optically transparent and/or optically active. + +A second reason for the large increase in research and development efforts was the ‘discovery’ of carbon nanotubes in the early 1990s [9]. Although more careful review has shown that nanotubes have been observed since the 1960s [10], it was only in the mid-1990s that they were made in the quantities required for property evaluation of composites. The properties of these carbon nanotubes, particularly strength and elec- trical properties, are significantly different from those of graphite and offer exciting possibilities for new composite materials. + +Third, significant development in the chemical processing of nanoparticles and in the in situ processing of nanocomposites has led to unprecedented control over the morphology of such composites. It has also created an almost unlimited ability to control the interface between the matrix and the filler. + +Thus, this is an exciting time to study nanocomposites, because of the unique com- binations of properties that are achievable and because of the high potential for suc- cessful commercial development. Although the technical community has made ad- vances in the processing of nanocomposites, we are just beginning to assemble the interdisciplinary teams required to understand, tailor, and optimize properties. We have at our fingertips, however, the ability to change the size, shape, volume frac- tion, interface, and degree of dispersion or aggregation. Thus, the opportunities may well become limitless when theory and experiment have assembled enough informa- tion to guide further development. + +A relevant question addressed throughout this chapter is: What is unique to nano- fillers compared to micrometer-scale traditional fillers, and how do the composites compare to their macroscopic counterparts? The most obvious difference is the small size of the fillers. For example, very small nanoparticles do not scatter light signifi- cantly, and thus it is possible to make composites with altered electrical or mechanical properties that retain their optical clarity. In addition, the small size means that the particles do not create large stress concentrations and thus do not compromise the ductility of the polymer. A similar concept applies for electrical breakdown strength. + +The small size of nanofillers can also lead to unique properties of the particles them- selves. For example, single-walled nanotubes are essentially molecules, free from de- fects, and have a modulus as high as 1 TPa and strengths that may be as high as 500 GPa. Single-crystal particles that are optically active, but are unmanageable on the macro scale can be combined in a polymer to achieve the optical gain of the material and the ease of processing afforded by the polymer. + +![](images/fc686567a86c919c3c8aa48cf104e26def4d9260424cdc36655c96f3878ab342.jpg) +Fig. 2.2 (a) Surface area per unit volume vs. particle size for spherical particles that are ideally dispersed, and (b) interparticle distance for spherical particles that are ideally dispersed (Thanks to B.J. Ash for preparation of this figure) + +In addition to the effect of size on particle properties, the small size of the fillers leads to an exceptionally large interfacial area in the composites. Figure $2.2\mathrm{a}$ shows the surface area per unit volume as a function of particle size for spherical particles that are ideally dispersed. The increase in surface area below $100\,\mathrm{nm}$ is dramatic. The interface controls the degree of interaction between the filler and the polymer and thus controls the properties. Therefore, the greatest challenge in developing polymer nanocompo- sites may be learning to control the interface. Thus, it seems relevant to define the interfacial region and discuss its properties. + +As defined in traditional composites, the interfacial region is the region beginning at the point in the fiber at which the properties differ from those of the bulk filler and ending at the point in the matrix at which the properties become equal to those of the bulk matrix [11]. It can be a region of altered chemistry, altered polymer chain mobility, altered degree of cure, and altered crystallinity. Interface size has been reported to be as small as 2 nm and as large as about $50\;\mathrm{nm}$ . Figure $2.2\mathrm{b}$ shows interparticle spacing as a function of particle size for an ideally dispersed nanoparticle composite: at low volume fractions the entire matrix is essentially part of the interfacial region. For ex- ample, for $15{\cdot}\mathrm{nm}$ particles at a filler loading of $10\;\mathrm{vol}\;\%$ , the interparticle spacing is only $10\:\mathrm{nm}$ . Even if the interfacial region is only a few nanometers, very quickly the entire polymer matrix has a different behavior than the bulk. If the interfacial region is more extended, then the polymer matrix behavior can be altered at much smaller loadings. Therefore, by controlling the degree of interaction between the polymer and the nanofiller, the properties of the entire matrix can be controlled. + +To implement the novel properties of nanocomposites, processing methods that lead to controlled particle size distribution, dispersion, and interfacial interactions are critical. Processing technologies for nanocomposites are different from those for composites with micrometer-scale fillers, and new developments in nanocompo- site processing are among the reasons for their recent success. + +This chapter highlights what we feel are important concepts for understanding and developing polymer nanocomposites. We start with a description of nanotube or na- nofiber fillers, plate-like fillers, and equi-axed fillers. We then introduce the interfacial region and attempt to give a broad understanding of how to control it, without delving too deeply into the chemistry involved. This is followed by a section on the processing of composites. Finally, we describe some of the properties that have been achieved in nanofilled polymer composites. + +# 2.2 Nanoscale Fillers + +Nanoscale fillers come in many shapes and sizes. For ease of discussion, we have grouped nanofillers into three categories (Figure 2.1). Fiber or tube fillers have a dia- meter ${<}100$ nm and an aspect ratio of at least 100. The aspect ratios can be as high as ${10}^{\mathrm{6}}$ (carbon nanotubes). Plate-like nanofillers (Figure 2.1) are layered materials typi- cally with a thickness on the order of $1\;\mathrm{nm}$ , but with an aspect ratio in the other two dimensions of at least 25. Three dimensional (3D) nanofillers are relatively equi-axed particles ${<}100\;\mathrm{nm}$ in their largest dimension. This is a convenient way to discuss polymer nanocomposites, because the processing methods used and the properties achieved depend strongly on the geometry of the fillers. + +# 2.2.1 Nanofiber or Nanotube Fillers + +# 2.2.1.1 Carbon Nanotubes + +Micrometer-size carbon tubes, which are similar in structure (but not in dimensions) to the recently discovered multi-walled carbon nanotubes, were first found in 1960 by Roger Bacon [12]. These nanosized, near-perfect whiskers (termed nanotubes) were first noticed and fully characterized in 1991 by Sumio Iijima [9] of NEC Corporation in Japan. He was investigating the surface of carbon electrodes used in an electric arc- discharge apparatus that had been used to make fullerenes. Several exciting develop- ments have taken place in this field since then, and several books document recent progress [13]. + +![](images/f6f8f32491bfdc22d478485b473e3270442d52518f1fb868db8faecea9fce762.jpg) +Fig. 2.3 (a) HRTEM image showing the SWNT in bundles, (b) HRTEM images of a MWNTalong its length and at the end, (c) schematic showing two examples of the helicity that occurs, zigzag and armchair + +The first nanotubes observed were multi-walled nanotubes (MWNT). MWNTs con- sist of two or more concentric cylindrical shells of graphene sheets coaxially arranged around a central hollow core with interlayer separation as in graphite $(0.34\:\mathrm{nm})$ [14]. In contrast, single-shell or single-walled nanotubes [15, 16] (SWNT) are made of single graphene (one layer of graphite) cylinders and have a very narrow size distribution $(1-$ $2\:\mathrm{nm}$ ). Often many (tens) single-shell nanotubes pack into larger ropes. Figure 2.3 shows electron micrographs of SWNT and MWNT. Both types of nanotubes have the physical characteristics of solids and are microcrystals, although their diameters are close to molecular dimensions. In nanotubes, the hexagonal symmetry of the car- bon atoms in planar graphene sheets is distorted, because the lattice is curved and must match along the edges (with dangling bonds) to make perfect cylinders. This leads to a helical arrangement of carbon atoms in the nanotube shells. Depending on the helicity and dimensions of the tubes, the electronic structure changes consid- erably [17, 18]. Hence, although graphite is a semi-metal, carbon nanotubes can be either metallic or semiconducting. Nanotubes are closed by fullerene-like end caps that contain topological defects (pentagons in a hexagonal lattice). The electronic char- acter of the ends of these tubes differs from the cylindrical parts of the tubes and is more metallic, due to the presence of defects in these regions [19]. + +The discovery of nanotubes has complemented the excitement and activities asso- ciated with fullerenes [20]. Although fullerenes have fascinating physical properties, their relevance in the nanocomposite field is limited, so we shall restrict our discussion of carbon nanostructures to nanotubes. + +![Tab. 2.1 Theoretical and experimentally measured properties of carbon nanotubes. ](images/3c49b8f479f7431fb9810872839819c9b54288401de56c7b401d8d6930a5ff12.jpg) + +The properties of carbon nanotubes are unique compared to other graphite fibers. Their structure remains distinctly different from that of traditional carbon fibers [21], which have been used industrially for several decades (e.g., as reinforcements in tennis rackets, airplane body parts, and batteries). Nanotubes represent the ideal, most per- fect, ordered carbon fiber, the structure of which is entirely known at the atomic level. Table 2.1 shows measured and theoretical properties of both SWNT and MWNT. A nice review of the mechanics of nanotubes was recently published [22]. The mechan- ical properties reported in Table 2.1 were determined directly on nanotubes by using various methods of loading with an AFM. Salvetat et al. [23] were able to place SWNT ropes across a silicon bridge and bend them using an AFM tip. They found that the modulus decreased dramatically as the size of the bundle increased (Figure 2.4). Wal- ters et al. [24] did a similar study and measured the complete stress–strain curve. The modulus of an SWNT rope, which, at this point for composites, is the relevant form of SWNT, was low at small strains and increased at higher strains. As further support of this, Wood et al. [25] found similar results with a more indirect measurement of the stress–strain curve. Because individual SWNTs are theorized to have a modulus as high as 1 TPa, this low modulus of larger ropes at low strains implies that slippage occurs between the individual nanotubes within the rope. The mechanics of this are not clear, but may be similar to that of rope mechanics [26]. If so, individual strands can slip until they are tightly packed, and then there is large friction between them; only then is the modulus of the individual fibers evident. When SWNT ropes were pulled directly by two AFM tips [27], a realistic modulus for the individual tubes within the bundle was achieved only if the SWNT in contact with the tips were + +![](images/b7908cf39bb608f0096af20c3011b53428c4ccab6eb2c11c8773b1827e9057d7.jpg) +Fig. 2.4 A plot of modulus vs. SWNT rope diameter showing the decrease in the modulus of SWNT ropes as a function of rope diameter. This highlights the poor load transfer between the SWNT within the rope and the need for separating SWNT from the ropes for use in com- posites. Reprinted with permis- sion from [23] copyright 1999, American Phyiscal Society + +counted as the cross sectional area. If the whole rope was used to determine the cross sectional area, the modulus was much lower. The important implication of this finding is that the modulus of SWNT can be realized only when the SWNT are isolated from + +![](images/4573db1fd7af6ab7a221809fd104d7dbb4832359e590754645ed81aed37055b2.jpg) +Fig. 2.5 Stress vs. strain curves for MWNT. The variation in the curves is partially a function of the difference in MWNT diameter. Reprinted with per- mission from [28] + +![](images/be301bd3812f36a6e68a96335adaa4663e893414767486df0b0cbec97c790b26.jpg) +Fig. 2.6 Transmission electron micrographs showing the tele- scopic failure that can occur in MWNT. Reprinted with permis- sion from [29] copyright 1998, American Institute of Physics + +the bundles and/or the bundles are small enough that matrix is in contact with each one. + +The most direct measurement of the mechanical properties of MWNT was con- ducted by Yu et al. [28] using two AFM tips (Figure 2.5). They found a range of mod- ulus and strength. The range in modulus may again be due to difficulty in determining the cross sectional area. Failure occurred via a sword-and-sheath mechanism (tele- scoping) first observed by Wagner et al. [29] (Figure 2.6), implying that only the out- ermost nanotube is carrying the load and the rest are slipping inside. Therefore, a smaller cross sectional area may be required for appropriate calculations of the mod- ulus. Thus, the variability in the number of layers in an MWNT probably accounts for the broad scatter in the modulus (Table 2.1). The important implication of this in composites, however, is that MWNT do not have the high modulus that is achievable with isolated SWNT. MWNT do have reversible deformation in compression, which may prove useful. Bower et al. [30] showed that MWNTs buckle under a compressive strain of about $4.7\,\%$ and that this buckling is reversible under very large strains, prob- ably due to the ability of the nanotubes to slide within each other. + +The fracture and deformation behavior of nanotubes is intriguing. Simulations have shown that highly deformed nanotubes switch reversibly into different morphological patterns with an abrupt release of energy. Nanotubes become flattened, twisted, and buckled as they deform [31]. They sustain large strains $(40\,\%)$ in tension without show- ing signs of fracture. These changes in deformation modes, such as buckling, have been recorded by TEM. The flexibility is related to the in-plane flexibility of the planar graphene sheet and the ability of the carbon atoms to rehybridize, with the degree of $\operatorname{sp}^{2}\!-\!\operatorname{sp}^{3}$ rehybridization depending on the strain. Such nanotube flexibility under me- chanical loading is important for their potential application as nanoprobes and in na- nocomposites. This behavior also offers an advantage against breakdown that could occur during composite processing. + +In addition to the deformation mechanics described, new modes of plastic behavior have been predicted in nanotubes at higher temperatures (relevant to nanotube/cera- mic composites, for example). Pairs of a 5–7 pair defect, called a Stone–Wales defect in $\mathrm{sp}^{2}$ carbon systems, are created at high strains in the nanotube lattice and become mobile. Such mobility leads to a step-wise size reduction (localized necking) of the nanotube. It also introduces changes in helicity in the region where the defects have moved (similar to a change in lattice orientation where a dislocation passes through a crystal) [32]. + +High-temperature fracture of individual nanotubes under tensile loading has been studied by molecular dynamics simulation. Elastic stretching elongates the hexagons until, at high strain, some bonds are broken. The local defect is then redistributed over the entire surface by bond saturation and surface reconstructions. The final result of this process is that instead of fracturing, the nanotube lattice unravels into a linear chain of carbenes (carbon atoms linked by double bonds into a chain) [31]. Such be- havior is unusual in crystals. It could play a role in increasing the toughness of na- notube-filled ceramic composites during high-temperature loading by increasing the energy absorbed during deformation. + +The electrical properties of nanotubes are also unique. Calculations have predicted that all the armchair tubes(Figure2.3c)aremetallic,whereas zigzag and helical tubes are either metallic or semiconducting [33]. The electronic conduction process in nanotubes is unique because the electrons are confined in the radial direction in the singular plane of the graphene sheet. Conduction in the armchair (metallic) tubes occurs through gapless modes, because the valence and conduction bands always cross each other at the Fermi energy [34]. In most helical tubes that contain a large number of atoms in the unit cell, the 1D band structure shows an opening of the gap at the Fermi energy, lending it semiconducting properties. This unique electronic behavior occurs only in small nanotubes. As the diameter of the tubes increases, the band gap (which varies inversely as the tube diameter) tends toward zero, yielding a zero-gap semiconductor that is equivalent to the planar graph ene sheet[35].InanMWNT,the electronic structure of the smallest inner tubes is overshadowed by the outer, larger, planar, graphene-like tubes. Experiments in which the band structure obtained from individual MWNT resembles that of graphite substantiate this phenomenon. In a later section, discussion of the electrical conduction and optical properties of nanotube-filled polymer compo- sites demonstrates the electronic importance of nanotube properties. + +# 2.2.1.2 Nanotube Processing + +The best methods to date for producing ideal nanotubes are based on the electric-arc and laser-ablation processes [36]. The material prepared by these techniques has to be purified by tedious chemical separation methods. Not one of these techniques is sui- table for producing the quantities needed for industrial applications (e.g., in compo- sites). In addition, this barrier has hindered research and development. In recent years, work has focused on developing chemical vapor deposition techniques using catalyst particles and hydrocarbon precursors to grow nanotubes [37]. In the past, such techniques were used to produce large quantities of hollow nanofibers of carbon. The drawback of catalytic CVD-based nanotube production is the inferior quality of the structures, which contain gross defects (twists, tilt boundaries, etc.), particularly be- cause the structures are created at much lower temperatures $(600\,{-}\,1000\,^{\circ}\mathrm{C})$ ) compared to the arc or laser process $(\sim\!2000\,^{\circ}\mathrm{C})$ . + +However, recently a major breakthrough has resulted in the efficient production of single-walled nanotubes by what is called a HiPCO process [38]. The technique, devel- oped at Rice University, has become the main source for commercially available high- purity SWNT in gram quantities. The technique uses the high-pressure (several atmo- spheres) disproportionation of CO gas in the presence of iron carbonyl catalyst vapor [39]. The plan proposed by the Rice group is to produce nanotubes in pound quantities by the year 2005 at a reasonable price (a few dollars per gram). Nanotubes obtained by this method are greater than $80\,\%$ pure by weight, and the $20\,\%$ impurity consists principally of iron particles from the catalyst. These particles should not have much effect on the mechanical properties of nanotubes when used as filler but could certainly have effects on their electrical, magnetic, and optical properties. + +In his first experimental report, Iijima [9] showed MWNT sticking to the ends of graphite electrodes that were used in the production of fullerenes. Fullerenes are formed in the vapor phase from the evaporation of graphite electrodes. MWNT are formed on the cathode surfaces used in such soot generation. A year after the discov- ery of MWNT, it was found that, if conditions are right, the evaporated carbon can be made to condense continuously on the cathode surface as a cigar-shaped deposit. This builds up into a boule a few centimeters long, made of a graphite shell packed with nanotubes and other forms of closed graphite nanoparticles [40]. The technique is similar to what Roger Bacon used almost 30 years ago to generate large (micrometer size) arc-grown carbon whiskers, although he used very different conditions in his experiment. The method is similar to the Kratschmer–Huffman method of generat- ing fullerenes [41]. To generate MWNT, a dc arc is normally used (with a modest voltage of $20\ \mathrm{V}$ and currents of ${<}100$ amp) in an inert atmosphere of 500 torr he- lium. Surprisingly, such perfect structures as nanotubes self-assemble in the plasma created in the interelectrode region, where the temperature is close to $3500\,^{\circ}\mathrm{C}$ . The time scale for formation of these nanostructures is extremely short; a MWNT 5 nm in diameter and $1000\,\mu\mathrm{m}$ long grows in about $10^{4}$ s [42]. + +The inside of the deposit formed in the electric arc-discharge contains a highly por- ous network of randomly oriented MWNT structures that are organized on a macro- scopic scale into pencil-like columns aligned in the axial direction of the deposit. Although MWNT grown by the electric-arc method has so far been the most perfect in terms of structure and properties, the technique suffers from drawbacks. Being a batch process limits the amount of material that can be produced, and the material in the deposit contains substantial amounts of nanoparticles that have polyhedral shapes and are low in aspect ratio. Several parallel attempts to make nanotubes through cat- alytic vapor deposition have overcome some of the problems of the arc process. In general, catalytic metal particles are exposed to a medium containing gaseous hydro- carbon species. Nanofibrils (twisted hollow fibers) are formed [43]. One good example + +![](images/7990ab3b0498e370ec0ca6959f4e8d17eea1cace3d2405b817a12a16dfb5aa28.jpg) +Fig. 2.7 Scanning electron mi- crographs showing MWNT grown by catalytic vapor depo- sition onto catalysts prefabrica- ted into patterned arrays resul- ting in well aligned nanotube assemblies + +is the Hyperion fibers that have been in commercial use for some time [44]. Control- ling the size of the catalyst seed has enabled achieving uniformity in the size of the fibers. The process can be scaled to produce large amounts of material. In some cases, when the catalysts are prefabricated into patterned arrays, well-aligned nanotube as- semblies are produced (Figure 2.7) [45]. Similarly, template-based approaches are also in use: the aligned pores of a nanoporous membrane (such as electro deposited porous alumina) are filled with carbon species through vapor deposition and later graphitized to produce nanotubes. The template membrane is removed to obtain aligned nanotube arrays [46, 47]. + +Single-walled nanotubes are made with a combination of catalyst and dense carbon vapor. Both are simultaneously introduced into an inert atmosphere by an electric arc [48] or through laser ablation [49]. In the former, the setup is similar to that used for MWNT synthesis, but a hole is drilled in the anode and packed with a mixture of the metal catalyst and graphite powder. Several metals and combinations of metals have been used to obtain good yield of nanotubes. The best so far consists of a mixture of Ni and Y with graphite in a 15:5:80 weight ratio. When the electric arc is struck with such a modified electrode, spectacular growth occurs in the reaction vessel, which becomes decorated with a network of webs that contain SWNT. Upon closer look, the webs contain ropes of nanotubes that consist of tens of SWNT (Figure 2.8). The maximum density of nanotubes is seen in the product that grows around the cathode like a collar, which consists of $\mathord{\left<>}50\,\%\right.$ wt. nanotubes. The rest of the carbon soot contains fullerenes, partially graphitized carbon (glassy carbon), amorphous carbon, and finely distributed catalyst particles. In the laser-ablation technique, a metal (Ni-Co) containing a small percentage ( $\scriptstyle\sum<1\%$ wt.) of graphite target placed in an oven is ablated with a strong laser pulse under an inert atmosphere. A felt of nanotube material is generated and is col- lected onto a water-cooled target by forced gas flow. Using two coordinated laser pulses breaks down the carbon formation better, producing a very high yield of nanotubes, with ${\tt>}90\,\%$ of the collected material being SWNT. + +![](images/0dbde39d60e3940ac20f1cd438fc984955fbc9f6a7f9e921ae48901d3a3b0749.jpg) +Fig. 2.8 Scanning electron micrograph showing SWNT grown by a catalyzed arc-discharge method + +# 2.2.1.3 Purity + +Nanotubes consisting of large impurity fractions are not useful for either laboratory studies or composite manufacture. For example, MWNT processed by the arc-dis- charge method have a significant volume fraction of other carbonaceous materials. In addition, they may contain a high content of scrolled layers instead of nano- tubes. Figure 2.9a shows typical material as produced. Because the extra carbonaceous material and catalyst can play a significant role in determining the composite’s proper- ties, they must be removed. In addition, we still need better control in manipulating (slicing, joining, and making larger structures from nanotube building units) nano- tubes, especially through generalized approaches such as chemistry. + +Nanotubes are insoluble, and hence all the purification procedures must use filtra- tion-based techniques. As a result, the nanotube products can never be as pure as, for example, fullerenes. All purification procedures follow certain essential steps: preli- minary filtration to get rid of large graphite particles, dissolution to remove fullerenes (in organic solvents) and catalyst particles (in concentrated acids), microfiltration, set- tling, and chromatography to separate MWNT and nanoparticles or SWNT from the amorphous carbon impurities [50]. Nanotubes must be kept separated when in sus- pension and are typically dispersed with a surfactant (e.g., sodium dodecyl sulfate) prior to the last stage of separation. Some reports claim that SWNT samples with + +![](images/907dc5000d4b1e1a3c07297950fef64413c8afe6fdd7beb00e98b343b4a6cab3.jpg) +Fig. 2.9 Scanning electron Micrographs showing the purity of MWNT (a) as received MWNTproduced by arc-discharge, (b) purified MWNT, (c) MWNT grown by CVD + +${\tt>}99\,\%$ purity can be made by repeating some of these steps [51]. Recent efforts [52] to use size-exclusion chromatography have yielded good separation between nanotubes and nanoparticles. Curran et al. [53] achieved a $20\,\%$ yield by using toluene. Oxidation of any remaining carbonaceous material in a $600\,^{\circ}\mathrm{C}$ oven [54] can follow. Figure $2.9\$ shows the purity that can be obtained with this method, but the yield is very low [55]. Other oxidation methods, such as plasma or acid oxidation, work but damage the nanotubes [56]. Processing MWNT by chemical vapor deposition or pyrolysis [57] is an alternative, which yields MWNT contaminated with catalyst but with little else in the way of impurities. A higher concentration of defects may result (Figure 2.9c). MWNT processed by laser-deposition techniques, including the nanofibers pro- duced by Hyperion [58], have a significant number of defects relative to pure MWNT but are produced purely. The effect of such defects on the mechanical properties is unclear, but they lead to early oxidation and probably reduce both the strength and the conductivity. + +It is now possible to cut nanotubes (SWNT) into smaller segments by extended sonication in concentrated acid mixtures [59]. The resulting pieces of broken nano- tubes (open pipes that are typically a few hundred nanometers long) form a colloidal suspension in solvents and can be deposited onto substrates or further manipulated in solution and functionalized at the ends. Such segments can perhaps be joined with appropriate chemical bridges to construct long nanotube chains resembling polymers. A whole gamut of possible chemistry based on nanotubes is just beginning to unfold. In the future, it may be possible to dissolve such functionalized nanotubes in organic solvents and later separate them to produce high-purity samples. + +Because size and helicity are two important parameters that affect nanotube proper- ties, selection on the basis of both these factors is important. This is mostly relevant to SWNT, because the MWNT shows average properties that tend to be semi-metallic only. Fortunately, the size distribution produced during the high-yield synthesis of SWNT is quite narrow, with an overwhelming majority of the nanotubes having dia- meters close to $1.4\:\mathrm{nm}$ . The nanotube diameter can be tuned in the range between 1.2 and $5\:\mathrm{nm}$ by altering the temperature (from $700\,^{\circ}\mathrm{C}$ to $1200\,^{\circ}\mathrm{C},$ ) under which they are formed. It is fortuitous that, when SWNT are formed in high yield, a majority of the tubes have armchair (or close to armchair) arrangement, and this arrangement is in- dependent of the conditions used in synthesis. This size and helicity have a strong bearing on the possible usability of nanotubes, especially in electronics. + +# 2.2.1.4 Other Nanotubes + +Other types of nanotubes require mention, and these could have some role in the fabrication of novel nanocomposite materials. The closest in structure and mechan- ical properties to carbon nanotubes are hexagonal boron/nitrogen nanotubes, which can be produced by arc-discharge, laser ablation, and CVD processes and can be fab- ricated into multiwalled and single-walled structures. The modulus and strength of BN nanotubes are very similar to those of their carbon counterparts [60]. The advantages of using them would be better oxidation resistance and their electrically insulating nature (for certain dielectric applications). The lattice of carbon nanotubes can be doped (to certain levels) with boron and nitrogen, providing a broad range of possible BCN na- notube structures [61]. Boron has an exceptionally interesting effect: insertion of boron into carbon nanotubes during growth allows selection of their helicity (zigzag nano- tubes are stabilized) [62]. The average length of nanotubes is also much higher if boron is added during growth, because boron acts as a surfactant near the growing ends, making it difficult for the structure to close. Nitrogen also can be inserted into the carbon nanotube lattice, but this insertion produces corrugated nanotube structures [63]. The presence of unsaturated nitrogen atoms in the lattice of nanotubes makes them more easily dispersed in solutions, and such nanotubes make good interfaces with certain polymers. Thus, doping nanotubes improves their surface reactivity. This could ultimately aid in the design of strong nanotube matrix (particularly polymer) interfaces, which is necessary in high-strength composites. These modified nanotube lattices also lead to modified electrical and optical properties and hence novel proper- ties for composites in which they are embedded. + +Most layered materials can be fabricated into nanotubes. Examples other than those noted above are nanotubes made of dichalcogenides $(\mathrm{MoS}_{2}$ , ${\mathbb{W}}{\mathbb{S}}_{2}$ , etc.) [64], nanotubes of several oxides $(\mathrm{V}_{2}\mathrm{O}_{5}$ , $\mathrm{MoO}_{3}$ , etc.) [65], and organic nanotubes [66]. These materials (including the BCN types) are not currently available in bulk quantities, and hence very little data is available on the mechanical and electrical properties of composites that contain these nanostructures. However, research is moving quickly in synthesizing several such nanotubular structures, which could ultimately find novel applications in multifunctional nanocomposites. + +# 2.2.2 Plate-like Nanofillers + +The most common 2D fillers are layered silicates. The most well known layered silicate is mica. Mica is made up of large sheets of silicate (on the order of centimeters or more) with relatively strong bonds between the layers. Smectic clays or phyllosili- cates, on the other hand, have relatively weak bonding between the layers, and the layers are small flakes. Each layer consists of two sheets of silica tetrahedra (corner shared) with an edge-shared octahedral sheet of either alumina (a lumi no silicates) or magnesia (magnesium silicates) [67]. Due to isomorphic substitution of alumina into the silicate layers $(\mathrm{Al}^{3+}$ for $\mathrm{Si}^{4+}$ ) or magnesium for aluminum $(\mathrm{Mg}^{2+}$ for $\mathrm{Al^{3+}}$ ), each unit cell has a negative charge between 0.5 and 1.3. The layers are held together with a layer of charge-compensating cations such as $\mathrm{Li^{+}}$ , $\mathrm{Na^{+}}$ , $\mathrm{K}^{+}$ , and $\mathrm{Ca^{+}}$ . These charge-com- pensating cations provide a route to the rich intercalation chemistry and surface mod- ification required to disperse clays at the nanoscale into polymers. The cation-ex- change capacity (CEC) defines the number of exchangeable interlayer cations and is usually described as $\mathrm{mEq}/100\,\mathrm{g}$ . Values range from 60–120 for smectic clays. A typical structure is shown in Figure 2.10 [68]. As they occur in nature or syntheti- cally, the layers are $20\mathrm{-}200\;\mathrm{nm}$ in diameter laterally and come in aggregates known as tactoids, which can be ${\sim}1\;\mathrm{nm}$ or more thick. Examples of smectic clays include montmorillonite (CEC ${\sim}76.4\!-\!119\,\mathrm{mEq}/100\,\mathrm{g})$ , which is an aluminosilicate; sapo- + +![](images/98a242ca7b7554e792f073b2a141367f4e5a15ba6460386270ee4a7fd8b1fbc8.jpg) +Fig. 2.10 Schematic of the crystal structure of 2:1 layered silicates (smectites). Van der Waals in- terlayer or gallery containing charge-compensating cations $(\mathsf{M}^{+})$ separates covalently bonded oxide layers, $0.96~\mathsf{n m}$ thick, formed by fusing two silica tetrahedral sheets with an edge-shared octahedral sheet of alumina or magnesia. Reprinted with permission from [68] + +nite, a synthetic material that closely resembles montmorillonite; hectrite, which is a magnesium silicate (CEC ${\sim}55\,\mathrm{mEq}/100\,\mathrm{g})$ , and its synthetic equivalent Laponite. Montmorillonite tends to have sheets that are up to $200\,\mathrm{nm}$ wide, and Laponite sheets are $25\,{-}\,30\,\mathrm{nm}$ wide. + +For these layered silicates to be useful as nanocomposites, the layers must be se- parated. Silicate clays are inherently hydrophilic, but polymers tend to be hydropho- bic. This presents an interesting challenge in terms of being able to disperse the si- licate layers in a polymer. Fukushima and Inagaki [69] demonstrated that an exchange reaction could be used to replace the inorganic exchange ions in the galleries between the layers with alkylammonium surfactants (di oct a dec yl dimethyl ammonium). This opens the galleries enough and makes them hydrophobic enough that a Nylon mono- mer could be intercalated between the layers, resulting in a clay/Nylon nanocomposite. Figure 2.11 [70] shows the increase in interlayer spacing, measured by x-ray diffrac- tion, as a decrease in the peak position for montmorillonite swelled with alkylammo- nium surfactants of various lengths. The intercalation also modifies the polarity of the layers by lowering the interfacial free energy of the silicate [67]. The number of onium ions that can pack into the galleries depends on the charge density of the clay and the cation-exchange capacity. This, as well as the length of the surfactant chain, deter- mines the distance between the layers. At lower charge densities, the surfactant packs in monolayers and, as the charge density increases, bilayers and trilayers can form (Figure 2.12) [4]. At very high CECs $(\geq\!120\,\mathrm{mEq}/100\,\mathrm{g})$ and long surfactants $({>}15$ car- bons), the packing can be ordered in a paraffin-type structure. The positively charged heads of the onium ions prefer to pack close to the clay layer, and the organic ends tend to fill the middle. Further processing of the layered silicates to form polymer compo- sites is discussed in the processing section below. + +In general, the properties of the clay layers are unknown, and estimates are made based on composite properties of similar materials in bulk. The properties are not unusual. For these fillers, it is the aspect ratio that is the most significant. Their + +![](images/a9cdfc62baf59b2e493c3bab0bafaf1778e8a8d5c6050906fbe7de8210aaf944.jpg) +Fig. 2.11 X-ray diffraction patterns of n-mont- morillonite as the length of the alkylammonium chain length increases. Reprinted with permission from [70] + +high aspect ratio theoretically makes them efficient for carrying loads. Such clay layers are almost impermeable to gases and water through the layer thickness and thus pro- vide an excellent barrier to gas and water in composites. + +Layered silicic acids provide an alternative to smectic clays. The intercalation chem- istry is similar to that of smectic clays, and they can be obtained with high purity and with structural properties that compliment those of clays [71]. Examples of layered silicic acids include kanemite $(\mathrm{NaHSi}_{2}\mathrm{O}_{5})$ ), makatite $\mathrm{(Na_{2}S i_{4})_{9}~\cdot~n H_{2}O)}$ , octosilicate $\mathrm{(Na_{2}S i_{8}O_{17}}$ · $n\mathrm{H}_{2}\mathrm{O})$ ), magadiite $(\mathrm{Na}_{2}\mathrm{Si}_{13}\mathrm{O}_{29}\ \cdot\ n\mathrm{H}_{2}0)$ ), and kenyaite $\mathrm{(Na_{2}S i_{2}O O_{41}}$ · $n\mathrm{H}_{2}\mathrm{O})$ . The thickness of the layers varies from $0.5\;\mathrm{nm}$ for makatite to $1.77\,\mathrm{nm}$ for kenyaite [72]. The general structure of layered silicic acids involves layers of $\mathrm{SiO_{4}}$ tetra- hedra with an abundant hydroxyl siloxane surface. Interlayer alkali ions can be ex- + +![](images/736d96a4ca16df298084b58737866b146b576e66c8271eac8177f3023fecd3ba.jpg) +Fig. 2.12 Schematic of the orientation of alkylammonium ions in the galleries of layered silicates with different layer charge densities. Reprinted with permission from [4] + +changed with the resulting active hydroxyl sites, leading to enhanced bonding with the intercalates. + +Layered double hydroxides (LDH), for example, $\mathrm{Mg_{6}A l_{2}(O H)_{16}C O_{3}\,\cdot\,4H_{2}O,}$ , have a positive (instead of negative) charge on the $\mathrm{Mg(OH)}_{2}$ layers. They are produced synthe- tically [73] and provide an opportunity for intercalation with organic anions. They have been used successfully to make nanocomposites [74, 75]. + +Transition-metal dichalcogenides are layered materials that can be intercalated with lithium. After intercalation, $\mathrm{MoS}_{2}$ and ${\mathbb{W}}{\mathbb{S}}_{2}$ can be placed in distilled water and exfo- liation occurs [76, 77]. Titanium and tantalum dichalcogenides have also been used to make nanocomposites [78, 79]. Essentially any layered material, including graphite [80], with appropriate chemistry can be expanded, creating the potential for intercala- tion of a polymer. + +# 2.2.3 Equi-axed Nanoparticle Fillers + +Production of nanoparticles with controlled size and degree of aggregation is the goal of many research efforts. The primary driver for this interest is the effect of particle size on their properties. For example, the optical absorption spectrum of Au changes with the size of the Au particles. The electroluminescence of semiconducting nano- particles is also size-dependent. One of the advantages of using nanoparticles in a composite structure is that the particle size and distribution can be stabilized. Materi- als that cannot be grown easily as single crystals can be used at the nanoscale and dispersed in a polymer to take advantage of the single-crystal properties. In addi- tion, the particles can lend properties to the polymers that they cannot achieve alone or with traditional fillers. For example, nanoparticle-filled polymers can lead to an increase in modulus and strength, but also maintain the polymer’s ductility. This is because the nanoparticles are such small defects relative to their micrometer-sized counterparts. + +Nanoparticles have been available for more than a century. Carbon black made by pyrolysis is available in a range of surface areas and degrees of aggregation. Silica made by a variety of methods, including the wet chemical process developed by Stober [81], the commercial Ludox Ø process [82], as well as flame processes, have been available for years. Although recent developments have been made [83], these are well-established processes. These nanoparticles have found significant applications in the rubber in- dustry, in catalysis, chemical mechanical polishing, water treatment, and other appli- cations. In recent years, significant progress in the diversity of processes for making nanoparticles has led to a more diverse set of nanoparticles and better control of par- ticle size, morphology, and surface properties. Recent developments in in-situ proces- sing of nanoparticles within a polymer matrix are discussed in the composite proces- sing section 2.4.3. In this section, we focus on the preparation of discrete particles that are subsequently incorporated into a polymer matrix. + +Aerosol methods usually result in the formation of nanoparticles by condensation from a gas phase [references in 84–87]. One example of a highly successful commer- + +![](images/710ec73b5bb0ef32b67244b4c2fbf979ec333b565d690a97f862b7399029e944.jpg) +Fig. 2.13 Micrographs showing (a) typical silica nanoparticle agglomerates and (b) titania made via gas condensation. (Thanks to T. Li for these micrographs) + +cial aerosol processes is flame hydrolysis. Here, a vapor precursor (such as silicon tetrachloride to make silica) is burned in a hydrogen/oxygen fuel mixture to produce the metal oxide. Titania, alumina, and zirconium oxide can all be produced this way. For silica, particle sizes range from 7 to $27\,\mathrm{nm}$ , and the surface areas range from 100 to $380\,\mathrm{m}^{2}\,\mathrm{g}^{-1}$ . The specific surface area and agglomerate structure can be controlled by adjusting the temperature and time of reaction. At higher temperatures, particle coa- lescence is fast, and low-surface-area materials result. At lower temperatures, coales- cence is slow relative to the collision rate, and fractal agglomerates result [88]. Figure 2.13a shows a typical silica agglomerate. + +Pyrolysis is another aerosol method and is commonly used to make carbon black, an amorphous form of carbon. The particles have a size range of $20\,{-}\,300\;\mathrm{nm}$ with surface areas of $20\!-\!500\,\mathrm{m}^{2}\,\mathrm{g}^{\!-\!1}$ . + +In a third method, called gas condensation [89, 90], a metal vapor is formed in either an oxygen atmosphere or an inert gas. In an oxygen atmosphere, the metal reacts with the oxygen to form metal oxides, which are collected by cooling the gas and condensing the nanoparticles. A large range of metal oxides can be processed this way, including $\mathrm{SiO}_{2}$ , $\mathrm{Al}_{2}\mathrm{O}_{3}.$ , $\mathsf{C u O}$ , $\mathrm{{CeO}_{2}}$ , $\mathrm{ZnO}$ , $\mathrm{ZrO}_{2}$ , $\mathrm{Fe}_{2}\mathrm{O}_{3}$ , and $\mathrm{Y}_{2}\mathrm{O}_{3}$ [91]. In an inert atmosphere, metal nanoparticles result. Advantages of this method are that the particle size dis- tribution is narrow, the particles are crystalline, and the surfaces are clean. Figure $2.13\mathsf{b}$ shows typical metal oxide particles that result from this process. They tend to be isolated, crystalline, and spherical. In the past dozen years or so, great progress has been made in understanding the processes by which such nanoparticles are made and the fundamental relationships between their sizes and structures and their special properties [92–95]. In the late 1990s these processes were improved and ton quantities of gas-condensed metal-oxide nanoparticles have been produced. + +Laser ablation processes are another form of gas condensation. Here, targets (metal, metal oxide, or semiconductor) are ablated with a laser (often a pulsed laser). An inert or reactive gas is supplied to cool, condense, and sometimes react with the target material. By varying the pulse frequency, gas flow rate, and total pressure, the size of the particles and particle aggregates can be controlled. For example, increasing the gas flow rate decreases the primary particle size, but increasing the laser fluence and total gas pressure produces larger primary particles. Significant quantities of na- noparticles can be produced this way. Many types of materials can be processed, in- cluding $\mathrm{SiO}_{2}$ , $\mathrm{MgO}$ , ${\mathrm{Fe}}_{3}\mathrm{O}_{4}.$ , $\mathrm{Mg_{2}S i O_{4}}$ , $\mathrm{CaTiO}_{3}$ , $\mathrm{MgAl_{2}O_{4}}$ [96], $\mathrm{Al}_{2}\mathrm{O}_{3}$ [96–98], $\mathrm{SiO}_{2}$ [97], $\mathrm{Fe}_{2}\mathrm{O}_{3}$ [97], and many others [99, 100]. + +Other methods for creating a vapor that condenses into nanoparticles include plas- ma and chemical vapor condensation, spray pyrolysis, electrospray, and plasma spray [101]. These processes can form a wide range of nanoparticles. One interesting varia- tion on the aerosol process developed recently combines sol–gel processing with the aerosol process. Hydrolysis and condensation of TEOS is carried out in a laminar flow chamber at temperatures ranging from room temperature to $100\,^{\circ}\mathrm{C}$ [102]. The advan- tage of this low-temperature process is that a large specific area $(400\,\mathrm{m}^{2}\,\mathrm{g}^{\ensuremath{\cdot}1})$ can be produced because of the low coalescence rate relative to the collision rate. In addi- tion, the agglomerates tend to be linear instead of highly branched. + +A host of wet-chemistry methods exist in which one or two precursors are placed in an appropriate solution and nanoparticles form. These can be stabilized with an emul- sion, surfactant, or a macromolecule that surrounds and protects the nanoparticles from agglomeration. Metal [103], metal-oxide [104], semiconducting [105–107], super- conducting, and magnetic particles [108] have all been made this way. Using $\mathrm{ZnO}$ as an example, zinc acetate $[\mathrm{(CH_{3}C O O)_{2}Z n\cdot2H_{2}O}]$ can be reacted with NaOH [109] in the presence of poly(vinyl pyr ro li done) to create very stable $\mathrm{ZnO}$ particles with an average diameter of $4\:\mathrm{nm}$ [110]. + +Metal nanoparticles can also be made by sonication chemistry [111]. This process takes advantage of the formation, growth, and implosive collapse of bubbles within a liquid to create localized spots with temperatures as high as $5000~\mathrm{K}$ and pressures as high as $1800\,\mathrm{atm}$ . Under appropriate sonication conditions, precursors such as ${\mathrm{Fe}}({\mathrm{CO}})_{5}$ and $\mathrm{Co}(\mathrm{CO})_{3}(\mathrm{NO})$ form nanoparticles. If sonication is done in the presence of an alkane, the particles can be separated, but sonication can also be done in the presence of a polymeric ligand to make stable colloids or in the presence of an inor- ganic support. + +Metal oxides an also be formed by taking advantage of hydrodynamic cavitation [112]. Here, hydrodynamic cavitation is produced by using a high-pressure fluid sys- tem in the presence of a sol–gel solution. The local high temperatures and pressures due to collapse of bubbles result in the creation of nanoparticles. The pressure and the exposure time can be adjusted to control the particle size. + +This section has clearly left out many details of nanoparticle processing, and many significant contributions have not been referenced. We introduce only the basics of nanoparticle processing here, because this topic is so broad and our focus is on na- nocomposites. + +# 2.3 Inorganic Filler–Polymer Interfaces + +As mentioned in the Introduction, nanocomposites have at least an order of magni- tude more interfacial area than traditional composites. This increase in interfacial area results in a 3D interfacial region within the polymer matrix that can have properties significantly different from those of the bulk polymer. The local chemistry, degree of cure, chain mobility, chain conformation, and degree of chain ordering or crystallinity can all vary continuously from the filler/matrix boundary to some point in the polymer bulk. To understand the significant effects nanoscale fillers have on polymer behavior, this interfacial region must be understood. The paragraphs below outline some of the literature describing the behavior of polymers near a surface. + +Consider first a pure thermoplastic polymer film interacting with either air or a very flat inorganic surface. We know that surface/polymer interactions greatly influence the glass transition temperature (mobility, relaxation spectra) of thin polymer films. For example, the presence of a free surface, as observed in free-standing ultra- thin films, reduces the glass transition temperature [113–116] (Figure 2.14), but an attractive surface increases the glass transition temperature [117, 118]. The mechan- ism causing these changes in Tg is currently under discussion in the literature. Mo- lecular dynamics simulations [119] show that the decrease in Tg for free-standing thin films is caused by a decrease in the density, which in turn increases the mobility of the polymer chains. Monte Carlo simulations show that the presence of a neutral surface can increase the density and decrease the mobility of the polymer chains [120], although experimental data suggest that the surface must be attracting before Tg in- creases. + +![](images/0a420590ccfa605532acda494d5af772e3187c73545d3d766ba9149c3c123fff.jpg) +Fig. 2.14 The glass transition temperature as a function of film thickness for low and high molecular weight polystyrene free-standing films. Reprinted with permission from [113] + +Composites, however, do not contain ideal 2D interfaces. In fact, the highly curved surfaces and the size scale close to that of a typical radius of gyration for a polymer chain make it difficult to apply 2D observations and measurement techniques directly to nanocomposites. Based on some early nanocomposite work on polymers filled with carbon black, one description of the interfacial region in nanofilled polymers is a bound polymer layer (where ‘bound’ implies a region of immobile polymer) with a distinct interface between the bound and unbound regions. The evidence for this is an insoluble fraction of polymer that results from the addition of nanoparticles [121]. In addition, the properties of the nanofilled polymers often scale with a para- meter somewhat larger than the average particle diameter [122]. The argument has been that the stronger the polymer–particle interaction, the larger the bound polymer fraction, up to $2\!-\!9\;\mathrm{nm}$ thick [123]. + +The discrete bound polymer layer could result in a second glass transition tempera- ture, according to some researchers [124]. Research on thin film, however, does not point to a discrete layer of affected material but to a far-field effect. Figure 2.14 [113, 118] shows the effect of film thickness on the glass transition temperature for free- standing polystyrene films of various molecular weights. The higher the molecular weight, the greater the film thickness at which the decrease in Tg begins. This sug- gests that some multiple of the radius of gyration may be a relevant parameter. Studies on diffusion using SIMS show that the effect can be up to 10 times the radius of gyration [125]. In addition, the thinner the film, the lower the Tg becomes. This in- dicates that the effect of the surface is a continuous function within the polymer. In addition, recent results [126] have shown that the nonlinear viscoelastic behavior of nanofilled polymer melts is strongly dependent on the surface treatment of the filler particles. The results suggest that the filler surface has a far-field influence on the chain motions. However, these effects are only apparent when finite shear–strain magnitudes are used to probe the behavior. Small-strain studies on the same compo- sites in the vicinity of the glass transition temperature show that, although the glass transition temperature is increased by filler addition, the curves of normalized loss modulus vs. frequency superpose perfectly. These studies suggest that the distribu- tion of relaxation times is shifted but otherwise unaltered. Taken together, these re- sults indicate that the bound polymer layer is essentially a monolayer and that the far- field influence is due to entanglement trapping. + +Further clarification reassures us that a discrete bound layer and a more diffuse region are not contradictory. First, a bound polymer layer , as originally defined, is sim- ply the amount of polymer that is ‘stuck’ to a particle surface; the term was not in- tended to completely describe the mobility within that region or the effect of that re- gion on the surrounding polymer. Thus, let us define a bound polymer layer as any polymer that is bonded to the particle surface. This sometimes occurs in nanocom- posites, but not always. Let us define the interfacial region as the region of polymer surrounding the nanoparticle and having altered chain conformation and/or mobility due to the presence of the filler. Clearly, polymer chains strongly bound to the surface result in a different interfacial region than chains either repelled by the surface or weakly interacting. The magnitude of the effect this interfacial region has on the prop- erties of a composite depends on the property being measured. For example, the effect + +![](images/861409bf1cc8bfbb02be75bb8eea3ac78406a47e5ef9b58975bee768e0d2d9ef.jpg) +Fig. 2.15 Glass transition temperature vs. volume ched. In addition, when the nanoparticles interact fraction of alumina particles for a PMMA matrix strongly with the polymer, the glass transition composite. The glass transition temperature does temperature does not decrease. Reprinted with not decrease until a critical volume fraction is rea- permission from [127] + +of the interfacial region on the glass transition temperature, which is the result of cooperative segmental motions, may be different from its effect on diffusion, which depends on larger-scale mobility. + +Consider for a moment, ignoring the specifics of their morphology and properties, the implications of both the bound polymer layer and the interfacial region on the behavior of nanofilled polymer composites. The film thickness at which a significant change in glass transition temperature occurs depends on the molecular weight, but can be as high as $80\,\mathrm{nm}$ . For a $30{\mathrm{-nm}}$ particle dispersed in a polymer matrix, the average interparticle spacing is $80\,\mathrm{nm}$ at filler loadings as low as 1.5 vol. $\%$ (Figure 2.2). Thus large effects on bulk properties can be expected at low filler loading. + +The glass transition temperature of a bulk part can be raised and lowered by the addition of nanoparticles [126–128]. Ash et al. [127] have shown that, for a non-wetting nanoparticle/polymer composite (alumina/PMMA), the glass transition temperature starts to decrease at a specific filler volume fraction corresponding to an interparticle distance of about $200\,\mathrm{nm}$ (Figure 2.15). On the other hand, if the PMMA is adsorbed on the particle surface, the glass transition temperature is stabilized. More commonly, an increase in Tg is observed [129, 130], which can be as large as $30^{\circ}$ [80]. + +The situation becomes even more complicated for composites in which the poly- mers are confined between two flat layers (as in clay composites) [131]. For exam- ple, NMR studies on PEO-intercalated nanoclay showed that confinement of the chains increases segmental mobility at low temperatures, but at higher temperatures a less mobile fraction is revealed. This suggests a broadening in the relaxation spectra. In addition, the glass transition temperature is suppressed. In thin films this is attrib- uted to a highly immobilized polymer layer. Krishnamoorti [131], armed with the knowledge that the rate of polymer intercalation into the clay layers can be as fast as the diffusion rate in the bulk, suggested that this phenomenon is due only to non- cooperative motion. This is clearly an area in need of further study and is discussed more below. + +In addition to changes in the mobility of a polymer chain, the average chain con- formation can be altered and/or the type and degree of crystallinity can change. Dhi- nojwala and coworkers recently reported measurements on a polystyrene (PS)/sap- phire interface [132, 133], which revealed a nearly perpendicular orientation of the phenyl rings to the PS/sapphire interface. It is unclear how these conformations change in the presence of a nanostructured surface, particularly one with a high ra- dius of curvature. Recent modeling, however, suggests that studies on flat surfaces may be relevant [134]; however, it is unclear how the many techniques used to study behavior on flat surfaces could be modified to measure this. Studies have focused on the effect of structural details of surfaces on protein structure and have found that the density of alkane chains on an alkylated surface directly affects the amount of protein adsorbed [135]. Other studies [136–138] showed that vitronectin (a mediator of osteo- blast adhesion) remained folded on monolithic alumina with micrometer-sized grains, but showed enhanced adsorption as well as conformational changes when it was ad- sorbed on nanograined alumina. Vitronectin interactions with nanoceramics were correlated with significantly enhanced osteoblast adhesion. The change in polymer chain conformation, in contrast to that of proteins, is likely to be much more local, but could still affect bulk properties. + +Crystallinity is also sensitive to surface interactions. This is seen dramatically in graphite fiber/polypropylene composites in which a trans crystalline layer develops perpendicular to the fiber [139]. Nanofillers can also change the degree of crystalli- nity, change the rate of crystallization, and/or suppress the formation of the thermo- dynamic crystal phase and stabilize the metastable phase [140, 141]. In silica-filled Nylon 11 that was thermally sprayed, the crystallinity increased due to the presence of the nanoparticles [142]. In montmorillonite/HDPE nanocomposites, the polymer crystals decreased in size and the crystallization rate decreased [143], whereas in poly- amide (Nylon 6) composites the metastable hexagonal phase formed instead of the typical monoclinic phase [144]. Another example in Nylon 6 occurs under high pres- sure: by annealing montmorillonite/ Nylon 6 at high pressures, a higher-temperature monoclinic phase forms $(a)$ , and the orthorhombic ( c ) phase is suppressed [145]. + +Finally, changes in the chemistry can occur in the interfacial region. A dramatic example of this is the preferential adsorption of a curing agent onto the nanofiller. When this occurs, the region surrounding the nanoparticle consists of a ‘layer’ of stoic hio metrically crosslinked thermoset with excess curing agent, surrounded by a ‘layer’ of depleted curing agent and thus less-than-stoichiometric crosslinking. This phenomenon has been observed in fiber-filled polymers [146] and is presumed to exist in nanoscale titania-filled epoxy [147]. More subtle changes could also occur, such as preferential adsorption of low-Mw material to the surface. + +In summary, the interfacial region is complex, and when the interfacial area is very large, the whole polymer matrix may essentially be interfacial region. This presents one of the essential challenges in polymer nanocomposites: to develop technology to control the interface, to describe the interface mathematically, and to be able to predict properties taking into account the interfacial region. Traditional composite theory, although very far advanced in describing properties that are relatively independent of the interface, is still in its infancy in taking into account the role of the interfacial region. Significant progress has been made in understanding the interaction of poly- mers with flat surfaces, but the interaction of polymers with highly curved surfaces and at scales similar to that of the radius of gyration is not well understood. The na- nocomposites community has their work cut out for them. + +# 2.4 Processing of Polymer Nanocomposites + +One of the key limitations in the commercialization of nanocomposites is processing. Early attempts at clay-filled polymers required processing that was not commercially feasible, but this situation has changed. Similarly, processing of other nanocomposites is becoming easier and more commercially viable as our understanding improves. A primary difficulty is proper dispersion of the fillers. Without proper dispersion and distribution of the fillers, the high surface area is compromised and the aggregates can act as defects, which limit properties. To facilitate discussion, we will define the state of aggregation in those nanocomposites. Distribution of a nanofiller describes the homo- geneity throughout the sample, and the dispersion describes the level of agglomera- tion. Figure 2.16 schematically illustrates good distribution but poor dispersion (a), poor distribution and poor dispersion (b), poor distribution but good dispersion (c), and good distribution and good dispersion (d). The state of aggregation is further defined in layered silicate/polymer nanocomposites later in this chapter. + +# 2.4.1 Nanotube/Polymer Composites + +The processing of nanotube/polymer composites is still in its infancy. Although na- notubes have been incorporated into composites commercially, the literature describ- ing the processes is limited. Significant issues remain to be solved about purification, dispersion, and bulk processing. + +The ability to disperse SWNT and MWNT into a polymer may be the most critical processing parameter for controlling properties. Nanotubes that are in clumps or are agglomerated with other carbonaceous materials create defect sites that will initiate failure. In addition, they limit the efficiency with which the nanotubes carry load. This limitation has been illustrated explicitly in both polymer and ceramic matrix com- posites [148, 149]. CVD-grown MWNT, which are easily dispersed and less agglom- erated, increased the modulus and strength of polystyrene without compromising the + +![](images/2be004fa9ac5b6bb8de5d51ce4d7db1cc6c16106e85dd59ddba7ef425170c021.jpg) +Fig. 2.16 A schematic illustrating the difference between dispersion and distri- bution and giving examples of good and poor for each + +strain-to-failure factor significantly. Other work on arc-discharge-grown MWNT, which were not fully purified and not as well dispersed, did not show the increase in toughness observed for well-dispersed MWNT. Similarly, the toughness of MWNT/alumina composites with excellent dispersion increased significantly com- pared to composites with somewhat worse dispersion. + +Dispersion has been achieved primarily by sonication of nanotubes in a solvent. The most focused efforts have been on the dispersion of SWNT. Chemical modification of the surface with the aid of surfactants [150], by function aliz ation of the end-caps with long aliphatic amines [151], or by function aliz ation of the sidewalls with fluorine [152] or alkanes [153] has resulted in stable suspensions of nanotubes. None of these meth- ods is ideal for composite processing. The use of surfactant results in an impurity in the composite. Functionalizing the ends limits further chemical modifications for controlling bonding with the matrix, and modification of the sidewalls can affect the mechanical properties [154]. As an alternative, the best solvents for direct disper- sion of SWNT were identified as NMP, DMF, hex a methyl phosphor amide, cyclopen- tane, tetramethylene sulfoxide, and $\varepsilon$ -caprolactone [154]. These solvents are all strong Lewis bases without hydrogen donors, although not all solvents with these character- istics were good solvents for SWNT. + +After dispersion, drying the dispersion on a glass slide and placing resin directly onto a thin film of nanotubes [29, 155, 156] can produce small-scale composites. Mix- + +![](images/ccccfd449b44995cdaa1abbaec993b27c6e2f7a0b815f8b5c467614ea5b21072.jpg) + +Fig. 2.17a Transmission electron micrograph of aligned singlewall carbon nanotube ropes bridging an elliptical hole in a polymer film. Reprinted with permission from [157] + +ing both the nanotubes and the polymer in the presence of a solvent [25, 30, 157, 158], often with the help of a surfactant [159], can also produce composites. Figure $2.17\mathrm{a}$ shows the wetting and the dispersion of SWNT dispersed in ethanol and then mixed with an epoxy resin [157]. Figure 2.17b [148] shows the dispersion achieved with CVD- grown MWNT, dispersed in toluene with dissolved polystyrene and casting films. Excellent dispersion was achieved. Nanotubes have also been dispersed directly into liquid urethane acrylate polymer [160], methyl meth a cry late monomer [161], and epoxy resin [162], followed by curing or polymerization. Damage to the nanotubes is a tradeoff that must be considered. + +![](images/0d5545b0d78f16b70de558e08a60e65aab1e2e9808c7e1c15e00930503348bdf.jpg) +Fig. 2.17b Treansmission elec- tron micrograph of MWNT-PS film in which the nanotubes are homogeneously distributed in the polystyrene matrix at a $\sim\!\mid\mu\mathfrak{m}$ length scale. The inset is the MWNT weight fraction as a function of length scale, which was determined by measuring the weight fraction of nanotubes at areas in the image. The error bars represent the standard de- viation of the experimental data, which reflects the homogeneity of the distribution. Reprinted with permission from [148] + +Nanofibers have been successfully melt-mixed with polyphenylene ether/polyamide matrices in a twin screw extruder. This process has led to a commercial product in conductive thermoplastics for electrostatic painting without the loss of mechanical properties [163]. + +An interesting composite processed from nanotubes is a macroscopic fiber that is a mixture of nanotubes and a traditional material used for carbon fibers, such as pitch or PAN. This was achieved with a pitch-based matrix and led to interesting improvements in properties, because the pitch fibers were not subjected to the typical high tempera- ture processing [164]. + +# 2.4.2 Layered Filler–Polymer Composite Processing + +Scientists have known for about 40 years that polymers interact strongly with mon- tmorillonite and that the clay surface can act as an initiator for polymerization [165, 166]. Patents for clay/Nylon 6 composites were not issued until the 1980s, at which point the clay/polymer nanocomposites were commercialized [167]. The improvement that led to commercialization was the appropriate dispersion of the clays at the nan- ometer scale. + +The first step in achieving nanoscale dispersion of clays in polymers is to open the galleries and to match the polarity of the polymer or monomer so that it will intercalate between the layers. This is done by exchanging an organic cation for an inorganic cation (Figure 2.18). The larger organic cations swell the layers and increase the hy- drophobic properties of the clay [168] (Figure 2.18), resulting in an organically mod- ified clay. The organically modified clay can then be intercalated with polymer by sev- eral routes. Solution processing involves dispersion of both the organically modified clay and the polymer in a common solution. Variations on this process include emul- sion or suspension polymerization [169, 170]. Highly polar polymers such as Nylon [171] and polyimides [8] are more easily intercalated than nonpolar polymers such as polypropylene, because polar polymers have a higher affinity for the polar clay gal- leries. In situ polymerization intercalates monomer directly into the organically mod- ified clay galleries, and the monomer can either adsorb onto the layer surface [172] or be anchored by free radical techniques [173]. Melt intercalation involves mixing the clay and a polymer melt, with or without shear. The success of melt intercalation is surprising, given that the gallery spacing is only about 2 nm and the radius of gyration of the polymer is significantly larger than this. Even more surprising is that the speed of melt intercalation is faster than that of self diffusion of polymers and scales with the inverse of the molecular weight [174]. Coarse-grained molecular dynamics studies predict a rate twice that of self-diffusion, with the rate scaling inversely with the mo- lecular weight. The results of molecular dynamics and experimental studies agree that the stronger the clay/polymer interaction, the slower the intercalation rate [175]. In addition, layer flexibility seems to control the mechanism of intercalation [176], per- haps due to a recently proposed mechanism called the ‘kink’ model of melt intercala- tion [177], in which sufficient shear force causes a kink to form in the clay sheet (a form + +![](images/22bdbb7dc4c4bc12843af408633ffd62f8c7672bf67836d34182da455acf1e9e.jpg) +Fig. 2.18 Schematic of the basic steps in processing clay-filled polymers + +of compression failure) (Figure 2.19). Polymer can then penetrate the new space be- tween the layers. This kink can propagate along the layer, and more polymer can inter- calate. This mechanism can explain both the fast intercalation rate (which is enhanced by the space created by kinking) and the dependence on layer flexibility. Layers with a lower modulus should kink more easily. A rigorous test of this model is still required. Models have also been developed to test the tendency for chains to melt-intercalate in the absence of shear [178, 179]. These studies showed that there is an optimal inter- layer separation (slightly larger than the pseudobilayer for intercalation of polystyrene) and an optimal chemical interaction (the greater the hydrophilicity of the polymer, the shorter the organic modifier should be). This minimizes unfavorable interactions be- tween the polymer and the aliphatic chain. Polar polymers are also more likely to intercalate. + +As the layer spacing increases, the process can be monitored by x-ray diffraction (XRD). Intense peaks between $3^{\circ}$ and $9^{\circ}$ indicate an intercalated composite, but if the peaks are extremely broad or disappear completely, this indicates complete exfo- liation. Figure 2.20 [180] shows the XRD pattern of organically modified montmor- illonite (d), an intercalated montmorillonite (c), and two exfoliated montmorillonite nanocomposites (a, b). + +![](images/feff748c393762494acedb37309c460a90df84409c512f3a3be8ef3aa3f65288.jpg) +Fig. 2.19 Schematic of the kink model of melt intercalation. $\mathsf{A g}$ denotes the agglomerated (immiscible) portion of the clay, and Int. denotes the intercalate state. Redrawn from Fig. 1(a) of [177] + +Once the clays have been intercalated with polymer, the nanocomposites can be processed by traditional melt-processing methods. This final processing of the com- posite is important in determining the final properties. Mixing facilitates nanoscale dispersion and can lead to clay and/or polymer chain alignment. The degree of shear during molding determines, not only the degree of clay layer alignment, but also the degree of crystallite alignment. For example, extruded Nylon sheet with a draw ratio of 4:1 had a higher modulus than a sheet processed by injection molding [181]. This may be due to a higher degree of platelet and crystallite alignment and is a common phenomenon in the drawing of Nylon. In some cases, the crystallinity increases + +![](images/91ef3dafdf99761efc404530465fad577f561340a0d9ced0ae8bd23fb1c2cea5.jpg) +Fig. 2.20 X-ray diffraction data showing the dif- fraction patterns that result from (a), (b) exfoliated clays, (c) intercalated clays, (d) organically modified clays. Reprinted with permission from [180] + +![](images/027f2a3aa34ed2cc6dc1731e02e908255c37901d53aa7906eff0fc5e0a983904.jpg) +Fig. 2.21 Schematic of the microstructures that can develop in clay-filled polymer composites: (a) a conventional composite with tactoids, (b) an intercalated nanocomposite, (c) an ordered exfoliated nanocomposites, (d) a disordered exfoliated nanocomposite + +$(36\,\%\!-\!38\,\%)$ , compared to $31\%$ for unfilled Nylon) and remains constant with filler content [140]. Other studies, however, have shown decreases in crystallinity [182]. In- creasing the pressure during processing to 0.1–0.6 GPa favors the $a$ phase [145]. The clay platelets also enhance alignment of the Nylon 6 chains and the crystallites [140], though the extent is governed by the degree of shear and can vary across the thickness of an injection-molded part [183]. Understanding and optimizing the mixing and tak- ing advantage of interesting rheology and other phenomenon such as strain hardening and rheopexy is in its infancy, and significant fundamental work still needs to be done [184]. + +The resulting nanocomposites can have several structures (Figure 2.21). The struc- ture of an intercalated nanocomposite is a tactoid with expanded interlayer spacing, but the clay galleries have a fixed interlayer spacing. Exfoliated nanocomposites are formed when the individual clay layers break off the tactoid and are either randomly dispersed in the polymer (a disordered nanocomposite) or left in an ordered array. + +The following sections give some details on processing clay/polymer nanocompo- sites. The discussion is organized by matrix type. + +# 2.4.2.1 Polyamide Matrices + +Nylon-6/Nylon-12/clay hybrid composites were the first exfoliated smectic clay com- posites made [187]. Montmorillonite, with a CEC of $119\,\mathrm{mEq}/100\,\mathrm{g}.$ , was intercalated with 12-aminolauric acid, which increased the intergallery spacing from 1.0 to $1.7\;\mathrm{nm}$ . This ‘12-montmorillonite’ was then mixed with e -caprolactam, which increased the intergallery spacing even further, to $4.0\:\mathrm{nm}$ , indicating that the $\varepsilon$ -caprolactam had intercalated into the galleries. Heating to $250\,^{\circ}\mathrm{C}$ led to polymerization, forming a clay/Nylon-6 nanocomposite. Further research [185] determined that e -caprolactam could intercalate directly into the galleries of montmorillonite in a hydrochloric acid solution and, upon intercalation, becomes oriented vertically in the galleries. The modified montmorillonite then mixed easily with additional molten e -caprolac- tam and 6-aminocaproic acid, yielding a Nylon-6 homopolymer/clay nanocompo- site. The montmorillonite was completely exfoliated. Recently [186], montmorillo- nite/Nylon 6 nanocomposites were processed by melt intercalation. Although the de- gree of exfoliation was not as high as in nanocomposites produced by the above meth- ods, at weight fractions less than 0.1 the composites were primarily exfoliated. + +# 2.4.2.2 Polyimide Matrices + +The preparation of polyimide matrix clay nanocomposites involves several steps [8, 187] (Figure 2.22). By intercalating montmorillonite with the ammonium salt of do- decylamine, it becomes soluble in dimethyl acetamide (DMAC). DMAC is also a sol- vent for $^{4,4^{\circ}}$ -di amino diphenyl ether and pyrometllitic dianhydride, the precursors for polyamic acid and, as such, polyimides. After intercalation of the ammonium salt of + +![](images/3a5d264b7a5fd2427063bb91d84aca3c88273b4b0296dcb95cd03a1b6819a1fe.jpg) +Fig. 2.22 Schematic of the synthesis of polyimide-clay hybrid film. Reprinted with permission from [187] + +dodecylamine, x-ray studies [188] showed that hectrite $({\mathrm{CE}}{\mathrm{C}}=55\,{\mathrm{mEq}}/100\,{\mathrm{g}})$ has one monolayer of organic material between the layers, whereas saponite, montmorillonite, and synthetic mica (all with $\mathrm{CEC}>100\,\mathrm{mEq}/100\,\mathrm{g})$ have two. After composite forma- tion, however, only the montmorillonite and the synthetic mica have exfoliated com- pletely, but the hectrite and saponite remain in a somewhat aggregated state. Lan et al. [189] found aggregates of montmorillonite after using a similar procedure. More re- cently, P-phylenediamine in an HCl solution was also found to form organic-modified montmorillonite that dissolves in DMAC [190]. This same study showed that the pre- sence of a small amount of nanoscale organoclay can decrease the imidization tem- perature by $50\,^{\circ}\mathrm{C}$ (from $300\,^{\circ}\mathrm{C}$ to $250\,^{\circ}\mathrm{C})$ ), and at $250\,^{\circ}\mathrm{C}$ the imidization time decreased by $15\;\mathrm{min}$ . The activation energy decreased by $20\,\%$ . Clearly, the organoclay surface is acting as a catalyst. + +# 2.4.2.3 Polypropylene and Polyethylene Matrices + +Nonpolar polymers are very difficult to intercalate into smectic clays, because the clays are strongly polar. This challenge has been met [191, 192] by first intercalating stear- ylamine into montmorillonite and synthetic mica. Melt-mixing the organoclays with maleic anhydride-modified polypropylene oligomers results in PP-MA intercalation. The modified organoclay is then melt-mixed with a polypropylene matrix. There is a balance between creating a polar oligomer with enough maleic anhydride to intercalate well, but nonpolar enough to mix with the polypropylene. Unfortunately, the oligomer limits the extent of property improvement achieved to date. Polyethylene has also been successfully melt-mixed with modified montmorillonite and saponite after ion ex- change with di oct a dec yl dimethyl ammonium bromide. The degree of dispersion is not excellent, and the layers are certainly not exfoliated; yet, significant modification of both the crystal structure and properties has been observed [143] . + +# 2.4.2.4 Liquid-Crystal Matrices + +Smectic clay/liquid crystalline nanocomposites exhibit interesting optical properties [193, 194]. Upon initial formation, the composites are opaque, but when an electric field is applied they became transparent. To process these composites, montmorillo- nite was intercalated with lauryl ammonium or 4-(4’-cyanobiphenyl-4-oxy) butyl am- monium (CBAM), or 4-cyano-( $\backprime4'$ -biphenyloxy) undecyl ammonium cation and solu- tion-mixed with a nematic crystal matrix such as 4-pentyl-4’-cyanobiphenyl (5CB) and nematic TFALC (a mixture of low-molar-mass liquid crystals). + +# 2.4.2.5 Poly methyl meth a cry late/Polystyrene Matrices + +The processing of clay/PMMA or clay/PS composites was first done by directly inter- calating the monomer into the clay, followed by polymerization [195]. This method was not successful in exfoliating the clays. At issue again is the compatibility between the clay and the monomer. One solution for PMMA has been to use appropriate ammo- nium salts [196, 197], which may be reactive [198]. Another solution is to use a como- nomer as a compatibilizer [199]. A similar solution was found for polystyrene by using the reactive cationic surfactant vinyl benz yl dimethyl do dec yl ammonium as the interca- lant [200]. Exfoliated graphite/polystyrene composites have been made by similar pro- cessing methods [80]. Recently, a commercially viable process was developed [180] for polystyrene in which montmorillonite intercalated with octadecyl trimethyl ammo- nium chloride was melt-mixed with a styrene methyl vinyl ox a zo line copolymer. This process resulted in complete exfoliation, which could not be achieved with pure polystyrene [201]. The hypothesis is that the hybridization is due to strong hydro- gen bonding between the oxazoline groups and oxygen groups in the silicate clays. + +# 2.4.2.6 Epoxy and Polyurethane Matrices + +Epoxy is a widely used thermoset, with applications ranging from household glues to high-performance composites. To improve performance, increasing the Tg of epoxy and improving its properties above the Tg are desirable. Adding clays and layered silicic acids to epoxy [6, 71, 202–204] can greatly improve its mechanical perfor- mance, particularly at temperatures above Tg. The processing has been studied in detail [71, 205]. In the smectic clay/epoxy composites, the length of the intercalated organic amine determines the ease of exfoliation, and only clays with primary and secondary onium ions form exfoliated nanocomposites [71, 206]. After intercalation of the organic amines, the epoxy resin or a combination of resin and curing agent can be intercalated into the smectic clays or layered silicic acids. If enough resin and curing agent are intercalated and the curing process is controlled, exfoliated nanocomposites result. Figure 2.23 shows a diagram of the process [207]. The acidic onium ions cat- alyze the intragallery polymerization or curing of the resin. If this reaction occurs more rapidly than extragallery curing, then the clay exfoliates. Otherwise, an intercalated nanocomposite results. Therefore, careful control of temperature and time is required [71], or the ratio of resin to curing agent must be significantly less than the stoichio- metric ratio [205] in order to achieve exfoliation. An approach similar to that used for epoxy composites was used to make intercalated montmorillonite/polyurethane com- posites [208]. + +![](images/a5eb4a58fe70bb84204895bcca6c373c7ec28eb21a0e726d28744b6e036567ce.jpg) +Fig. 2.23 Proposed pathway for formation of an cointercalation of epoxide and curing agent. (C) epoxy-exfoliated magadiite nanocomposite: (A) In- Rapid intragallery formation of a polymer gel and itial organomagadiite with a paraffin-like gallery expansion of the gallery height beyond a lipid-like structure of onium ions and neutral amine. (B) layer. (D) Silicate nanolayers are completely exfo- Reorientation of the alkylammonium ions into a liated in a fully crosslinked epoxy polymer network. lipid-like bilayer structure to accommodate the Reprinted with permission from [207] + +# 2.4.2.7 Polyelectrolyte Matrices + +Poly electrolytes can be used in electrochemical devices such as solid-state batteries, electrochromic devices, and sensors [209]. The addition of layered silicates to polyelec- trolytes increases the conductivity, improves the mechanical stability, and improves the interfacial stability with electrode materials. Poly electrolytes are characterized by a large number of ionizable groups and thus are highly polar. This makes them excellent candidates for intercalation into smectic clays. Polyvinyl pyr i dines are of par- ticular interest because of the variety of processing methods available [210]. Interca- lated nanocomposites can be formed easily from the water-soluble hydrobromide salt of the 1,2 or 1,6 polyelectrolyte (1,2 or 2,6 polyvinyl pyr id in i um cations). However, only a single layer of polymer intercalates, and exfoliation does not occur. A slower, but ultimately more effective process, uses neutral poly-4-vinylpyridine and results in an exfoliated composite. A second method involves intercalation of 4-vinylpyridinium salts, followed by polymerization. + +Poly(ethylene oxide) (PEO) matrix composites have also been processed both by intercalating PEO in solution into organically modified smectic clays [209] and by melt- mixing clay with PEO and PEO/PMMA mixtures [211,212]. In neither case does an exfoliated composite result. Aranda and Ruiz-Hitzky [209] dissolved PEO in acrylonitrile and found that the structure of the PEO changed when the interlayer cation was changed. Use of $\mathrm{Na}^{+}$ montmorillonite or $\mathrm{NH_{4}}^{+}$ montmorillonite resulted in either a helical PEO or a bilayer zigzag PEO structure in the galleries. The PEO ar- rangement was reversible with exchange of the interlayer cations. + +# 2.4.2.8 Rubber Matrices + +Several applications of rubbers might benefit from inclusion of exfoliated clays. Their greatly reduced permeability [213] would be useful for the inner liners of tires and inner tubes [214]. In addition, modification of the glass transition temperature and/or the loss modulus might be useful in a variety of damping applications. Mon- tmorillonite has been ion-exchanged with a protonated form of butadiene and acry- lonitrile copolymer. This was subsequently mixed with nitrile butadiene rubber in the presence of crosslinking agents and resulted in highly dispersed nanocompo- sites. Nanocomposites have also been prepared from di oct a dec yl dimethyl ammo- nium-exchanged montmorillonite in poly(styrene $\cdot b$ -butadiene) matrices [215]. + +Silicone rubbers have also been successfully intercalated into smectic clays. Burn- side and Giannelis [216] intercalated $\mathrm{Na^{+}}$ montmorillonite with dimethyl d it allow am- monium bromide and then sonicated in a silanol-terminated poly(dimethyl silo x a ne) $(\mathrm{Mw}\,=\,18{,}000)$ ) with a tetraethyl orthosilicate and tin 2-ethylhexanoate crosslinking agent. Hex a dec yl tri methyl ammonium bromide has also been used successfully as the intercalating agent [217]. Recently, a latex method was developed in which unmo- dified clays were mixed with a rubber latex, leading to improved properties over solu- tion intercalation [218]. + +# 2.4.2.9 Others + +Clay/polymer nanocomposites that include poly( e -caprolactone) have been made via in-situ polymerization [7]. Composites that include poly(p-p hey lene vinyl ene) have been made via intercalation of poly(xy lyle ne dimethyl s ulf on i um bromide) and subse- quent elimination of the dimethylsulfide and HBR [219]. Those including cyclic poly- carbonate [220] or polyethylene terephthalate have been made via monomer intercala- tion and subsequent polymerization [221]; and those including polyaniline via in-situ polymerization of aniline monomer [222]. + +# 2.4.3 Nanoparticle/Polymer Composite Processing + +There are three general ways of dispersing nanofillers in polymers. The first is direct mixing of the polymer and the nanoparticles either as discrete phases or in solution. The second is in-situ polymerization in the presence of the nanoparticles, and the third is both in-situ formation of the nanoparticles and in-situ polymerization. The latter can result in composites called hybrid nanocomposites because of the intimate mixing of the two phases. + +# 2.4.3.1 Direct Mixing + +Direct mixing takes advantage of well established polymer processing techniques. For example, polypropylene and nanoscale silica have been mixed successfully in a two-roll mill [5], but samples with more than 20 wt. $\%$ filler could not be drawn. This is typical and is a limitation of this kind of processing method. Nanoscale silica/PP composites have been processed in a twin-screw extruder, but the dispersion was successful only after modification of the silica interface to make it compatible with the matrix [223]. A Brabender high-shear mixer has been successfully used to mix nanoscale alumina with PET, LDPE [224]. Thermal spraying has also been successful in processing nanopar- ticle-filled Nylon [142]. When these traditional melt-mixing or elastomeric mixing methods are feasible, they are the fastest method for introducing new products to market, because the composites can be produced by traditional methods. This has been successful in many cases, but for some polymers, the viscosity increases rapidly with the addition of significant volume fractions of nanofiller, which in turn can limit the viability of this processing method. + +In addition to viscosity effects, nanoparticles can either enhance or inhibit polymer degradation. One method for measuring degradation is to place the polymer in a high- shear mixer and measure the torque as a function of time and temperature. As the material crosslinks, the torque begins to increase (at a constant speed), and when chain scission begins, the torque decreases. This leads to a peak in the torque, whose posi- tion is often used as a measure of the degradation time. A recent study on $\mathrm{ZnO/LDPE}$ composites showed that, for nanoparticle/matrix mixtures, the time at the peak in- creased by a factor of almost 2 [224]; however, micrometer-size particles decreased the degradation time. In other instances, the catalytic nature of the nanoparticles greatly decreases the degradation time [225]. Whether degradation is inhibited or en- hanced depends on the particle surface activity and the increased interfacial area (par- ticle size). + +# 2.4.3.2 Solution Mixing + +Some of the limitations of melt-mixing can be overcome if both the polymer and the nanoparticles are dissolved or dispersed in solution. This allows modification of the particle surface without drying, which reduces particle agglomeration [226]. The na- noparticle/polymer solution can then be cast into a solid, or the nanoparticle/polymer can be isolated from solution by solvent evaporation or precipitation. Further proces- sing can be done by conventional techniques. + +# 2.4.3.3 In-Situ Polymerization + +Another method is in-situ polymerization. Here, nanoscale particles are dispersed in the monomer or monomer solution, and the resulting mixture is polymerized by stan- dard polymerization methods. One fortunate aspect of this method is the potential to graft the polymer onto the particle surface. Many different types of nanocomposites have been processed by in-situ polymerization. A few examples are silica/Nylon6 [227, 228], silica/poly 2-hydroxy ethyl meth a cry late [229], alumina/poly methyl meth a cry late [230], titania/PMMA [231], and $\mathrm{CaCO_{3}/P M M A}$ [232]. The key to in-situ polymeriza- tion is appropriate dispersion of the filler in the monomer. This often requires mod- ification of the particle surface because, although dispersion is easier in a liquid than in a viscous melt, the settling process is also more rapid. + +# 2.4.3.4 In-Situ Particle Processing – Ceramic/Polymer Composites + +An interesting method for producing nanoparticle-filled polymers is in-situ sol–gel processing of the particles inside the polymer. The process has been used successfully to produce polymer nanocomposites with silica [233] and titania [234] in a range of matrices [235]. The overall reaction for silica from te tr ethyl ortho silicate (TEOS) is shown below. + +# $\mathrm{Si}(\mathrm{OC}_{2}\mathrm{H}_{5})_{4}+\mathrm{excess~H}_{2}\mathrm{O}\rightarrow\mathrm{SiO}_{2}+4\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH}$ + +The composites can be formed in one of several ways. In the first method, a copolymer of the matrix polymer and silica precursor are mixed, and the sol–gel reaction is al- lowed to progress. Figure 2.24 shows an example of the process for silica/polystyrene composites [233]. During drying, the polymer blocks phase-separate and the silica regions coalesce. A second approach mixes a silica precursor such as TEOS [236] with a polymer such as polyvinyl acetate [237], polyetherimide [238], or polymethylme- thacrylate [239]. For example, a recent paper on the processing of TiO2/poly(styrene maleic anhydride) resulted in excellent dispersion of $\mathrm{SiO}_{2}$ in PSMA [234]. Direct addi- tion of $\mathrm{SiO}_{2}$ to a PSMA solution resulted in serious aggregation. Therefore, PSMA was dissolved in THF, and then tetrabutyl titanate was added under the appropriate condi- tions. Because the uncondensed TiOH and maleic anhydride could react, the polymer + +![](images/457cf2974a99554f688a019a881531436729426d08d3258a0f82d94e0f07d250.jpg) +Fig. 2.24 Example of in-situ synthesis of silica particles in polystyrene. Reprinted with permission from [233] + +coated the titania particles as they formed and prevented agglomeration. A combina- tion of these first two methods has been used for silica/poly dimethyl silo x a ne (PDMS) composites in a two step process. First, an unfilled PDMS network was formed, with TEOS as the end-linking agent. The network was then swelled with TEOS and the sol– gel reaction was catalyzed [240]. Significant work on determining the parameters affec- ting particle size has been accomplished with this method [241]. + +Finally, the polymer and ceramic can be polymerized simultaneously, and compo- site materials ranging from a few percent to $100\,\%$ inorganic phase can be formed [242]. For example, silica/polyacrylate nanocomposites were formed with the simul- taneous polymerization of HEMA (2-hydroxyethyl methacrylate) and tetra eth oxys i lane (TEOS) [229]. This resulted in a continuous inter penetrating network of silica and polyacrylate. There is a rich chemistry here that has been reviewed elsewhere on the processing of materials that range from glasses to polymers [243, 244] in which there is little if any phase separation between the organic and inorganic. We focus here on systems with discrete, isolated phases. + +An interesting variation on in-situ formation of nanoparticles and polymerization is CVD deposition of nanocomposite films [245]. Here, the chemical precursors are mod- ified separately and are mixed only just before deposition. Dichloro- $\cdot p$ -xylylene was used as a precursor for poly(dichloro- $\cdot p$ -xylyene), and diacetoxydi-tert-butoxysilane was used a precursor for silica. The resulting microstructure has discrete polymer and silica phases. This method is promising for the creation of interlayers with a low dielectric constant. + +Barium titanate/polymer nanocomposites have also been prepared in situ by mixing a titanium alkoxide with a polymer and then casting and drying films. The films are then reacted with an aqueous solution containing barium, resulting in nanoscale bar- ium titanate particles within a polymer matrix [246, 247]. Making the precursor and the polymer compatible improves the degree of dispersion, and using compounds that tend to form micelles can introduce controlled heterogeneity [246]. + +Nanosized calcium phosphate particles have been processed by an in-situ deposition technique in the presence of polyethylene oxide (PEO) or PEO/PVAc mixtures [248]. + +In this method, a polymer-calcium chloride complex is blended with trisodium phos- phate. As the phosphate and chloride ions diffuse through the polymer, a gel-like precipitate forms. This can be separated; the size and morphology can be varied by changing the polymer and/or the polymer concentration. + +# 2.4.3.5 In-Situ Particle Processing – Metal/Polymer Nanocomposites + +Metal/polymer nanocomposites have also been processed via in-situ formation of me- tal particles from suitable metal precursors. The reaction occurs in the presence of a protective polymer, which limits the size of the particles. Figure 2.25 [249] shows a reduction scheme that uses a gold precursor and polypyrrole. Once a stable suspen- sion of metal particles is prepared in the presence of a polymer, the composite can be cast, or additional monomers, of the same or a different polymer type, can be added to form a nanocomposite. + +Mayer [250] wrote an excellent review paper on the processing of metal/polymer nanocomposites via in-situ methods in 1998. She reviewed the parameters that affect particle size, stability, and morphology. Several primary parameters controlled the particle size, including the choice of metal precursor and the metal/polymer interac- tion. For example, if $\mathrm{P}\mathrm{d}\mathrm{Cl}_{2}$ is compared with $(\mathrm{NH}_{4})_{2}\mathrm{PdCl}_{4}$ , the former tends to form halogen-bridged complexes and thus tends to form agglomerates of nanoparticles, but the latter does not. The interaction of the metal precursor with the polymer is also important. If the polymer has a stronger interaction with the precursor, then the par- ticle size tends to be reduced [251], because the metal precursors are prevented from phase separating. In general, because most polymers are hydrophobic, precursors with a more hydrophobic character result in stronger interactions and thus smaller particle size. Complicating this situation is the ability of the polymer to form a stable suspen- sion in solution. The rate of reduction affects the particle size, with faster reduction methods resulting in smaller particles [252, 253]. + +Another method giving more specific control over particle size and morphology is the use of either micelles formed from amphiphilic block copolymers (ABC) [254] or crosslinked/gelled matrices. For example, through the use of crosslinking and gelation + +![](images/863aaeca9ce564d3fcef96343530601699169c1737611412c2ccc92515a904e4.jpg) +Fig. 2.25 Schematic of reaction leading to in-situ gold particles in a polymer. Reprinted with permission from [249] + +![](images/af3f7a5c4e068d1c288ef8cf3c9f8da821f14f63f7528c7c0101f0d526e2a568.jpg) +Fig. 2.26a Overall morphologies of block copolymer-metal systems involving spherical micelle formation of the amphiphilic block copolymers: (a) cherry morphology, (b) raspberry morphology, (c) strawberry morphology, (d) propo- sed red currant morphology. Reprinted with permission from [250] + +reactions, dendritic gold particles can be formed [255]. Amphiphilic block copolymers provide another avenue of control. Here, block copolymers are used to form micelles. Metal salts then either penetrate into the micelles or are stable in the micelle corona. The reducing agent can be added, and metal particles form either within the micelles or in the corona, resulting in several morphologies shown schematically in Figure 2.26a [250]. Figure 2.26b [294] shows an example of the raspberry morphology for Pd colloids in a polystyrene-b-poly (4-vinylpyridine) block copolymer matrix that was reduced with hydrazine. Changing the block length can control the particle size. + +This same principle is applicable to other block copolymer morphologies, such as rods and layers, and leads to interesting morphologies. One thing that is not clear, and needs to be addressed by theory, is which of these morphologies will lead to the best properties. Because of the range of metal particles that can be formed, including gold, silver, palladium [256], platinum, semiconductors [257], and metal oxides [234], tre- mendous opportunity exists to tune the properties of these systems. + +Other methods are available for producing metal nanoparticle-filled polymers, such as deposition of a metal film onto the polymer [258, 259] and subsequent annealing to form particles, or electrochemical methods [260], both of which lead to highly con- trolled thin-film composite structures. + +![](images/0c111bb748f30fcec3104fc635983cea965761dfac2ba45e2569331325003e94.jpg) +Fig. 2.26b Electron micrographs of Pd colloids synthesized in Ps- b P4VP block copolymers via reduction with hydrazine. Reprinted with permission from [253] + +# 2.4.4 Modification of Interfaces + +Literature on the modification of interfaces of traditional composite fillers is extensive. Many of the methods, however, are not directly applicable to nanocomposites for sev- eral reasons. Carbon and glass fibers are usually coated as long fibers on a spool, which is not practical for nanofibers. In addition, methods that require drying the particles are not appropriate for metal oxide and metal nanoparticles, because during drying (even at only $100\,^{\circ}\mathrm{C})$ , they may agglomerate significantly. Freeze-drying methods are more appropriate. Although describing the relevant chemical processes is impor- tant for understanding nanofiller interface modification, this section focuses primarily on the results of modification, instead of on the detailed chemistry. + +# 2.4.4.1 Modification of Nanotubes + +Several reasons exist for modifying nanotubes. The first is to render them more dis- persible in solution, thus providing a route toward further function aliz ation or use in applications. The second is to improve the interaction of the nanotubes with polymer matrices. Modifying the surface activity of MWNT has focused primarily on doping them with other elements. Doping MWNT with boron has resulted in changes in the electronic states [261, 262] and formation of a p-type semiconductor. In addition, bor- on substitution, as well as other electron-rich substitutions, may very well lead to out- of-plane bonding configurations that increase the reactivity of the surface with a poly- mer matrix. Nitrogen doping of MWNT creates an n-type semiconductor and leads to a bamboo-like structure having ordered regions connected via disordered regions [263]. The nitrogen creates nitrogen-rich cavities in the otherwise graphitic nanotube. Pre- liminary work has shown that nitrogen doping changes the interface enough to im- prove the dispersion of nanotubes and results in enough bonding with epoxy matrices to increase the glass transition temperature by $20^{\circ}$ as well as in increased modulus and strength [264]. In addition, acid treatments and subsequent chemical reactions can lead to attachment of epoxy groups or even particles to MWNT [265, 266]. + +SWNT are assembled as ropes or bundles. To maximize their interaction in a poly- mer matrix, they need to be chemically modified and separated from the bundles. One way to increase the chemical activity is to use an SWNT with a smaller radius of cur- vature [267]. A second method is to add functional groups to the ends of SWNT. Car- boxylic groups have been linked to SWNT ends, and then the SWNT have been teth- ered to nanoparticles via thiol linkages [268–270]. SWNT have been placed in solvent through the addition of octadecylamine groups on the ends and a small amount of dich l oro carbenes on the sidewalls [271]. The first successful method for adding func- tional groups to the sidewalls involved fluorination at elevated temperatures. The tubes could then be solvated in alcohols and reacted with other species, particularly strong nucleophiles such as alkylithium reagents [272]. A simpler route based on a process used on other carbon species [273, 274] was developed, in which an aryl diazonium salt was reduced electro chemically, resulting in a free radical that can attach to the carbon surface of small diameter tubes [271] (Figure 2.27). + +![](images/0ccc5adff2603891c677d9e42a26d9eec99b8fc3cd0b7327853bf8255d125ce5.jpg) +Fig. 2.27 Electrochemical re- duction of an aryl diazonium salt to yield, giving a reactive radical that covalently attaches to a carbon surface. Reprinted with permission from [271] + +The most exciting development in terms of composites, however, is the dissolution of full-length SWNT that have been separated from the bundles [275]. Given the issues of slipping of nanotubes within a bundle discussed earlier, their use in composites requires separation from the bundles. The process involves, first, partially breaking up the SWNT bundles in an acidic solution. Carboxylic acid groups are then added to the ends of the tubes. Finally, the SWNT are placed in an oct a dec yl ammonium (ODA) melt for several days. This results in exfoliated nanotubes. This important re- sult should eventually lead to SWNT/polymer composites with the SWNT separated in the polymer, which will provide an avenue for taking advantage of the extraordinary properties of SWNT. + +The development of methods to attach polymers to nanotubes has progressed sig- nificantly. First, separation of the nanotubes from their aggregates is required. This separation can be achieved by dispersion in the appropriate solvents with or without the aid of a surfactant [276] and/or by choosing an appropriate synthesis method. A polymer chain can then be attached to a bonding site by sonication [277], plasma ac- tivation [278], chemical etching of the tube ends [279], or chemical adsorption [280]. These advances should lead in the near future to some very interesting copolymers with SWNT as one block in the polymer and a rich interfacial chemistry. Some evi- dence suggests that polymers can be patterned by the nanotube surface, providing a method for templating polymer structures near the surface [281]. It is not clear yet what role this interface control will have in terms of properties of composites, but to develop technology that fully exploits the properties of nanotubes in composites, control over the interface is required. + +# 2.4.4.2 Modification of Equi-axed Nanoparticles + +A recent review by Frank Caruso [282] provides extensive background on the modi- fication of nanoparticle surfaces. He cites two primary methods for modifying an in- organic nanoparticle surface with organic molecules. The first requires connecting a short-chain molecule such as a siloxane onto the surface through grafting or strong hydrogen bonding. The second involves application of a coating by polymerizing a polymer onto the nanoparticle. In addition, inorganic coatings can be applied. + +# 2.4.4.3 Small-Molecule Attachment + +Both metal and metal-oxide nanoparticles are readily modified with small molecules. They tend to have hydroxyl groups on the surface, although the number of hydroxyl groups and the strength of the metal-OH bond varies. The most common method + +![](images/13ffd36748ceec8b13d751170be335752a7505dfb8dfbc14a31b6fe9572cb433.jpg) +Fig. 2.28 Schematic summarizing many of the possibilities for modifying equi-axed nanoparticles + +takes advantage of a long history of modifying metal-oxide glass fiber or filler surfaces. The hydroxyl group on the nanoparticle surface reacts with a silane coupling agent. The relative ease of the reaction depends on the nature of the nanoparticle surface (basicity vs. acidity). The silane coupling agent can have a large variety of functional- ities attached to the silanol to modify the surface. The chains can be long or short, hydrophilic or hydrophobic, and linear or bulky. This provides tremendous flexibility in choosing a coupling agent. Changing the coupling agent can lead to control over the strength of the interaction between the filler and the matrix, from covalent bonding to repulsion. The change in bonding leads to increases or decreases in both glass transi- tion temperature and the modulus, but large effects occur only for particles much less than $100\,\mathrm{nm}$ [128]. + +Of course, reacting with molecules other than silanes is also common. Nanoscale titania has been coupled with chelating agents such as lauryl sulfate [283] and with carotenoids (natural pigments) containing a terminal carboxylate group that reacts with OH groups on the surface [284]. + +One creative study underway provides a method for placing one, two, or several active functional groups on the surface of nanoparticles, followed by passivation of the rest of the particle. This will result in controlled aggregation of the particles into dimers, chains, or branched chains [285] (Figure 2.28). + +# 2.4.4.4 Polymer Coatings + +Historically, polymer coatings have provided control over the compatibility of the poly- mer with the matrix and the strength of the interaction. The coating can be grafted onto the surface of the particle or just highly adsorbed. After the particles are coated they can be dispersed easily in a matrix of similar chemistry to make low-volume-fraction composites. The exciting news is that controlling interfacial interactions has moved well beyond single-component coatings. The possibility for assembling coated nano- particles directly into useful structures, the core-shell approach, has driven significant development in multicomponent organic and inorganic coatings. The possibilities of this approach seem almost endless. We currently know how to graft or strongly adsorb several types of polymers onto a surface. In addition, we know how to polymerize polymers with very monodisperse molecular weight and we know how to make block (di and tri) copolymers [286]. Progress has also been made in placing inorganic coat- ings onto particles. This provides flexibility in our ability to engineer the surface of the particles being placed in composites. No longer will the coating be only for controlling the extent of the interaction, for it may contain several layers. The role of the first layer may be to bond with the surface. The second layer may provide electrical, mechanical, or optical functionality, and the third layer may make the particle compatible with the matrix and help in controlling the properties. The final layer may have some reactive sites that could be used to bond particles together and lead to controlled aggregation of the nanoparticles (Figure 2.28). + +In principle, polymer coatings can be produced on organic and inorganic particles by a variety of polymerization methods. Monomer adsorption and subsequent polymer- ization, heterocoagulation–polymerization [287], and emulsion polymerization [288] have been demonstrated on micrometer-scale filler surfaces [287], although not all have been applied to nanoparticles. Monomer adsorption has been used successfully in several studies, and initiation has occurred both chemically [289] and via irradiation [290]. One recently reported method can be used to place nanoparticles (in this case fumed silica) into a monomer such as styrene or methyl meth a cry late. The mixture is + +![](images/f4ec3fea85bc60ec173772c26132aaa6f2c483c075e5a30c6bcfc7331d3075b7.jpg) +Fig. $\pmb{2.29a}$ Synthetic scheme for structurally well defined polymer-na- noparticle hybrids. Reprinted with permission from [292] + +![](images/8bc22ed1b1ed705bd559ddc9fe74a108d746272cd81fd7ab958b1e4892c0e4b1.jpg) +Fig. 2.29b Proposed approach to formation of 3-D polymer nanocomposite material. Stage A: synthesis of the core-shell latex particles with hard func- tionalized cores and soft inert shells. Stage B: assembly of latex particles in a 3-D close-packed structure. Stage C: heat treat- ment of the 3-D compact struc- ture, which leads to flow of soft shells and formation of a nano- composite polymers. Reprinted with permission from [294] + +then irradiated [290]. The monomer penetrates the silica aggregates, and after grafting, the surfaces are more hydrophobic and blend more easily with polymers such as poly- propylene. This treatment leads to more uniform distribution of the particles and greatly enhanced composite ductility. In some cases, the polymer is not directly grafted onto but interacts strongly with the surface via hydrogen bonding. For exam- ple, carboxylic acids are strongly adsorbed to alumina. If the acid contains a polymer- izable group, then after adsorption, a polymer can be grown from the molecule at- tached to the surface. This has been shown for maleic acid adsorbed onto a surface and was used to attach AIBN-initiated 1-octadecene polymers to the surface [291]. + +Another method that leads eventually to a composite with an ordered array of na- noparticles involves grafting an initiator onto the surface of the nanoparticles and then polymerizing a grafted polymer onto the surface [292]. If living radical polymerization is used, then the particles can be processed into a composite without additional matrix, to make high-volume-fraction composites with particles in an ordered array. Figure $2.29\mathrm{a}$ shows this for silica particles with grafted polystyrene packed into an ordered matrix [292]. This process lends itself nicely to block copolymer coatings and even triblock copolymers. Another method that can lead to multilayer structures is the use of self-assembled polymer layers in a ‘layer-by-layer colloid templating strategy’ [293]. Here, charged polymers are adsorbed onto charged particles, resulting in a coated particle with surface charge. A second charged polymer can then be exposed to the previously charged particles. When this polymer is adsorbed, the surface charge of the particle is reversed, and the particles have a bilayer coating. This method can be repeated and enables careful control of the coating thickness and functionality. + +A similar core–shell approach has been used to coat a low-Tg polymer onto a high- Tg latex particle. The coated polymer particles were then taken out of solution and annealed to create an ordered array of polymer in polymer [294] (Figure 2.29b). The concept applies to coated nanoparticles other than latex as well. + +# 2.4.4.5 Inorganic Coatings + +Inorganic coatings have also been applied to submicrometer particles via either pre- cipitation of the inorganic coating onto the particles or deposition via a sol–gel type process. With these methods, silica [295], yttria [296], titania [297], titanium nitride [298], and zirconia [299] have all been applied to support particles, usually other metal oxides. The advantage of such coatings is that the nanoparticles (both organic and inorganic) can be coated with inorganics for further function aliz ation. For exam- ple, conducting coatings can be applied, and the optical properties can be tuned. + +Sonochemistry can also be used to apply inorganic coatings in much the same way as described above for the processing of nanoparticles. For example, $\mathrm{ZnS}$ coatings $(1-$ $5\:\mathrm{nm}$ ) on colloidal silica have been prepared in this way [300]. + +2.5 Properties of Composites + +2.5.1 Mechanical Properties + +One of the primary reasons for adding fillers to polymers is to improve their mechan- ical performance. For example, the addition of high-modulus fillers increases the mod- ulus and the strength of a polymer. In traditional composites, unfortunately, this often comes at the cost of a substantial reduction in ductility, and sometimes in impact strength, because of stress concentrations caused by the fillers. Well-dispersed nano- fillers, on the other hand, can improve the modulus and strength and maintain or even improve ductility because their small size does not create large stress concentrations. In addition, the large interfacial area of nanocomposites provides an opportunity for altering the matrix properties in unique ways. This section highlights the potential effect of nanofillers on the modulus, failure stress/strain, toughness, glass transi- tion, and wear behavior of polymers. + +# 2.5.1.1 Modulus and the Load-Carrying Capability of Nanofillers + +In traditional composites, the modulus can be predicted, within bounds, and the me- chanism that leads to a change in modulus is load transfer. For example, if a high- modulus filler is added to a lower-modulus polymer, then load transfers from the matrix to the filler, leading to an increase in modulus. One important question for nanocomposites is how load is transferred from the matrix to the filler. This is a com- plicated question, because in some nanocomposites the polymer chain and filler are almost the same size, and in others, the fillers are atomically smooth, with few func- tional groups to which the polymer can bond. The importance of load transfer is easily illustrated with the shear lag discussion commonly applied to graphite fiber/polymer composites. Figure 2.30a shows a stress–strain curve for a typical high-modulus fiber, a thermoplastic matrix, and the resulting composite. At a given applied composite strain, the fiber carries more load than the matrix; therefore stress must be transferred from the matrix to the fiber. Figure $2.30\flat$ shows the stress-transfer behavior on the micromechanical level. At the fiber end (or at a fiber break) the fiber does not carry stress. Stress is transferred via a shear stress at the filler/matrix interface, and the rate of load transfer depends on the shear stress. This highlights the role of the interface in controlling the modulus of filled polymers. For example, the higher the interfacial shear stress and the shorter the load transfer length, the more efficient the filler is in carrying load and the higher the composite modulus. In addition, the higher the aspect ratio of the filler, the longer the length that carries the maximum load, and the higher the composite modulus. + +To fully understand the modulus of nanocomposites, a second mechanism in addi- tion to load transfer must be considered. For example, the filler can constrain the mobility of the polymer chains as well as their relaxation spectra [120], which can change the glass transition temperature [120, 121] and modulus of the matrix. Nano- + +![](images/d4d8dd75568b65498435a0475313b717ad48f4618e027347306fd1f56ec185ae.jpg) +Fig. 2.30 Schematics of (a) stress vs. strain curve for a typical high modulus fiber, a thermosetting polymer, and the resulting composite, (b) transfer of strain from the matrix to the fiber near the fiber end + +fillers can also change the degree or type of crystallinity in a polymer, which also changes the modulus [140, 141]. These two mechanisms of reinforcement are ad- dressed for each geometry of nanofiller below. + +# Nanotubes + +Consider nanotube-filled polymer composites. The modulus of nanotubes may be significantly higher than that of any graphite fiber, and therefore, they have the po- tential to yield higher composite moduli at the same filler volume fraction than a graphite fiber composite. In addition, because of their small size, if properly dis- persed, nanotubes may not decrease the strain-to-failure. But, how is load transferred to a nanotube? Qian et al. [148] and Yu et al. [28] have shown, as first described by Wagner [29], that MWNTs fail via a sword-and-sheath mechanism (see section 2.2.1). This implies that, even if load is transferred to the outermost nanotube in a MWNT, the inner tubes slide within the concentric cylinder and do not carry much load. This situation limits the efficiency of MWNT/polymer composites, be- cause only a small portion of the volume fraction of the MWNT carries load. Evidence for this was shown for MWNT/polystyrene composites, in which the effective mod- ulus of the MWNT in the composite was only $500\;\mathrm{GPa}$ [148] but the measured mod- ulus of the MWNT is close to 1TPa. Therefore, to minimize the number of layers not carrying load, 2–3 layers are preferable. + +For SWNT composites, the SWNT are in a bundle and, as was suggested earlier, individual SWNTs may slip within the bundle. Work by Yu et al. [27] showed that, if only the nanotubes on the outer edge of a SWNT in a bundle are used to calculate the modulus, it is close to the predicted 1 TPa. However, if the whole area of the bundle is used, the calculated modulus is considerably lower. This suggests that, until the SWNT are isolated from the bundles or the bundles are crosslinked, the modulus of composites made from these materials will be limited. Fortunately, progress is being made in obtaining significant volume fractions of exfoliated nanotubes [279]. + +A final issue of relevance is the waviness of the nanotubes. If they are not straight when placed in the composite, then, as recently reported [301], the modulus of the composite is significantly decreased. + +# Clays + +Plate-like fillers can also increase the modulus of a polymer significantly. A compar- ison of the upper bound modulus prediction (assuming in compress i bil it y) shows that the modulus for well-aligned platelets can be three times that for well-aligned fibers [302]. In addition, there is direct evidence that nanoclay carries load; for example, as the aspect ratio of the clay layers increases, the modulus increases. Some indirect evidence [303] also shows that the stiffness of the clay layers affects the modulus (e.g., a stiffer filler leads to a higher composite modulus). Neither of these observa- tions is surprising if the clay platelets are carrying load. Other studies, however, sug- gest that the modulus increase is not entirely due to the load-carrying ability of the platelets, but is caused by the volume of polymer constrained by the platelets [174]. This suggests that, to optimize the increase in modulus, the degree of dispersion must be optimized to maximize the degree of matrix/filler interaction. Work on PP nanocomposites [304], in which adding maleic anhydride (MA) to the matrix chan- ged the degree of filler dispersion, supports this suggestion. Despite the plasticizing effect of MA, the modulus improved, due to enhanced dispersion of the clay. Addi- tional evidence [305] (Figure 2.31) relates the interlayer spacing to the modulus. Lan and Pinnavaia [202] found that, as the degree of exfoliation increased by changing the length of the alkylammonium intercalating chain, the modulus and strength im- proved. As the polymer intercalates and swells, the layers and the area of interaction between the polymer and the filler increase, and the modulus increases significantly. + +![](images/d86522a1519a5404c4818a83513c9db019b70c2ae9880aea372b311e2f7b285c.jpg) +Fig. 2.31 Relationship between storage modulus and interlayer spacing during the melt intercalation process. Reprinted with permission from [305] + +The specific role of the clay/polymer interface in controlling the modulus is also becoming clearer. An interesting study on the modulus of clay/Nylon 6 composites examined the modulus as a function of the ionic strength (as measured via $\mathrm{^{15}N}$ -NMR chemical shifts) of the clay [182] (Figure 2.32). Another study determined the types of matrix/filler interactions responsible for modulus changes [306] (Figure 2.33). Using short-chain alkylammonium ions minimized the type-B interactions, allowing more of the polymer (epoxy) to interact directly with the clay surface (type-A interactions). Di- rect interaction of the matrix with the clay basal planes led to a larger increase in the modulus. It is not clear if the modulus increase caused by the type of interaction and/ or the ionic strength is due to an increase in the interfacial shear stress (the load-bear- ing efficiency of the clays), the ability of the clay to constrain the polymer, or increases in the degree of crystallinity. These effects are all related, and the exact mechanism is still an open question. + +![](images/b7f50aa986f878cf6c72bfcc4ece3ba9cacac1caeffea4b30d1bc7867011f788.jpg) + +![](images/99bc7952df21d9b664cd16fd431cca71292fc39f3dbecffae4a84927425eb649.jpg) +Fig. 2.34 Relative dynamic storage modulus of polypropy- lene clay hybrids as a function of temperature at filler loadings of 2–5 wt. percent. Reprinted with permission from [304] + +Further evidence that the polymer/clay interaction plays a significant role in con- trolling mechanical behavior is that the improvement in properties tends to be higher above the glass transition temperature than below it [307]. Figure 2.34 shows this explicitly for montmorillonite/modified PP matrix composites. Below Tg, the modu- lus increases by a factor of 2, and above Tg it increases by a factor of 2.5. This behavior is also seen for epoxy matrices. When a brittle epoxy with a Tg well above room tem- perature was reinforced with organically modified montmorillonite (OMMT), the in- crease in modulus at room temperature was modest. In contrast, when an epoxy with a Tg below room temperature was used as the matrix, the modulus (and strength) im- proved by a factor of 10 at 15wt. $\%$ of filler [202]. A strong clay/polymer interaction affords more opportunity for immobilization above Tg than below Tg. In addition, if the surface area for interaction decreases, the modulus decreases. This result was also observed in OMMT/polyimide composites. The modulus increased by 1.4 over that of unfilled polyimide, but at higher volume fractions it started to decrease, presumably due to aggregation of the filler [308]. + +# Equi-axed nanoparticles + +The size scale for equi-axed nanoparticle/polymer composites ranges from hybrid nanocomposites, in which the matrix and filler are so intimately mixed that they are no longer truly distinct, to discrete particles in a continuous matrix. We focus here on discrete fillers more than approximately $5\;\mathrm{nm}$ in diameter. The matrix sig- nificantly influences changes that occur in modulus and strength. For example, in alumina/PMMA composites, the modulus decreased for very weak particle/polymer interactions [127]. In silica/polystyrene composites in which the particle was grafted to a polymer chain, however, the modulus increased [129]. In addition to the reinforcing effect that the particles can contribute, the increase or decrease in modulus is also related to the effect of the particles on the polymer’s mobility. Changes in the Tg are strong evidence that polymer mobility is significantly altered. Tg increases for strong polymer/filler interactions and decreases for weak interactions [129]. Given the sensitivity of the polymer modulus to the temperature relative to Tg, changes in Tg can lead directly to changes in modulus. A third mechanism, namely localized yielding near the particles, may also decrease the modulus at very low and seemingly macroscopic elastic strains. + +# 2.5.1.2 Failure Stress and Strain – Toughness + +The addition of rigid micrometer-scale fillers to a polymer often increases its strength, but decreases the toughness. This tradeoff is a significant technical problem for com- mercial applications of filled polymers. The reason for the decrease in ductility is straightforward: the fillers or agglomerates act as stress concentrators, and the defects initiated at the filler quickly become larger than the critical crack size that causes fail- ure. + +Well-dispersed nanoparticles are much smaller than the critical crack size for poly- mers and need not initiate failure. Thus, they provide an avenue for simultaneously toughening and strengthening polymers. Proper dispersion is critical for achieving this. For example, Hasegawa et al. [307] studied the dispersion of clays in polypropy- lene. When they achieved exfoliation, the strain-to-failure ratio remained high $(>\!200\,\%)$ at loadings as high as $3\,\%$ , but even a small amount of aggregation decreased the strain-to-failure ratio to $5\,\%\!-\!8\,\%$ . Another example is polyimide matrices, in which the exfoliation is not complete and the strain-to-failure ratio decreases by $72\%$ with just $1\%$ of OMMT [308]. Further addition of OMMT causes further aggre- gation and a precipitous drop in the strain-to-failure ratio. Many authors report an optimum volume percent of filler and claim that the decrease in strain-to-failure ratio above the optimum is due to agglomeration [147]. + +Although the small size of nanoparticles, based on the explanation above, should not decrease the strain-to-failure, it also should not cause the observed increases in the strain-to-failure [127]. Clearly, there is a second mechanism operating. The lack of discussion on toughening polymers with rigid particles means we must turn to the literature on rubber toughening of polymers to obtain hints about the second me- chanism. An excellent review on rubber toughening [309] provides the background required for this discussion. First, the toughness of a material can be defined as the energy to cause failure. This is related to the deformation in the material, but also to the volume of material undergoing deformation. Even brittle amorphous poly- mers such as polystyrene and poly methyl meth a cry late have the potential for large deformation, but the volume of material undergoing deformation is small. Material must first strain-soften after yielding and eventually strain-harden. If the material does not strain-soften, then defects in the material lead to stress concentrations that cause + +![](images/c8d2afaf1a8b16b0859c9c0ce2d599cc5011ce6cae4c25274f4dba878775b454.jpg) +Fig. 2.35 Schematic of Consi- dere’s construction. True stress is plotted against extension ra- tio, and tangents are drawn through the origin. The first tangent defines the strain at which the tensile load reaches a maximum (the engineering yield stress, $\sigma_{y})$ . The second tangent marks the point at which strain hardenening stabilizes the neck (the natural draw ratio $\lambda_{\mathsf{d}})$ ). Fai- lure at point A between the two tangents gives a low extension to break (curve a). Failure beyond the second tangent gives curve b. Reprinted with permission from [309] + +catastrophic brittle failure. If strain softening occurs prior to strain hardening, then the strain-hardened region draws a larger volume of material into the deformation zone and a large strain-to-failure results. If strain softening occurs, but not strain hardening, the initially strain-softened material propagates the crack. Figure 2.35 [309] illustrates all three scenarios with Considere’s construction. + +Plastic deformation and strain hardening can occur via two different mechanisms: crazing and shear yielding. Crazing is the result of dilatational strains that cause an increase in volume and subsequent drawing of the material between the voids (fibrils). Thus, strain softening occurs due to dilatational strains in front of a craze tip and at the craze/matrix boundary, and strain hardening occurs, which allows drawing and strain hardening of the material in the craze. In principle, this allows the crazes to thicken and the volume of deformed material to increase significantly. Unfortunately, in many brittle amorphous polymers, failure initiates at the craze/bulk polymer interface and the volume of deformed material is small. + +Shear yielding also results in strain softening and strain hardening and can result in stable neck formation and significant toughness. Due to the large change in shape that shear stress causes, a triaxial stress state forms in the center of the sample, which can lead to void formation and crack growth. + +In rubber particles having a shear modulus ${<}0.1$ that of the matrix, the yield stress decreases due to the stress concentration caused by the soft rubber (which is essen- tially a void in terms of stiffness). If the particle does not cavitate, then the only effect of the addition is to delocalize the stress and allow a significant volume of material to yield and to limit craze formation. This effect leads to modest improvements in strain- to-failure and toughness and may occur at a critical interparticle spacing [310]. If, on the other hand, the rubber particle cavitates, then the stress state in the matrix between particles is biaxial or uniaxial and can lead to significant drawing (strain hardening) of the matrix. By this mechanism, large volumes of matrix are drawn into the deforma- + +![](images/b2d1ff1391f799e8b3bd98090a0e5395176131d4077c730fb0d6ad5349613f00.jpg) +Fig. 2.36 (a) True stress–true strain curves at room temperature for PP-based materials. (b) Corresponding volume strain versus elongation for the PP-based materials. Reprinted with permission from [311] + +![](images/eb77ca79981a368b4a279e7f92ba1776c64839b3c974a1fe12fce0304f689dfd.jpg) +Fig. 2.37 Field-emission electron micrograph showing cavitation surrounding weakly bound $39\;\mathsf{n m}$ alumina particles in PMMA. (Thanks to B. Ash for this micrograph) + +tion zone and toughening occurs. In addition, the rubber particles strain-harden and thereby strengthen the most highly deformed regions, allowing a large volume of material to participate in the deformation process. + +How does this apply to nanoparticle-filled polymers? First of all, if the particle/poly- mer interaction is weak, then the nanoparticles can essentially act as voids, which initiate yield and allow for shear bands to form and then thicken. Evidence of this mechanism is provided by several studies. Video extensometry was used to monitor the volume strain as well as the true stress-vs.-strain behavior of clay-filled PP and nylon 6 [311]. The results are fascinating. The typical failure mechanism for nylon is shear banding, but the addition of well-dispersed montmorillonite encouraged ca- vitation. This added another mechanism of deformation and increased the volume of material involved in deformation. Figure 2.36 shows a graph of the volumetric strain, as well as a scanning electron micro graph of the fibrillation. Interestingly, this result occurred only for weak polymer/clay interactions or for strong interfacial interactions at temperatures above Tg. In addition, crystalline alumina-filled PMMA [127] with weak particle/polymer interactions exhibited an order of magnitude increase in the strain-to-failure. Evidence for cavitation was seen via fractography (Figure 2.37). + +Another example in which nanoparticle inclusions lead to significant improvements in strain-to-failure and/or toughness is rubber matrices. Here, even if the interaction between the particle and polymer is strong, the soft material around the particle can easily lead to interface failure. In rubbery epoxy, the strain-to-failure is maintained for exfoliated nanocomposites [202] and can be improved by using magadiite to reinforce relatively ductile epoxy [207]. The addition of OMMT to polyurethane improved the strain-to-failure by a factor of two [208]. This may be due to a plasticizing effect of + +![](images/4e1a0818c16fb8d842737203f52c8a0f9e031a3f71190b43b265890369c44bbc.jpg) +Fig. 2.38 Stress–strain curves for (a) a pristine polyurethane elastomer, and (b) a polyur- ethane-clay nanocomposite at a strain rate of $1.0~{\mathsf{i n}}/{\mathsf{m i n}}$ . Re- printed with permission from [208] + +the alkylammonium ions, but did not result in a decrease in strength or modulus. Figure 2.38 shows the stress–strain curve for a representative sample. Other inves- tigators have found similar results [312], which can be maximized when the intercalat- ing agent acts as a crosslinking agent. + +# 2.5.1.3 Glass Transition and Relaxation Behavior + +As discussed in section 2.1, the interfacial region is extremely large in nanocompo- sites. The interaction of the polymer with the nanoparticles gives significant oppor- tunity for changing the polymer mobility and relaxation dynamics. For example, poly- styrene chains intercalated between the layers of a smectic clay have more mobility locally than in the bulk polymer [313]. This greater mobility may be due to an ordering that occurs between the layers, which creates low- and high-density regions, thus pro- viding the opportunity for mobility in the low-density regions [314]. If, however, the polymer is tethered to the clay, the relaxation spectra broaden to include slower relaxa- tion times, suggesting that the clay can reduce the mobility locally. Recent work on silica/polyvinyl acetate nanocomposites indicates that, for equi-axed particles, the over- all effect of the filler is to create a weak network structure that contains both physical entanglements in the matrix phase and ‘trapped’ entanglements arising from tempor- ary bonds to the filler particles. This suggests a greater far-field effect of altering chain dynamics than simply a local immobilization of the polymer chain [318]. Another example of altered mobility was found for gold particles dispersed in poly( tert -butyla- crylate). For particle sizes ${<}10\;\mathrm{nm}$ , the viscosity of the low molecular weight polymer at 30 vol. $\%$ filler increased by a factor of 4 [315]. The authors claim this result is due to bridging of the polymer chains between the particles. The bridges have relatively long lifetimes and decrease segmental motion at the interface compared to the bulk. + +Although the specific mechanics of chain dynamics discussed above are not yet completely understood, it is very clear that the rheology/glass transition temperature of a polymer can be controlled by changing the polymer mobility with nanocomposite interfaces. For example, the glass transition temperature can be eliminated for clay nanocomposites with intercalated polymer chains. This indicates a limited ability for cooperative chain motion when the polymers are confined between the layers [131]. If the clays are exfoliated, and thus the polymer is not confined between layers, the Tg does not change significantly. + +Without the help of confinement, the Tg can be increased if the polymer is tightly bound to the nanofiller. Increases of $10^{\circ}$ have been observed for clay-filled Nylon, PMMA, and PS [196], as well as for polyimide composites [308]. The behavior is not limited to clays. A Tg increase of almost $30^{\circ}$ was observed for exfoliated gra- phite/polystyrene composites [80], as was a $20^{\circ}$ increase for nitrogen-doped carbon nanotube/epoxy composites [264]. The Tg also changes in nanoscale equi-axed parti- cle-filled composites. The addition of ${\mathrm{CaCO}}_{3}$ to PMMA resulted in a $35^{\circ}$ increase Tg at 6 wt. $\%$ filler when the composites were prepared via an in-situ process [232]. Smaller increases were observed for silica/PMMA [128] nanocomposites. + +The Tg can decrease if the interaction between the filler and the matrix is weak, as has been observed with silica [128] and alumina-filled PMMA [316]. This decrease in Tg was recently given more attention in nanoalumina/PMMA composites, where it was found that the Tg did not decrease until a critical volume fraction was reached. Furthermore, changing the particle surface and making it compatible with the matrix eliminated the Tg depression. + +The explanation for the increase or decrease in Tg in the thermoplastics may be related to the thin film results discussed in section 2.3. + +Of course, chain dynamics affect much more than the glass transition temperature. Recent studies have attempted to understand the role of filler/polymer interaction, to begin to understand the role of nanofillers in controlling damping, melt rheology, and other dynamic processes [131, 317, 318]. + +# 2.5.1.4 Abrasion and Wear Resistance + +Although it is well known that the abrasion resistance of filled polymers depends on particle size, the incorporation of nanoscale fillers has led to unexpected results. For filler particles that are larger than the abrasive particles, most of the filler particles are stable and increase the abrasion resistance of the composite. As the filler size is de- creased to a size similar to that of the abrading particles, filler particles are removed, and the abrasion resistance is compromised. This does not happen on the nanoscale, however. For example, the addition of nanoscale ${\mathrm{CaCO}}_{3}$ to PMMA resulted in a sig- nificant decrease (factor of 2) in material loss due to abrasion at only 3 wt. $\%$ of filler [232]. In addition, nanoparticles can simultaneously improve wear resistance and de- crease the coefficient of friction. With larger-scale fillers, increased wear resistance is accompanied by an increased coefficient of friction, which has been observed for si- lica/Nylon [319] as well as for poly ether ether ketone matrix composites filled with $\mathrm{ZrO}_{2}$ , SiC, $\mathrm{SiO}_{2}$ , and $\mathrm{Si}_{3}\mathrm{N}_{4}$ [320–323] and for sheet silicates in polyimide [324]. The coefficient of friction measured as dry sliding against steel decreased monotoni- cally with increasing weight percent of fillers. The authors attributed this finding to an improved and tenacious transfer film that was generated on the counterface, an area that needs further exploration. + +# 2.5.2 Permeability + +The reduced gas and liquid permeability of nanofilled polymers makes them attractive membrane materials. The large change in permeability of liquid or gas through a composite material can be explained from simple predictions [325, 326]. Given a plate-like material with an aspect ratio of $L/W$ dispersed parallel in a matrix, Nielson calculated the tortuosity factor $\tau$ as + +$$ +\tau=1\,+\,(L/2\,W)\,V_{f} +$$ + +where $V_{f}$ is the volume fraction of plate-like filler. The relative permeability coefficient $P c/P p$ , where $P c$ and $P p$ are the permeability coefficients of the composite and polymer matrix, respectively, is given by + +$$ +P c/P p=1\mathrm{~/~}(1\mathrm{~+~}(L/2W)\,V_{\!f\!)} +$$ + +A slightly more rigorous analysis for a random in-plane arrangement of plates yields + +$$ +P c/P p=1\mathrm{~/~}(1+\mu(L/W)^{2}(V_{f}^{2}/(1\mathrm{~-~}V_{\!f\!)} +$$ + +where $\upmu$ depends on the distribution of the plate-like material. In either case, the hig- her the aspect ratio of the filler, the larger is the decrease in permeability. Yano et al. [188] compared Eq. 2 to data on clay-filled polyimide (Figure 2.39) and found a good fit. The large reductions in permeability (Figure 2.39) occur at only 2 wt. $\%$ for exfoliated composites (Figure 2.40) [8]. The barrier, however, is sensitive to the degree of disper- sion and the alignment of the plates. Significant reductions in permeability were also reported for exfoliated clay/PET [327], Nylon [328], and poly ca pro lac tone [7] compo- sites. The reduction is linear with the wt. $\%$ of filler for poly ca pro lac tone [7], but for polyimide [8] (Figure 2.40), the decrease in permeability with wt. $\%$ of filler is nonli- near. + +![](images/95fe2bda5ff863fe450b6cf7c2ff95d79a3872e8ba4dc56ece6aa46ea49ef897.jpg) +Fig. 2.39 Effect of clay layer size on relative per- meability compared with the prediction from Eq. 4.2.2. for clay/polyimide nanocomposites. Reprin- ted with permission from [188] + +![](images/b4ae25b9035c9b6af88342a456b82e43f2b882fa0828eadb645f84751aa9d442.jpg) +Fig. 2.40 Effect of montmorillonite content on relative permeability coefficient of water vapor in polyimide nanocomposites. Reprinted with per- mission from [8] + +Several nanocomposites are being produced commercially for barrier applications [327]. They include a vermiculite clay/PET composite with oxygen permeation values of ${\leq}1\;\mathrm{cc}\,\mathrm{m}^{\cdot2}\,\mathrm{d}^{\cdot2}$ , produced by ICI Corporation, which is significantly less than that of pure PET. Ube Industries produces a Nylon/clay hybrid whose permeability is less than that of Nylon by a factor of 2 at 2 wt. $\%$ filler. + +Apparently, significant improvements in barrier properties are also achievable with nonplate-like nanoparticles [329]. Nano Material Inc. reports that a PVA/EVOH matrix composite with 7 nm silica and titania nanoparticles exhibits a gas permeability of 1 cc $\mathrm{m}^{\cdot2}\ \mathrm{d}^{\cdot1}\,\mathrm{atm}^{\cdot1}$ and moisture permeability of less than $1\mathrm{\g\megam^{\cdot2}\d^{\cdot1}}$ . Although this is achieved at very high loadings, the material is melt processable. + +The absorption of water into composites is significant. For example, one of the lim- itations of Nylon is the reduction in mechanical properties that accompanies the ab- sorption of water. The addition of exfoliated montmorillonite increases the resistance to water permeation after 30 min from $2\%$ to $1\%$ at 5 wt. $\%$ of filler [330]. The me- chanism of the reduction is attributed to the constrained region of the Nylon. If the constrained region is taken into account, the diffusion coefficient follows a rule of mixtures. Figure 2.41 shows the change in diffusion coefficient of water in Nylon in response to clay content. + +![](images/d21b6772c4585b71be96fa6ada496df53b79573cf8d2b75747166fbbce6d761c.jpg) +Fig. 2.41 Dependence of diffusion coefficient of water on clay content for montmorillonite with a layer width of $100\;\mathsf{n m}$ , and saponite with a layer width of $50\;\mathsf{n m}$ . Reprinted with permission from [330] + +# 2.5.3 Dimensional Stability + +Dimensional stability is critical in many applications. For example, if the layers of a microelectronic chip have different thermal or environmental dimensional stabilities, then residual stresses can develop that cause premature failure. Poor dimensional stability can also cause warping or other changes in shape that affect the function of a material. + +Nanocomposites provide methods for improving both thermal and environmental dimensional stability. The first mechanism by which nanofillers can affect the coeffi- cient of thermal expansion (CTE) of a polymer is also observed in traditional fillers. The composite CTE changes according to the volume average of the two CTEs. For traditional fillers, at low volume fractions of filler, this change in CTE is almost linear. Figure 2.42 shows the CTE for polyimide filled with exfoliated montmorillonite [8]. In contrast to traditional fillers, the dependence is nonlinear. This leads to the second mechanism effecting CTE – the interfacial region. It may be that the nonlinear effect is due to the increase in volume of polymer that is behaving similar to a thin film and has a lower CTE than the bulk matrix. Observations that, in thin films that interact with a substrate, the CTE decreases as the film thickness decreases below $10\;\mathrm{nm}$ [118] sup- port this possibility. + +Nonlinear dimensional changes are also observed for nanofilled rubbers that swell in the presence of a liquid. This is a well-known phenomenon [331] and is again due to the interfacial region around the particles, as discussed in section 2.3. The effect of this layer increases as the strength of the particle/polymer interaction increases [332]. The behavior is also observed in recently developed nanofilled rubbers. In comparisons of the behavior of montmorillonite/PDMS composites with other conventional alumino- silicates or carbon black [216], the relative swelling of the montmorillonite/PDMS composites was about half that of both unfilled PDMS and carbon black/PDMS. Mean- while, kaolinite increased the swelling. The authors argued that the decrease in swel- ling is due to a bound polymer layer and that the large surface area and very strong polymer/filler interaction enhance swelling resistance. Reduced swelling was ob- served recently in nanoparticle titania/epoxy composites [147]. + +![](images/b7d24802e22c038aa62c199544ac26524de24f990d53494ca0f5fdd83f839867.jpg) +Fig. 2.42 Effect of montmorillonite content on thermal expansion coefficient of polyimide clay hybrids at (a) ${\mathsf{I}}\,50\,^{\circ}{\mathsf{C}},$ (b) $200\,^{\circ}\mathsf{C},$ (c) $250\,^{\circ}\mathsf{C},$ , (d) $295\,^{\circ}\mathsf{C}$ . Reprinted with permission from [8] + +A recent paper described the addition of OMMT to the type of PMMA used in dental work [333]. Here, PMMA is added to MMA monomer, and the resulting mixture is polymerized under ultraviolet light. The shrinkage occurs largely during polymeriza- tion, which causes internal stresses and reduces the lifetime of the filling. The addition of OMMT to the PMMA/MMA mixture, however, reduces shrinkage considerably and thus lowers the stresses that form. This may provide an excellent method for increas- ing longevity. + +# 2.5.4 Thermal Stability and Flammability + +The first indications that nanoclays could increase the thermal stability of polymers was demonstrated in 1965 for montmorillonite/PMMA composites. The degradation temperature (defined as $50\,\%$ mass loss) increased about $50\,^{\circ}\mathrm{C}$ for a composite with only 10 wt. $\%$ PMMA (90 wt. $\%$ filler) [334]. More recently, the degradation tempera- ture of PDMS was found to increase about $140\,^{\circ}\mathrm{C}$ with 10 wt. $\%$ montmorillonite for a delaminated nanocomposite [216]. A $50\,^{\circ}\mathrm{C}$ increase was found for intercalated clay/PE composites [143]. The dispersion of the clays is critical to increasing the degradation temperature. Delaminated composites have significantly higher degradation tempera- tures than intercalated nanocomposites or traditional clay composites [335]. Some speculate that this increase in stability is due to the improved barrier properties of the composites. If oxygen cannot penetrate, then it cannot cause oxidation of the resin [216]. In addition, the inorganic phase can act as a radical sink to prevent polymer + +![](images/8d94a466316c8cf11ee58d8cc653c89714f86dc043b7f12f8638783024cc43f8.jpg) +Fig. 2.43 Comparison of the heat release rate (HRR) plot for Nylon-6, Nylon-6 silicate-nanocomposite (mass fraction 5 percent) at $35~\mathrm{kW}/\mathsf{m}^{2}$ heat flux, sho- wing a 63 percent reduction in HRR for the nanocomposite. Reprinted with permission from [338] + +chains from decomposing. The improved thermal stability of some composites may be limited by the lower thermal stability of alkylammonium ions. For example, in inter- calated clay/polystyrene composites, the intercalating agent decomposes at about $250\,^{\circ}\mathrm{C}$ . Bonding the intercalating ion to the polystyrene matrix noticeably improved the thermal stability. Polyimide [308] and poly methyl meth a cry late [333] also become more thermally stable with the addition of OMMT. + +In addition to thermal stability, the flammability properties of many clay/polymer nanocomposites are also improved [336, 337, 338]. Figure 2.43 shows typical plot of the heat release rate (HRR) vs. time for Nylon-6 and a clay/Nylon-6 nanocomposite with only a few wt. $\%$ of filler. The HRR increases at a lower temperature for the nanocom- posite, but levels off quickly. Clay nanocomposites commonly show a $60\,\%$ reduction in peak HRR relative to unfilled polymer (Table 2.2). In addition, the mass loss rate is the same for the two materials until a char forms, and then the nanocomposite exhibits a significantly lower mass loss rate. Observation of the char suggests that the layered silicate collapses during combustion and forms a uniform layered structure (interlayer spacing about $1.3\;\mathrm{nm}$ ). The layers act to reinforce the char and reduce the permeability of the char, reducing the rate of volatile product release. The structure of the char was fairly uniform for several different matrices and independent of the initial microstruc- ture of the composite (intercalated or delaminated). + +Combining traditional flame retardants with intercalated or exfoliated clays can re- sult in further improvements in flame retardance [339, 340]. An additional advantage + +![Tab 2.2 Cone calorimeter data. Heat flux: $35~\mathrm{kW}/\mathsf{m}^{2}$ , Hc: heat of combustion, peak heat release rate, mass loss rate, and specific extinction area (SEA) data are reproducible to within $\pm10\,\%$ . The carbon monoxide and heat of combustion data are reproducible to within $\pm15\,\%$ ](images/31bc3f510197affc0a34df8739599390035e4a0f429270b89ed5ca4ada290614.jpg) + +of nanoclay composites relative to traditional flame retardants is the improvement in other properties, such as the heat distortion temperature and the bending modulus. + +The flammability resistance of clay-filled polymers indicates that their ablation re- sistance might also be excellent. As a material is heated during ablation, the surface of the material reacts and forms a tough char. If the char is not reinforced, it fails and is removed from the surface, exposing more material [341]. Traditional composites re- quire a significant weight fraction of filler (more than 30 wt. $\%$ ) to achieve significant ablation resistance. On the other hand, 2–5 wt. $\%$ nanoclay-filled Nylon 6 exposed to a mock solid rocket motor firing rig formed a layer of char on the surface that was tough and significantly retarded further erosion [341]. In addition, oxygen plasma forms a passivation layer on Nylon 6/layered silicate nanocomposites, which significantly re- tards further erosion of the composite surface [342]. This behavior is not a strong function of the organic molecules used to modify the clay or the strength of the clay/polymer interaction [342], but is a function of the degree of exfoliation [341]. + +# 2.5.5 Electrical and Optical Properties + +The electrical and optical properties of nanofilled polymers are exciting areas of re- search. This is particularly true because of the possibility of creating composites with unique combinations of functionalities, such as electrically conducting compo- sites with good wear properties that are optically clear. Such properties can result because nanoparticles, with diameters distinctly below the Rayleigh scattering lim- it, still display their solid-state physical properties when embedded in transparent ma- trices. + +Optical composites have been defined as composites consisting of optically active nanoparticles embedded in a transparent host material, often a polymer [343]. Optical composites take advantage of the optical properties of materials that are hard to grow in single-crystal form or that require protection from the environment and give them the ease of processing afforded many polymers. In addition, sometimes the material must be used at the nanoscale to achieve specific optical properties, and the matrix is used just to hold the particles together and provide processability. For example, high-grade optical composites, with properties otherwise obtainable only in optical glasses, be- come accessible through the use of polymer molding techniques. + +The following sections briefly review some of the literature in this area. + +# 2.5.5.1 Resistivity, Permittivity, and Breakdown Strength + +Electrical properties are expected to be different when the fillers get to the nanoscale for several reasons. First, quantum effects begin to become important, because the electrical properties of nanoparticles can change compared to the bulk. Second, as the particle size decreases, the interparticle spacing decreases for the same volume fraction. Therefore, percolation can occur at lower volume fractions. For example, the high aspect ratio of carbon nanotubes or exfoliated graphite leads to percolation at a lower volume fraction and thus increases in electrical conductivity of polymers at a + +![](images/fbdfc21918fb3fc29fd7cffda2dac73eb3fd51de21a29bc485d539e1a75cfe1d.jpg) +Fig. 2.44 Resistivity vs. volume fraction for nanoscale and micrometer scale ZnO filler in low density polyethylene. (Thanks to J. Hong for this figure) + +lower volume fraction than in nonexfoliated graphite [80]. This can be understood from geometric considerations. Figure 2.44 [344] shows an example of this behavior for $\mathrm{ZnO}$ -filled low-density polyethylene. In addition, the rate of resistivity decrease is lower than in micrometer-scale fillers. This is probably due to the large interfacial area and high interfacial resistance. + +The permittivity of polymers can also be increased with the addition of many metal oxide fillers, on both the micro- and nanoscale. The use of micrometer-scale fillers for this application, however, results in a significant decrease in breakdown strength due the field concentration created by the particles. One of the significant advantages of using nanoscale fillers instead of micrometer-scale fillers may be a reduction in the loss in breakdown strength. The breakdown strength decreases for micrometer-scale fillers because of the field concentration the particles create. Nanoscale fillers do not lead to as large a reduction in breakdown strength. Preliminary results using $\mathrm{ZnO}/\$ low-density polyethylene composites show that at 50 wt. $\%$ of filler, the breakdown strength of the nanofilled LDPE is about $10\%$ higher than that of LDPE filled with submicrometer $\mathrm{ZnO}$ [344]. In addition, fundamental to controlling the breakdown strength of insulating polymers is the cohesive energy density [345] and the free vo- lume or mobility of the polymer [346]. + +Nanotubes can also be used to change the resistivity. Recently, such a behavior was demonstrated in a conjugated luminescent polymer, poly( m -phenylenevinylene-co- 2,5-dioctoxy $\cdot p$ -phenyl ene vinyl ene) (PPV), filled with MWNT and SWNT. Nanotube/ PPV composites have shown large increases in electrical conductivity compared to the pristine polymer, of nearly 8 orders of magnitude, with little loss in photolumi- nescence/electroluminescence yield. In addition, the composite is far more robust than the pure polymer with respect to mechanical strength and photo-bleaching prop- erties (breakdown of polymer structure due to thermal buildup). Preliminary studies indicate that the host polymer interacts weakly with the embedded nanotubes but that the nanotubes act as nanometric heat sinks. Recent experimentation with composites of conjugated polymers such as PPV and nanotubes, show a promising new phenom- enon. Apparently, the polymer chains wrap around nanotubes suspended in dilute solutions of the polymer. Microscopy suggests that coiled polymer chains can interact strongly with nanotubes and assemble on the nanotubes, leading to the creation of unique interfaces affecting the electronic and optical properties of the polymer [347]. + +An interesting example of how nanocomposites may affect electronics is the devel- opment of photo-patternable hybrid inorganic–organic polymers with negative resist behavior. They are composed of inorganic oxidic structures that are cross-linked or substituted by organic groups. They are synthesized from organosilane precursors reacted by sol–gel processing in combination with organic cross-linking of mono- mers. The processing is often integrated with micro-patterning technologies to fabri- cate dielectric and passivation layers in microelectronic systems and devices as well as cladding for optical applications [348]. + +# 2.5.5.2 Optical Clarity + +A limitation in the development of optical composites or transparent nanocomposites with improved mechanical or electrical performance is minimizing light scattering due to the presence of the particles. The scattering power for light propagation through a collection of scattering particles can be predicted by Rayleigh scattering: + +$$ +P_{\mathrm{scat}}\:-\:24\:\pi^{4}\ P_{\mathrm{o}}\:\rho\ (\ (n^{\prime}\:-\:n)/n^{2})\ (V^{2}/\lambda^{4}) +$$ + +Where $P_{\circ}$ is the incident power, $\uprho$ is the concentration of particles, $n'$ is the refractive index of the particles, $n$ is the refractive index of the matrix, $V$ is the volume of a single particle, and $\uplambda$ is the wavelength of light. Therefore, to minimize scattering, the par- ticles must be as small as possible with an index of refraction as close as possible to that of the matrix material. This equation assumes, however, that the particles are much smaller than the wavelength of light, and if this is not true, Rayleigh–Debye calcula- tions are required [349]. + +Relatively good optical clarity has been obtained in many nanocomposites, particu- larly at low volume fractions. Studies on PMMA modified with n-do dec yl meth a cry late during polymerization were shown to maintain optical clarity up to 10 wt. $\%$ of ben- tonite [199]. In addition, when exfoliation was achieved in polyimide matrix compo- sites, the 2 wt. $\%$ hybrid was relatively transparent [8] compared to nonexfoliated clay. A comparison between smectic clays and magadiite (a layered silicic acid) shows that magadiite results in much better optical clarity than smectic clay, probably due to index of refraction matching [203]. Optically clear nanofilled polyurethanes have also been prepared [208]. The addition of UV-absorbing nanoparticles have been used to create a clear coating on polymers to prevent degradation Figure 2.45 shows a quantitative + +![](images/1e4d45ba858db7c23944f430136c71e6ba35d96274eca53f1400d69c99990e88.jpg) +Fig. 2.45 Optical transmittance vs. wavelength for a nanoscale alumina-filled gelatin at 17 wt. percent filler. (Thanks to G. Irvin, Q. Chen for this figure [350]) + +example of the transmission achieved in nanoscale alumina/gelatin nanocomposites as a function of wavelength [350]. + +# 2.5.5.3 Refractive Index Control + +Preparation of clear nanocomposites with refractive indices over the entire range of ${<}1$ to ${>}3.9$ is possible, and these values are by far the lowest and highest ever achieved for any polymer system. An excellent example of the use of nanoparticles to obtain a spe- cific index of refraction was presented by Zimmerman et al. [351]. They found that the index of refraction scaled closely with the volume fraction of lead sulfide nanoparticles in a gelatin matrix. Indices of refraction as high as 3.9 were obtained [352]. The addi- tion of nanoscale iron sulfide to polyethylene increased the index of refraction to be- tween 2.5 and 2.8 [353]. + +# 2.5.5.4 Light-Emitting Devices + +After the discovery of electroluminescence from conjugated polymer materials (such as PPV), polymer-based light emitting diodes (LEDs) have attracted much attention. The practical advantages for polymer-based LEDs are low cost, low operating voltage, ease of fabrication, and flexibility. Organic light emitting diodes (OLEDs), however, suffer from low quantum efficiency and low environmental stability (against oxida- tion, etc.). + +Function aliz ation of nanotubes and doping of chemically modified nanotubes in low concentration into photoactive polymers such as PPV can affect the hole-transport mechanism and hence the optical emission of the polymer. Small loadings of nano- tubes are used in these polymer systems to tune the color of emitted light from organic LEDs. Although there have been some initial attempts at quantification of nanotube/ conducting polymer interactions, little has been done to understand how excitonic transport in a polymer might be modified by the presence of a high-aspect 1D metal such as a carbon nanotube. Recently, Ago et al. [354] proposed ‘hole collecting’ proper- ties of multi-wall carbon nanotubes from a conjugated polymer at the composite inter- facial region. Such hole collecting properties seem to be common in some conjugated polymers. Specifically, SWNTs in a polymer matrix trap holes injected from the anode in OLEDs [355]. + +The addition of layered silicates may combat the low quantum efficiency and poor stability to oxygen and moisture of OLEDs [356]. Devices made from organically mod- ified clay and (poly[2-methoxy-5-( $2^{,}$ -ethyl-hexyloxy)-1,4-pheneylenevinylene]) (MEH- PEV) posses an external quantum efficiency 100 times that of the pure polymer $({\sim}0.38\,\%$ photons/electron) [357]. The composite also shows increased photolumines- cence efficiency, due to increased excitation within the two-phase composite. + +The addition of equi-axed nanoparticles to electro luminescent polymers also in- creases current densities, radiances, and power efficiencies. When nanoscale titania and silica were added to MEH-PPV, the current density and light output increased by an order of magnitude at low driving voltages [358]. Lifetime was sacrificed slightly. The explanation for the improvement at low voltages was that the nanoparticles created a high surface area contact with the cathode and thus improved charge injection into the polymer. The improvement at high voltages may be due to the thin regions of MEH-PPV that exist between the nanoparticles. This creates regions of higher current density and thus higher light output. Nanoparticles can also lead to changes in the color of emitters. For example, CdSe/polymer nanocomposites emit blue light [359]. + +# 2.5.5.5 Other Optical Activity + +The dispersion of metal particles into liquids or polymers leads to interesting absorp- tion spectra that can be controlled by changing the metal used, the particle size, degree of dispersion, and the polydispersity of the particle size [360]. This has provided an impetus to look for new applications of nanocomposites in areas such as optoelectro- nic devices, nonlinear optical devices, and color filters. In these applications, the metal particle and/or the polymer matrix can be active optical components. + +If the particle is the optically functioning component,then the optical properties of the composite material can be tuned by adjusting the particle size and the average separation between particles. The polymer in these situations plays the role of stabilizing the nanoparticles. It is often possible to create ordered arrays (2D or 3D) of nanoparticles inthe polymer matrix.This involves size-selective depositionof nanoparticles, as well as surface modification of the particles to prevent agglomeration during processing. An example of a transparent, mechanically stable nanocomposite film is a continuous periodic array of self-assembled thiol-passivated Ag nanocrystals in a polystyrene ma- trix. The film is formed at an air–water interface by the Langmuir–Blodgett technique. The spatial distribution of particles in the film is altered by changing thethickness of the film. The films so produced have applications in optical filters [361]. + +The addition of relatively monodisperse silver particles to polyethylene, followed by drawing them into films has led to anisotropic absorption of light and a color that depends on the polarization of incident light [362]. In addition, annealing the com- posite led to further changes in color, giving and added dimension of control. The silver particles formed pearl-necklace type strings upon drawing, which gave rise to anisotropy. The films were easy to process, strong, and flexible. + +Interesting nonlinear optical activity has also been obtained with nanoparticle-filled polymers. For example, by using Cr:forsterite $(\mathrm{Cr};\mathrm{Mg}_{2}\mathrm{SiO})$ in a polymer with a matched index of refraction, relative optical amplification was achieved [343]. Unfor- tunately, large particles, cracks in the films, and poor coupling to the electronics lim- ited the gain obtained. The authors suggested that cubic materials with isotropic op- tical properties might be most appropriate for such applications. + +The optical limiting behavior of polymer nanocomposites has been investigated, a good example being nanotube filled polymer composites [363]. Polymerization of + +![](images/7588d85f88251564a76a14627bd8312695142d7fcf58c74ef21ab3870c3b177a.jpg) +Fig. 2.46 Typical change in transmittance of clay-filled nematic liquid crystals (cell gap $12\,\mu\mathsf{m}$ ) due to various fields: (a) electric field and shearing liquid at liquid crystalline temperature (about $25\,^{\circ}\mathsf{C})$ , (b) thermal change. Reprinted with per- mission from [193] + +phenylacetylene in the presence of short nanotubes yields a solvated product of tubules wrapped with poly(phenylacetylene). The optical limiting properties of this composite are excellent and far better than those of the pristine polymer and of polymer filled with other carbon structures. These composites also fare well in photovoltaic applications, suggesting that such nanocomposites could find an array of potential applications in optics-related and laser-based technologies [363]. + +Liquid crystalline materials have an interesting optical behavior that changes with electric field and with temperature. The addition of intercalated smectic clay to ne- matic liquid crystals added a memory effect to their behavior [193, 194]. Upon imposi- tion of an electric field, the initially opaque composite became transparent. When the electric field was removed, the composite remained transparent (with a slight decrease in transparency). When heated so as to return to the initial state, the composite instead went into a new state (Memory State II) with light transmittance between that of the initial state and the first memory state (Figure 2.46). + +# 2.6 Summary + +The primary purpose of this chapter was to introduce common types of fillers used in nanocomposites, to emphasize the role of the interfacial region, to provide a basic level of understanding of how nanocomposites are currently processed, and to excite the reader about the multifunctional and as yet incompletely explored properties of nano- filled polymers. + +Today, our ability to control the structures and properties of nanocomposites is limited only by our current understanding of how to manipulate these nanoscale struc- tures. This area of research will clearly lead to fruitful commercial applications, with significant economic effect, driven by materials with new combinations of properties. + +# Acknowledgements + +Sincere thanks go to the funding agencies that gave me the opportunity to enter this field, including the National Science Foundation, the Office of Naval Research, the Army Natick facility, Eastman Kodak Inc., General Electric, ABB, and Albany Inter- national. In addition, sincere thanks to several people for helpful discussions and re- views as well as for contribution of figures: Benjamin J. Ash, Elena Petrovicova, San- ford S. Sternstein, Brian C. Benicewicz, Richard W. Siegel, Keith Nelson, Jung Il Hong, Tao Li, and Quan Chen. In addition, I would like to thank several editors, in- cluding Kathy Westbrook, Thomas Feist, Margaret Schadler, and Harvey Schadler. + +# References + +1 A. M. Bueche, J. Polym. Sci., 25, 139 (1957). 2 B. Kuriakose, S. K. De, S. S. Bhagawan, R. Sivaramkrishnan, S. 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Over the past 10 years, materials scien- tists and chemists have made considerable efforts to create synthetic analogs of bio- logical materials by attempting to mimic biology and, importantly, to learn the design rules of biological systems. Considerable efforts have also been made to use what has been learned from biology to create new materials with properties not found in bio- logical systems. All this effort has been made because natural materials unquestion- ably have exquisite properties not found in synthetic materials. And additionally, bio- logical systems can produce these exquisite materials at or near room temperature in aqueous environments, whereas most synthetic schemes that produce materials often inferior to natural biomaterials require elevated temperature and pressure and harsh chemicals. + +Biological nanocomposite materials can be entirely inorganic, entirely organic, or a mixture of inorganic and organic materials. Even where the final material may be entirely one class of material, multiple classes of materials may have been involved in the synthetic process, which may or may not remain in the final structure. A good example of a biological nanocomposite in which the organic material does not remain in the final product is the enamel of the mature human tooth, which is $95\,\%$ by weight hydroxyapatite. During tooth formation, enamel consists of a com- posite of proteins (primarily amelogenin and enamelin) and hydroxyapatite; however, the proteins are removed as the tooth develops. The presence of the proteins, and the self-assembled structures they form with other biological macromolecules, do however help generate the mineral cross-ply structure of the enamel, which plays a large part in its observed toughness. The best known example of an inorganic/organic structural composite for which both phases remain in the final product is the aragonitic nacreous layer of the abalone shell, which is exceptionally strong because of its organic/inor- ganic layered nanocomposite structure, in which crystalline ceramic layers are sepa- rated by highly elastic organic layers. However, there is much more to the exceptional properties of the abalone shell than simply its layered structure, as many researchers are discovering. Synthetic efforts have been made for about 10 years to create such a structure in the laboratory, and although some synthetic structures resemble the aba- lone shell, to date their properties have been inferior. Thus the practical value of the synthetic structures is limited. With the many exciting scientific discoveries that have been made, there is great potential for new materials in the not too distant future. + +Many more examples of attempts to copy biology have not been as successful in generating engineering materials as one would have hoped. In hindsight, this is not surprising. Biological materials generally form over a period of days to years, use a limited set of elements, and are designed to be used within a limited tempera- ture range. Practical engineering materials must be made rapidly (hours or minutes) and generally must operate over a wide range of temperature and other environmental conditions. The disconnect between the needs of engineering materials and of biolo- gical materials has led many scientists to conclude that, rather than attempting to directly copy biology, a much better philosophy is to learn from biology and use this knowledge to create synthetic materials. This may or may not involve the use of some biological molecules, but no attempt is made to ‘copy’ specific biological pro- cesses. + +This chapter covers natural nano bio composites, biomimetic nanocomposites, and biologically inspired synthetic nanocomposites. Natural nano bio composites are ex- actly what the name implies, that is, natural composite materials with structure on the nanoscale; biomimetic nanocomposites are synthetic nanocomposite materials formed through processes that mimic biology as closely as possible; and biologically inspired nanocomposites are composite materials with nanoscale order created through processes that are inspired by a biological process or a biological material, but without attempting to mimic, or directly copy, the mechanism of formation of the biological material. There are many areas I do not attempt to cover, including mixtures of two or more materials in which the materials form a homogeneous mix- ture (and thus are not composites) or in which the organic molecule is simply dis- persed in an organic host, such as organic molecules in a sol–gel matrix. I also only minimally cover the widely publicized work on mesoporous silica and related materials, because several very good reviews on this subject already exist [1–5]. + +The formation of solids containing defined nanometer-scale structures and features is a challenging goal of great interest in the synthesis of novel materials. Significant efforts have been made to generate nanostructured materials by techniques ranging from scanning microscopy to molecular self assembly [6–14]. Through careful control of molecular architecture, self-assembly can yield many different nanostructured or- ganics. The potential for creating new materials by mineralizing, polymerizing, or otherwise replicating these organized structures is very interesting. For example, mineral growth within and on lipid aggregates has resulted in unusual morpholo- gies, including mineralized tubules and disks [15, 16]. Another approach to control- ling mineral growth through self-assembly involves the coassembly of molecular spe- cies and mineral-phase precursors into mesoporous structures. This methodology has been successful in the now widely studied formation of mesoporous oxides [17–23]. + +# 3.2 Natural Nanocomposite Materials + +In biology, examples abound of structures with nanoscale dimensions, and virtually all the functionality provided by these materials is a direct consequence of the nanoscale dimensions of the structure. A few examples of nanoscale materials in biology are lipid cellular membranes, ion channels, proteins, DNA, actin, spider silk, and so forth. In all these structures, the characteristic dimension, at least in 1D, and often in 3D, is on the order of a few nanometers. Although a materials scientist does not typically consider such materials to be composites, in truth, the properties of many biological materials are driven by structure on the nanoscale and, in the sense that the larger material is composed of discrete nanoscale building blocks, most of these biological materials can be considered nanocomposite materials. In their active form, that is, when folded, proteins are composed of domains with varying hydrophilic and hydrophilicity, as well as domains with such structural features as alpha helixes, beta sheets, and turns. The assembly of proteins into these complex structures containing nan- ometer-sized domains of varying chemical properties gives proteins, as well as many other biomolecules, their intriguing properties. Because these chemically di- verse regions can exhibit acidic, basic, hydrogen-bonding, hydrophilic, or hydropho- bic behavior, they can interact in exceedingly diverse ways with precursors for mineral compounds and the final mineral product. The inorganic/organic composite materials formed by these exceedingly complex biomolecular structures are the primary focus of the rest of this section. + +However, before discussing inorganic/organic nanocomposites, a short discussion of at least one completely organic nanocomposite with outstanding properties, namely spider silk, is in order. Dragline spider silk, which makes up the spokes of a spider web, is five times tougher than steel by weight and can stretch $30\,\%\!-\!40\,\%$ without breaking. However, to be fair to steel, I should point out that, although the work- to-failure is greater for silk than steel, the elastic modulus of silk is significantly less than that of steel. For applications in which flexibility and toughness are the pri- mary need, a synthetic route to creating a material with properties equivalent to spider silk would be exceedingly valuable. The strong core of spider silk is composed primar- ily of two protein components that self-assemble into crystalline and amorphous re- gions. The crystalline regions consist of alternating alanine-rich crystalline forming blocks, which impart hardness [24–26], and glycine-rich amorphous blocks, which are believed to provide elasticity [27]. This nanocomposite structure of organic nano- crystals in an amorphous organic matrix generates the exceptional properties of spider silk. Spiders cannot be kept in close quarters and harvested, because they eat one another; thus the only route to creating quantities of spider silk sufficient for applica- tion will be synthetic. + +Knowledge of the molecular structure of spider silk is not sufficient for creating a synthetic material with its properties. Inside the spider, the silk precursor exists as a lyotropic liquid crystal that is approximately $50\,\%$ silk. As the silk is excreted, the pro- tein molecules that make up silk fold and are aligned as they approach and then pass through the spinneret, forming a complex insoluble nanostructured fiber. In fact, not only is the spider silk made up of crystalline and amorphous regions, but the crystal- line regions are in turn composed of both highly oriented crystals and less oriented, but still crystalline, regions [28, 29]. These poorly oriented crystalline regions may serve to mechanically couple the highly oriented crystalline structure with the amor- phous domains. This three-phase morphology presumably has properties exceeding those of a similar two-phase system because of better coupling between the phases, due to the smaller mismatch in mechanical properties at the interface between the various domains. + +Recently, a synthetic spider silk-like material with properties approaching those of natural spider silk has been created. Because of their high complexity, the silk precur- sors could not be created by conventional organic synthesis and thus were created by expressing two of the dragline silk genes in mammalian cells [30]. The resulting so- luble recombinant dragline silk proteins were wet-spun into fibers of diameters ran- ging from 10 to 40 micrometers. After a postspinning draw, fibers with mechanical properties approaching those of natural spider silk were obtained. The fibers were highly birefringent, indicating a high degree of chain orientation, but it is not yet known if the internal nanostructure of the fibers was similar to that of natural spider silk. Work is now ongoing to build up a herd of transgenic goats that express spider silk proteins and excrete them in their milk, which should result in the production of sufficient silk protein for manufacturing. + +Much like the nanostructure of natural silk, which self-organizes in the biological organism, natural organic/inorganic nanocomposites are also formed through self- assembly. The two extremes of the mechanism for formation of natural nanocompo- sites are when the organic matrix forms first, followed by mineralization, and when the organic and inorganic materials coassemble into the nanostructured composite. There do not appear to be any examples in which the inorganic structure forms first, followed by organic structure formation. Most biological composites appear to form through the first route; a process whereby an organic structure first forms, followed by the biolo- gically directed nucleation and growth of a mineral phase. However, it is also clear that the organic matrix restructures and reorganizes continuously as the mineral deposits, which resembles the second mechanism. With both mechanisms, the organic material is generally composed of an amazing diversity of macromolecules, including proteins and other biopolymers. Considerable effort has been made to understand the mechan- ism of formation of these composite materials, a process that requires an understand- ing of both the structure and function of the organic matrix and the interactions be- tween the organic matrix and the inorganic precursors and product. + +The simplest examples of biological nanocomposites are those in which the mineral phase is simply deposited onto or within an organic structure. The next level of com- plexity is shown by examples in which the structure of the mineral phase is clearly determined by the organic matrix. The greatest complexity is where the mineral is intimately associated with the organic phase to create a structure with properties that are superior to those of either the mineral or organic phase. In the first and sec- ond examples, the organic phase forms first, but in the third, the organic and inorganic phases almost certainly coassemble into the final structure. + +# 3.2.1 Biologically Synthesized Nanoparticles + +For the simplest example of a biological nanocomposite, one needs to look no farther than the grasses. Many species of grass precipitate $\mathrm{SiO}_{2}$ nanoparticles within their cellular structures. Perhaps due to some internal structure in their cells, the silica nanoparticles generated within the cells can be found in sheet-like, globular, or rod-like morphologies, with characteristic dimensions ranging from a few nan- ometers to tens of nanometers (Figure 3.1 ) [31]. The exact reason for the silica deposi- tion is not known, although it may simply be a way for the plant to sequester silica, which it continuously takes in as dissolved silicic acid and other silica-containing so- luble species. If the plant could not sequester silica, it would build up and potentially limit growth. At high levels, silica may make the plant undesirable as food [32], for example, the dry weight of horsetails consists of $20\,\%\!-\!25\,\%$ silica. + +Another example of a biologically simple organism that can generate internal na- nocomposite structures is bacteria, the most famous of which are magnetic bacteria, which contain an internal chain of magnetite $(\mathrm{Fe}_{3}\mathrm{O}_{4})$ ) nanocrystals running down their long axis (Figure 3.2). These nanocrystals are about $40\;\mathrm{nm}$ in diameter and give the bacteria a net magnetic moment, which aligns them with respect to the earth’s mag- netic field. The presence of this magnetic chain of nanocrystals plays a role in defining + +![](images/4bcd9ebe22b4370cdc87ae8b0ed874a429b298961803d13625fe7d23d23d58a9.jpg) +Fig. 3.1 Transmission electron micrograph of rod-like amorphous silica extracted from a grass. The rod-like nature is due at least in part to the local environment inside the plant cells. Adapted from [31] + +the direction of locomotion. Because an individual bacterium, on average, points in a specific direction, as it moves it continuously explores fresh territory. In contrast, the movement of a bacterium lacking a directing force follows a random walk, resulting in a substantially lower probability for movement into fresh territory. Besides the ${\mathrm{Fe}}_{3}\mathrm{O}_{4}.$ - containing bacteria, other bacteria deposit chains of nanoparticles of the magnetic mineral greigite $(\mathrm{Fe}_{3}\mathrm{S}_{4})$ within themselves. This mineral chain appears to serve the same function as the ${\mathrm{Fe}}_{3}\mathrm{O}_{4}$ chain in the previous bacteria. + +# 3.2.2 Biologically Synthesized Nanostructures + +Biological organisms commonly synthesize nanostructured and nanocomposite ma- terials of much greater complexity than nanoparticles. These structures can come in many forms, ranging from needles and plates to complex 3D nanostructures, and can exhibit many unique and interesting mechanical, chemical, and optical properties. Although simplifying all biological nanostructure formation routes into a single me- chanism is not possible, several general synthetic tenets tend to hold true. The first is that the biological structure forms first, and then the inorganic phase begins to form. Although truly cooperative assembly is possible, it is much more common that the organic matrix forms first, regulating the growth of the inorganic material. We should keep in mind, however, that the organic matrix is plastic and can deform or reshape in response to the growing inorganic material. + +Besides forming nanoparticles, bacteria can form more complex nanostructures. Bacteria in general are responsible for a vast amount of mineral deposition and in fact can contribute greatly to mineral deposits on the bottoms of lakes and other aqua- tic environments. Mineral formation in the vicinity of bacteria is generally driven by a biologically induced change in the pH or ionic strength around individual bacteria. This change in the local environment reduces the solubility of certain mineral com- pounds, leading to precipitation of minerals. Generally, this results in an unfeatured mineral, so is not of interest here. However, in specific cases, the structure of the precipitated mineral is highly regulated by proteins on the surface of the bacteria, resulting in the formation of complex nanostructures. Specifically, some bacteria have on their exterior a layer of proteins called an S-layer. As these bacteria re- spire, the local pH and ionic strength change, driving, for example, the mineralization of gypsum $(\mathsf{C a S O}_{4})$ ). Rather than being deposited as an unfeatured solid on the exterior of the bacteria, the growing gypsum is templated by the organic S-layer into a complex nanostructured form that very closely replicates the structure of the protein S-layer (Figure 3.3) [33, 34]. The exact mechanism of the mineral templating is not known, but clearly preferred sites must exist on the S-layer for heterogeneous nuclea- tion of mineral, or a regular structure would not form. As these mineral nuclei grow, they impinge on one another, forming a thin, periodically structured film of gypsum. One reason for the complexity of this structure is that the resulting pores in the in- organic layer may allow the bacterium to exchange nutrients with its surroundings even after the mineral phase forms. If the gypsum had formed as a dense solid film, it would rapidly stop the bacterium’s respiration, thus killing it. + +![](images/1fe30c4a373ed49ae29cb84cc2b2498a210753900c36755fa8be28b60364f68f.jpg) +Fig. 3.3 Transmission electron micrograph of (left) bacterial S-layer, which serves as a protein template for the formation of (right) a thin film of me- sostructured gypsum. Adapted from [33] + +Although this specific S-layer templated structure as yet has no known applications, inorganic nanostructures of these length scales are of great interest for many high- technology applications. Patterning on these length scales is exceedingly difficult, so if bacteria could be genetically engineered to express proteins that result in S-layers having technologically significant nanostructures, there would be the potential to bio- mineralize materials for advanced applications. In fact, initial strides have been made in this regard and are covered later in this chapter. + +Higher organisms also generate inorganic/organic composite structures. An excel- lent example is the sea urchin. The sea urchin spine is a single crystal essentially composed of calcite, containing only about $0.02\,\%$ glycoproteins trapped within the crystal lattice of the spine [35]. However, this very small amount of organic matter leads to the extraordinarily high toughness and unique fracture properties of the spine. The spine is much tougher than a synthetic equivalent composed of polycrystal- line or single-crystalline calcite. Most surprising is the fracture mode of the spine. Rather than fracturing along a specific crystallographic direction, as occurs for most single crystals, it fractures with conchoidal cleavage, that is, with a smooth rounded fracture surface, similar to that observed for glassy materials (Figure 3.4 ) . The fact that it is a single crystal but does not fracture along any specific crystallo- graphic direction is quite unlike almost all other crystalline ceramic materials. The mechanism for the toughening is still unknown but is clearly due at least in part to the nanoscale dispersion of protein molecules in the crystal lattice. + +![](images/b029941b346c57c4ae97f1984c836a5d0d24bff765be6c06f7b7482fdd6ca6d5.jpg) +Fig. 3.4 Scanning electron micrograph of the fracture surface of a single-crystal spine from a sea urchin, demon- strating conchoidal cleavage. Adapted from [35] + +![](images/77b80a0ced7e8af450aa63c29859c19759641c7dbf7ff4e593d0141c0bfbd6f0.jpg) +Fig. 3.5 Scanning electron micrograph of a fracture surface of the nacre of the red abalone, showing a side view of the aragonite tablets. Adapted from [162] + +Another sea creature that creates an organic/inorganic composite that is vital to the creature’s survival is the abalone, as already mentioned. The nacreous (mother-of- pearl) layer of the abalone shell consists of alternating layers of 500-nm-thick arago- nite platelets and ${\sim}30$ -nm-thick sheets of an organic matrix ( Figure 3.5 ) . The matrix forms first, then the mineral plates develop between the layers. The matrix defines the thickness of the crystals, and certainly also adds to the strength of the composite struc- ture by forcing fractures to dissipate their energy as they travel a tortuous path through this elastic structure. The matrix also controls the crystallographic orientation of the mineral plates, which must also be important for strength. The resulting composite structure has a fracture toughness that is about 3000 times greater than that of inor- ganic aragonite [32]. Along with defining the size and shape of the aragonite plates, the organic matrix also likely serves to define their crystallographic orientation. The me- chanism for formation of nacreous aragonite is not entirely clear, although it is known that the organic matrix forms first into a hollow layered structure, and then the voids in this structure are mineralized. + +Another excellent example of a biological nanocomposite with complex structure and function is bone. Bone must perform multiple functions. It must have high strength, yet low weight; it must support remodeling in response to applied stres- ses, yet not deform under applied stress; it must contain pores to allow oxygen and nutrients to reach the cells within the bone, yet these same pores cannot lead to fractures; finally, it must act as a reservoir for minerals, but not allow itself to de- mineralize and thus weaken. Bone exists in several forms; lamellar bone provides the structural properties of long bones in mammals and other species and is the subject of the following discussion, adapted from [36, 37] (Figure 3.6). The basic structure of bone is a mineralized collagen fibril consisting of on average $65\,\%$ mineral, the re- mainder being organic material and water. The collagen phase of bone is composed of individual fibrils about $80\!-\!100\;\mathrm{nm}$ in diameter. These fibrils are in turn composed of assemblies of a triple helix of polypeptide chains with an average diameter of $1.5\;\mathrm{nm}$ and length around $300\,\mathrm{nm}$ . The internal arrangement of this triple helix structure in the collagen fibril is not random: the helix aligns with its long axis parallel to the long axis of the bone and has additional fine structure. The mineral phase of the bone, + +![](images/f1b04b18d45ad4053d272e5c309ef18adc3c34e2e3beb334bedf9351d496d851.jpg) +Fig. 3.6 The 7 hierarchical levels of organization of the bone family of ma- terials, according to Weiner [37]. Level 1: Transmission electron micrographs of individual mineral crystals from human bone (left side) and an unmineralized, unstained collagen fibril from turkey tendon embedded in vitreous ice (right side). Level 2: Transmission electron micrograph of a mineralized collagen fibril from turkey tendon. Level 3: Transmission electron micrograph of a thin section of mineralized turkey tendon composed of multiple fibrils. Level 4: Four fibril packing motifs found in the bone. Level 5: Scanning electron micrograph of a single osteon from human bone. Level 6: Light micrograph of a fractured section through a 5500 year old fossilized human femur. Level 7: Whole bovine bone (scale: $10\,\mathsf{c m}$ ). Adapted from [37] + +carbonated apatite $(\mathrm{Ca}_{5}(\mathrm{PO}_{4},\mathrm{CO}_{3})_{3}(\mathrm{OH}))$ , forms as 50-nm-long, 25-nm-wide platelets that are on average $1.5\,{-}\,4\;\mathrm{nm}$ thick (Figure 3.7). These platelets lie within the collagen fibrils, with only a small number of polypeptide chains between each platelet. The long axis of the platelets is generally parallel to the outer surface of the bone, most likely to maximize the compressive strength of the bone. + +The mineral structure forms in bone after the collagen fibrils form. However, as the mineral forms, the collagen structure may reorganize to some extent. Thermodyna- mically, the carbonated apatite platelets should coarsen into larger single crystals over time (Ostwald ripening), but clearly this is prevented by the collagen matrix. Thus, although the collagen matrix reorganizes to some extent, it clearly cannot be comple- tely pushed out of the way by the growing mineral. This level of control by biology over structural development during mineralization currently exceeds that found in syn- thetic materials. Additionally, after a bone is broken, new structures form which gen- erate material that is as strong as the original bone; yet, the mineralization process stops when the healing process is complete. Bone thus is an excellent example of a self-healing biological nanocomposite in which an organic host phase is formed, followed by highly regulated mineralization processes. The mechanism for this pro- cess is not entirely clear, but substantial strides in understanding it are being made. + +Clearly, biological systems can create a tremendous diversity of inorganic materials; however, biological systems are also very limited in terms of material selection. Metals, as well as many polymers, semiconductors, and ceramics cannot be synthesized by biological processes, because the precursors are not available in the natural environ- ment, the materials or precursors are toxic, and/or the product is not stable in the + +![](images/d3dfdfb4723b6304cbe4c7e165df062a2767e3608f0d576d471cbb7a039d0a9c.jpg) +Fig. 3.7 Transmission electron micrograph of a fractured piece of mineralized bone from a 50-year-old human male femur. (a) Many mineral crystals are oriented edge-on and are parallel to the long direction of the collagen fibrils. (b) Higher magnification view of the bone fragment at the location in (a) marked with an arrowhead. Here, crystals are oriented face-on. The flat, plate-like nature of the mineral crystals in bone can be observed in these figures. Adapted from [163] + +presence of water and/or oxygen. Because of this, there is serious interest in using elements of biological systems to create synthetic nanocomposites from materials not found in biology, thus achieving the best of both the biological and synthetic worlds. + +# 3.3 Biologically Derived Synthetic Nanocomposites + +Because of the limited variety of materials available by purely biological routes, there is great interest in utilizing the sophistication offered by biological systems in concert with synthetic procedures to create materials with otherwise unobtainable nanostruc- tures and thus, potentially, unique properties. Biology offers a unique selection of building blocks that would be difficult or impossible to synthesize in the labora- tory. These include proteins, DNA, RNA, and small but highly functional molecules. + +# 3.3.1 Protein-Based Nanostructure Formation + +As already mentioned, protein S-layers on the surface of bacteria create complex na- nostructures. The protein S-layer present on the surface of some bacteria has now also been used to create entirely synthetic nanostructures. The diversity of S-layer struc- tures, coupled with the potential for chemical function aliz ation, make them ideal start- ing points for nanostructure synthesis. To review, S-layers are 2D protein crystals that have oblique, square, or hexagonal lattice symmetry with lattice constants between 3 and $30\;\mathrm{nm}$ . Interesting for potential nanostructure formation, and to enable bacterial respiration, S-layers almost universally contain pores of identical size and identical surface chemistry – properties that make S-layers useful for nanostructure and na- nocomposite fabrication. In an early example [38, 39], an S-layer was used to template a periodic structure into a thin metal film. Specifically, a 1-nm-thick metal $(\mathrm{Ta}/\mathrm{W})$ film containing 15-nm holes periodically arranged in a triangular lattice with a lattice con- stant of $22\;\mathrm{nm}$ was created by S-layer templating. This was accomplished by first de- positing a suspension of S-layer fragments onto an amorphous carbon support film. After S-layer deposition, a 1.2-nm thick film of Ta/W was evaporated onto the S-layer at an angle of $40^{\circ}$ from the normal of the substrate surface. When examined in the trans- mission electron microscope (TEM), the resulting film shows contrast indicating some thickness variation in the metal, but the contrast is not significant. To improve the contrast and to open up holes in the metal film, the film was argon-ion-milled for a short time. The result was a metal film that contained a periodic array of holes with the same symmetry as in the S-layer template [38, 39]. This work was done more than 15 years ago, yet creating such a nanostructure today would still be a chal- lenge, even with a modern e-beam lithography system. + +Much more recently [13], an S-layer was used to create nanostructured semiconduc- tor films. The S-layer in this study had an oblique 2D lattice (space group $p_{1}$ ; $a=9.8$ nm, $b\,{=}\,7.5\,\mathrm{nm}$ , $\theta\!=\!80^{\circ}$ ). Depending on the mechanism of S-layer deposition, either its negatively charged inner face or its charge-neutral outer face was exposed to a $10\,\mathrm{mM}$ $\mathrm{{Cdd}}_{2}$ solution. After drying, the cadmium-ion-doped S-layer was exposed to $\mathrm{H}_{2}\mathrm{S}$ gas, resulting in formation of a nanocrystalline film of the semiconductor CdS on the S- layer. In the TEM, the CdS film exhibited a superlattice structure that was a direct copy of the structure of the S-layer [13]. The S-layer may have survived the entire process, and thus the actual mineral structure formed may be a nanocomposite of CdS and protein. It is still unclear what applications might be found for such materials, although robust metal or semiconductor nanoporous structures may find application as filter membranes, sensors, and optoelectronic devices. + +Another protein assembly that has been used to form nanocomposite materials is ferritin (Figure 3.8 ) . In its native form, ferritin consists of a supramolecular arrange- ment of proteins around an iron oxide core. The iron oxide core can be selectively dissolved without damaging the structure of the protein shell, yielding a hollow ball of protein about $10\,\mathrm{nm}$ in diameter. The demineralized protein (apoferritin) can be refilled with iron oxide, demonstrating that its structure was not greatly af- fected by the de mineral iz ation process. Interestingly, apoferritin can also be filled with other mineral nanoparticles. In one example [32], the iron oxide core of ferritin was converted to FeS by treating the ferritin with $\mathrm{H}_{2}\mathrm{S}$ . Nanoparticles of MnOOH, $\mathrm{{UO}}_{3}$ , ${\mathrm{Fe}}_{3}\mathrm{O}_{4}$ , and CdS can also be formed inside apoferritin through the appropriate chemi- + +![](images/6e4b195d581f57cf7316970968217c68292b88abc1f92b8c68eeacf454c6386e.jpg) +Fig. 3.8 Transmission electron micrographs of magnetite $(\mathsf{F}\mathsf{e}_{3}\mathsf{O}_{4})$ - and maghemite ( c - $\mathsf{F e}_{2}\mathsf{O}_{3})$ -filled ferritins. (a) 260 Fe atoms/molecule, unstained; only the discrete electron-dense inorganic cores can be seen. (b) 260 Fe atoms/molecule, after staining with uranyl acetate, showing encapsulation of inor- ganic cores by intact protein shell (white halo around each particle). (c) 530 Fe atoms/mo- lecule, unstained. (d) 1000 Fe atoms/molecule, unstained. (e) 2040 Fe atoms/molecule, un- stained. (f) 3150 Fe atoms/mo- lecule, unstained. Scale bars in all figures $=50\ \mathsf{n m}$ . Adapted from [164] + +cal treatment. Because the protein shell is not disrupted by the mineralization process, the final product is truly a nanocomposite of protein and inorganic material. See [32] for an extended discussion of the use of apoferritin as a template for synthesizing nanocomposite materials. + +# 3.3.2 DNA-Templated Nanostructure Formation + +DNA offers great potential as a building block for nanocomposite materials. It can be tethered to a wide range of substrates, can direct assembly with specificities that greatly exceed that of any synthetic molecule, is relatively robust, can be synthesized in re- latively large quantities, and can be functionalized with tags such as fluorescent mo- lecules to enable rapid detection of binding events. + +The use of DNA to assemble and create nanostructures and nanocomposite materi- als is only in its infancy, but even the preliminary work done so far indicates the great potential of DNA-based assembly techniques. A few of the approaches that have been studied to date include the mineralization of DNA, the use of DNA to assemble na- noparticles, and the use of DNA to assemble much larger colloidal particles. These three approaches are outlined in this chapter; however, the number of possibilities is vast, and significant work on DNA-mediated assembly of nanostructures is cur- rently being done. + +The possibility of using plasmid DNA as a template for mineralization was first explored in 1996 [40]. In this work, single strands of a 3455-basepair circular plasmid DNA were mineralized with CdS nanoparticles, by mixing the plasmid and cadmium perchlorate in solution, followed by spin coating this solution onto a polylysine-coated glass slide. The DNA/cadmium perchlorate-coated glass slide was then exposed to $\mathrm{H}_{2}\boldsymbol{S}$ , converting the cadmium perchlorate to CdS, which preferentially mineralized the DNA. This procedure results in ring-like structures consisting of DNA embedded within a 5–10-nm-thick CdS strand that can be directly observed in the TEM (Figure 3.9). The diameter of the ring formed by the CdS-encrusted DNA is directly related to the diameter of the DNA plasmid, and the thickness of the CdS/DNA composite strand is about $10\;\mathrm{nm}$ . This represents the first example of DNA templating of semi- + +![](images/2994a52fb3a947d5aab4d9edee4e763fc132966dba947ab1f6c9618129173814.jpg) +Fig. 3.9 Transmission electron micrograph of a nanostructure obtained by mineralization of a cir- cular plasmid DNA with CdS nanoparticles. The ring diameter closely matches the predicted dia- meter of the plasmid DNA. Adapted from [40] + +conductors but, given the vast array of structures that can be formed from DNA, should not be the last. + +Another example of DNA-based nanostructure development can be seen in the work on DNA-based nanoparticle assembly. Several approaches have been explored, with the majority based on the Watson–Crick base-pairing of DNA strands attached to various nanoparticles. The first approach involved the function aliz ation of gold nano- particles with multiple strands of thiol-terminated type-A or -C DNA in separate reac- tions [41] (where ‘type’ is used as a label for different nucleotide sequences). The thiol terminal group is important, because it enables covalent attachment of the DNA strand to the gold nanoparticles. Importantly, the A and C strands were designed so that they would not hybridize. The result, after mixing the solutions of each kind of DNA-func- tionalized nanoparticles, is a simple mixture of DNA-functionalized gold of type A and DNA-functionalized gold of type C. Then a single strand of DNA with one end com- plimentary to the type-A DNA and the other end complimentary to the type-C DNA was added. We label this strand B-D. Upon hybridization of the B-D strand with both nanoparticles of type A and nanoparticles of type C, a DNA-linked aggregate of na- noparticles was formed. Aggregate formation is thermally reversible by increasing the temperature above the melting point of the DNA, at which point the strands de- hybridize. The assembly could be monitored by UV/visible spectroscopy, because the UV/visible absorption of gold nanoparticles changes greatly when they are close to- gether, due to changes in the surface plasmon. + +The formation of controlled aggregates, although interesting, is really a starting point for the creation of much more complex structure through a marriage of biolo- gical macromolecules and synthetic materials. In addition to placing multiple DNA molecules on a nanoparticle, it is possible to place a single or a finite number of DNA molecules on a nanoparticle. However, this requires a significant degree of syn- thetic effort, because one usually starts with a population of nanoparticles with various numbers of attached DNA molecules. Through careful separation, nanoparticles with exactly one or exactly two single-stranded DNA molecules on their surface can be recovered. These DNA-functionalized particles can then be assembled in highly con- trolled fashions through DNA hybridization reactions, by adding single-stranded DNA that is complimentary to the single-stranded DNA on the nanoparticles ( Figure 3.10 ) . This is similar to the approach used to make DNA-linked aggregates; however, because each nanoparticle contains only one DNA strand, not multiple strands, the result after hybridization is not the formation of an aggregated structure, but rather, individual dimers of gold nanoparticles. Trimers of gold nanoparticles were also created by a similar approach, except that nanoparticles with one strand of DNA of three different sequences was mixed with a strand of DNA complimentary to all three of the DNA strands on the surface of the nanoparticles [42, 43]. + +Recently, it was demonstrated that DNA can assemble much larger colloidal parti- cles into 3D assemblies of controlled shape. The basic approach is very similar to the assembly of nanoparticles; however, here the results after hybridization are assemblies of colloidal particles with defined connectivity [44]. + +The overall power of DNA-based assembly of nanostructures is based on several powerful aspects of DNA hybridization. First, hybridization is reversible with tempera- + +![](images/0d7c13ec1d388828369efd31faf3696feb3643b3662ce68780acfdf8bcac99b6.jpg) +Fig. 3.10 DNA allows for precise manipulation of backbone. See examples A, B, and C on the left and the order and spatial arrangement of nanoparticles. the resulting nanostructures on the right. Adapted The large and small nanoparticles can be designed from [43] to hybridize or react with a single site on the DNA + +ture; thus it is possible to anneal structures that at least partially eliminate kinetic traps that may be present in the assembly process. For example, if during nanostructure assembly, a particle does not hybridize in the correct location, the structure can be gently heated, releasing the particle, and then the particle can be retrapped by cooling below the melting point of the DNA. Second, DNA hybridization is very specific. Un- like most organic linking chemistries, which cannot distinguish small changes in molecular structure, DNA can be designed to hybridize only to its exact complimen- tary strand. Thus it may be possible to combine many DNA-functionalized nanopar- ticles into very specific arrays by hybridizing to a long strand of DNA that is encoded so as to react with each DNA functionalized nanoparticle at only a very specific location. + +# 3.3.3 Protein Assembly + +To achieve a level of complexity similar to what can be obtained by DNA-based assem- bly, one can engineer or select biological organisms that recognize and bind with high specificity to specific minerals, semiconductors, and metals. In the first example [45], Escherichia coli containing genetic sequences specific for recognizing and binding iron oxide but not other metal surfaces was identified and multiplied. Bacteria that would specifically adhere to iron oxide were identified via serial enrichment from a popula- tion of bacteria. The experimental procedure was as follows: A population of geneti- cally diverse E. coli was exposed to iron oxide particles. The bacteria that bound to the iron oxide were collected, and the remainder were discarded. After repeating this pro- cedure for several generations, bacteria with high degrees of specificity for iron oxide were obtained. The reason only some of the bacteria bind iron oxide is that E. coli , like many other bacteria, express proteins on their outer surface whose sequences are a function of the unique genetic makeup of the bacteria. By starting with a library of ${\sim}10^{7}\;E.$ . coli that express surface proteins of slightly different sequences, the probabil- ity that at least one of the E. coli has a highly specific protein is reasonable. Through enrichment, it is possible to extract and multiply just those bacteria that bind to the surface of interest. + +It is also possible to select for protein sequences that attach to specific metal sur- faces, and presumably also to metal nanoparticles. Through such biological experi- ments, a library of ${\sim}10^{7}$ different polypeptides 14 or 28 amino acids long was cre- ated. Then, the polypeptides that adhere to the metal surface were isolated, and their sequences were determined [46]. This approach is exceedingly powerful because of the vast number of different molecular sequences that can be studied in parallel in a single experiment. + +From a technological standpoint, it is interesting to consider the possibility of re- cognizing semiconductor surfaces and semiconductor nanoparticles with this proce- dure. If an appropriate genetic sequence can be identified, it would be possible to assemble nanoparticles via biological organisms that are designed to present the ap- propriate biological macromolecules on their surface. Phage display enables the simul- taneous testing of many peptide sequences for specificity to a given surface. A phage library containing ${10}^{9}$ polypeptide sequences was exposed to a surface, and all the phages that did not bind to the surface were washed away. The phages that stuck were then removed from the surface by lowering the pH and were amplified by in- fecting E. coli bacteria with them. This process was repeated until only phages that stuck strongly to the surface were present. The DNA of these phages was then se- quenced to determine the specific peptide sequence(s) that bind with such high affi- nities to the specific surface of interest [47]. + +A phage that identifies and binds to a surface of a specific material also binds strongly to nanoparticles of this material. The result of mixing a nanoparticle-binding phage with nanoparticles is ‘decoration’ of the phage with nanoparticles, which occurs only at the end of the phage that contains the specific binding polypeptide, and no- where else on the phage [48]. Depending on the experimental conditions and phage design, it may even be possible to bind single nanoparticles to an individual phage or to assemble the nanoparticles into defined structures, although in the present state of the art, multiple nanoparticle are bound in a fairly random fashion to each phage. + +# 3.4 Biologically Inspired Nanocomposites + +The properties of biocomposites and synthetic pathways for their formation have in- spired wide ranging research. However, early on it was recognized that it is not always necessary or even desirable to use biologically derived materials for many applications and that it may be possible simply to use biology as an inspiration for totally synthetic nanocomposite systems. It is interesting to consider biological systems as an inspira- tion for nanocomposite materials, because biological systems exhibit many character- istics that would be attractive in synthetic materials, but it is also clear that direct mi- micking of biology will be limited to a specific small subset of materials and to specific nanostructures. However, many lessons can be learned from biology on how to form complicated nanostructures and on the potential properties of these synthetic nanos- tructured materials, should one be successful in synthesizing them. Of course, just because a synthetic material resembles a natural process and the process of forming that material resembles a natural synthesis does not always mean that the scientists and engineers who performed the work were inspired by biology. Often it is not stated whether biology was an inspiration for the work, and thus some care must be taken before assuming that, just because something appears biologically inspired, it is bio- logically inspired. In this chapter, I do not attempt to make this distinction and by necessity assume that, if the work has a biological analog, then it can be considered ‘biologically inspired’, although certainly in some cases this assumption may be wrong. In this chapter I take the liberty to lump such work under the heading of ‘Biologically Inspired Nanocomposites’. + +Much can be learned from biological systems to further the development of syn- thetic approaches to the formation of complex inorganic structures. Each of the routes to nanostructure formation that is discussed in this section – liquid crystal templating, colloidal particle templating, block copolymer templating, and surfactant-inorganic self-assembly (mesoporous silica being the most famous of this approach to nanos- tructure formation) – invokes many of the tenets of biologically directed mineral growth. As already discussed, biological systems rely on self assembly and mineraliza- tion in the synthesis of hard inorganic structures such as shells, teeth, and bone [11], and their approaches to materials fabrication can provide guidance and direction to synthetic systems. Often, the term ‘biomimetic’ has been applied to any approach using self assembly in the synthesis of nanocomposite materials. As mentioned above, biological systems do indeed use self-assembling molecules, and high levels of molecular organization are a very important part of an organism’s inorganic struc- ture development. However, it would be naive to expect to simulate this process in the laboratory except on the most basic level. Biological processes are extremely dynamic, involving huge numbers of very specific proteins and other molecules being generated and transported to very specific locations, with temporal control. The best synthetic systems are indeed very simple approximations of living systems and generally are much too simple to be considered to be truly mimicking biology. Although the syn- thetic systems are only simple approximations of life, still much can be learned by attempting to mimic living systems [8], even if biomimetic is perhaps not the best term. For example, many attempts centered around the synthesis of mineral phases in a self-organized matrix have indeed mimicked the mineralization processes of many biological systems. Examples of matrix-mediated biological mineralization processes include the reliance of bacteria, plants, shells, and even mammals on organically mediated growth of mineral phases to eliminate byproducts (bacteria and plants) [31, 33, 34], create exoskeletons (shells) [49], and grow teeth (mammals) [35]. If during the synthetic process, organic molecules are incorporated into the mineral phase, the resulting material may resemble the spicule of a sea urchin, which contains much less than $1\%$ protein intercalated into the crystal lattice [35]. + +Despite the low degree of sophistication of synthetic systems, they do have very distinct advantages over biologically based schemes. Biological systems operate with only a limited subset of elements and compounds, but synthetic systems can be designed to use a wider range of elements and compounds, many of which would be toxic to most living organisms. Biological systems form and operate near room temperature in the presence of water and oxygen (for the most part), but synthetic systems can be formed and operate under a wide range of temperature and condi- tions. Finally, biological materials generally take days to years to form, but synthetic systems may be formed rapidly. For biology, systems that form slowly in response to external stimuli have significant advantages, for example, bone remodels to meet the demands of applied loads. However, in general, the long time scales required for for- mation of biological nanocomposites limit the application of direct biological synthesis of engineering materials to a few very specific cases. + +Through self-assembly of organic molecules, biological systems have succeeded in synthesizing a wide range of composite and inorganic nanostructures. Biological sys- tems contain large quantities of lipids, or soap-like molecules, which self-assemble (along with many other biological macromolecules, including proteins) to form the external membranes of cells, as well as smaller vesicles within the cell. These mem- branes serve to protect the interior contents, as well as to provide synthetic microreac- tors for biological processes. The concept of performing syntheses in self-assembled microreactors is in this sense biologically inspired, has been exploited by many re- searchers to create nanoparticles and composite nanostructures, and is the topic of this section. + +The simplest examples of nanomaterials are zero-dimensional materials, known as ‘nanoparticles’. The study of small semiconductor particles in general is of significant interest, because the electronic and optical properties of semiconductor nanoparticles change drastically as the characteristic dimension of a particle is reduced to the na- noscale regime, largely due to the quantum confinement of electrons within the par- ticle, although surface-area effects may also play a role (Figures 3.11, 3.12) [50–52]. The most popular nanoparticles for basic studies have been II–VI semiconductor particles, because of their scientifically interesting and technologically important prop- erties. In this context, synthetic methodologies for the formation of metal sulfide and selenide quantum dots and their assembly into higher-order structures have been widely studied [53–57]. For nanostructuring to dominate the properties of metal sul- fides and other semiconductors, it is generally necessary for the characteristic dimen- sion of the nanoparticle to be ${<}10\;\mathrm{nm}$ [58]. Semiconducting nanoparticles are gener- ally synthesized through one of three routes. The first, grinding of large chunks is rarely done for nanoparticle preparation, because the grinding process is poorly regu- lated, generally generating a very polydisperse population of particles, and grinding introduces too many contaminates for most applications. The other two methods, gas-phase and solution-phase synthesis, are much more common. Gas-phase synth- esis is essentially a vaporization and condensation process – a crucible containing the desired semiconductor (or other material) is heated until it starts to sublime, and then an inert carrier gas is flowed over the material. The carrier gas then heads into a cool region where the gaseous semiconductor atoms or molecules condense into nanopar- ticles and are collected [59]. Although this method is fairly versatile, it operates only under conditions of high temperature and vacuum and generally produces solid sphe- rical particles. + +Solution-based synthetic routes for nanoparticle formation have ranged from simple precipitation reactions to much more complex self-assembly-based routes. In general, simple precipitation results in agglomerates of nanoparticles, and the size distribution generally varies widely. These problems led to research into synthetic procedures that would result in nanoparticles that are stable against aggregation and have narrow size distributions. A primary route to preventing both these problems is to use self-assem- bly-based techniques, which in many respects resemble nanostructure development in biological systems, including bio mineral iz ation, cell membrane development, and other biological structure formation. Solution-phase synthesis of semiconductors is often preferred over other techniques, because it is generally mild (even being carried out at room temperature and pressures) and can be used to create reasonable volumes of materials. Solution-phase synthesis has been widely used to grow semiconductor + +![](images/7138c1f04f5e3021e367d62fe49e3011f30adc38884ac726ccae3d17f685cdbf.jpg) +Fig. 3.11 Theoretical calcula- tion of band-gap energy as a function of particle diameter for several different semiconduc- tors. Adapted from [58] + +![](images/363110023b69c797fd843e2c906c232d82d15f3a99a06dfaa7a9ec0f60053089.jpg) +Fig. 3.12 $\mathsf{C d}_{3}\mathsf{P}_{2}$ quantum dots. Particle size increases in the direction of the arrows. The white particles are about $1.5~\mathsf{n m}$ , and the black particles are larger than $5~\mathsf{n m}$ . Adapted from [50] + +quantum dots, yielding particles with low poly disp ers i ties and novel optical properties [50, 51, 56, 57, 60]. Solution-based chemical syntheses are very attractive because they allow for direct control over the actual concentrations of the chemical precursors. De- pending on actual conditions, it is even possible to cap the surface of the particles with organic molecules, which allows for further solution-based processing [53, 55]. + +The more conventional route to creating nanostructured materials is of course through top-down lithographic methods. Some examples of top-down techniques for generating very small features are extreme UV $(\lambda<<200\ \mathrm{nm})$ ) lithography [61, 62], electron beam writing [63], focused ion-beam lithography [64], x-ray-lithography [65], scanning probe lithography [66], and microcontact printing [67]. In general, all these techniques can form structures on the scale of tens to hundreds of nanometers, although generally only on very flat substrates, and they can be quite slow and often very expensive. The self-assembly-based route to nanostructure formation has the sig- nificant advantage that it is not limited to feature generation on flat surfaces, and it can be massively parallel. Of course, the general problem with self-assembly is that it is not possible to highly regulate the exact spatial position of the nanostructure, and thus we are still many years away from creating highly functional self-assembled electronic circuits. + +In the micellar routes to nanoparticle formation, micelles are self-assembled from surfactant molecules and a solvent that contains at least one of the precursors for the inorganic nanoparticles in solution. The result is a solution that contains vast numbers of discrete nanoreactors which individually contain only a finite number of precursor species for the inorganic phase. When these ions are converted to mineral, generally by reduction or oxidation, the result can be one nanoparticle per micelle. The polydisper- sity in nanoparticle diameter is thus directly related to the polydispersity in the initial micelle size. If it were possible to create a suspension of monodisperse micelles, it would be a straightforward process to create nanoparticles with a very narrow size distribution. Taking a lesson from biology, this might be possible through the use of complex macromolecules that organize into particles of only a specific size. A very good biological example of a potential nanoreactor with a tight size distribution is a virus particle. It may be possible to use virus particles to synthesize nanoparticles with tight size distributions if a way to load the interior of the virus particles with precursors for nanoparticles can be developed. + +Most of the early nanoparticle synthesis routes produced solid semiconductor par- ticles with morphologies never far from spherical. The creation of nanoparticles with complex morphologies is most interesting, because even such sophisticated patterning techniques as e-beam lithography are limited to ${\sim}10{\cdot}\mathrm{nm}$ features, which are often too large to result in the desired quantum confinement and other properties [63]. Nano- particles having certain complex morphologies cannot be created through e-beam pat- terning or other top-down processing routes. The possibility of generating complicated + +![](images/217812f646cd37e735be658d0a6968e0b33c5fb00622332f8bb341e8d0efecdc.jpg) +Fig. 3.13 Transmission electron micrograph of nanocrystalline CdS structure grown under an arachidic acid monolayer at room temperature. Scale bar $=200$ nm. Adapted from [165] + +![](images/369ba91fdfbe57478294506f44cf96ed2b6355cc60eaedcfbedec2db6e8b7207.jpg) +Fig. 3.14 Examples of CdSe nanoparticles with complex shape and form that can be created by solution synthesis from a mixture of surface passivating agents. Adapted from [70] + +morphologies was examined in studies of CdS synthesized under Langmuir mono- layers, in which dendritic structures were generated (Figure 3.13 ) [68]. Subse- quently, a range of strategies have resulted in rod-like and even complex nanoparticles with complex morphologies that are based on self-assembly regulating the growth of semiconductor particles. + +As examples of self-assembly processes which create nanoparticles with complex shapes, Alivisatos and coworkers [69, 70] demonstrated the formation of CdSe nanor- ods with aspect ratios of 30:1, as well as arrow-, teardrop-, tetrapod-, and branched tetrapod-shaped nanocrystals of CdSe (Figure 3.14). These highly shaped nanoparti- cles result from using a mixture of hexylphosphonic acid and trio c tyl phosphine oxide as passivating agents in the synthesis. Apparently, it is possible to block the growth of specific crystallographic faces while encouraging the growth of other faces. Thus, par- ticles that have one crystallographic direction as their long axis, and their short axis as another crystallographic direction can be formed. The advantage of high aspect ratio particles over normal, spherical particles has not yet been proven, but the electronic or physical properties of the particles might be impacted by the change in shape. In addition, the particles might be able to self-assemble into higher-order structures be- cause of their high aspect ratios; for example, they may form liquid crystalline phases, much like molecules with high aspect ratios [69, 70]. + +Semiconductor nanostructures may also have potential nonquantum confinement- based properties not found in the bulk equivalent. Besides quantum confinement, one way to greatly modify the properties of nanostructured semiconductors may be to design synthetic methodologies that result in dispersion of organic molecules within the inorganic phase at the molecular level. These composite materials could exhibit novel properties significantly enhanced over those of either the inorganic or organic phase alone, as has actually been observed in a wide range of materials. Composite materials that are tougher [71–73], have increased thermal stability [74], are electro- nically more sophisticated [75,76], or have enhanced chemical selectivity [77] than either of the constituent parts have been created. Even without the incorporation of organic material, periodically nanostructured semiconductors have great potential in solid-state science and technology, for example because of their potential for both electronic and catalytic activity. For example, a periodically nanostructured semicon- ductor might behave as an array of antidots (a material with a regular array of scattering centers spaced closer than the mean free path of electrons traveling through them) [78, 79]. A nanostructural material is necessary because, if the mean free path of the elec- trons is shorter than the spacing between the scattering centers, the antidot lattice does not operate. At high magnetic fields $^{(>2}$ Tesla), quantum steps in the conductivity as a function of electric field may be observed if the lattice spacing of the antidots is on the order of the cyclotron diameter of the electrons. In contrast to forming a solid of quan- tum dots, the formation of an antidot lattice requires the semiconducting structures to be continuous, with a periodic array of nanocavities. + +Early on, the greatest emphasis was on creating nanoparticles with narrow size dis- tributions, not on creating superlattice structures. However, through careful control of size distribution and chemical functionality, CdSe nanocrystals, and now many other nanocrystals, have been observed to order into superlattice structures [55]. These struc- tures may present properties beyond simple quantum effects. The individual crystal- lites in this system do not form a continuous mineral structure, but are actually se- parated by thin layers of organic molecules, which are composed of the self-assem- bling molecules used in the synthesis to regulate the diameter and polydispersity of the particles. During synthesis, the organic self-assembles into a shell around the nanoparticles, imparting organic solubility to the nanoparticles, which enables them to be processed similar to organic compounds. Because of the high degree of regularity in size and shape, these organically coated nanoparticles assemble into a crystal of nanoparticles with long range periodicity, much in the way that or- ganic molecules and atoms can crystallize. + +The next level of complexity in nanostructure formation is the creation of nanos- tructures with complex, predefined morphologies. Here, the biological concepts of self-assembly and nanostructure formation become most applicable. For example, in biology, it is common to have complex predefined structures on the nanometer scale, yet this length scale is exceedingly difficult to regulate in synthetic materials. However, if the power of self-assembly is coupled with the materials synthesis stra- tegies known today, there is great potential for the formation of complex composite nanostructures. + +Liquid-crystal templating of inorganic nanostructures, an approach in which the periodic structure of liquid crystals is imparted to a mineral phase, is one such route. Liquid crystals present an ideal matrix for the creation of nanoscale composite materials, because the characteristic $1\!-\!10\;\mathrm{nm}$ length scales most often expressed in liquid crystals are similar to the size scale of interest for semiconductor nanostruc- tures. Furthermore, the periodic structure in liquid crystals can be quite long- range, and thus the periodic nanostructure also has the potential for long-range or- der, which is of exceptional interest for many applications. A very important goal of several research groups is in fact the creation of long-range nanoperiodic order in materials such as semiconductors through liquid-crystal templating. + +Semiconductors with long-range periodic composite structures could come in sev- eral forms, for example as a particle containing a periodic array of embedded second- phase material, as a thin film with a periodic topography, or as a periodically porous material. Highly porous, periodically nanostructured semiconductor particles could be quite interesting for solution-based chemistry. For example, the photochemical nature of their semiconducting phase and zeolite-like pore structure could make them highly applicable for photochemical degradation of toxic compounds or for performing shape-selective chemistry, that is, chemistry that operates only on molecules of spe- cific shape and size. Thin films may be even more technologically important, given the wide range of potential uses for both supported and freestanding thin films. The ability to predefine a nanoperiodic array of features in semiconductor thin films may open up many applications, including electronic devices, sensors, and filter membranes. Three-dimensional semiconductor structures may have unique optical or electronic properties, depending on the characteristic length scale of the structure. As the length approaches hundreds of nanometers, the materials may even exhibit photonic band- gap effects. Because direct top-down patterning of ${<}10$ -nm-long nanostructures is difficult or impossible, and the patterning of 3D nanostructures of almost any length scale is difficult, templating of nanostructures through self-assembly processes has significant promise. Essentially, self-assembly-based templating can take place in either 2 or 3D, and the templating agent may or may not be removed, as desired. Here, we discuss several templating methodologies with potential for nanocomposite formation, the properties of such templated materials, and their potential applications. + +# 3.4.1 Lyotropic Liquid-Crystal Templating + +Lyotropic liquid-crystal templating is an approach for nanostructure and nanocompo- site formation that utilizes the self-assembled structure of a liquid crystal to regulate the structure of a growing inorganic material. When processed correctly, the structure of the inorganic phase directly replicates the structure of the liquid crystal; thus the liquid crystal is a ‘template’ for the inorganic. The most important aspect of liquid- crystal templating of inorganic material is the lyotropic liquid crystal; thus, we should to review some of the basics of lyotropic liquid crystals. Lyotropic liquid crystals are composed of at least two covalently linked components, one of which is usually an amphiphile, which is a molecule that has two or more physically distinct compo- nents, and the other a solvent. Typically, one of the components making up the am- phiphile is hydrophobic, and the other component is hydrophilic, as in common soaps, although this is not the necessary distinction. The dual solvent properties of an am- phiphile lead to the interesting self-assembly of these molecules in solution by means that include surface segregation, formation of micelles [80,81] and vesicles [82,83], as well as the formation of a wide range of LC structures [84,85]. + +Most amphiphilic molecules contain two or occasionally three segments, of which at least one is water soluble. Clearly, if half a molecule dissolves in a solvent and the other half does not, interesting self-assembled structures may result. Within the general class of amphiphilic molecules are four distinct subclasses: cationic, anionic, zwitter- ionic, and nonionic. Cationic, anionic, and zwitterionic amphiphiles all contain a for- mally charged polar moiety, typically called the head group, and a nonpolar moiety, typically termed the tail. As the names imply, cationic amphiphiles contain a cationic head group, such as a quaternary ammonium salt, and anionic amphiphiles, an an- ionic head group such as a sulfonate salt. Zwitterionic amphiphiles contain a head group having both positive and negative charge, for example tri methyl ammonium phosphonate. Nonionic amphiphiles do not contain charged functional groups, but instead contain polar segments such as oligo(ethylene oxide) or oligo(vinyl alco- hol). Examples of anionic and nonionic amphiphiles, sodium octanoate and oli- go(ethylene oxide) oleyl ether, respectively, are shown below. + +![](images/a10be98bbcfa5eb810c3cd8bd310a025e685e94f57c4cb54e55c2abf3452788a.jpg) + +Amphiphiles exhibit very rich, complex phase behavior as a function of solvent con- centration. In the dilute amphiphile limit, structures such as micelles form, and in the concentrated amphiphile limit (even solvent free), some amphiphiles show liquid crys- talline or crystalline phases. Between these endpoints, a wide range of phases and structures can form. A typical nonionic amphiphile, oligoethylene oxide (10) oleyl ether $\mathrm{{[(EO)~}_{\overline{{10}}}}$ oleyl], forms micelles, micellar rods, and hexagonal, cubic, and lamellar liquid crystals as the amphiphile:water ratio increases. This rich self assembling be- havior occurs because the polar segment is readily solubilized by water, and the non- polar tail is not. For a single molecule in solution, other than curling up on its self (which is entropically very unfavorable), there is no physical way to reduce the unfa- vorable interactions, but if multiple molecules are allowed to associate, or self-assem- ble, the unfavorable interactions can be reduced. Determination of the structure of this minimum-energy aggregate is beyond the scope of this book, but essentially one can calculate the free energies of each possible phase for a particular concentration and temperature and then select the lowest-energy phase [86]. + +The three most common phases observed in mixtures of water and amphiphile are hexagonal, lamellar and cubic. Correctly stated, the cubic phase encompasses a wide range of potential phases, all exhibiting cubic symmetry but with differing degrees of continuity in the hydrophilic or hydrophobic phases. Similar to the structures ob- served for block copolymers [87], the structures observed in lyotropic liquid crystals can be rationalized as a function of the volume fraction of the various components. Figure 3.15 shows representative phase diagrams for two nonionic amphiphiles: the first liquid crystalline phase observed as the amphiphile concentration is increased is the hexagonal phase, which is in essence the hexagonal close packing of rod-like mi- celles (Figure 3.16). These rod-like micelles develop in solution as the concentration of the amphiphile increases, but below a critical concentration do not pack closely. As the concentration of amphiphile is increased further, the spacing between the rod centers decreases, and eventually a critical point is reached where a bicontinuous cubic phase is formed (Figure 3.17). Finally, at high enough concentration of amphiphile, a lamel- lar phase forms (Figure 3.18). In some systems, especially those composed of nonionic triblock amphiphiles, a close-packed cubic phase is observed, usually at lower concen- trations of amphiphile than the hexagonal phase (Figure 3.19) [88, 89]. + +Rather then relying on coassembly, which is the process by which nanostructures develop in many mesoporous oxide systems, many biological processes utilize the order present in a preformed structure to form nanostructured inorganics. Substan- tial efforts have been made to utilize the order present in an organic mesophase to + +![](images/c4cccb44ee58de2a7d5993336dae2e5fbd385ef7a169263d3046b1301ebed239.jpg) +Fig. 3.15 Phase diagrams for nonionic amphiphile/water systems. (a) $\mathsf{C}_{12}\mathsf{E}_{6}$ ( ${\mathsf{C}}_{12}$ repre- sents the number of carbons in the tail, and $\mathsf{E}_{6}$ represents the number of ethylene oxide groups in the head), (b) N-Methyl-N-do decano y lg luca mine. Adapted from [84] + +![](images/b4880e58c9480be3c470cbc62509507ddfdaf05199b22ba885b47202f3e3b467.jpg) +Fig. 3.16 Schematic illustra- tion of the hexagonal phase. The polar head group of the surfac- tant points out into the water phase, and the hydrophobic tail inserts into the center of the rod- like micelles. Adapted from [166] + +directly template the growth of an inorganic phase; liquid crystal templating has been one of the most successful of these approaches. Early efforts in this area resulted only in oblong or cubic crystallites or microporous reticulated structures [90–92]. More recently, through liquid-crystal templating, the successful synthesis of periodically nanostructured semiconductors that copied directly the symmetry and dimensionality of the precursor liquid crystal was demonstrated [11, 93–96]. Liquid-crystal templating appears to be a general route to the synthesis of semiconductor nanostructures. The general concept of liquid-crystal templating is to first form a liquid crystal that contains at least one of the precursors of the mineral phase, and then to induce a mineral phase to precipitate in only one chemical region of the liquid crystal by applying an outside + +![](images/2370ad040b8df831ea3d9f80758984fb71ae4617d9eeef8bc094039e7d4ab3a7.jpg) +Fig. 3.17 Schematic illustration of a bicontinuous phase. The light gray region in the center of each circle indicates the hydrophobic segments, and the dark gray, the hydrophilic segments + +![](images/aa486e67676b76d90cab6762eb8e5f16f4e0ac39b7da896011973995e70353d5.jpg) + +Fig. 3.18 Schematic illustra- tion of the lamellar phase. Adapted from [166] + +perturbation. Obviously, it is very important to select the correct synthetic conditions, mesophase, and mineral phase to be successful in this process. The versatility of this process has been demonstrated by the fact that materials have been formed in liquid- crystal phases, including the hexagonal, lamellar, and cubic phases, and the materials have included the already mentioned II–VI semiconductors; periodically nanostruc- tured metals, both as thin films and in bulk [97–102]; as well as films of the chaco- genides selenium and tellurium [103, 104]. Not only are the characteristic dimensions of the materials synthesized by liquid-crystal templating smaller than those obtainable + +![](images/92dc9b00d31a716119c6a47dc10862548f56d6dd6470501d92f425f9f66ebde0.jpg) +Fig. 3.19 Schematic illustra- tion of the simple cubic phase. Adapted from [166]. + +by lithographic techniques, but they are often attainable through bulk synthesis, which is obviously not possible via lithography. + +The direct templating of materials in the preordered environment of a nonionic amphiphilic mesophase generates semiconductor/organic superlattices containing both the symmetry and the long-range order of the precursor liquid crystal. In the templated growth of II–VI semiconductors, the semiconductor is grown in a water-containing liquid crystal by reaction of $\mathrm{H}_{2}\boldsymbol{S}$ or $\mathrm{H}_{2}\mathrm{Se}$ with a dissolved salt such as $\mathsf{C d}(\mathsf{N O}_{3})_{2}$ . Both the chemical nature and the structure of the amphiphile are important for direct templating. For example, the order obtained in the nanostruc- tured systems was even observed to be dependent on the counterion for the metal [95]. + +One advantage of direct templating with liquid crystals as a route to nanostructure formation, as was already discussed, is that there are a large number of amphiphilic liquid crystals, with lattice constants ranging from a few nanometers to tens of nan- ometers, and which include lamellar, hexagonal, cubic, and bicontinuous phases [84, 105, 106]. Potentially, many of these systems can be mineralized, generating materials with an array of novel structures and properties. + +Specifically, II–VI semiconductors have been directly templated by hexagonal liquid crystals based on $\mathrm{{(EO)}\ \frac{\partial}{\partial l o}}$ oleyl [11, 93, 95], and a lamellar liquid crystal formed from oligo(vinyl alcohol) (23) oleyl ester [93]. As expected, the hexagonal liquid crystal + +![](images/eaf2333f05965bf8c192f6677e42f57d468d014fa72de5798e4cec5457490295.jpg) +Fig. 3.20 The structure on the left in the transmission electron micrograph (bottom) is nanostructured CdS viewed with the hexagonally packed cylindrical templated pores parallel to the electron beam; the one on the right has the cylindrical structures per- pendicular to the electron beam. Top: model of the nanostructure as a hexagonal ar- rangement of cylindrical pores of low electron density, corresponding to organic material in a solid matrix of semiconductor. The left and right schematic representations corre- spond to the adjacent micrographs; the central view shows the cylindrical assemblies in an intermediate state of rotation + +yielded product with a hexagonal nanostructure, and the lamellar liquid crystal yielded a lamellar structure. $\mathrm{{(EO)}\,}_{\overline{{10}}}$ oleyl can also form a lamellar liquid crystal, which also was successful in templating a lamellar CdS product. Other lyotropic phases, such as a variety of bicontinuous and cubic liquid crystals [84, 105, 107], may also yield inter- esting mineral nanostructures. In this regard, CdS has been grown in a body-centered cubic phase by using a triblock copolymer of poly(oxyethylene)-poly(oxypropylene)- poly(oxyethylene) $[\mathrm{(EPO)}_{\mathrm{~\overline{{{106}}}~}}\mathrm{~(PO)}_{\mathrm{~\overline{{{70}}}~}}\mathrm{~(EO)}_{\mathrm{~\overline{{{106}}}}}]$ as the amphiphile. When mixed with water, the PO segment is only weakly solvated, but the EO is highly solvated. Thus, when this molecule is hydrated it forms micelles which pack closely, forming a cubic phase [106]. The result of precipitation in this cubic phase was the formation of hollow nanospheres of CdS, which are covered later in this chapter. + +It is remarkable that a soft organic liquid crystal can directly template a hard covalent mineral phase. As a specific example, a hexagonal mesophase consisting of 50 vol. $\%$ aqueous $0.1~\mathrm{M}~\mathrm{{Cd}(O A c)_{2}}$ and 50 vol. $\%$ (EO) $\overline{{10}}$ oleyl templated an inorganic/organic nanocomposite of CdS and amphiphile when exposed to $\mathrm{H}_{2}\boldsymbol{S}$ gas (Figure 3.20) [93]. The composite material contained an internal nanostructure that replicated the sym- metry and dimensions of the liquid crystal in which it was grown. Interestingly, CdS and $Z\mathrm{n}S$ exhibited a superlattice morphology when formed in a liquid crystal from their respective nitrate salts and $\mathrm{H}_{2}\mathrm{S}$ , but $\mathrm{Ag}_{2}\mathrm{S}$ , $\mathsf{C u S}$ , HgS, and $\mathrm{Pb}\,\mathrm{s}$ did not (Figure 3.21). CdSe is also nanostructured by this method (Figure 3.22). An additional virtue of the (EO) $\overline{{10}}$ oleyl system is that, when combined with $\mathrm{H}_{2}\mathrm{O}$ , it forms a hexagonal me- sophase at $25\,^{\circ}\mathrm{C}$ over the range of ${\sim}35\$ to ${\sim}65$ vol. $\%$ amphiphile and a lamellar me- + +![](images/639383b8c79b7d54803b68ae635dd0f69b3e9b093dbbe5b2ff330b7203a29192.jpg) +Fig. 3.21 Transmission electron micrographs of mineralized structures grown in hexagonal mesophases from their respective nitrate salt and $\mathsf{H}_{2}\mathsf{S}$ (bar $=50\ \mathsf{n m}$ , except for $\mathsf{P b S}$ where the bar is $250\ \mathsf{n m}$ ) + +Fig. 3.22 Transmission electron micrographs of CdSe mineralized in hexagonal mesophases doped with (a) cadmium nitrate and (b) cadmium acetate + +![](images/d44a5a8dd208eefb6637308a044c70bc7626482c039ce1a2f8ae3f3e03fc091c.jpg) + +![](images/c07339dfd088b588ce881b344de6d33855e19f6b1e406c29214d7bbbcd6e5e8c.jpg) + +sophase from ${\sim}70$ to ${\sim}85$ vol. $\%$ amphiphile [108, 109], and it can directly template the growth of mineral over almost this entire range. + +As shown in Figure 3.21, the nanostructures of the semiconductors CdS and $Z\mathrm{n}S$ synthesized by precipitation in hexagonal mesophases doped with their respective nitrate salts have hexagonal symmetry with a periodicity and dimensionality commen- surate with that of the template. The hexagonal nanostructure is not always evident in TEM micrographs, due to random orientation of particles in the field of view. Presum- ably, if properly oriented, all the particles would show hexagonal symmetry. This was partially observed by tilting the samples on the TEM stage, revealing many more par- ticles with hexagonal nanostructures. + +CdS is also templated when grown from its acetate salt, although the order of its nanostructures is not nearly as well defined as in the nitrate systems (Figure 3.23) [95]. When $\mathrm{ZnS}$ is generated from its acetate salt, only spherical polycrystalline par- ticles with a porous appearance are formed. Another difference between the product obtained from the acetate and nitrate salts is the average particle diameters of the semiconductor product: both CdS and $Z\mathrm{n}S$ grown from their respective nitrate salts are approximately five times larger than when grown from their acetate salts. This size + +![](images/8d01762679527e0c241669e438af2b30e94fb5575e51b984b94c1b06d28fa54b.jpg) +Fig. 3.23 Transmission electron micrographs of CdS grown in identical hexagonal mesophases except for precursor salt. Note the significantly improved order when cadmium nitrate is used as the precursor over the order obtained with cadmium acetate + +difference can be clearly observed in low magnification electron micrographs. As con- trols, CdS was also grown in aqueous environments from both the nitrate and acetate precursors, and as expected, no nanostructure was generated. The counterion of the metal did not affect the templating of the other mineral systems studied: $\mathrm{Ag}_{2}\mathrm{S}$ , $\mathsf{C u S}$ , $\mathrm{HgS}$ , and $\mathrm{Pb}\,\mathrm{S}$ were not generated with a superlattice morphology when grown in a hexagonal mesophase, irrespective of whether the acetate or nitrate salts were used. The reason may be that the byproduct of the synthesis from the nitrate salt is nitric acid, whereas the byproduct of the synthesis from the acetate salt is acetic acid. Nitric acid is a much stronger acid, and apparently enables the mineral phase to reform around the template during growth so as to remove any structural defects. + +As already mentioned, in addition to the hexagonal mesophase, a lamellar meso- phase of (EO) $\overline{{10}}$ oleyl also can template a precipitated mineral (Figure 3.24). The la- + +![](images/147f041b55301b6871536879f1d46ef0c369567eb89fdcc9dfb20f8fda31d87d.jpg) +Fig. 3.24 Schematic representation of the lamellar organic template and transmission electron micrograph of the resulting product after mineralization. The dark bands are CdS that mineralized in the hydrophilic region of the template liquid crystal; mineral growth is prohibited in the hydrophobic regions + +mellar periodicity in the resulting CdS is ${\sim}7~\mathrm{nm}$ , which agrees very well with the periodicity of the lamellar template. The lamellar morphology can be confirmed by careful tilting of particles within the TEM. If a particle is tilted about an axis perpen- dicular to the stripes, no change is observed in the pattern. However, if it is tilted on an axis parallel to the stripes (and perpendicular to the electron beam), the pattern quickly disappears. These observations constitute strong evidence for a lamellar morphology within the particles and agree closely with results obtained from lamellar nanocom- posites formed in a poly(vinyl alcohol)-based liquid crystal [94]. As shown in Figure 3.24, the mineralized product consists of disk-like particles with a long axis in the plane of the layers that is ${\sim}1.5$ times the maximum width perpendicular to the layers. Very interestingly, particles having lamellar morphology do not disperse, even with re- peated ultrasonication, perhaps due to mineral or organic bridging between the CdS layers. Presumably, CdS nucleates within a hydrophilic layer of the mesophase and grows rapidly in the plane, but concurrently an occasional finger forms perpen- dicular to this layer, piercing the hydrophobic region. This finger then nucleates an- other layer of CdS, resulting in a mineral bridge between layers. Another possibility is tethering of organic molecules in the mineral phase, tying the layers together. Of course, growth in the plane of the layers is faster than growth perpendicular to the layers, generating the disk-like habit observed. Not surprisingly, since it seems to en- hance the templating effect, lamellar-nanostructured CdS was seen only when the nitrate salt was used, while the acetate salt afforded only small particles. + +The layered structure formed by templating with a lamellar liquid crystal is in fact reminiscent of the layered abalone shell structure; however, in the templated material the characteristic dimension is just a few nanometers, not hundreds of nanometers to micrometers. It still remains to be seen if the properties of the layered material are significantly improved over those of the solid equivalent, but if the abalone shell is any guide, advanced mechanical properties are a possibility. + +As described, mineral growth in the hexagonal and lamellar phases yields interest- ing, controlled nanostructures. Mineral growth in the cubic phase formed from (EO) $\scriptstyle{\overline{{{106}}}}\,\left(\mathrm{PC}\right)\,{\overline{{{70}}}}\,\left(\mathrm{EO}\right)\,{\overline{{{106}}}}$ was also done, using cadmium acetate and $\mathrm{H}_{2}\boldsymbol{S}$ as precursors [110]. Although the mechanism is not entirely understood, the result was hollow spheres $20\mathrm{-}200\;\mathrm{nm}$ in diameter (Figure 3.25), which can be observed by both TEM and SEM. When the sample is tilted in the stage, the shape and observed structure do not change, as expected for a hollow sphere (Figure 3.26). If the particles had been corpuscular in shape, their appearance would change as a function of sample tilt. The strongest evidence for their hollow nature is the dark edges of the spheres ob- served in TEM micrographs. If each sphere were solid, TEM would show greater scat- tering from the center than the edges, making the center appear darker. In the SEM, the spheres also scatter the most electrons from their edges and appear somewhat transparent in their centers, providing further proof of their hollow nature (Figure 3.25 ) . Unlike materials produced by direct templating in hexagonal and lamellar li- quid crystals, the hollow spheres are not of a size commensurate with the structure of the liquid crystal, but are rather 1–10 times the size of the characteristic dimension of the liquid crystal in which they were formed. + +![](images/3e7047e15ac87acdee9b3e43acb430be3b4a62d7731038c166127ea23e2ec545.jpg) + +Fig. 3.25 Scanning electron micrograph of hollow spheres of CdS mineralized within a cubic meso- phase + +Because the characteristic dimension (diameter) for hollow CdS spheres obtained from the $\mathrm{{Cd}(C H}_{3}\mathrm{{CO}}_{2})_{2}$ -doped cubic phase is $20\mathrm{-}200\,\mathrm{nm}$ , (1–10 times the diameter of the micelles making up the cubic phase), it does not appear that the mineral nanos- tructure is directly templated by the liquid crystal. In addition, this nanostructure is entirely absent from the CdS when the cubic phase is doped with $\mathsf{C d}(\mathsf{N O}_{3})_{2}$ as the precursor salt. This result is important when taken in the context of previous results in which the use of the nitrate salt led to a nanostructure with enhanced order [95]. In essence, the nitrate salt allowed the growing mineral to access a thermodynamic ally lower-energy morphology, which was a nanostructure commensurate with the struc- ture of the liquid crystalline matrix. The inability of the nitrate salt to ‘sharpen’ the order in the cubic system is not surprising, because, as observed in previous stu- dies, the nitrate salt only improves the registry between the nanostructure and the liquid crystal and does not result in a new nanostructure. In the cubic system, the spheres are not a copy of the liquid crystal, so there is no registry to improve. The mineralization of the cubic phase must lead to local rearrangements of the liquid crystal, leading to the hollow-sphere morphology observed; however, the detailed me- chanism is still not understood. + +![](images/cd2368bbf290903ae017af4d9d613e4288ec1f182c42ba148357265d31f84955.jpg) +Fig. 3.26 Transmission electron microscope tilt ning from upper left to lower right of each micro- series of hollow CdS spheres grown in an acetate- graph. (a) $-45^{\circ}$ , (b) $0^{\circ}$ , (c) $45^{\circ}$ doped cubic phase. The tilt axis was diagonal run- + +A potentially significant difference between the cubic phase and the other phases studied (hexagonal and lamellar) is the connectivity of the hydrophilic and hydropho- bic portions of the liquid crystal. In the lamellar and hexagonal phases, both these regions are continuous in at least one direction. In the cubic phase, however, the hy- drophobic regions are confined to discrete micelles. This confinement may cause a difference in templating ability relative to the other phases. For all templating phases, as the mineral nucleates and grows, it is necessary to expel some molecules from the volume occupied by the mineral. For both the lamellar and hexagonal phases, a molecule can diffuse away from the growing mineral without ever exposing its hy- drophobic (oleyl) or hydrophilic (EO) segments to domains of the opposite nature. In contrast, in the cubic phase, when a molecule is forced away from the growing mineral it must leave its micelle and expose its hydrophobic (PO) segment to the polar sur- roundings $(\mathrm{EO}+\mathrm{H}_{2}\mathrm{O})$ , a high-energy situation. As already stated, the nitrate salt al- lows the mineral phase to access a lower-energy configuration, which in the lamellar and hexagonal systems results in a high degree of fidelity between the template and the semiconductor nanostructure. That a hollow-sphere morphology is not observed in the cubic system when the nitrate salt is used is not surprising, given that the hollow morphology is not directly templated by the liquid crystal and that formation of the spheres requires the mesophase to go through a high-energy intermediate state. The full reason for the spherical morphology when $\mathrm{{Cd}(C H}_{3}\mathrm{{CO}}_{2})_{2}$ is used as the semiconductor precursor must be due to a subtle kinetic balance that is not yet understood. At the very least, the energy difference between the lamellar, hexago- nal, and cubic systems is rather small, and thus the fact that templating is successful in the first two systems but unsuccessful in the last indicates that some fairly specific interactions are necessary for direct templating. This is similar to the action of many biological systems, in which very specific interactions between proteins and other macromolecules and growing inorganic phases are exceedingly important for structural development. + +In addition to simple one-component systems, it is interesting to consider the result of templating binary mixtures of the precursor salts within a liquid crystal. In one system, a hexagonal mesophase containing $0.05\mathrm{~M~}$ $\mathsf{C d}(\mathsf{N O}_{3})_{2}$ and 0.05 M $\mathrm{Zn(NO}_{3})_{2}$ was mineralized. The resulting semiconductor $\mathrm{{'Cd}}_{x}\mathrm{{Zn}}_{1.x}\mathrm{{S}}$ ; $x\sim\!0.5$ ) product’s nanos- + +![](images/6cc2106ac48c375ad2be726050c9bd43386586738a1a2f4fb97dcdb44fda35a9.jpg) +Fig. 3.27 Transmission electron micrograph of the product obtained from precipitation in a hexagonal mesophase doped with $0.05\,\textsf{M}\mathsf{C d}(\mathsf{N O}_{3})_{2}$ and 0.05 M $\mathsf{Z n}(\mathsf{N O}_{3})_{2}$ . The templated product is a solid solution of CdS and ZnS + +![](images/eab3f7ced432e014fe7e5a3fd033988e0541995967d594eac22ca470939e21f8.jpg) +Fig. 3.28 Transmission electron micrograph of a composite product of PbS and ZnS grown in a hexagonal mesophase doped with $0.05~\mathsf{M}$ $\mathsf{P b}(\mathsf{N O}_{3})_{2}$ and 0.05 M $\mathsf{Z n}(\mathsf{N O}_{3})_{2}$ . The single crystal cube at the core of the particle is PbS, and the shell is a periodically nanostructured solid consisting of ZnS + +tructure exactly matched that of the template (Figure 3.27). In a system of $0.05\mathrm{~M~}$ $\mathrm{Pb}(\mathrm{NO}_{3})_{2}$ and $0.05~\mathrm{M}$ $\mathrm{Zn(NO}_{3}\mathrm{)}_{2}.$ , the result was a very different nanostructure, which consisted of a single-crystal core of $\mathrm{Pb}\,\mathrm{S}$ surrounded by a shell of nanostructured $Z\mathrm{n}S$ (Figure 3.28). From this it appears that, when mixed metal products are synthesized, there is a direct correspondence between the behavior of the constituent solids and the mixed solid. $\mathrm{Cd}_{x}\mathrm{Zn}_{1.x}\mathrm{S}$ was nanostructured (as were CdS and $Z\mathrm{n}S$ , Figure 3.21), but precipitation from $\mathrm{Pb}^{2+}$ and $Z\mathrm{n}^{2+}$ ions resulted in a single crystal of PbS surrounded by nanostructured polycrystalline $\mathrm{ZnS}$ , corresponding with the result seen for PbS and $\mathrm{ZnS}$ when grown discretely (Figure 3.21). The formation of mixed metal precipitates gives an insight into the growth processes and also opens the possibility of engineering a property. The system of $\mathrm{{Cd}}_{x}\mathrm{{Zn}}_{1.x}S$ offers the possibility of band-gap engineering, although, due to the somewhat lower solubility of CdS than $Z\mathrm{n}S$ in water, the particles may be cadmium-rich in their center and zinc-rich on their exterior. The differing solubilities of $\mathrm{Pb}\,\mathrm{S}$ and $Z\mathrm{n}S$ in water play a very important role in the structure of the particles formed from this mixed system. PbS has a much lower solubility than $\mathrm{ZnS}$ and, as a result, upon exposure of the doped mesophase to $\mathrm{H}_{2}\mathrm{S}$ , single-crys- tal $\mathrm{Pb}\,\mathrm{S}$ cubes nucleate and grow. Then, after most of the $\mathrm{Pb}^{2+}$ ions are locally ex- hausted, $Z\mathrm{n}S$ heterogeneously nucleates on the $\mathrm{Pb}\,\mathrm{S}$ particles. As expected, because $\mathrm{ZnS}$ is templated by liquid crystals, the shell of $\mathrm{ZnS}$ around the PbS single crystal contains a periodic superlattice structure. + +Direct templating of an inorganic by an organic liquid crystal may depend on many factors, the most important of which is probably the thermodynamic stability of the mesophase throughout the mineral growth process. The mesophase must be stable to the addition of mineral precursors, and the mineral precipitation process must not disrupt the order of the liquid crystal. In studying direct templating, researchers have observed that the textures observed by polarized optical microscopy are the same for the pure mesophase and for a mesophase that contains the precursor salt, indicating that the doping did not lead to radical disruption of the order in the mesophase. Nuclear magnetic resonance (NMR) can also be utilized to verify the structure of liquid crystalline mesophases. To verify that the characteristic mole- + +$^2\mathrm{H}$ cular order of the mesophase was not disrupted by ion doping, broadline NMR spectra were obtained from both cadmium ion-doped and undoped mesophases + + [96]. For both mesophases, the same quadrupole splitting was observed (Figure + + 3.29). If ionic doping had perturbed the structure of the mesophase, the splitting would have decreased [111, 112]. As additional proof of molecular order in the me- sophase, x-ray diffractograms were collected to characterize both the mesophase’s long period and symmetry. For systems containing $35\,\%$ , $40\%$ , $50\,\%$ , and $60\,\%$ am- phiphile, the 100, 110, and 200 reflections are clearly observed, indicating that the liquid crystalline structure is hexagonal (Figure 3.30). A mesophase containing $78\,\%$ amphiphile forms a lamellar liquid crystal, as indicated by the presence of 100 and 200 reflections and the absence of a 110 reflection (Figure 3.31). As ex- pected, a strong correlation was found between the phase diagrams as determined by optical analyses and the x-ray data. Similar experiments have been performed + +![](images/fa686d14e1ac17f9e192f28f6f169335e44a0e89cdd1dbecbf7278215d12d767.jpg) +Fig. 3.29 Quadrupole splitting for a hexagonal mesophase (a) doped with 0.1 M cadmium acetate and (b) undoped. The splitting indicated by an arrow is the same for both samples + +![](images/3c37e29a64b3422bcaf679ecfd12e4c577031edc5cf499ae13b1e78785fa3588.jpg) +Fig. 3.30 Small-angle $\mathsf{X}$ -ray scattering of hexagonal aqueous mesophases containing (a) $35\,\%.$ , (b) $40\,\%,$ , (c) $50\,\%$ , and (d) $60\,\%$ amphiphile by volume + +for other liquid-crystal templating systems, as well as for the formation of mesoporous silica. + +In addition to the metal sulfides, which can be successfully templated as already mentioned, several sulfide materials, including $\mathrm{Ag}_{2}\mathrm{S}$ , CuS, HgS, and PbS, were not templated by the liquid crystal in which they were grown, irrespective of the coun- terion. Design of amphiphiles with proper structures and binding constants for both the inorganic precursors and the inorganic product may enable a wide variety of in- organic and organic compounds to be templated in the future. + +To better understand the scientific underpinnings of liquid-crystal templating, it is instructive to consider a few additional experiments. For example, liquid-crystal tem- plating at elevated temperature sheds some light on the mesophase–ion–product interactions. As already mentioned, it appears the periodically nanostructured materi- + +![](images/b6f8e88714a661a06c4e11208ab27ffb7c4e893da7cf54fcc967c6899156289d.jpg) +Fig. 3.31 Small-angle x-ray scattering of a lamellar mesophase containing $78\,\%$ amphiphile and $22\,\%$ water by volume + +als are thermodynamic ally stable with respect to their solid equivalent. This implies that there is a critical energy balance between the energy gained by reducing the sur- face area of the mineral phase and the energy lost due to disruption of the mesophase structure. To study this further, CdS was precipitated in mesophases at both $35\,^{\circ}\mathrm{C}$ and $50\,^{\circ}\mathrm{C}$ . These temperatures are both below the isotropization temperature of the doped mesophases, so the reactions were carried out in a self-assembled medium. The CdS produced from the reaction at $35\,^{\circ}\mathrm{C}$ did express the order of the mesophase, albeit poorly when compared with the order obtained at room temperature $(22\,^{\circ}\mathrm{C})$ . Precipi- tation at $50\,^{\circ}\mathrm{C}$ resulted in mineral with no periodic order, clearly indicating that the energetic difference between periodically nanostructured and disordered product is small and that it only takes a small perturbation to result in non nano structured pro- duct. + +As additional evidence for the direct templating mechanism, samples composed of $35\,\%$ , $40\,\%$ , $50\,\%$ , and $60\,\%$ by weight (EO) $\overline{{10}}$ oleyl were mineralized with CdS. By varying the amphiphile content of the mesophase, the spacing between the cylindrical aggregates of amphiphilic molecules making up the hexagonal mesophase was varied. Assuming that CdS is directly templated by the liquid crystal in which it is grown, the hexagonal symmetry and associated length scale should be nearly identical to that found in the precursor hexagonal mesophase, which is what indeed happened. The result (Figure 3.32), demonstrating that the superlattice dimension in the preci- pitate can be varied by changing the lattice constant of the mesophase, is very strong evidence for direct templating. If the nanostructure present in the templated inorganic phase had been the result of a cooperative self-assembly process, much as occurs for + +![](images/1a3115ee6bd61900dc2b927d604362590d6483039d20b98bbf9715a8e262cdfe.jpg) +Fig. 3.32 Center-to-center spacing in a cylindrical assembly of amphiphilic molecules of hexagonal mesophases as determined by x-ray diffraction $(\bullet)$ , and center-to-center pore spacing in templated CdS as measured by transmission electron microscopy $(\spadesuit)$ . Both are plotted as a function of the water content in the mesophase + +most mesoporous silica systems, the result would have been that the periodicity of the nanostructure would not have changed with varying water content. + +# 3.4.2 Liquid-Crystal Templating of Thin Films + +The templating of thin films by self-organized organic structures should find signifi- cant application in both technology and scientific study. Thin-film templating is struc- turally related to the bulk templating of inorganic materials, which generally results in periodically structured particles; however, because the result is a periodically struc- tured thin film, the potential for application is clearer. Templating with organic struc- tures is especially intriguing because of the potential to create features much smaller than those feasible by almost any top-down technique, because it utilizes the nanoscale molecular order inherent in self-assembled organic structures to define the structure of the resulting thin film. A few key points must hold true for successful thin-film templating by liquid crystals. First, clearly, for templating to be successful, the self- assembled matrix must be compatible with the substrate. Then, via some process, the inorganic material must be deposited or grown on the substrate. Realistically, only chemical and electrochemical routes for materials deposition can meet these requirements. Other conventional methods of thin-film deposition require high va- cuum, which is incompatible with lyotropic liquid crystals and furthermore cannot operate through a thick overlying layer of liquid crystal. + +The synthetic routes for liquid-crystal templating of thin films are relatively straight- forward. To date, most studies have used electrochemical techniques to drive the ma- terial deposition. First, a precursor containing lyotropic liquid crystal is interfaced with the substrate. Then, under an applied potential, material is electro chemically depos- ited at the liquid-crystal/substrate interface. Nanostructured materials that have been created through this process include a variety of metals, selenium, and tellurium [97, 99–104, 113]. It may also be possible to electrodeposit other interesting materials including semiconductors; however, no publications have yet appeared demonstrat- ing success. + +Throughout, it is quite interesting that all the successful templating experiments have relied on liquid crystals formed of nonionic amphiphiles. However, in biol- ogy, most preformed matrices are formed via ionic macromolecules. In part, this is likely because biology makes use of very specific interactions to create mineral struc- tures, but synthetic systems do not have this degree of sophistication. Thus, the fact that nonionic amphiphiles are much more stable to varying concentrations of soluble salts is actually an advantage. Ionic amphiphiles (as well as biomolecules) are affected much more strongly by salts, because a single salt ion can bind to one or more of the polar amphiphile headgroups, greatly reducing their polarity. This was indeed ob- served for several of the anionic amphiphiles studied, and most likely was the reason that ionic systems were not successful in templating the growth of a mineral phase. In biology, where the molecular structures are designed to interact specifically with one salt under very specific conditions, the strong interaction of ionic groups with dis- solved species is an advantage. However, for generalized synthetic systems, this may not be an advantage. + +# 3.4.3 Block-Copolymer Templating + +Block copolymers are a widely studied class of materials that organize into both 2D and 3D structures at slightly longer length scales than observed for liquid crystals. Similar phase behavior is observed, with systems transitioning between lamellar, hexagonal and cubic phases as the relative volume fraction of the two blocks changes [87]. Much more complex morphologies can form in triblock systems. The characteristic periodi- city in block copolymer systems ranges from ${\sim}5\;\mathrm{nm}$ primarily depends on the molecular weight of the block copolymer. As expected, as the molecular weight of the block copolymer increases, the characteristic length scale in- creases. It was realized that if there was a way to impart this nanoscale order into a substrate, one might have a powerful technique for patterning materials with a per- iodic array of nanometer-sized structures. Unlike lyotropic liquid crystals, block copo- lymers are generally solvent-free and can be taken to elevated temperatures and under vacuum without destabilizing the self-assembled structure; thus high-vacuum mate- rial deposition and processing approaches can be used. Usually the chemistry of the block copolymer is designed so that one of the two bocks can be removed via a dry etch with ozone or other reactive compound to generate the porous structure, which will subsequently serve as a template for nanostructure formation. + +The procedure for block copolymer templating of nanostructures usually is as fol- lows. A thin film of some block copolymer is spun-coated from solvent onto a substrate and allowed to self-assemble. After this, one of the blocks of the polymer is removed by ozone etching. The result of this etching procedure is a substrate coated with a thin polymer film containing a periodic nanoscale void structure. After removal of any solid polymer film that overlies the void structure, the polymer film is used as a mask. Material can be evaporated through the polymer film onto the substrate, material can be electro chemically grown from the substrate up through the polymer film, or the polymer film can be used as an etch mask. In all cases, the result is material structured to be a replica of either the polymer film (when the polymer is used as an etch mask) or the pore structure of the polymer film (when material is deposited in the pores). Usually, at the end, the polymer film is removed with solvent or reactive ion etch, leaving behind nanostructured templated features on a substrate. With this ap- proach it is possible to create features as small as ${\sim}20\;\mathrm{nm}$ holes or dots in a periodic array on a substrate of a wide range of materials, including oxides, semiconductors, magnetic materials, and of course the polymer itself (Figure 3.33) [114–121]. The power of block copolymer templating is further enhanced due to the facts that both the size and spacing of the feature can be modulated simply by varying the mo- lecular weight and composition of the polymer and that the lattice structure can be modulated by varying the relative length of the two blocks. For example, both the lamellar and hexagonal phases can template nanometer-scale lines if they are oriented + +![](images/875de56074377322c4b4361dba86c0728b2a9aa28b6e44fe72c471b455e7aa63.jpg) +Fig. 3.33 (a) Scanning electron micrograph of a polystyrene/polyisoprene block copolymer template after it has been partially etched with ozone and the continuous polystyrene layer on the top has been removed. The now empty PI domains are now holes and are darker in the micrograph. (b) Scanning electron micrograph of hexagonally ordered arrays of holes in silicon nitride on a thick silicon wafer. This pattern is formed by using as a template a copolymer film such shown in (a). The darker regions are ${\sim}20{\cdot}\mathsf{n m}$ -deep holes in the silicon nitride layer, which are formed by etching through the overlying template. Adapted from [114] + +properly on the substrate. I should point out that the periodic arrays of dots have at- tracted the greatest attention for applications such as magnetic storage media. + +# 3.4.4 Colloidal Templating + +In any discussion of biologically inspired nanocomposite materials, one must include recent developments on colloidal crystal templating of photonic materials. The basic premise behind this approach is to use the 3D periodic structure of synthetic opals to direct the structure of a second phase material. This approach is not biologically in- spired, but should more accurately be described as ‘naturally inspired’, because opals, although natural, are geological, not biological, in origin. Furthermore, although the lower limit to the characteristic length scale of the material generated is $10\,{-}\,20\,\mathrm{nm}$ , the characteristic dimension is often relatively large, on the order of $500\,\mathrm{nm}$ . This is in fact intentional, since most of the applications for these materials are optical, and thus the characteristic length should not be much smaller than the wavelength of the light that one desires to modulate. Commonly, however, much smaller features are embedded within the templated structure, and these features may be as small as a few nan- ometers. Because of the clear natural inspiration for the colloidal templating of ma- terials and the relationship of the templating process to many biological processes, I cover this approach in this chapter. + +The interest in microperiodic 3D structures has grown tremendously due to the exciting potential of such materials, particularly in the area of photonics [122]. Such 3D structures, often termed photonic crystals, are the extension of the well- known dielectric stack into three dimensions. Although the colors that occur in opals, which stem from diffraction of white light by planes of highly ordered submic- rometer silica spheres, are our inspiration; for practical application, synthetic ap- proaches are needed to create materials and structures with the necessary refractive index and periodicity to meet the requirements for most optical applications, which opals simply do not have. + +A particularly interesting class of optical structures are the so-called photonic band- gap materials. For example, a microperiodic material consisting of low-refractive-index spheres arranged in a face-centered-cubic array in a matrix with a high index of re- fraction, and having a lattice constant on the order of the wavelength of light (visible or infrared), could be such a photonic band-gap material [123]. Similar to how a di- electric stack has a stop-band for light in a given frequency range, this material would not allow light in a given frequency range to travel through it in any direction . In es- sence, it would be an omnidirectional, perfectly lossless, mirror. + +The synthesis of these structures however is exceedingly difficult. Layer-by-layer fabrication of photonic crystals using state-of-the-art VLSI tools, e.g., deep UV photo- lithography, chemical vapor deposition (CVD), chemical-mechanical polishing, has been demonstrated [3], but formidable processing difficulties limit the formation of large area and truly 3D structures. + +When appropriately formed, self-assembled colloidal crystals are natural candidates for the construction of photonic crystals. Good crystal quality is achieved only with colloids that have very low size polydispersity $(<\!5\,\%)$ , which currently limits the choice of materials to $\mathrm{SiO}_{2}$ or polymers, both of which have a fairly low index of refraction around 1.5, which is much smaller than that required for most optical applications. This has led researchers to take a two-stage templating approach. In a first step the desired microperiodic structure is assembled by using colloids. In a second step this structure is used as a template to build a complementary structure with a material having a higher index of refraction [124]. + +Colloidal crystal templating is a very promising approach for production of high- resolution, micrometer-scale, 3D periodic photonic crystals, but as conventionally ap- plied to the fabrication of photonic band gap structures it has serious optical limita- tions, unless materials with the necessary optical properties can be used. Typically, the most important point is to infill with a material with a sufficiently high refractive index to generate an optically interesting material. A range of approaches have been sug- gested to maximize the index contrast, including sol–gel [125–127], chemical vapor deposition [128–131], imbibing of nanoparticles [132–134], reduction of $\mathrm{GeO}_{2}$ to Ge [135], electroless [136] and electrochemical deposition [137], and melt imbibing [138]. In addition, polymers have been used to infill colloidal crystals, and in one report, the colloidal particles were less than $100\,\mathrm{nm}$ in diameter, which, although perhaps not interesting from an optical standpoint, may have potential for separation membranes and confined chemical reactor spaces [139]. Although they are not the focus of the following discussion, the colloidal templates used in these attempts are commonly polycrystalline and can contain unacceptably high numbers of defects; thus, substan- tial effort has also gone into creating colloidal crystals with low defect densities. Each of these infilling techniques has various advantages and disadvantages, which are dis- cussed below. In general, they all consist of approaches to infill the interstitial space of the colloidal template, after which the colloidal template is generally removed. + +The sol–gel infilling of colloidal crystal templates is intriguing to consider as a route to 3D porous materials [125–127], although it is somewhat limited in application for photonic materials for several reasons. First, the refractive index of most materials that can be formed via sol–gel is $<2$ (with the exception of $\mathrm{SiO}_{2}$ , which can have a re- fractive index of ${\sim}2.5$ ), second, there is considerable reduction in volume during the conversion of the sol to solid material, and third, the refractive index of most sol–gel-derived material is substantially less than that of a single crystal of the same material. The net effect is that most sol–gel-derived macroporous materials have relatively low refractive index contrast, and their long-range order is somewhat disrupted due to the uneven contraction of the matrix. However, if one is not interested in photonic materials, but rather is attempting to make a ceramic macroporous ma- terial, sol–gel infilling of colloidal crystals may be a very good route. The contraction of the matrix may in fact be an advantage, in that it may be possible to make structures with pore diameters on the order of $50\,\%$ of the diameter of the template. Because it is difficult to make colloidal crystal templates from spheres smaller than a few hundred nanometers, this may be valuable for creating nanoperiodic structures. + +Another pathway to macroporous materials is to fill the interstitial space of a col- loidal crystal with nanoparticles, followed by removal of the colloidal template. This has some advantages over sol–gel infilling, in that the contraction of the structure + +![](images/44093f4a64dbdfbe1b1f7b63acfcacf9d2f9dc55a96d7f54ba07fc66066fea04.jpg) +Fig. 3.34 Left, transmission electron micrographs of $3\,\%$ erbium-doped hydrothermally synthesized titania nanoparticles. Right, scanning electron mi- crographs of macroporous titania structure formed + +upon removal of the template from a nanoparticle filled colloidal crystal is significantly less than that seen upon removal of the template from a sol-gel filled system, and a much larger subset of materials can be prepared as nanoparticles, including semicon- ductors, metals, and ceramics. The first example of semiconductor nanoparticle in- filling of colloidal template used II–VI semiconductor nanoparticles [132]; since then, Er-doped $\mathrm{TiO}_{2}$ nanoparticles, for example, have been filled into a colloidal tem- plate, followed by removal of the template to generate a macroporous solid (Figure 3.34) [140]. + +The use of CVD as a pathway to filling colloidal crystals at first may seem counter- intuitive. After all, CVD generally is most efficient at coating planar surfaces, and it would seem almost impossible to fill structures with deep pores, such as the interstitial space of a 3D colloidal crystal. However, significant strides have been made in the past few years, and now virtually complete infilling of colloidal structures with both Si and Ge via CVD has been demonstrated [128–131]. After dissolution of the colloidal tem- plate, the result is an inverse structure with the necessary refractive index and struc- tural conditions to exhibit a complete photonic band gap. + +Electrode position-based infilling is intriguing for several reasons and has the poten- tial to be general with respect to both characteristic lattice constant and material (Fig- ure 3.35). The potential for high-refractive-index materials, large area structures, and the complete infilling of thick 3D colloidal templates, as well as the low cost of elec- trodeposition have led to interest in this area. To date, three different classes of ma- terials that have been electro deposited into self-assembled colloidal crystals, namely semiconductors, polymers, and metals. + +Semiconductors are interesting candidates for photonic crystals, primarily because of their high refractive indices and generally robust nature. For example, CdS has a refractive index of 2.5, and materials such as GaP, Si, and Ge have indices of 3.4, 3.5, and 4.0, respectively. However, routes to creating periodic macroporous structures + +![](images/a063259218734558334f330f41d902619aa7f443e8474ccd84cc2221c3142c3a.jpg) + +Fig. 3.35 Generalized procedure for creating 3D periodic macroporous materials by colloidal tem- plating and electrode position. Monodisperse col- loids sediment onto a conducting substrate, self- assembling into a crystal. The sample may be dried and sintered before electrolyte is added. A coun- terelectrode allows electrode position of the desired material (semiconductor, polymer, metal) into the interstitial spaces. In a final step the electrolyte and the templating colloid are removed. For polymeric colloids this can be done by treatment at elevated temperature or by dissolution with a solvent. For silica colloids, aqueous HF is effective for dissol- ving the template + +Fig. 3.36 Schematic representation of the ex- perimental setup for potentiostatic deposition of CdSe through the interstitial space of a colloidal crystal + +Fig. 3.37 Scanning electron micrographs of po- ten tio statically deposited CdSe (a) and galvanos- tatically deposited CdS (b, c) after removal of the polystyrene colloidal template. In the overdeposited system (b), the overlying solid CdS film can be clearly seen on the right. The apparent lack of periodic pore structure in the underdeposited system is not due to disorder in the colloid, but occurs because the nodular surface of the semi- conductor cuts through multiple lattice planes of the template + +![](images/43c7f03fc80ba033d23a339e2937c8d4646175a8a3e8c6afcfbd18fde811ee80.jpg) + +![](images/78c6c0548bf3cce68391797cd4499106587d35bf42ebfce1fc0a9859a983a639.jpg) + +from such materials are limited because of their very high melting points and low solubility in common solvents. + +To date, the II–VI semiconductors CdS and CdSe [137, 141], and $\mathrm{ZnO}$ [142], have been electro chemically grown through colloidal templates, resulting, after dissolution of the template, in macroporous semiconductor films. For all systems, a conducting oxide film on glass was used as the substrate. Macroporous CdS films were generated by galvanostatic deposition through the interstitial spaces of a colloidal crystal formed from $1\;\mu\mathrm{m}\ \mathrm{SiO}_{2}$ spheres, and CdS and CdSe macroporous films were generated by potentiostatic deposition through a colloidal template generated from $466\;\mathrm{nm}$ polystyr- ene spheres (Figure 3.36). After electrodeposition, the $\mathrm{SiO}_{2}$ and polystyrene colloidal templates were removed with aqueous HF and toluene, respectively. Because of the high rigidity of the semiconductor network, contraction upon removal of the template was limited to a few percent at most. The fine and gross morphologies of the electro- deposited semiconductors are shown in Figure 3.37. + +Macroporous $\mathrm{ZnO}$ films were formed by potentiostatic deposition through a colloi- dal crystal formed from $368\,\mathrm{nm}$ polystyrene spheres, and the spheres were removed with toluene. Careful control of the electrode position conditions was necessary: if electrode position was done at a potential less negative than $-1.0\:\mathrm{V}$ vs. $\mathrm{Ag/AgCl}$ , large crystalline grains of $\mathrm{ZnO}$ formed, which disrupted the structure of the colloidal tem- plate. Using a deposition potential more negative than $-1.0\;\mathrm{V}$ suppressed the forma- tion of large-grain $\mathrm{ZnO}$ , and the colloidal template was not disrupted. + +All the electro deposited semiconductor films are reported to be opalescent; however, detailed optical spectroscopy has yet to be performed. Real progress in optically inter- esting materials may await the electrochemical deposition of materials such as GaP, Ge, and Si, which, because they have refractive indices ${>}3$ , may result in materials with 3D photonic band gaps. Routes to the electrode position of such materials have been demonstrated [143], but problems, such as hydrogen gas evolution and generally harsh conditions, need to be solved before success in these areas is likely. + +Electrode position of conducting polymers (electro polymerization) through self-as- sembled colloidal crystals, followed by removal of the colloidal template, is a promising route to achieving active macroporous materials. Several significant advancements over the past few years have begun to demonstrate the potential of conducting poly- mer-based microperiodic photonic structures. Inherently, because of the low refractive index of polymeric materials, it is quite unlikely that a 3D photonic band-gap material will result from a polymer-based photonic crystal; however, conducting polymers have advantageous properties as compared to conventional polymers or inorganic materi- als: their optical properties can be electro chemically modulated, fine control over prop- erties can be obtained through organic chemistry, and they are often mechanically flexible. + +Electrochemical growth of conducting polymers is a fairly well developed field, and many procedures for growing solid films have been published [144]. There are, how- ever, only a few reports on the growth of porous conducting polymer films. Fibers of polypyrrole, poly(3-methylthiophene), and polyaniline were formed in the early 1990s by electrode position from the appropriate monomer solution through a porous mem- brane [145]. The first example of electrochemical deposition of a conducting polymer around a colloidal template was in 1992, when polypyrrole was grown around latex particles [146]. However, no attempt was made to remove the colloidal particles, and the optical properties of the resulting films were not measured. Only in the past few years have researchers been exploring the possibility of the templated growth of conducting polymers for photonic applications. + +To date, three reports on colloidal templating of conducting polymers have ap- peared, all of which followed the general procedure of (1) colloidal crystal formation on a conducting substrate, (2) electrochemical deposition from solution, and (3) dis- solution of the colloidal template with an appropriate solvent. In the first example, polypyrrole was grown potenti o statically from a solution of pyrrole in acetonitrile through a colloidal crystal composed of $\mathrm{SiO}_{2}$ spheres with a mean diameter of $238\,\mathrm{nm}$ assembled on F-doped $\mathrm{SnO}_{2}$ -coated glass, followed by removal of the colloidal template with aqueous HF [147]. Macroporous polypyrrole, polyaniline, and poly- bithiophene films have been potenti o statically polymerized through a colloidal crystal assembled from $500\;\mathrm{nm}$ and $750\,\mathrm{nm}$ polystyrene spheres, on a substrate of gold- coated glass. The polystyrene template was then removed with toluene [148]. In the most recent example, polypyrrole and polythiophene macroporous films were po- ten tio statically grown through colloidal crystals assembled from $150\,\mathrm{nm}$ and $925\;\mathrm{nm}$ polystyrene spheres, respectively, on glass coated with indium tin oxide; the polystyr- ene was removed with tetrahydrofuran [149]. A preliminary optical characterization [149] showed a weak dip in transmittance that appeared to be correlated with the per- iodic structure. + +One significant issue is the contraction of the period structure upon removal of solvent for electro deposited macroporous polymers. This was most clearly observed in polystyrene-templated systems, in which significant contraction, ranging from $13\,\%$ to $40\,\%$ , was observed for the macroporous polypyrrole and polyaniline. How- ever, very little contraction was observed in polystyrene-templated macroporous poly- bithiophene or when $\mathrm{SiO}_{2}$ spheres were used as the template. For example, poly(ethylene di oxy thi oph ene) was templated by silica colloidal particles (Figure 3.38). The primary difference is that organic solvents are used to remove the polystyr- ene spheres and an aqueous HF solution is used to remove the $\mathrm{SiO}_{2}$ spheres. This suggests that the organic solvent softens the electro deposited polymer, allowing it to contract; however, there may be other systems similar to polybithiophene, in which contraction of the macroporous matrix does not occur. This is less of a problem for macroporous metals and semiconductors. + +Metallic macroporous ordered replicas of colloidal assemblies are of potential inter- est for a wide range of applications including filtration, separation, and catalysis. In addition, they might have interesting electrical, magnetic, or optical properties. The tools and techniques for electro chemically plating metals have been well established for thin films and even bulk materials. It is thus fairly straightforward to develop re- cipes to backfill the interstitial space of a colloidal self-assembled crystal with almost any metal. + +From a photonic standpoint, the properties of bulk metals are very poor, although of course templated structures may have many other applications. The imaginary com- ponents of the dielectric constants of bulk metals are large, hence they readily absorb + +![](images/e2d5fd3de4be45b879de40bceda55ef2c2e5eb70fd656d8e7c44dcfa23948251.jpg) +Fig. 3.38 Scanning electron micrograph of poly(ethylene di oxy thi oph ene) electro deposited around a colloidal crystal template, after dissolution of the template + +light. When the metallic structures become small enough, however, strong optical resonances associated with plasmon frequencies of the conduction electron in the metals can lead to qualitatively new phenomena. A well-known example of this is the red color of a nanosized dispersion of gold colloid. A more recent manifestation of unexpected behavior is the anomalously high light transmission through small holes $(<\!200\ \mathrm{nm})$ in thin metallic films [150]. + +Theoretical calculations [151–153] on ordered3D arrays of metallo-dielectric spheres show that these are promising for the construction of materials with full photonic band- gap in the visible part of the optical spectrum. The advantage of metallo-dielectric structures over purely dielectric structures is that it should be easier to achieve a full band-gap in the visible. A full band-gap in the visible is exceedingly difficult, if not impossible, to create with purely dielectric structures, because very few dielectric materials have an indexof refraction ${>}3$ and very low absorption in the visible.Thishasled to the development of synthesis routes to produce metallo-dielectric colloidal core-shell particles with sizes in the sub micrometer range[154,155]and metallic shell thicknesses or cores that are small enough to show resonance effects. + +Vos et al. [156] made gold replicas of colloidal crystals made of silica (radius $113\,\mathrm{nm}$ ) and polystyrene (radius $322\ \mathrm{nm})$ ). Prior to electrode position of the gold, the silica spheres were sintered by heat treatment at $600\,^{\circ}\mathrm{C}$ . After electrode position of the gold, the silica template was removed by etching with aqueous HF, and the polystyr- ene spheres were removed by combustion at $450\,^{\circ}\mathrm{C}$ (Figure 3.39). There are no dimen- sional changes between the dried, sintered colloid and the final replica, although some cracking is observed during the original drying and sintering process, indicating that the electro chemically formed gold is dense and structurally robust. This is a definite improvement over other methods of infilling macroporous structures with high-di- electric materials, e.g., liquid-phase or sol–gel chemistry [126, 127], and infiltration with nanosized particles [132], in which considerable contraction of the matrix is ob- served, which leads to serious crack formation and warping of the colloidal structure. Other electro deposited materials include Ni, Pt, and a SnCo alloy [157, 158], Pd, Pt, and Co [159]; electroless deposition has also been attempted [136]. + +As just described, recent work has demonstrated that templating of the interstitial space of highly ordered colloidal crystals has promise for creating macroporous photo- nic crystals from a diverse set of materials including oxides, semiconductors, metals, polymers, and glasses. The resulting 3D macroperiodic materials have been formed with close-packed macropores ranging in diameter from $100\,\mathrm{nm}$ to a few micrometers, giving the potential to modulate light ranging from deep UV to the infrared. However, problems with the infilling process still need to be overcome before this approach to photonic structures comes to fruition. As outlined, filling the 3D interstitial space of a colloidal crystal with a high-index material has been problematic, because many tech- niques either only deposit material in the top few layers of a colloidal crystal or do not fully fill the colloidal crystal with a material of high enough refractive index. + +Another route to infilling of colloidal crystals to generate a high-refractive-index structure is melt-imbibing of a chalcogenide glass such as selenium, followed by dis- solution of the silica template. Selenium was selected because it has a high refractive index of 2.5, and thus can provide a nearly complete 3D photonic gap (Figure 3.40), a very low optical loss coefficient between 1 and $10\,\mu\mathrm{m}$ [160], low melting point $(217\,^{\circ}\mathrm{C})$ , and relatively low surface tension $({\sim}100$ dynes $\mathrm{cm^{\cdot1}}$ ), which reduces the force neces- sary to infill the structure. Importantly, selenium vitrifies easily, forming an optically isotropic glass [160]. Other chalcogenide glasses certainly could be used to infill col- loidal crystals; however, they have higher softening points and thus are not suitable for initial investigation. Through melt imbibing, essentially complete infilling of a colloi- dal crystal was demonstrated ( Figure 3.41 ) . + +![](images/c44c4e21868aa0f538d51b2977f1e6b2dcecaf779fb80ae306d0848005f48e05.jpg) +Fig. 3.39 (a) Scanning electron micrograph of a crystal of air spheres (radius 111 nm) in gold, made with a silica template. The inset is a Fourier transform of the image. (b) Scanning electron micrograph of macropores (radius $322\,\mathsf{n m}.$ ) in gold, made with a latex template. The structure has short- range order, but no long-range order. Adapted from [156] + +![](images/7aa27bbb4aa83a1633b504191a3067b64f187a54947abec3097b0679f45ea025.jpg) +206 3 Natural Nano bio composites, Biomimetic Nanocomposites, and Biologically Inspired Nanocomposites + +Fig. 3.41 Scanning electron micrographs of a selenium photonic crystal cut and polished to ex- pose specific crystallographic faces. (a) Polished (001) facet, which is perpendicular to the settling direction. The contact points between the spheres on adjacent layers are clearly visible as pores, but within the (001) plane the voids do not appear to be interconnected. This pore morphology is the result of a $2\%$ mismatch between the sphere diameter and the template size. (b) (110) face, proof that the selenium photonic crystal is indeed fcc. Again, the voids are generally not interconnected within the (001) plane, but are interconnected between (001) planes. (c) Low-magnification scanning electron micrograph of the (110) face, presenting a cross section of the entire thickness of the selenium photonic crystal. The sample was mounted in epoxy prior to polishing, which is seen below the photonic crystal; solid selenium is present above the photonic crystal. Importantly, the structure is highly ordered throughout the entire crystal, including the top layer. Because of an $\sim\!1.5^{\circ}$ misorientation of the polished sample with respect to the (110) face, the size of the pores appears to change from left to right across the sample. Adapted from [138] + +![](images/199f20ad05d521bfd1d9f252c1c0c1aa26bbcbdbb334db4e3e2914c7f0d2febf.jpg) + +![](images/d4908bbff65125ea271b2a56fec9d8508bf8169e595731d17368f19e022c2c74.jpg) + +![](images/1e1f5475144c970599ec04a835d6cfe784b18eccf7a7fca04ea5ea0edc759dc8.jpg) + +Infilling was accomplished by imbibing molten selenium under high pressure into the colloidal crystal, followed by quenching to vitrify the selenium. Subsequently, the colloidal crystal template was removed with HF, resulting in a macroporous selenium/ air structure with a high contrast in refractive index. + +# 3.5 Summary + +The confluence of nanoscience, biotechnology, and materials chemistry offers great potential for discovery and fabrication of advanced composite materials. Great amounts of information still need to be gleaned from the study of biological sys- tems, but we have now reached a point where the current body of knowledge on how biological systems can create highly functional nanocomposites is starting to en- able the creation of advanced materials. For example, natural systems widely exploit self-assembly to create a great diversity of interesting and highly functional materials, and today we are beginning to also create synthetic systems by similar processes. We must always remember that biological organisms work with a limited subset of ma- terials and take up to years to create nanocomposite structures, so, rather than attempt- ing to create materials by direct mimicking of biology, it will likely be much more productive to create materials by exploiting the design rules expressed by biological systems and applying them to synthetic systems. 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The fillers may be spheroid, such as ceramic nanoparticles or carbon black; plate-like, such as layered silicates; or fiber- like, such as nanotubes. + +Nanocomposite materials exhibit properties that are significantly different from those of the corresponding microcomposite. Examples are discussed at length in other chapters of this book. Polymers filled with nanoparticles have significantly en- hanced stiffness and strength compared to similar materials filled with regular micro- meter-size ceramic particles. Most importantly, a gain in these properties does not compromise ductility, as typically occurs with conventional materials. In other sys- tems a significant ductility enhancement was measured with no decrease in strength [1, 2]. Also, ceramic nanocomposites exhibit increased fracture toughness. Other ex- amples involve fluids such as water or glycol which, when filled with metal nanopar- ticles, exhibit thermal conductivity enhancement [3] that cannot be explained by clas- sical approaches valid for standard composites [4]. This clearly indicates that, whereas a body of knowledge derived from standard continuum-level modeling approaches provides a good description of microcomposites, it is inadequate in its current form to describe properties and behaviors of nanocomposites. + +The development of these new materials, and particularly an understanding of their properties, is still in its infancy. Consequently, to date, proofs of concept have been repeatedly and consistently made, but material optimization is still an elusive goal. For example, good filler dispersion has been demonstrated to be critical for achieving properties superior to those of microcomposites, but it is still unclear how other para- meters such as filler distribution and filler surface function aliz ation influence the overall properties. Optimization of these materials will be possible only when the phy- sics controlling the enhanced properties is properly understood, and when the proces- sing routes leading to the desired nano- and microstructures are established. The cur- rent approach to investigating these issues is experimental and based on trial and error. Hence, modeling the material response would provide valuable information on the underlying physics and the desired structure–property linkages. Further- more, once the desired structure is established, modeling has the capability to assist the development of material processing. + +# 4.2 Current Conceptual Frameworks + +The properties of the matrix, the distribution and properties of the filler, and the nat- ure of their interface control the behavior of a typical microcomposite. The interface plays an important role in controlling several microcomposite properties such as its thermal conductivity, load-carrying capacity, toughness, and transport properties. However, for a characteristic filler particle sizes on the micrometer scale and lar- ger, the main parameter controlling the macroscopic properties is still the filler vo- lume fraction. In these materials, the interface may be viewed in the standard way, as a 2D surface separating the constituents. By contrast, when the matrix is filled with nanosized particles, a significant fraction of the material is either at the interface or within its immediate vicinity. For ceramic composites, one can argue that the inter- face width is still small relative to the filler size or the filler-to-filler distance; thus extension or modification of standard composite theory or modeling approaches may be sufficient. For polymer nanocomposites, the modification of the matrix may extend over several radii of gyration from the particle surface. In turn, this situa- tion results in novel macroscopic properties that are dictated qualitatively, as a trend, and quantitatively, by the behavior of the confined polymer that forms the modified matrix. + +Several attempts have been made to provide a conceptual framework for justifying the observed enhanced properties of polymer nanocomposites. One such theory, al- luded to above, the ‘bonded polymer layer’ [5], is based on the observation that, at the same volume fraction, the total interfacial area in a nanocomposite is much larger than in a microcomposite. The total volume of ‘bonded polymer’ (confined polymer next to the filler wall) is conjectured to represent a large fraction of the volume of the nano- composite, and therefore, the properties of this confined material determine the macroscopic properties of the system. However, the available experimental results on the existence of a bonded polymer layer are contradictory [6, 7]. It is also unclear how thick the layer surrounding each particle is and, most importantly, what the prop- erties of the polymer in the interfacial region are. + +More recently, the ‘double network’ theory was proposed. This idea is based on the observation that the reduction in particle size entails a similar reduction in the inter- particle spacing and that special macroscopic properties are obtained when the average distance between particles becomes comparable to the bulk radius of gyration of the polymer chains. These events create conditions for a chain to connect two or more particles, hence forming a network that provides additional strength by increasing the number of effective entanglements in the system [8]. The intuitive appeal of this theory is high, but quantitative predictions that may be compared with experimen- tal results require a thorough understanding of the structure of chains confined be- tween spherical fillers. + +# 4.3 Multiscale Modeling + +Considering the multiple time and length scales characterizing nanocomposites, and polymer nanocomposites in particular, a comprehensive approach to modeling these systems requires multiscale modeling. Whereas multiscale modeling across all length scales, from electronic level to bulk-matter sizes, and all time scales is a future chal- lenge, the various components of composites have however undergone extensive in- dependent study. The dynamics of polymers, which control the mechanical and di- electric responses of these materials, have been studied for many years. Much litera- ture exists on both molecular and continuum aspects of this problem. Several classical models of stress production and chain dynamics dominate current thinking. Follow- ing an idea introduced by Guth and Mark [9], the chains are regarded as entropic springs in tension. This concept is rooted in the observation that, as the material de- forms, the chains are stretched and their entropy decreases. The change in entropy entails a free energy variation, leading to a retractive force. The force required to stretch the chain proves proportional to the change in the end-to-end distance, which suggests that the chain may be compared with a linear spring. The force is always retractive, and in vacuum, the chain should collapse into itself. The corollary is that all chains are in tension. Furthermore, the force is proportional to the absolute temperature, which is the signature of its entropic nature. The energetic component of the free energy is usually neglected in this theory. This view gained support from early experimental observations that established unequivocally that the nature of stress in rubbers is indeed entropic. + +Two main models describe chain dynamics, and hence stress and dielectric relaxa- tion. The Rouse model represents the dynamics of chains shorter than the average distance between entanglements, but long enough to behave as entropic springs [10]. This model is in agreement with the entropic spring description of a chain. The principal interactions accounted for are those along the chain, and the surround- ing medium is represented in a homogenized way, by means of a frictional force acting on the representative chain. Chains longer than the entanglement length exhibit slower dynamics, which is described by the reptation model [11]. In this model, the effect of nonbonded interactions between monomers not directly connected by backbone covalent bonds is conjectured to lead to constraints that force the chain to diffuse primarily along its contour. Several other representations have been pro- posed [12–14]. + +A rich literature exists on the development of continuum constitutive models of polymer behavior. These models are mostly phenom eno logical and are developed based on macroscopic mechanical testing of both neat and filled polymeric materi- als. Several excellent treatises exist on this topic; [15] is a general overview. + +Polymer interfaces and, in particular, the polymer chain behavior close to flat inter- faces, have been studied by simulations and have a long history of theoretical treat- ment. The study of the conformations and dynamics of linear polymeric chains in the vicinity of flat walls or free surfaces [16–29] is motivated by the many applications in which the behavior of confined chains is important. Such applications include, but are not limited to, coatings, flow of molecular solutions in capillaries, and their interaction with filters and membranes. + +In nanocomposites, the chain–filler interactions control to a large extent the matrix and macroscopic properties. Therefore, understanding chain conformations and dy- namics of the confined chains next to filler surfaces is crucial. The physical picture that emerges from the cited studies is summarized below. + +The conformations of a macromolecule located close to an impenetrable interface are restricted. They are determined by the configurational entropy of the chain, by the energetic interactions between polymers (cohesive interactions in bulk), by the nature and strength of the interaction between polymers and the confining wall, and by global system parameters such as temperature and density. When the chain approaches the wall, the configurational entropy decreases due to the reduced number of accessible chain configurations. This generates an entropic driving force that retracts the chain from the interface, with the force increasing linearly with temperature. This effect generates a low-density polymer layer in the immediate vicinity of the interface [30–32]. The variation in density has a significant effect on chain mobility as well as on the diffusion of small molecules. The retraction is balanced by packing in the bulk polymer. An attractive energetic interaction between polymers and the wall also promotes chain adsorption. The interplay of these forces determines the actual configuration. + +Another entropic force promotes movement of chain-end monomers to the inter- face [21, 32]. This force is due to the fact that the reduction in the number of accessible chain configurations induced by confinement is less dramatic if the chain comes in contact with the wall with only one of its ends. Hence, chains having their center of mass within two radii of gyration of the wall adjust so as to replace chain-inner-mono- mer contacts with chain-end contacts. + +In the immediate vicinity of a flat impenetrable wall, the polymer chains are pre- ferentially aligned on multiple scales in a direction parallel to the interface. This leads to changes in the glass transition temperature and to a broader relaxation spectrum. The degree of alignment depends on the details of the energetic interaction with the wall. It is generally considered that a chain may adopt a ‘docking’ type configuration at high temperatures and relatively weak attractive interactions and may completely ad- sorb if the attraction is strong [33]. Chain collapse in the interface leads to pronounced alignment on the bond and larger scales. + +These details are important for understanding the physics of polymer-filler interac- tions in systems filled with layered silicates, in which interfaces are mostly planar [34]. Another issue in these composites regards the dynamics of polymer intercalation be- tween silicate layers. In polymers filled with nanoparticles and nanotubes, the inter- face is curved, and it is necessary to ask to what extent the curvature modifies the physical picture described above. This issue was studied [35, 36], resulting in the con- clusion that, for realistic curvatures, the polymer structure next to a wall is close to that next to a flat surface in the presence of similar energetic interactions. Interestingly, these models showed that, even in the presence of strong hydrogen bonding of poly- mers to the curved wall, the shape and size of polymer coils having their center of mass close to the wall remains essentially similar to that in the bulk. However, the ellipsoidal coils undergo a ‘docking transition’, rotating with their long semiaxis in the direction tangential to the curved wall. + +The connection between the physics on the atomic and molecular scales and the meso and macroscopic material behavior requires the development of continuum models. These models must incorporate the smaller-scale physics. Steps along this line have been taken for neat polymer and are reviewed in [37]. In nanocompo- sites, additional difficulty resides in the fact that the mesoscale continuum model needs to describe deformation on the scale of the inter-filler spacing. Because the fillers have dimensions comparable to those of the polymeric chains, and because the fields vary on the same scale, the homogeneity assumption for the representative volume element is no longer mandated. Under these conditions of high structural and field gradients, nonlocal constitutive laws are required. The development of such laws that incorporate molecular and atomistic information is an emerging field. + +On a larger scale, nanoparticles tend to cluster before and during processing of the composite. Currently, it is generally accepted that good dispersion is required for enhanced macroscopic properties. Obtaining the appropriate dispersion in practice is largely a matter of experimental ability. Modeling may be used in this connection to elucidate two issues. On the one hand it is necessary to understand the lower limit of dispersion required for imposed macroscopic properties, and on the other, process modeling that would guide the routes for achieving the desired dispersion is highly desirable. Developing these types of models requires reliable constitutive equations. + +On the scale of filler agglomerates and larger, the material may be described in the continuum sense by using the methodologies developed for classical microcompo- sites. Homogenization theory and asymptotic expansion techniques are useful tools in this endeavor. The standard homogenization theory leads to poor approximations of local fields in the presence of high-gradient regions. In fact, the use of this technique is possible only within the assumption of homogeneous fields over the representative volume considered. In practice, the adequacy (or lack of it) of the homogenization theory can be assessed on the basis of the uniform validity of the double-scale asymp- totic expansion [38]. + +These techniques may be used to study the effect of filler distribution. Good dis- persion of fillers, but nonuniform distribution, improved wear properties and scratch resistance of the composite. Most properties depend on filler distribution, and mate- rial optimization would require the development of models that may provide the gross behavior while accounting for nonuniform distributions of fillers. + +# 4.4 Multiphysics Aspects + +One of the principal advantages of polymer nanocomposites is multi functionality. Through multi functionality, the material exhibits enhanced properties with regard to several physical phenomena. The composite may be transparent to the visible spec- trum and opaque to ultraviolet while being wear- and scratch-resistant; it may simul- taneously provide good strength and diffusion barrier properties. All models predict- ing the macrosopic behavior should capture the relevant multiple physical processes with respect to which the material is being optimized. + +Various physical phenomena are intimately coupled at the atomic and molecular scales and, as such, are captured by discrete models. In continuum representa- tions, the physics is traditionally decoupled. Nanocomposite models should overcome this limitation. Hence, when going from discrete to continuum models, one has to ensure that discretely represented processes can be described by conventional partial differential equations, that the constitutive law selected for the ‘smallest-scale conti- nuum’ is able to represent the (mean) ‘fields’, and that the time scale cutoff, which is associated with this transition, does not alter the essential physics studied. Accurate modeling of discrete-continuum transition hinges on an affirmative answer to all these questions. + +Multiphysics coupling in the continuum sense may be performed either by adding additional terms in the continuum balance equations (‘equation coupling’) or by re- quiring the phenom eno logical coefficients entering the constitutive laws to depend on the fields to be coupled (‘constitutive law coupling’). The coupling of fields (stress/ strain, temperature, solute/reactants concentration) occurs through diverse phenom- ena, some of which are shown in the interaction matrix of Table 4.1. The interaction matrix is ‘nonsymmetric’, with cell $(i,j)$ representing the phenomenon induced by the + +![Tab. 4.1 Coupling of physical phenomena in continuum models. ](images/21dc474de98491deb37bc46d9cda1b787fb5a36278b446486caa7753837ae2fd.jpg) + +process corresponding to field i and which influences field j . For instance, cell (1, 2) represents heating due to deformation, and cell (2, 1) corresponds to thermal expan- sion and thermal stresses. A fully coupled continuum analysis would consider all pro- cesses shown in the matrix, but an uncoupled approach would only consider the di- agonal entries. As mentioned, the coupling of various physical processes in the math- ematical model can be carried out either by considering additional terms in the field equations or by allowing the constitutive law to depend on the interacting field. An example from the first category is the effect of the temperature gradient on the stress field, which can be captured by adding a thermal stress term to equilibrium equations. A less familiar example within the same class of problems is the effect of diffusion (clustering of diffusants) and chemical reactions (reaction products) on the stress field, which can be accounted for by the eigenstrain formulation [39]. An example from the second category is the influence of temperature upon diffusion, which can be captured by simply considering the diffusion coefficients to be temperature dependent. + +# 4.5 Validation + +Development of faithful predictive multiscale models of nanocomposites will require a systematic validation effort. The multiscale multiphysics models must be calibrated and validated at multiple scales against experiments that resolve the finer scales. For polymer nanocomposites, the experimental data obtained by nuclear magnetic resonance (NMR), dielectric spectroscopy (DS) and quantitative dynamic mechanical analysis (QDMA) can be used to verify the ability of the developed models to predict the conformation and kinetics of polymer chains. NMR measurements probe segmental motions of the main chain and side chain movement over a frequency domain from several $\mathrm{Hz}$ to several MHz. Longer-time and larger-scale dynamic effects, which can be modeled by coarse-grained discrete and continuum approaches, would be validated against DS and QDMA measurements. By these techniques, short- and long-time relaxation modes are probed by mechanical or electric excitations. + +The challenge to the experimental work will be to perform measurements on well characterized systems, such as those with well defined particle size and shape distri- bution and particle dispersion. Ideally, one would like to control and vary individual parameters so as to asses their roles. Some successes in this area involve the ability to change surface chemistry, thus controlling the strength of the particle-matrix interac- tions. However, in experiment, it is often very difficult or even impossible to control various parameters and characteristics independently of one another. The hope is that the combined experimental–modeling effort will allow the acquisition of better and deeper understanding of the structure–property–processing relationships in the quest for guided design of nanocomposites with desired, optimized properties. + +# References + +1 C. B. Ng, B. J. 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Edwards, The Theory of Poly- mer Dynamics, Clarendon, Oxford, 1986. 12 M.F. Herman, B. Panajotova, K.T.J. Lorenz, J. Chem. Phys., 105, 1153 (1996). 13 R.C. Picu, G. Loriot, J.H. Weiner, J. Chem. Phys., 108, 4984 (1998). 14 R.C. Picu, Atomic level stress in polymeric systems: a review of principal results, In Recent Research Developments in Macro- molecules, Ed. S.G. Pandalai, Research Si- gnpost, 2003. 15 L. E. Nielsen, R. F. Landel, Mechanical Properties of Polymers and Composites, Marcel Dekker, New York, 1994. 16 M. Stamm, Adv. Poly. Sci. 100,357 (1992). 17 B.J. Factor, T.P. Russel, M.F. Toney, Ma- cromolecules 26, 2847 (1993). 18 J.A. Forrest, K. Dalkoni-Veress, J.R. Stevens, J.R. Dutcher Phys. Rev. Lett. 77, 2002 (1996). + +19 S. Sen, J.M. Cohen, J.D. McCoy, J.G. Curro, J. Chem. Phys. 101, 9010 (1994). 20 D.N. Theodorou, Macromolecules 21,1391 (1988). 21 P.G. de Gennes, Adv. Colloid Interface Sci. 27, 189 (1987). 22 J.M.H.M. Scheutjens, G.J. Fleer, J. Phys. Chem. 83, 1619 (1979). 23 M.H.M. Scheutjens, G.J. Fleer, Macromo- lecules 18,1882 (1985). 24 Y.Zhan, W.L. Mattice, Macromolecules 27, 7056 (1994). 25 K.F. Mansfield, D.N. Theodorou, Macro- molecules 24, 6283 (1991). 26 S.K. Kumar, M. Vacatello, D.Y. Yoon, Ma- cromolecules 23, 2189 (1990). 27 A.J. Yethiraj, Chem. Phys. 101, 2489 (1994). 28 E. Eisenriegler, K. Kremer, K. Binder, J. Chem. Phys. 77, 6296 (1982). 29 I.A. Bitanis, G. ten Brinke, J. Chem. Phys. 99, 3100 (1993). 30 J.H. Jang, W.L. Mattice, Polymer 40, 4685 (1999). 31 J. Baschnagel, K. Binder, Macromolecules 28, 6808 (1995). 32 P. Cifra, E. Nies, F.E. Karasz, Macromole- cules 27, 1166 (1994). 33 A. Bellemans, J. Orban, J. Chem. Phys. 75, 2454 (1981). 34 E. Manias, H. Chen, R. Krishnamoorti, J. Genzer, E.J. Kramer, E.P. Ginnelis, Macro- molecules 33, 7955, 2000. 35 M.S. Ozmusul, R.C. Picu, Polymer 43, 4657 (2002). 36 R.C. Picu, M.S. Ozmusul, M.S., submitted to Macromolecules, 2002. 37 R.B. Bird, C.F. Curtis, R.C. Armstrong O. Hassager, Dynamics of Polymeric Liquids: Kinetic Theory, Wiley-Interscience, New York, 1987. 38 J.Fish, P.Nayak, M.H.Holmes, Comp. Mech. 14, 323–338 (1994). 39 J. Fish, Q. Yu, K.L. Shek, Int. J. Numer. Methods Eng. 45(11), 1657–1679 (1999). + +# Index + +# a + +abalone shell 155, 162 abrasion 132 absorption spectra 66 actin 157 active membranes 67 additives 20 aerogels 8 aerosol method 93 agglomeration + + – biocomposites 173 + + – magnetic nanocomposites 64 + + – modeling 219 + + – polymer nanocomposites 100 ff, 117 aggregates, DNA-linked 168 AgS 186 alanine 157 alcohol soaking 8 aliphatic amines 101 alkylammonium 91, 124 AlN 47 alumina + + – hard coatings 24 + + – metal nanocomposites 4, 7 f + + – nanoporous membranes 56 + + – polymer-filled nanocomposites 94 + + – templates 35 alumina-filled PMMA 130 alumina-gelatin nanocomposites 141 a lumi no silicates 90 ff amino acids 169 ammonium salts 107, 179 amorphous structures 7 amphiphilic block copolymers (ABC) 114 amphiphilic molecules 178 apatite 164 applications 44 ff, 57 ff aragonitic nacreous layer 155 asymtotic expansion techniques 219 + +# b + +bacteria 159 ball milling 6 BaTiO3 49 bilayers 91 binary systems 29 biocomposites/biometric nanocomposites 155–214 block copolymer templating 171, 195 f Bohr exciton 47 bonded polymers 216 bone 162 borides 29 boron-cabon-nitrogen nanostructures 35 boron nitride nanotubes 89 bound polymer layer 97 breakdown 77, 138 brittle failure 19 buckling 40, 84 bulk ceramic nanocomposites 18 ff bulk metal nanocomposites 1–76 + +# c + +e -caprolactam 107 carbenes 85 carbides 29 carbon 55 carbon black + + – modeling 215 + + – polymer-filled nanocomposites 77, 93, 135 carbon-carbon nanocomposites 29 carbon composites 10 carbon fibers 117 carbon nanotubes 31 ff, 78 ff carbonated apatite 164 carboxylic acid groups 118 casting 14 catalytic nanocomposites 68 cation exchange capacity (CEC) 90 ff + +cavitation processes 95 CdS 167 f, 176 ff, 185 ff CdSe 201 ceramic metal composites 5 ceramic nanocomposites 176 ceramic polymer composites 112 chain dynamics 217 chain mobility 79 ff, 96 chalcogenides 182, 205 chemical activity, coupling 220 chemical attack 64 chemical drilling 53 chemical etching 118 chemical vapor deposition (CVD) + + – biocomposites 197 + + – hard coatings 29 + + – polymer-filled nanocomposites 85 chlorinated precursors 26 clay/nylon composites 103 clays 90 ff, 124 clusters + + – metal nanocomposites 4 + + – modeling 220 + + – nanocomposite particles 45 coarse grain molecules 103 coatings 5, 24 f cobalt-containing salts 28 cobalt nitride films 60 coercitivity + + – magnetic nanocomposites 63 + + – metal nanocomposites 6 f + + – nanowires 44 + + – particle-dispersed nanocomposites 58 coherency strain 25 collagen fibrils 163 colloidal consolidation sintering 20 colloidal systems, inorganic 46 colloidal templating 171, 197 color, gold colloids 204 composite systems, inorganic 50 ff condensation 173 condensed phase electrolytic processing conducting polymers 202 conduction band 85 conductivity + + – inorganic nanocomposites 51 f + + – polymer-filled nanocomposites 82 Considere’s construction 128 constitutive law coupling 220 continuum models 218 ff cooling + + – metal nanocomposites 12 + + – particle-dispersed nanocomposites 58 coprecipitation methods 4 + +CoPt 63 cosputtering 60 coupling, continuum model 220 coupling agents 119 Cr2/NiCr composites 13 crack tip bridging 22 cracking 19, 67 + + – biocomposites 204 + + – hard coatings 27 + + – polymer filled nanocomposites 127 ff creep resistance 3, 22 cromium forsterite 143 crystallinity, polymer-filled nanocomposites 79 ff, 96 ff, 106 cubic phases, amphiphiles 179 CuS 186 + +# d + +dangling bonds + + – encapsulated composite nanosystems 37 + + – polymer-filled nanocomposites 81 decomposition + + – hard coatings 28 + + – metal nanocomposites 9 + + – spinodal 29 defects 81, 84 deformation + + – coupling 220 + + – encapsulated composite nanosystems 42 + + – polymer-filled nanocomposites 84 see also: plastic deformation degradation + + – inorganic nanocomposites 47 + + – nanoparticle/polymer composites 111 + + – polymer filled nanocomposites 136 delamination 136 densities 8 dendritic gold particles 115 deposition techniques 23 f diamond-like carbon 19, 24 dichalcogenides 90 die casting 18 diffusion, strain-controlled 220 dimensional stability 135 dimethyl a cet amine (DMAC) 107 direct mixing 111 direct templating 183, 190 discrete models 220 dislocations, hard coatings 27 dispersions 4, 101 displacement reactions 6 DNA 157, 165 ff docking transitions 219 dodecylamine ammonium salts 108 + +doping 117, 199 double network theory 216 drilling 53 dry etching 196 drying + + – polymer composites 117 + + – supercritical 8 ductility + + – metal matrix nanocomposites 17 + + – modeling 215 + + – polymer filled nanocomposites 122 + +# e + +elastic moduli + + – polymer-filled nanocomposites 82 + + – thin film nanocomposites 23 electric arc processes 85, 88 f electrical applications 49 ff electrical properties 85, 138 ff electrochemical growth 202 electrode position-based infilling 199 enamel 155 encapsulated composite nanosystems 36 ff end-cap function aliz ation 101 endohedral fullerenes 35 entropic springs in tension model 217 epoxy resins + + – nanofillers 125 + + – polymer nanocomposites 102, 109 equation coupling 220 equi-axed fillers, polymeric 80 f, 93 ff, 118, 126 erbium-doped titania 199 Escherichia coli 169 etching 118 ethanol 102 exciton Bohr radius 47 exfoliated clays 133 f, 137 f, 140 exfoliated nanocomposites 106, 109 + +# f + +failure 77, 84, 122, 127 $\mathrm{Fe_{80}P_{20}}$ metallic glass nanocomposites 34 feedstock variables, metal nanocomposites 12 FePt 63 FermiDirac distribution 51 Fermi energy 85 ferritin 166 ferrocene 42 ferroelectric materials 4 fibers + + – fillers 77, 80 ff + + – metal matrix nanocomposites 3, 14 filled carbon nanotubes 35 + +fillers + + – inorganic nanocomposites 50 ff + + – modeling 216 + + – polymer 77153 flame process 93 forsterite 143 fracture + + – bulk ceramic nanocomposites 22 + + – metal nanocomposites 3 f + + – polymer-filled nanocomposites 84 freeze drying 117 friction coefficient 132 fullerenes + + – endohedryl 35, 39 + + – polymer-filled nanocomposites 81 ff, 86 ff fumed silica 77 functional groups 45 functional low-dimensional nanocomposites 35 ff function aliz ation 141 + +# g + +gallium arsenide 46 gas condensation 94 gas phase synthesis 173 gas pressure sintering 20 gelatin matrix 141 gelation chemistry 8 giant magnetoresistance (GMR) 64 glass fibers 117 glass-transition temperature 96 ff, 121, 131 f glycine 157 gold particles, dendritic 115 gold replicas 204 grain boundaries + + – metal nanocomposites 3 + + – sliding 21 grain growth rates 61 grain sizes 17 granular films 23 f granular solids 59 graphite + + – fiber nanofillers 122 + + – pyrolytic 18 graphite-encapsulated metal/ceramic structures 36 graphite-polystyrene matrices 109 greigite 160 grinding 173 group IIVI semiconductors, biocomposites 172 f, 182 f, 199 gypsum, biocomposites 160 gyration radius 97, 218 + +# + +hard coatings 24 f hardness + + – bulk ceramic nanocomposites 22 + + – carbon nanotube-based composites 33 + + – metal matrix nanocomposites 4, 17 heat production 220 heat release rate (HRR) 136 hectrite + + – clay layers 133 + + – polymer nanocomposites 91, 108 Hertzian elastic response 29 heterocoagulationpolymerization 120 hexagonal phases, amphiphiles 179 HgS 186 high-energy ball milling 6 high-temperature laser assisted deposition 43 high-temperature properties 3 high-velocity oxy fuel process (HVOF) 12 homogenization theory 219 hot pressing 20 hybrid nanocomposites 46, 111 hybridization 169 hydrazine 116 hydrocarbons + + – encapsulated composite nanosystems 41 + + – hard coatings 31 + + – low-dimensional nanocomposites 35 + + – polymer-filled nanocomposites 86 hydrodynamic cavitation 95 hydrolysis 95 hydroxyapatite 155 hydroxyl groups 118 hydroxyl siloxane 92 Hyperion fibers 87 + +# i + +in-situ polymerization 112 indium oxide 47 indium phosphide 44 infilling methods 198 ff, 207 inorganic coatings 121 f inorganic filler-polymer interfaces 96 ff inorganic nanocomposites + + – electrical applications 49 ff + + – optical applications 46 ff inter/intragranular designs, metal nanocomposites 3 intercalation + + – nanofillers 124 + + – polymer nanocomposites 103 f interfaces + + – inorganic polymers 96 ff + + – polymer-filled nanocomposites 78 ff, 117 – polystyrene sapphire 99 interlayer spacing 124 intrinsic magnetic properties 62 ion channels 157 ion coimplantation 46 ion-conducting materials 4 iron nanowires, encapsulation 44 iron oxide core 166 ff iron powders 7 + +j jet variables 12 + +# k + +kanemite 92 kenyate 92 kink model 104 Kr at sch mer Huffman n method 86 + +l lamellar phases, amphiphiles 179 Lang muir B lodge tt technique 142 Langmuir monolayers 174 laponite 91 laser ablation + + – hard coatings 29 + + – polymer-filled nanocomposites 85 f, 95 f lattice defects 81 lattice mismatch 25 layer-by-layer fabrication 197 layered double hydroxides (LDH) 93 layered fillers 103 ff layered silicates 90 ff, 219 layers 59 Lewis bases 102 light-emitting devices 141 f lipid cellular membranes 157 liquid crystal matrices 108 liquid crystal templating 171, 177 f, 194 ff lithographic methods 174 load carrying capability 122, 127 loadings 10 low-dimensional nanocomposites 35 ff low-friction coatings 27 Ludox process 93 luminescence 48, 66 lyotropic liquid crystal templating 178 + +# + +machinability 3 magadiite 92, 140 magnesia 4 magnesium silicates 91 magnetic alloys 38 + +magnetic applications 57 ff magnetic bacteria 159 magnetic properties 17 magnetite 159 Magnetospirillum magnetotactium 159 magnetothermal analysis (MTA) 34 magnetron sputtering 27 makatite 92 maleic anhydide 124 matrices 107 ff mechanical alloying + + – metal matrix nanocomposites 4 ff, 14 + + – particle-dispersed nanocomposites 58 mechanical properties + + – bulk ceramic nanocomposites 18, 22 + + – metal nanocomposites 3 f + + – polymer filled nanocomposites 122 ff media manufacturing 59 ff melt processing 105 melt spinning 58 membranes, lipid 157 mesoporous oxides 156 metal matrix nanocomposites (MMCs) 14 ff metal nanocomposites 1–76 metal nanoporous membranes 56 metal-polymer composites processing 114 metal-semiconductor junctions 42 metallic glass nanocomposites 34 mica 133 mica type silicates (MTS) 78 micellar routes 174 micelles 114 microcracks 27 see also: cracking microstructures 59 milling 6 mineralization 158 ff mixing 111 ff mobility 96 modeling nanocomposites 215–222 moduli 122 monolayers 91 monomer adsorption 120 monosilicic acid 64 montmorillonite + + – clay layers 133 + + – organically-modified 126, 130 + + – polymer-filled nanocomposites 90 ff, 103, 107 montmorillonite/HDPE nanocomposites 99 + + $\mathrm{MoS}_{2}$ 24 + + $\mathrm{MoSi_{2}}$ 68 mullite, SiC/zirconia toughened 19 multilayer films 23 f + +multilayer nanocomposites 59 ff multiphase nanowires 42 multiscale modeling 217 ff multiwalled nanotubes (MWNT) + + – alumina 32 + + – polymer 81 ff, 123 + +nano bio composites 155–214 nanocermets 7 nanocomposites applications 44 ff, 57 ff nanocracks 27 see also: cracking nanofibrils 86 nanofillers 122 – polymeric 80 f nanoparticle polymer composites 111 ff nanoporous structures 53 ff nanotube carbon composites 31 ff nanotubes + + – modeling 215 + + – polymer composites 117, 122 + + – processing 85 ff, 100 nanowires 35 natural nano bio composites 155–214 nitrides 29 nucleation + + – encapsulated composite nanosystems 36 + + – magnetic nanocomposites 60 Nylon 91–107, 125–136 + +# o + +octadecylamine groups 117 ff oleyl ester 183 ff oligo(ethylene oxide) 179 oligo(vinyl alcohol) 179 oligomers 54 optical applications 46 ff optical properties 66, 138 ff organic matrices 158 organic nanotubes 90 organo transition metal complexes 10 organosilanes 10 organosilicon polymers 54 Ostwald ripening 164 oxalic acid 56 oxidation resistant coating 68 + +# p + +pairing, biocomposites 168 palladium colloids 116 particle dispersed magnetic nanocomposites 57 f + +particulate-reinforced metal matrix nanocomposites 14 ff PbS 186 PDMS 135 percolation effects + + – bulk ceramic nanocomposites 22 + + – inorganic nanocomposites 50 ff permeability 78, 133 permittivity 138 perovskite dispersed composites 67 phages 170 phase diagrams + + – amphiphiles 179 f + + – Si/B//C/N 5 phosphoric acid 56 photo electrochemical properties 65 photo luminescence + + – encapsulated composite nanosystems 42 + + – inorganic nanocomposites 46 + + – silica aerogels 10 photonic bandgap materials 197 phyllosilicates 90 physical properties 122 ff physical vapor deposition (PVC) 26, 29 piezoelectric materials 4 pitches 31 plasma activation 118 plasma-assisted chemical vapor deposition (PACVD) 26 plasma resonance frequency 48 plasma spraying 12 plastic deformation + + – bulk ceramic nanocomposites 21 + + – hard coatings 27 + + – metal nanocomposites 7 + + – polymer filled nanocomposites 128 platelets 3 platelike fillers 77 ff, 90 ff, 124 poly(ethylene oxide) (PEO) 110 poly acetonitrile 28 polyamide matrices 107 polyaniline 203 poly ca pro lac tone 78, 134 polyelectrolyte matrices 110 poly ether ether ketone matrix 132 polyethylene + + – matrices 108 + + – zirconia-filled 139 poly hydro methyl sila zane pyrolysis 5 polyimides 78, 103, 132–137 – matrices 107 polyisoprene 196 polylysine 167 + +polymer-based/filled nanocomposites 77–153 polymer coatings 119 f polymer films 196 polymer precursors + + – hard coatings 28 + + – processing 20 polymeric membranes, nanoporous 53 poly methyl a cry late 127, 137 poly methyl a cry late/polystyrene matrices 108 poly methyl phenyl sila zane pyrolysis 20 polypeptide chains 163, 170 polypropylene 103, 108 polypyrrole 203 polystyrene 123, 127, 203 + + – biocomposites 203 + + – films 97 ff polystyrene-polyisoprene block copolymer templates 196 polyurethane matrices 109 pore sizes 53, 56 powder metallurgy 4, 14 precursors + + – biocomposites 156 ff, 174, 187 ff + + – bulk ceramic nanocomposites 19 ff + + – hard coatings 26, 31 + + – metal nanocomposites 8 ff, 114 + + – nanoporous membranes 54 + + – polymer-filled nanocomposites 86, 95 processing polymer nanocomposites 100 protein assemblies 169 ff protein layers 160 pulsed-laser deposition + + – bulk ceramic nanocomposites 19 + + – hard coatings 27 purity 88 ff pyrolysis + + – poly hydro methyl sila zane 5 + + – poly methyl phenyl s ilia zane 20 pyrolytic graphite 18 PZT powder 49 + +# q + +quadrupole splitting 191 quantum confinement + + – biocomposites 174 + + – optical applications 46 f + + – semiconductors 66 quantum dots + + – biocomposites 174 + + – magnetic nanocomposites 61 + +# r + +rapid solidification process (RSP) + +– carbon nanotube-based nanocomposites 33 + + – metal matrix nanocomposites 15 Rayleigh scattering138, 140reaction sintering 20 reactive magnetron sputtering 27 refractive indices + + – biocomposites 205 + + – inorganic nanocomposites 48 rehybridization 84 reinforcement 14 ff relaxation spectra + + – dielectric 217 + + – polymer-filled nanocomposites 96, 131 f resins 31 resistivity 82, 138 resorcinol fomaldehyde monomers 8 rf magnetron sputtering 60, 63 rf plasma deposition 69 rheocasting 14 rheopexy 106 RNA 165 f Rouse model 217 rubber matrices 110 + +# + +S layers 160, 165 ff saponite 133 sapphire-polystyrene interface 99 saturation magnetization 7 sea urchin 161 segregation 29 selenium chalcogenides 182, 205 self-assembled quantum dot arrays 61 self-assembly routes 173 self-propagating combustive reactions 6 self reinforced silicon nitrides 18 semiconductors + + – biocomposites 172 f, 182 f, 199 + + – nanoporous membranes 56 shear modulus 25 silane coupling 119 silica + + – biocomposites 159, 171 + + – coatings 64 + + – fumed 77 + + – matrix 46 + + – metal nanocomposites 8 + + – modeling 219 + + – polymer-filled nanocomposites 93 silica aerogel composites 10 silica-Nylon 132 silica-polyvinyl acetate nanocomposites 131 silicates 90 ff silicon, nanoporous membranes 56 + +silicon-boron-carbon-nitrogen system 5 silicon carbide 4 silicon carbide-zirconia-toughed mullite 19 silicon composites 10 silicon nitrides + + – metal nanocomposites 4 + + – self-reinforced 18 silicone rubber matrices 110 silver-polystyrene matrix 142 single-walled nanotubes (SWNT) 39, 78–87, 123 sintering 4, 20 small molecule attachment 118 smectic clays 90 ff, 140 sodium octanante 179 sol-gel methods + + – biocomposites 198 + + – inorganic coatings 121 + + – metal nanocomposites 4, 8 ff + + – polymer-filled nanocomposites 95 solution mixing 112 solution phase synthesis 4, 173 sonication 95, 118 spider silk 157 spinning 58 spinodal decomposition 29 splat cooling 58 spraying + + – metal matrix nanocomposites 14 + + – metal nanocomposites 11 ff + + – nanoparticle/polymer composites 111 sputtering 60 squeeze casting 14 stability 135 StoneWales defect 85 straight pore alumina 56 strain-controlled diffusion 220 strain hardening 106 strain-to-failure ratio 123, 130 strength + + – metal nanocomposites 3, 17 + + – polymer-filled nanocomposites 82 strengthening 22 stress production model 217 stress-strain curves 122, 127 strong field behavior 49 subgrains 7 substrate variables 12 sulfuric acid 56 supercritical drying 8 superlattices 183 super para magnetic fraction 7 superparamagnetism 62 superplasticity 3 + +surface area 79 surfactants 50 + + – amphiphiles 181 + + – biocomposites 171, 181 f + + – polymer nanocomposites 101 swelling 135 synthesized nanoparticles, biologically 159 f synthetic mica clay layers 133 + +# t + +Ta/W films 160 tactoids 90 f, 106 telescopic failure 84 tellurium chalcogenides 182 templates + + – alumina 35 + + – biocomposites 161197 tensile strength 78 ternary systems, hard coatings 29 tetra ethyl ortho silicate (TEOS) 69, 95, 112 TGA analysis 5 thermal expansion coefficient 135 thermal shock resistance 3 thermal spraying 11 ff, 111 thermal stability + + – magnetic nanocomposites 59 + + – polymer filled nanocomposites 136 thermoelectric junctions 43 thermoplastic polymers 78, 96, 215 thin films 23 f – liquid crystal templating 194 ff thiol 142, 168 Ti/B alloys 15 Ti/Si alloys 15 TiC 24 TiC/Al 16 TiC/TiCN 19 TiN 24 titania + + – biocomposites 199 + + – colloids 65 + + – epoxy composites 135 + + – polymer-filled nanocomposites 94 toluene + + – biocomposites 203 + + – polymer-filled nanocomposites 89 torch variables 12 tortuosity factor 133 + +toughness + + – metal matrix nanocomposites 17 + + – polymer filled nanocomposites 122, 127 transfer mechanisms 122 transition metal composites + + – encapsulated 41 + + – silica aerogels 10 transition metal dichalcogenides 93 transparent dielectrics 48 transport phenomena 50 ff tribological applications 19 tribological properties 25 trilayers 91 tube fillers, polymeric 80 ff twisted fibers 86 + +# + +valence band 85 validation 221 vapor phase infiltration 9 vaporization 173 Vickers hardness 33 void formation 129 volume fractions 138 + +# + +WatsonCrick pairing 168 WC/C, hard coatings 24 WC/Co coatings 13, 28 WC/TiC/Co coatings 28 weak field behavior 49 wear properties + + – polymer filled 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Chivers, BSc, PGCEd, MSc, MBCS Rhymney Consulting UK + +Jane Sleightholme, MSc, MBCS Kings College London UK + +British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library + +Library of Congress Control Number: 2005931518 + +ISBN-10: 1-84628-053-2 eISBN 1-84628-054-0 Printed on acid-free paper ISBN-13: 978-1-84628-053-5 + +$\copyright$ Springer-Verlag London Limited 2006 + +Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. + +The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. + +The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. + +Printed in the United States of America (MVY) + +Springer Science + Business Media springeronline.com + +# Acknowlegement + +The material in the book has evolved firstly from our combined experience of working in Computing Services within the University of London at + +• King's College, IDC (1986–2002) and JS (1985 to date) • Chelsea College, JS (1978–1985) • Imperial College, IDC (1978–1986) + +in the teaching, advice and support of Fortran and related areas, and secondly in the provision of commercial training courses. The following are some of the or- ganisations we've provided training for: + +• AWE, Aldermaston. • Centre for Ecology and Hydrology, Wallingford. • Environment Agency, Worthing. • The Met Office, Bracknell and Exeter. • QinetiQ, Farnborough. • Rolls Royce, Derby. • Veritas DGC Ltd., Crawley. • Westland Helicopters, Yeovil. + +The examples in the book are based on what will work with compilers that support the Fortran 90 and 95 standards and also support ISO TR 15580 and 15581. At the time of writing this book there are no compilers that fully support the Fortran 2003 standard. + +Thanks are due to: + +• The staff and students at King's College, Chelsea College and Imperial College. + +• The people who have attended the commercial courses. Its been great fun teaching you and things have been very lively at times. • The people on the Fortran 90 list and comp.lang.fortran. Access to the ex- pertise of several hundred people involved in the use and development of Fortran on a daily basis across a wide range of disciplines is inestimable. • The people at NAG for the provision of the Fortran 95 compilers and Nag Tools on the enclosed cd. • The staff and facilities at PTR Associates. It is a pleasure training there. • The patience of our families during the time required to develop the courses upon which this book is based and whilst preparing the cam- era-ready copy. • Finally Rebecca Mowat, Joanne Cooling, Helen Desmond and Beverley Ford at Springer for their enthusiasm and encouragement! + +Our King's home page is: + +• http://www.kcl.ac.uk/fortran + +All of the program examples can be found there. If you would like to contact us our email addresses are: Ian D Chivers: ian.chivers@chiversandbryan.co.uk Jane Sleightholme: jane.sleightholme@kcl.ac.uk + +# Contents + +1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Introduction to Computer Systems . . . . . . . . . . . . . . . . . . 9 + + 2.1 The core of a computer system . . . . . . . . . . . . . . . . . . 10 + + 2.1.1 Central processor unit — CPU. . . . . . . . . . . . . . . . . . . 10 + + 2.1.2 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 + + 2.1.3 Bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 + + 2.2 Other components of a computer system . . . . . . . . . . . . . . 11 + + 2.2.1 Disks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 + + 2.2.2 Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 + + 2.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 + + 2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 + + 2.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Introduction to Operating Systems . . . . . . . . . . . . . . . . . 15 + + 3.1 History of operating systems. . . . . . . . . . . . . . . . . . . . 16 + + 3.1.1 The 1940s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 + + 3.1.2 The 1950s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 + + 3.1.3 The 1960s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 + + 3.1.4 The 1960s and 1970s . . . . . . . . . . . . . . . . . . . . . . . 16 + + 3.1.5 The 1970s, 1980s, and 1990s . . . . . . . . . . . . . . . . . . . 17 + + 3.2 Networking. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 + + 3.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 + + 3.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Introduction to Using a Computer System. . . . . . . . . . . . . . 19 + + 4.1 Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 + + 4.2 Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 + + 4.3 Single-user systems . . . . . . . . . . . . . . . . . . . . . . . . 20 + + 4.4 Networked systems . . . . . . . . . . . . . . . . . . . . . . . . 20 + + 4.5 Multiuser systems. . . . . . . . . . . . . . . . . . . . . . . . . 21 + + 4.6 Other useful things to know . . . . . . . . . . . . . . . . . . . . 21 4.7 Common methods of using computer systems to develop Fortran programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 + + 4.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Introduction to Problem Solving . . . . . . . . . . . . . . . . . . 25 + + 5.1 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . 26 + + 5.2 Artificial language . . . . . . . . . . . . . . . . . . . . . . . . 27 + + 5.2.1 Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 + + 5.3 Resumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 + + 5.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 + + 5.4.1 Top-down . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 + + 5.4.2 Bottom-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 + + 5.4.3 Stepwise refinement. . . . . . . . . . . . . . . . . . . . . . . . 29 + + 5.4.4 Modular programming . . . . . . . . . . . . . . . . . . . . . . 29 + + 5.4.5 Object oriented programming . . . . . . . . . . . . . . . . . . . 29 + + 5.5 Systems analysis and design . . . . . . . . . . . . . . . . . . . . 30 + + 5.5.1 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . 30 + + 5.5.2 Feasibility study and fact finding . . . . . . . . . . . . . . . . . 30 + + 5.5.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 + + 5.5.4 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 + + 5.5.5 Detailed design . . . . . . . . . . . . . . . . . . . . . . . . . . 31 + + 5.5.6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 31 + + 5.5.7 Evaluation and testing. . . . . . . . . . . . . . . . . . . . . . . 31 + + 5.5.8 Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 + + 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 + + 5.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 + + 5.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6 Introduction to Programming Languages . . . . . . . . . . . . . . 35 + + 6.1 Some early theoretical work . . . . . . . . . . . . . . . . . . . . 36 + + 6.2 What is a programming language? . . . . . . . . . . . . . . . . . 36 + + 6.3 Program language development and engineering . . . . . . . . . . 36 + + 6.4 The early days . . . . . . . . . . . . . . . . . . . . . . . . . . 36 + + 6.4.1 Fortran — The Early Days . . . . . . . . . . . . . . . . . . . . 37 + + 6.4.2 Fortran 77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 + + 6.4.3 Cobol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 + + 6.4.4 Algol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 + + 6.5 Chomsky and program language development . . . . . . . . . . . 39 + + 6.6 Lisp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 + + 6.7 Snobol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 + + 6.8 Second-generation languages . . . . . . . . . . . . . . . . . . . 40 + + 6.8.1 PL/1 and Algol 68 . . . . . . . . . . . . . . . . . . . . . . . . 40 + + 6.8.2 Simula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 + + 6.8.3 Pascal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 + +6.8.4 APL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 + + 6.8.5 Basic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 + + 6.8.6 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 + + 6.9 Some other strands in language development . . . . . . . . . . . . 42 + + 6.9.1 Abstraction, stepwise refinement and modules . . . . . . . . . . . 42 + + 6.9.2 Structured programming. . . . . . . . . . . . . . . . . . . . . . 42 + + 6.9.3 Standardisation . . . . . . . . . . . . . . . . . . . . . . . . . . 42 + + 6.10 Ada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 + + 6.11 Modula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 + + 6.12 Modula 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 + + 6.13 Other language developments . . . . . . . . . . . . . . . . . . . 44 + + 6.13.1 Logo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 + + 6.13.2 Postscript, TeX and LaTeX . . . . . . . . . . . . . . . . . . . . 45 + + 6.13.3 Prolog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 + + 6.13.4 SQL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 + + 6.13.5 ICON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 + + 6.14 Object orientated programming — OOP . . . . . . . . . . . . . . 46 + + 6.14.1 Oberon and Oberon 2 . . . . . . . . . . . . . . . . . . . . . . . 46 + + 6.14.2 Smalltalk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 + + 6.14.3 $\mathrm{C++}$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 + + 6.14.4 Java . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 + + 6.14.5 Visual Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 + + 6.14.6 C# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 + + 6.15 Fortran 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 + + 6.16 Fortran 1995 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 + + 6.17 ISO technical reports TR15580 and TR15581 . . . . . . . . . . . 52 + + 6.18 Fortran 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 + + 6.19 DTR 19767 enhanced module facilities. . . . . . . . . . . . . . . 53 + + 6.20 Internet resources . . . . . . . . . . . . . . . . . . . . . . . . . 54 + + 6.20.1 Standards information . . . . . . . . . . . . . . . . . . . . . . . 54 + + 6.20.2 Fortran discussion lists . . . . . . . . . . . . . . . . . . . . . . 55 + + 6.20.3 Other sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 + + 6.21 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 + + 6.22 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7 Introduction to Programming . . . . . . . . . . . . . . . . . . . . 63 + + 7.1 Language strengths and weaknesses . . . . . . . . . . . . . . . . 64 + + 7.2 Elements of a programming language . . . . . . . . . . . . . . . 64 + + 7.2.1 Data description statements . . . . . . . . . . . . . . . . . . . . 65 + + 7.2.2 Control structures . . . . . . . . . . . . . . . . . . . . . . . . . 65 + + 7.2.3 Data-processing statements . . . . . . . . . . . . . . . . . . . . 65 + + 7.2.4 Input and output (I/O) statements . . . . . . . . . . . . . . . . . 65 + + 7.3 Variables — name, type and value . . . . . . . . . . . . . . . . . 68 7.4 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 + + 7.5 Some more Fortran rules . . . . . . . . . . . . . . . . . . . . . 71 + + 7.6 Fortran character set . . . . . . . . . . . . . . . . . . . . . . . 72 + + 7.7 Good programming guidelines . . . . . . . . . . . . . . . . . . . 73 + + 7.8 Compilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 + + 7.9 Program development . . . . . . . . . . . . . . . . . . . . . . . 74 + + 7.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 + + 8.1 Rounding and truncation . . . . . . . . . . . . . . . . . . . . . 81 + + 8.2 Time taken for light to travel from the Sun to Earth. . . . . . . . . 83 + + 8.3 The PARAMETER statement . . . . . . . . . . . . . . . . . . . 84 + + 8.4 Range, precision and size of numbers . . . . . . . . . . . . . . . 85 + + 8.5 Health warning: optional reading, beginners are advised to leave until later . . . . . . . . . . . . . . . . . . . . . . . . . . 88 + + 8.5.1 Selecting different INTEGER kind types . . . . . . . . . . . . . . 90 + + 8.5.2 Selecting different REAL kind types . . . . . . . . . . . . . . . . 91 + + 8.5.3 Specifying kind types for literal integer and real constants. . . . . . 91 + + 8.5.4 Positional number systems. . . . . . . . . . . . . . . . . . . . . 92 + + 8.5.5 Bit data type and representation model . . . . . . . . . . . . . . . 92 + + 8.5.6 Integer data type and representation model . . . . . . . . . . . . . 93 + + 8.5.7 Real data type and representation model . . . . . . . . . . . . . . 93 + + 8.5.8 IEEE 754. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 + + 8.5.9 Testing the numerical representation of different kind types on a system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 + + 8.5.10 Binary representation of different integer kind type numbers . . . . 98 + + 8.5.11 Binary representation of a real number . . . . . . . . . . . . . . 100 + + 8.5.12 Summary of how to select the appropriate kind type . . . . . . . . 101 + + 8.6 Variable status. . . . . . . . . . . . . . . . . . . . . . . . . . 101 + + 8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 + + 8.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 + + 8.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9 Arrays 1: Some Fundamentals . . . . . . . . . . . . . . . . . . . 107 + + 9.1 Tables of data . . . . . . . . . . . . . . . . . . . . . . . . . . 108 + + 9.1.1 Telephone directory . . . . . . . . . . . . . . . . . . . . . . . 108 + + 9.1.2 Book catalogue . . . . . . . . . . . . . . . . . . . . . . . . . 108 + + 9.1.3 Examination marks or results. . . . . . . . . . . . . . . . . . . 109 + + 9.1.4 Monthly rainfall . . . . . . . . . . . . . . . . . . . . . . . . . 109 + + 9.2 Arrays in Fortran . . . . . . . . . . . . . . . . . . . . . . . . 110 + + 9.3 The DIMENSION attribute. . . . . . . . . . . . . . . . . . . . 110 + + 9.4 An index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 + + 9.5 Control structure. . . . . . . . . . . . . . . . . . . . . . . . . 111 + + 9.6 Monthly rainfall . . . . . . . . . . . . . . . . . . . . . . . . . 111 + +9.6.1 Example 1: Rainfall . . . . . . . . . . . . . . . . . . . . . . . 112 + + 9.7 People's weights . . . . . . . . . . . . . . . . . . . . . . . . . 113 + + 9.7.1 Example 2: Setting array size with a parameter . . . . . . . . . . 114 + + 9.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 + + 9.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 10 Arrays 2: Further Examples . . . . . . . . . . . . . . . . . . . . 119 + + 10.1 Varying the array size at run time . . . . . . . . . . . . . . . . 120 + + 10.2 Higher-dimension arrays . . . . . . . . . . . . . . . . . . . . . 121 + + 10.2.1 A map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 + + 10.2.2 Example 3: Sensible tabular output . . . . . . . . . . . . . . . . 123 + + 10.2.3 Example 4: Average of three sets of values . . . . . . . . . . . . 124 + + 10.2.4 Example 5: Booking arrangements in a theatre or cinema . . . . . 125 + + 10.3 Additional forms of the DIMENSION attribute and DO loop statement . . . . . . . . . . . . . . . . . . . . . . . . . . 126 + + 10.3.1 Example 6: Voltage from $-20$ to $+20$ volts . . . . . . . . . . . . 126 + + 10.3.2 Example 7: Longitude from $-180$ to $+180$ . . . . . . . . . . . . . 127 + + 10.3.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 + + 10.4 The DO loop and straight repetition. . . . . . . . . . . . . . . . 127 + + 10.4.1 Example 8: Table of temperatures . . . . . . . . . . . . . . . . 127 + + 10.4.2 Example 9: Means and standard deviations . . . . . . . . . . . . 128 + + 10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 + + 10.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 11 Whole Array and Additional Array Features. . . . . . . . . . . . 133 + + 11.1 Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . 134 + + 11.1.1 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 + + 11.1.2 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 + + 11.1.3 Extent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 + + 11.1.4 Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 + + 11.1.5 Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 + + 11.1.6 Conformable . . . . . . . . . . . . . . . . . . . . . . . . . . 134 + + 11.1.7 Array element ordering. . . . . . . . . . . . . . . . . . . . . . 134 + + 11.2 Whole array manipulation . . . . . . . . . . . . . . . . . . . . 135 + + 11.2.1 Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 + + 11.2.2 Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 + + 11.3 Array sections . . . . . . . . . . . . . . . . . . . . . . . . . . 138 + + 11.3.1 Rank 1 array example . . . . . . . . . . . . . . . . . . . . . . 138 + + 11.3.2 Rank 2 array example . . . . . . . . . . . . . . . . . . . . . . 138 + + 11.4 Array constructors . . . . . . . . . . . . . . . . . . . . . . . . 140 + + 11.4.1 Rank 1 array example — explicit values . . . . . . . . . . . . . 140 + + 11.4.1.1 Rank 1 array example and implied DO loop. . . . . . . . . . . . 141 + + 11.4.1.2 Rank 1 array example and the DOT_PRODUCT intrinsic . . . . . 141 + + 11.4.2 Rank 1 example with step size of 2 in implied DO loop . . . . . . 143 11.4.3 Rank 1 array and the SUM intrinsic function . . . . . . . . . . . 144 + + 11.4.4 Rank 2 arrays and the SUM intrinsic function . . . . . . . . . . . 145 + + 11.5 Masked array assignment and the WHERE statement . . . . . . . 146 + + 11.5.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 + + 11.6 The FORALL statement and FORALL construct . . . . . . . . . 147 + + 11.6.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 + + 11.6.2 Array element ordering and physical and virtual memory . . . . . 148 + + 11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 + + 11.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 + + 11.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 149 12 Output of Results . . . . . . . . . . . . . . . . . . . . . . . . . 151 + + 12.1 Integers — I format or edit descriptor. . . . . . . . . . . . . . . 152 + + 12.2 Reals — F format or edit descriptor. . . . . . . . . . . . . . . . 155 + + 12.2.1 Metric and imperial conversion . . . . . . . . . . . . . . . . . . 156 + + 12.2.2 Overflow and underflow . . . . . . . . . . . . . . . . . . . . . 156 + + 12.3 Reals — E format or edit descriptor . . . . . . . . . . . . . . . 158 + + 12.3.1 Simple E format example . . . . . . . . . . . . . . . . . . . . 159 + + 12.4 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 + + 12.5 Characters — A format or edit descriptor . . . . . . . . . . . . . 160 + + 12.5.1 Headings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 + + 12.6 Mixed type output in a FORMAT statement. . . . . . . . . . . . 162 + + 12.7 Common mistakes . . . . . . . . . . . . . . . . . . . . . . . . 162 + + 12.8 OPEN (and CLOSE) . . . . . . . . . . . . . . . . . . . . . . . 163 + + 12.8.1 The OPEN statement . . . . . . . . . . . . . . . . . . . . . . 163 + + 12.8.2 Writing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 + + 12.9 Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 + + 12.10 Some more examples . . . . . . . . . . . . . . . . . . . . . . 167 + + 12.11 Implied DO loops and array sections for array output . . . . . . . 168 + + 12.12 Formatting for a line printer . . . . . . . . . . . . . . . . . . . 170 + + 12.12.1 Mechanics of carriage control . . . . . . . . . . . . . . . . . . 171 + + 12.12.2 Generating a new line on both line printers and terminals . . . . . 172 + + 12.13 Timing of writing formatted files . . . . . . . . . . . . . . . . . 173 + + 12.14 Timing of writing unformatted files. . . . . . . . . . . . . . . . 174 + + 12.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 + + 12.16 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 13 Reading in Data . . . . . . . . . . . . . . . . . . . . . . . . . . 179 + + 13.1 Reading from the terminal or keyboard versus reading from files . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 + + 13.2 Fixed fields on input . . . . . . . . . . . . . . . . . . . . . . . 180 + + 13.2.1 Integers and the I format . . . . . . . . . . . . . . . . . . . . . 180 + + 13.2.2 Reals and the F format . . . . . . . . . . . . . . . . . . . . . . 181 + + 13.2.3 Reals and the E Format . . . . . . . . . . . . . . . . . . . . . 182 + +13.3 Blanks, nulls and zeros . . . . . . . . . . . . . . . . . . . . . 185 + + 13.4 Characters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 + + 13.5 Skipping spaces and lines . . . . . . . . . . . . . . . . . . . . 187 + + 13.6 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 + + 13.7 File manipulation again . . . . . . . . . . . . . . . . . . . . . 188 + + 13.8 Reading using array sections . . . . . . . . . . . . . . . . . . . 189 + + 13.9 Timing of reading formatted files . . . . . . . . . . . . . . . . . 190 + + 13.10 Timing of reading unformatted files. . . . . . . . . . . . . . . . 191 + + 13.11 Errors when reading . . . . . . . . . . . . . . . . . . . . . . . 192 + + 13.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 + + 13.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 + +# 14 Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 + +14.1 Data files in Fortran . . . . . . . . . . . . . . . . . . . . . . . 196 + + 14.2 Summary of options on OPEN . . . . . . . . . . . . . . . . . . 198 + + 14.3 More foolproof I/O . . . . . . . . . . . . . . . . . . . . . . . 200 + + 14.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 + + 14.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 15 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 + + 15.1 An introduction to predefined functions and their use . . . . . . . 204 + + 15.1.1 Example 1: Simple function usage . . . . . . . . . . . . . . . . 205 + + 15.2 Generic functions . . . . . . . . . . . . . . . . . . . . . . . . 205 + + 15.2.1 Example 2: The ABS generic function . . . . . . . . . . . . . . 206 + + 15.3 Elemental functions . . . . . . . . . . . . . . . . . . . . . . . 206 + + 15.3.1 Example 3: Elemental function use . . . . . . . . . . . . . . . . 206 + + 15.4 Transformation al functions . . . . . . . . . . . . . . . . . . . . 206 + + 15.4.1 Example 4: Simple transformation al use . . . . . . . . . . . . . 207 + + 15.4.2 Example 5: Intrinsic DOT_PRODUCT use . . . . . . . . . . . . 207 + + 15.5 Notes on function usage . . . . . . . . . . . . . . . . . . . . . 207 + + 15.6 Example 6: Easter . . . . . . . . . . . . . . . . . . . . . . . . 208 + + 15.7 Complete list of predefined functions . . . . . . . . . . . . . . . 210 + + 15.7.1 Inquiry functions . . . . . . . . . . . . . . . . . . . . . . . . 210 + + 15.7.2 Transfer and conversion functions . . . . . . . . . . . . . . . . 210 + + 15.7.3 Computational functions . . . . . . . . . . . . . . . . . . . . . 211 + + 15.7.4 Array functions . . . . . . . . . . . . . . . . . . . . . . . . . 211 + + 15.7.5 Predefined subroutines . . . . . . . . . . . . . . . . . . . . . . 211 + + 15.8 Supplying your own functions . . . . . . . . . . . . . . . . . . 212 + + 15.8.1 Example 7: Simple user defined function . . . . . . . . . . . . . 212 + + 15.9 An introduction to the scope of variables and local variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 + + 15.10 Recursive functions . . . . . . . . . . . . . . . . . . . . . . . 214 + + 15.10.1 Example 8: Recursive factorial evaluation. . . . . . . . . . . . . 215 15.11 Example 9: Recursive version of GCD . . . . . . . . . . . . . . 216 + + 15.12 Example 10: After removing recursion . . . . . . . . . . . . . . 217 + + 15.13 Pure functions . . . . . . . . . . . . . . . . . . . . . . . . . . 218 + + 15.14 Elemental functions . . . . . . . . . . . . . . . . . . . . . . . 218 + + 15.15 Internal functions . . . . . . . . . . . . . . . . . . . . . . . . 218 + + 15.15.1 Example 11: Stirling's approximation . . . . . . . . . . . . . . . 218 + + 15.16 Resumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 + + 15.17 Function syntax . . . . . . . . . . . . . . . . . . . . . . . . . 220 + + 15.18 Rules and restrictions . . . . . . . . . . . . . . . . . . . . . . 220 + + 15.19 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 + + 15.20 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 221 + + 15.20.1 Recursion and problem solving . . . . . . . . . . . . . . . . . . 222 16 Control Structures . . . . . . . . . . . . . . . . . . . . . . . . . 223 + + 16.1 Selection among courses of action . . . . . . . . . . . . . . . . 224 + + 16.1.1 The BLOCK IF statement . . . . . . . . . . . . . . . . . . . . 225 + + 16.1.2 Example 1: Quadratic roots. . . . . . . . . . . . . . . . . . . . 227 + + 16.1.3 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 + + 16.1.4 Example 2: Date calculation . . . . . . . . . . . . . . . . . . . 228 + + 16.1.5 The CASE statement. . . . . . . . . . . . . . . . . . . . . . . 229 + + 16.1.6 Example 3: Simple calculator. . . . . . . . . . . . . . . . . . . 230 + + 16.1.7 Example 4: Counting vowels, consonants, etc.. . . . . . . . . . . 231 + + 16.2 The three forms of the DO statement . . . . . . . . . . . . . . . 232 + + 16.2.1 Example 5: Sentinel usage . . . . . . . . . . . . . . . . . . . . 232 + + 16.2.2 CYCLE and EXIT . . . . . . . . . . . . . . . . . . . . . . . . 234 + + 16.2.3 Example 6: e\*\* x evaluation . . . . . . . . . . . . . . . . . . . 234 + + 16.2.4 Example 7: Wave breaking on an offshore reef . . . . . . . . . . 235 + + 16.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 + + 16.3.1 Control structure formal syntax . . . . . . . . . . . . . . . . . . 237 + + 16.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 + + 16.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 240 17 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 + + 17.1 Character input . . . . . . . . . . . . . . . . . . . . . . . . . 243 + + 17.2 Character operators . . . . . . . . . . . . . . . . . . . . . . . 244 + + 17.3 Character substrings . . . . . . . . . . . . . . . . . . . . . . . 245 + + 17.4 Character functions . . . . . . . . . . . . . . . . . . . . . . . 247 + + 17.5 Collating sequence. . . . . . . . . . . . . . . . . . . . . . . . 248 + + 17.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 + + 17.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 18 Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 + + 18.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 + + 18.2 Complex and kind type . . . . . . . . . . . . . . . . . . . . . 256 + +18.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 + +19.1 I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 + + 19.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 + + 19.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 20 User Defined Types . . . . . . . . . . . . . . . . . . . . . . . . 263 + + 20.1 Example 1: Dates . . . . . . . . . . . . . . . . . . . . . . . . 264 + + 20.2 Type definition . . . . . . . . . . . . . . . . . . . . . . . . . 264 + + 20.3 Variable definition . . . . . . . . . . . . . . . . . . . . . . . . 265 + + 20.4 Example 2: Address lists . . . . . . . . . . . . . . . . . . . . . 265 + + 20.5 Example 3: Nested user defined types. . . . . . . . . . . . . . . 266 + + 20.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 + + 20.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 268 21 An Introduction to Pointers . . . . . . . . . . . . . . . . . . . . 269 + + 21.1 Some basic pointer concepts . . . . . . . . . . . . . . . . . . . 270 + + 21.2 The ASSOCIATED intrinsic function . . . . . . . . . . . . . . . 272 + + 21.2.1 CVF 6.6C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 + + 21.2.2 Intel, Windows, 8.1 . . . . . . . . . . . . . . . . . . . . . . . 272 + + 21.2.3 Lahey, Windows 5.70f . . . . . . . . . . . . . . . . . . . . . . 272 + + 21.2.4 NAG, Windows, 4.2 . . . . . . . . . . . . . . . . . . . . . . . 273 + + 21.2.5 Salford 4.6.0 . . . . . . . . . . . . . . . . . . . . . . . . . . 273 + + 21.3 Referencing A and B before assignment. . . . . . . . . . . . . . 273 + + 21.3.1 CVF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 + + 21.3.2 Intel, Windows 8.1. . . . . . . . . . . . . . . . . . . . . . . . 274 + + 21.3.3 Lahey, Windows 5.70f . . . . . . . . . . . . . . . . . . . . . . 275 + + 21.3.4 NAG, Windows 4.2 . . . . . . . . . . . . . . . . . . . . . . . 275 + + 21.3.5 Salford 4.6.0 . . . . . . . . . . . . . . . . . . . . . . . . . . 275 + + 21.4 The NULL intrinsic . . . . . . . . . . . . . . . . . . . . . . . 275 + + 21.5 Assignment via $=$ . . . . . . . . . . . . . . . . . . . . . . . . 276 + + 21.6 Singly linked list. . . . . . . . . . . . . . . . . . . . . . . . . 278 + + 21.7 Reading in an arbitrary quantity of numeric data. . . . . . . . . . 280 + + 21.8 Arrays of pointers . . . . . . . . . . . . . . . . . . . . . . . . 283 + + 21.9 Arrays of pointers and variable sized data sets — 1 . . . . . . . . 284 + + 21.10 Arrays of pointers and variable sized data sets — 2 . . . . . . . . 285 + + 21.11 Memory leak examples. . . . . . . . . . . . . . . . . . . . . . 285 + + 21.12 Nonstandard pointer examples . . . . . . . . . . . . . . . . . . 288 + + 21.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 22 Introduction to Subroutines . . . . . . . . . . . . . . . . . . . . 295 + + 22.1 Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 + + 22.1.1 Defining a subroutine . . . . . . . . . . . . . . . . . . . . . . 298 + + 22.1.2 Referencing a subroutine . . . . . . . . . . . . . . . . . . . . . 299 22.1.3 Dummy arguments or parameters and actual arguments . . . . . . 299 + + 22.1.4 Intent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 + + 22.1.5 Local variables . . . . . . . . . . . . . . . . . . . . . . . . . 299 + + 22.1.6 Local variables and the SAVE attribute . . . . . . . . . . . . . . 300 + + 22.1.7 Scope of variables . . . . . . . . . . . . . . . . . . . . . . . . 300 + + 22.1.8 Status of the action carried out in the subroutine. . . . . . . . . . 300 + + 22.2 Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 + + 22.3 Example 3 — Quadratic example with interface blocks . . . . . . 301 + + 22.4 Example 4 — Quadratic example and the CONTAINS statement . . 304 + + 22.5 Why bother? . . . . . . . . . . . . . . . . . . . . . . . . . . 306 + + 22.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 + + 22.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 23 Subroutines: 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 + + 23.1 More on parameter passing. . . . . . . . . . . . . . . . . . . . 310 + + 23.1.1 Explicit-shape array . . . . . . . . . . . . . . . . . . . . . . . 310 + + 23.1.2 Assumed-shape array . . . . . . . . . . . . . . . . . . . . . . 310 + + 23.1.3 Deferred-shape array. . . . . . . . . . . . . . . . . . . . . . . 310 + + 23.1.4 Automatic arrays . . . . . . . . . . . . . . . . . . . . . . . . 310 + + 23.1.5 Assumed-size array — Fortran 77 style . . . . . . . . . . . . . . 310 + + 23.1.6 Adjustable arrays — Fortran 77 style . . . . . . . . . . . . . . . 311 + + 23.2 Common code example . . . . . . . . . . . . . . . . . . . . . 311 + + 23.3 Explicit-shape example. . . . . . . . . . . . . . . . . . . . . . 311 + + 23.4 Assumed-shape example . . . . . . . . . . . . . . . . . . . . . 313 + + 23.4.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 + + 23.5 Characters arguments and assumed-length dummy arguments . . . 315 + + 23.6 Rank 2 and higher arrays as parameters . . . . . . . . . . . . . 316 + + 23.6.1 Explicit-shape dummy arrays . . . . . . . . . . . . . . . . . . . 316 + + 23.6.2 Assumed-shape dummy array arguments . . . . . . . . . . . . . 319 + + 23.6.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 + + 23.6.4 Using the intrinsic functions MATMUL and TRANSPOSE . . . . 321 + + 23.7 Automatic arrays and median calculation . . . . . . . . . . . . . 322 + + 23.7.1 Internal subroutines and scope . . . . . . . . . . . . . . . . . . 325 + + 23.7.2 Timing the selection sort algorithm . . . . . . . . . . . . . . . . 325 + + 23.7.2.1 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 + + 23.8 Alternative median calculation algorithm . . . . . . . . . . . . . 327 + + 23.8.1 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 + + 23.9 Recursive subroutines — Quicksort. . . . . . . . . . . . . . . . 332 + + 23.9.1 Note — Interface blocks. . . . . . . . . . . . . . . . . . . . . 336 + + 23.9.2 Note — Recursive subroutine . . . . . . . . . . . . . . . . . . 337 + + 23.9.3 Note — Flexible design . . . . . . . . . . . . . . . . . . . . . 337 + + 23.9.4 Note — Timing information . . . . . . . . . . . . . . . . . . . 337 + + 23.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 + +23.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 + + 23.12 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 340 + + 23.13 Commercial numerical and statistical subroutine libraries . . . . . 340 24 An Introduction to Modules . . . . . . . . . . . . . . . . . . . . 341 + + 24.1 Modules for global data . . . . . . . . . . . . . . . . . . . . . 342 + + 24.2 Modules for precision specification and constant definition. . . . . 343 + + 24.2.1 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 + + 24.3 Modules for sharing arrays of data . . . . . . . . . . . . . . . . 345 + + 24.4 Modules for derived data types . . . . . . . . . . . . . . . . . . 346 + + 24.4.1 Person data type . . . . . . . . . . . . . . . . . . . . . . . . . 347 + + 24.5 Modules containing procedures — Quicksort example . . . . . . . 349 + + 24.6 Modules containing procedures — Statistics example . . . . . . . 353 + + 24.7 The solution of linear equations using Gaussian elimination . . . . 356 + + 24.7.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 + + 24.7.1.1 Module for kind type . . . . . . . . . . . . . . . . . . . . . . 361 + + 24.7.1.2 Deferred-shape arrays . . . . . . . . . . . . . . . . . . . . . . 361 + + 24.7.1.3 Intrinisic functions MAXVAL and MAXLOC. . . . . . . . . . . 361 + + 24.8 Notes on module usage and compilation . . . . . . . . . . . . . 361 + + 24.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 + + 24.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 + + 24.11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 363 25 Converting from Fortran 77 . . . . . . . . . . . . . . . . . . . . 365 + + 25.1 Deleted features . . . . . . . . . . . . . . . . . . . . . . . . . 366 + + 25.2 Obsolescent features . . . . . . . . . . . . . . . . . . . . . . . 366 + + 25.2.1 Arithmetic IF . . . . . . . . . . . . . . . . . . . . . . . . . . 366 + + 25.2.2 Real and double precision DO control variables . . . . . . . . . . 366 + + 25.2.3 Shared DO termination and non-ENDDO termination . . . . . . . 366 + + 25.2.4 Alternate RETURN . . . . . . . . . . . . . . . . . . . . . . . 367 + + 25.2.5 PAUSE statement . . . . . . . . . . . . . . . . . . . . . . . . 367 + + 25.2.6 ASSIGN and assigned GOTO statements . . . . . . . . . . . . . 367 + + 25.2.7 Assigned FORMAT statements . . . . . . . . . . . . . . . . . . 367 + + 25.2.8 H editing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 + + 25.3 Better alternatives . . . . . . . . . . . . . . . . . . . . . . . . 367 + + 25.4 Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 + + 25.5 Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 + + 25.6 Commercial conversion tools . . . . . . . . . . . . . . . . . . . 379 + + 25.6.1 NAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 + + 25.6.2 Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 + + 25.6.3 Original Fortran 66. . . . . . . . . . . . . . . . . . . . . . . . 407 + + 25.6.4 Fortran 77 Version. . . . . . . . . . . . . . . . . . . . . . . . 407 + + 25.6.5 Fortran 90 Version. . . . . . . . . . . . . . . . . . . . . . . . 408 + + 25.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 25.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 26 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 + + 26.1 Using linked lists for sparse matrix problems . . . . . . . . . . . 412 + + 26.1.1 Inner product of two sparse vectors . . . . . . . . . . . . . . . . 413 + + 26.2 Solving a system of first-order ordinary differential equations using Runga–Kutta–Merson. . . . . . . . . . . . . . . 417 + + 26.2.1 Note: Alternative form of the ALLOCATE statement . . . . . . . 424 + + 26.2.2 Note: Automatic arrays. . . . . . . . . . . . . . . . . . . . . . 424 + + 26.2.3 Note: Dummy procedure arguments. . . . . . . . . . . . . . . . 425 + + 26.2.4 Keyword and optional arguments . . . . . . . . . . . . . . . . . 425 + + 26.3 Generic procedures . . . . . . . . . . . . . . . . . . . . . . . 427 + + 26.4 A function that returns a variable length array . . . . . . . . . . . 434 + + 26.5 Operator and assignment overloading . . . . . . . . . . . . . . . 436 + + 26.6 A subroutine to extract the diagonal elements of a matrix . . . . . 437 + + 26.7 Perfectly balanced tree . . . . . . . . . . . . . . . . . . . . . . 439 + + 26.8 Pure function example . . . . . . . . . . . . . . . . . . . . . . 442 + + 26.8.1 Pure constraints . . . . . . . . . . . . . . . . . . . . . . . . . 442 + + 26.9 Elemental function example . . . . . . . . . . . . . . . . . . . 443 + + 26.9.1 Elemental constraints . . . . . . . . . . . . . . . . . . . . . . 444 + + 26.10 Elemental subroutine example . . . . . . . . . . . . . . . . . . 445 + + 26.11 Date class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 + + 26.12 Graphics example — dislin. . . . . . . . . . . . . . . . . . . . 461 + + 26.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 + + 26.14 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 470 + +# 27 ISO TR 15580 — IEEE Arithmetic. . . . . . . . . . . . . . . . . 473 + +27.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 + + 27.2 IEEE 754 Specifications . . . . . . . . . . . . . . . . . . . . . 476 + + 27.2.1 Single precision floating point format . . . . . . . . . . . . . . . 477 + + 27.2.2 Double precision floating point format . . . . . . . . . . . . . . 479 + + 27.2.3 Two classes of extended floating point formats . . . . . . . . . . 479 + + 27.2.4 Accuracy requirements . . . . . . . . . . . . . . . . . . . . . 479 + + 27.2.5 Base conversion — Converting between decimal and binary floating point formats and vice versa . . . . . . . . . . . . . . . . . . . 479 + + 27.2.6 Exception handling . . . . . . . . . . . . . . . . . . . . . . . 480 + + 27.2.7 Rounding directions . . . . . . . . . . . . . . . . . . . . . . . 480 + + 27.2.8 Rounding precisions . . . . . . . . . . . . . . . . . . . . . . . 480 + + 27.3 Resumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 + + 27.4 ISO TR 15580. . . . . . . . . . . . . . . . . . . . . . . . . . 481 + + 27.4.1 IEEE_FEATURES module . . . . . . . . . . . . . . . . . . . . 481 + + 27.4.2 IEEE_EXCEPTIONS module . . . . . . . . . . . . . . . . . . 481 + + 27.4.3 IEEE_ARITHMETIC module . . . . . . . . . . . . . . . . . . 483 + + 27.4.3.1 IEEE data type selection . . . . . . . . . . . . . . . . . . . . . 484 27.4.3.2 General support enquiry functions . . . . . . . . . . . . . . . . 484 + + 27.4.3.3 Rounding modes. . . . . . . . . . . . . . . . . . . . . . . . . 485 + + 27.4.3.4 Number classification . . . . . . . . . . . . . . . . . . . . . . 485 + + 27.4.3.5 Arithmetic operations . . . . . . . . . . . . . . . . . . . . . . 487 + + 27.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 + + 27.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 488 + + 27.6.1 Web-based sources . . . . . . . . . . . . . . . . . . . . . . . 489 + + 27.6.2 Hardware sources . . . . . . . . . . . . . . . . . . . . . . . . 490 + + 27.6.3 Operating Systems . . . . . . . . . . . . . . . . . . . . . . . . 491 + + 27.6.4 Java and IEEE 754. . . . . . . . . . . . . . . . . . . . . . . . 491 + + 27.6.5 C and IEEE 754 . . . . . . . . . . . . . . . . . . . . . . . . . 492 28 ISO TR 15581 Allocatable Enhancements . . . . . . . . . . . . . 493 + + 28.1 Allocatable dummy array example . . . . . . . . . . . . . . . . 494 + + 28.2 Allocatable function result example. . . . . . . . . . . . . . . . 497 + + 28.3 Allocatable structure component example . . . . . . . . . . . . . 499 + + 28.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 + + 28.5 Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 29 Fortran 2003 and the Enhanced Module Facility . . . . . . . . . . 501 + + 29.1 Derived type enhancements . . . . . . . . . . . . . . . . . . . 502 + + 29.2 Object oriented programming support . . . . . . . . . . . . . . . 502 + + 29.3 Data manipulation enhancements . . . . . . . . . . . . . . . . . 502 + + 29.4 Input/output enhancements . . . . . . . . . . . . . . . . . . . . 503 + + 29.5 Interoperability with the C programming language. . . . . . . . . 503 + + 29.6 Procedure pointers . . . . . . . . . . . . . . . . . . . . . . . . 504 + + 29.7 Scoping enhancements . . . . . . . . . . . . . . . . . . . . . . 504 + + 29.8 Support for IEC 60559 (IEEE 754) exceptions and arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 + + 29.9 Support for international usage: (ISO 10646) . . . . . . . . . . . 504 + + 29.10 Enhanced integration with the host operating system. . . . . . . . 505 + + 29.11 The ASSOCIATE construct . . . . . . . . . . . . . . . . . . . 505 + + 29.12 Enhanced modules facility . . . . . . . . . . . . . . . . . . . . 505 + + 29.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 30 Parallel Programming . . . . . . . . . . . . . . . . . . . . . . . 507 + + 30.1 MPI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 + + 30.2 Co–array Fortran . . . . . . . . . . . . . . . . . . . . . . . . 508 + + 30.3 Openmp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 + + 30.4 PVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 + + 30.5 HPF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 + + 30.6 Parallel programming and high-performance computing . . . . . . 509 + + 30.6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 31 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 + +31.1 Program development and software engineering . . . . . . . . . . 512 + + 31.1.1 Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 + + 31.1.2 Programming style — Programs should be easy to read . . . . . . 513 + + 31.1.3 Programming style — Programs should behave well. . . . . . . . 514 + + 31.2 Data structures. . . . . . . . . . . . . . . . . . . . . . . . . . 514 + + 31.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 + + 31.4 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 + + 31.5 Structured programming and the GOTO statement . . . . . . . . . 515 + + 31.6 Efficiency, space-time trade-off. . . . . . . . . . . . . . . . . . 516 + + 31.7 Program testing . . . . . . . . . . . . . . . . . . . . . . . . . 516 + + 31.8 Simple debugging techniques. . . . . . . . . . . . . . . . . . . 516 + + 31.9 Software tools . . . . . . . . . . . . . . . . . . . . . . . . . . 517 + + 31.9.1 Cross referencing . . . . . . . . . . . . . . . . . . . . . . . . 517 + + 31.9.2 Pretty print . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 + + 31.9.3 NAGWare f90 Tools. . . . . . . . . . . . . . . . . . . . . . . 517 + + 31.10 Numerical software sources . . . . . . . . . . . . . . . . . . . 517 + + 31.10.1 Numerical Algorithms Group. . . . . . . . . . . . . . . . . . . 518 + + 31.10.2 Visual Numerics. . . . . . . . . . . . . . . . . . . . . . . . . 518 + + 31.10.3 Netlib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 + + 31.11 Coda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 + + 31.12 Bibliography: All sources (bar one) taken from comp.software-eng. . . . . . . . . . . . . . . . . . . . . . . . 518 + + 31.12.1 Software engineering. . . . . . . . . . . . . . . . . . . . . . . 518 + + 31.12.2 Programming style. . . . . . . . . . . . . . . . . . . . . . . . 519 + + 31.12.3 Software testing . . . . . . . . . . . . . . . . . . . . . . . . . 519 + + 31.12.4 Fun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 A Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 B Sample Program Examples. . . . . . . . . . . . . . . . . . . . . 530 C ASCII Character Set. . . . . . . . . . . . . . . . . . . . . . . . 534 D Intrinsic Functions and Procedures . . . . . . . . . . . . . . . . 535 E English and Latin Texts . . . . . . . . . . . . . . . . . . . . . . 568 F Coded Text Extract . . . . . . . . . . . . . . . . . . . . . . . . 569 G Formal syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 H Compiler Options . . . . . . . . . . . . . . . . . . . . . . . . . 575 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 + +# + +Overview + +“I don't know what the language of the year 2000 will look like, but it will be called Fortran.” + +C.A.R. Hoare + +# Aims + +The aims of the chapter are to provide a background to the organisation of the book. + +# 1 Overview + +The book aims to provide coverage of a recommended subset of the full Fortran language. The subset we have chosen is one that fits most closely with the theory and practice of structured programming, data structuring and software engineering. + +This book has been written for both complete beginners with little or no program- ming background and experienced Fortran programmers who want to update their skills and move to a modern version of the language. + +Chapters 2–4 provide a short background to computer systems and their use: + +• Chapter 2 looks at the basics of computer systems from the hardware point of view. • Chapter 3 provides a short history of operating system developments and looks at some commonly used operating systems. • Chapter 4 looks at some of the fundamentals of using a computer system. + +These three chapters provide information that will be very helpful in the longer term for the successful use of computer systems for programming. + +Chapters 5 and 6 provide a coverage of problem solving and the history and devel- opment of programming languages. Chapter 5 is essential for the beginner as the concepts introduced there are used and expanded on throughout the rest of the book. Chapter 6 must be read at some point but can be omitted initially. Pro- gramming languages evolve and some understanding of where Fortran has come from and where it is going will prove valuable in the longer term: + +• Chapter 5 looks at problem solving in some depth, and there is a coverage of the way we define problems, the role of algorithms, the use of both top-down and bottom-up methods, and the requirement for formal systems analysis and design for more complex problems. • Chapter 6 looks at the history and development of programming lan- guages. This is essential as Fortran has evolved considerably from its ori- gins in the mid-1950s, through the first standard in 1966, the Fortran 77 standard, the Fortran 90 standard, the Fortran 95 standard, TR 15580 and TR 15581, Fortran 2003 and beyond. It helps to put many of the current and proposed features of Fortran into context. Languages covered include Cobol, Algol, Lisp, Snobol, PL/1, Algol 68, Simula, Pascal, APL, Basic, C, Ada, Modula, Modula 2, Logo, Prolog, SQL, ICON, Oberon, Oberon 2, Smalltalk, $\mathrm{C++}$ , C# and Java. + +Chapters 7 through 11 cover the major features provided in Fortran for numeric programming in the first instance and for general purpose programming in the sec- ond. Each chapter has a set of problems. It is essential that a reasonable range of problems is attempted and completed, as it is impossible to learn any language without practice: + +• Chapter 7 provides an introduction to programming with some simple Fortran examples. For people with a knowledge of programming this chapter can be covered fairly quickly. • Chapter 8 looks at arithmetic in some depth, with a coverage of the vari- ous numeric data types, expressions and assignment of scalar variables. There is also a thorough coverage of the facilities provided in Fortran to help write programs that work on different hardware platforms. • Chapter 9 is an introduction to arrays and DO loops. The chapter starts with some examples of tabular structures that one should be familiar with. There is then an examination of what concepts we need in a programming language to support manipulation of tabular data. • Chapter 10 takes the ideas introduced in chapters 8 and 9 and extends them to higher-dimensioned arrays, additional forms of the DIMENSION attribute and corresponding form of the DO loop, and the use of looping for the control of repetition and manipulation of tabular information with- out the use of arrays. • Chapter 11 looks at more of the facilities offered for the manipulation of whole arrays and array sections, ways in which we can initialise arrays us- ing constructors, look more formally at the concepts we need to be able to accurately describe and understand arrays, and finally look at the differ- ences between the way Fortran allows us to use arrays and the mathemati- cal rules governing matrices. + +Chapters 9 through 11 provide a coverage of some of the more important features and uses of arrays in the field of numerical problem solving. The framework pro- vided here is drawn upon in later chapters in the book with more complex and realistic examples. + +Chapters 12, 13 and 14 look at input and output (I/O) and file handling in Fortran. An understanding of I/O is necessary for the development of so-called production, non interactive programs. These are essentially fully developed programs that are used repeatedly with a variety of data inputs and results: + +Chapter 12 looks at output of results and how to generate something that is more comprehensible and easy to read than what is available with free format output and also how to write the results to a file rather than the screen. + +• Chapter 13 extends the ideas introduced in Chapter 12 on output to cover input of data, or reading data into a program and also considers file I/O. • Chapter 14 provides a coverage of files. + +Chapter 15 introduces the first building block available in Fortran for the construc- tion of programs for the solution of larger, more complex problems. It looks at the functions available in Fortran, the so-called intrinsic functions and procedures (over 100 of them) and covers how you can define and use your own functions. + +It is essential to develop an understanding of the functions provided by the lan- guage and when it is necessary to write your own. + +Chapter 16 introduces more formally the concept of control structures and their role in structured programming. Some of the control structures available in Fortran are introduced in earlier chapters, but there is a summary here of those already covered plus several new ones that complete our coverage of a minimal working set. + +Chapters 17 through 21 complete our coverage of the facilities for data typing and structuring provided by Fortran, both predefined and user defined. Fortran has now caught up with some of the major developments in the data-structuring area of the last 20 years, which have been available in other languages for some time: + +• Chapter 17 looks at the character data type in Fortran. There is a coverage of I/O again, with the operators available — only one in fact. • Chapter 18 looks at the last numeric data type in Fortran, the complex data type. This data type is essential to the solution of a small class of problems in mathematics and engineering. • Chapter 19 looks at the logical data type. The material covered here helps considerably in increasing the power and sophistication of the way we use and construct logical expressions in Fortran. This proves invaluable in the construction and use of logical expressions in control structures. • Chapters 20 looks at user-defined data types. This introduces another ma- jor new feature of Fortran. Previous versions of the language lacked any facilities in this area. This meant that in many applications earlier versions of Fortran were not the language of first choice for many people. • Chapter 21 looks at the dynamic data-structuring facilities now available in Fortran. Examples are drawn from a range of sources. + +These chapters conclude coverage of the data-structuring facilities provided by Fortran. There are problems that will require facilities not provided, but it is sur- prising what can be achieved with the set now provided in Fortran. The material covered is extended into more realistic examples when we look at the construction of larger and more complex programs in the last few chapters in the book. + +The next two chapters look at the second major building block in Fortran — the subroutine. Chapter 22 provides a gentle introduction to some of the fundamental concepts of subroutine definition and use and Chapter 23 extends these ideas. + +Chapter 24 introduces the concept of a module and the range of things that it brings to Fortran. + +Chapter 25 looks at converting to modern Fortran. A number of examples are used and several software tools are examined. + +Chapter 26 has a number of case studies helping to pull together the ideas pre- sented in the earlier chapters. + +Chapter 27 looks at ISO TR 15580 — IEEE Arithmetic. + +Chapter 28 deals with ISO TR 15581 — Allocatable Enhancements + +Chapters 29 covers the new features of Fortran 2003 and ISO/IEC DTR 19767, Enhanced Module Facilities. + +Chapter 30 examines parallel Fortran. + +Chapter 31 ties up some loose ends. It looks at program development and software engineering, modules, programming style, data structures, algorithms, structured programming, recursion and recursion removal, efficiency in space and time, pro- gram testing, simple debugging techniques, software tools and numerical software sources. There is also coverage of the various internet resources available for For- tran. + +Many of the chapters have annotated bibliographies. These often have pointers and directions for further reading. The coverage provided cannot be seen in isolation. The concepts introduced are by intention brief, and fuller coverage must be sought where necessary. + +There are several appendices: + +• Appendix A — This is a glossary which provides coverage of both the new concepts provided by Fortran and a range of computing terms and ideas. • Appendix B — Provides an example of a simple program in a number of the languages described in the chapter on program language development. There is also coverage of the standards that apply. • Appendix C — The ASCII character set. + +• Appendix D — Contains a list of all of the intrinsic procedures in Fortran and includes a full explanation of each procedure with a coverage of the rules and restrictions that apply and examples of use. • Appendix E — Contains the English and Latin text extracts used in one of the problems in the chapter on characters. • Appendix F — Contains the coded text extract used in one of the prob- lems in Chapter 17. • Appendix G — Formal syntax • Appendix H — Sample compiler options + +This book is not and cannot possibly be completely self-contained and exhaustive in its coverage of the Fortran language. Our first intention has been to produce a coverage of the features that will get you started with Fortran and enable you to solve quite a wide range of problems successfully. + +Fortran, like most languages, has features that are of relatively little use or make the construction of larger-scale programs more difficult, especially when moving between hardware platforms. We have deliberately avoided these features. + +Another problem is backwards compatibility with Fortran 77. Existing Fortran 77 programs have to be maintained, and there is much in that language that is depre- cated or obsolescent in terms of Fortran 95 and Fortran 2003. + +We have aimed to introduce a working subset of the new language that emphasises the better constructs provided in Fortran over its predecessors, Fortran 77 and For- tran 66. + +All in all Fortran is an exciting language, and it has caught up with language de- velopments of the 1970s, 1980s, and 1990s. + +A range of hardware platforms, operating systems and Fortran compilers were used. These include: + +• DEC VAX under VMS and later Open VMS using the NAG Fortran 90 compiler. • DEC Alpha under Open VMS using the DEC/Compaq Fortran 90 com- piler. • PC under DOS and Windows, DEC/Compaq Fortran 90. • PC under DOS and Windows, DEC/Compaq/HP Fortran 95. • PC under DOS and Windows, NAG/Salford Fortran 90. • PC under DOS and Windows, Lahey Fujitsu Fortran 95 PRO 5.7. + +• PC under DOS and Windows, Intel. • PC under DOS and Windows, NAGWare f95. • Sun UltraSparc under Solaris using NAGWare F90. • Sun UltraSparc under Solaris using NAGACE F90. • Sun UltraSparc under Solaris using NAGWare F95. • Sun UltraSparc under Solaris using Sun F90. • Intel Linux, NAGWare f95. • Intel Linux, Lahey Fujitsu Fortran 95 PRO, 6.1. • Intel Linux, Intel. + +Our recommendation is that you use at least two compilers in the development of your code. Moving code between platforms teaches you a lot. + +We are the current owners of the Fortran 90 list, and quoting the introduction “ This list covers all aspects of Fortran 90 and HPF, the new standard(s) for Fortran. The emphasis should be on the \*new\* features of Fortran 90. It welcomes contri- butions from people who write Fortran 90 applications, teach it in courses, want to port programs and use it on (super)computers.” + +Visit: + +• http://www.jiscmail.ac.uk/lists/comp-fortran-90.html + +for more information. + +Ian Chivers is also Editor of Fortran Forum, the SIGPLAN Special Interest Publi- cation on Fortran, ACM Press. + +# Introduction to Computer Systems + +“Don't Panic.” + +Douglas Adams, The Hitch Hiker's Guide to the Galaxy + +# Aims + +The aims of this chapter are to introduce the following: + +• The components of a computer — the hardware. • The components of a complete computer system — the other devices that you need to do useful work with a computer. • The software needed to make the hardware do what you want it do. + +# Chapter 2 + +# 2 Introduction to Computer Systems + +A computer is an electronic device and can be thought of as a tool like a lever or a wheel, which can be made to do useful work. At the fundamental level it works with bits (binary digits or sequences of zeros and ones). Bits are generally put to- gether in larger configurations, e.g., 8, 16, 32, or 64. Hence computers are often referred to as 8-bit, 16-bit, 32-bit, or 64-bit machines. + +# 2.1 The core of a computer system + +The heart of most computer systems comprises a motherboard, CPU, memory, one or more busses and a power supply. We will look at the CPU, memory and bus in more depth below. + +# 2.1.1 Central processor unit — CPU + +This is the brains of the computer. All of the work that the computer does is organ- ised here. + +# 2.1.2 Memory + +The computer also has a memory. Memory on a computer is a solid state device that comprises an ordered collection of bits/bytes/words that can be read or written by the CPU. A byte is generally 8 bits (as in 8-bit byte ), and a word is most com- monly accepted as the minimum number of bits that can be referenced by the CPU. This referencing is called addressing . The memory typically contains programs and data. + +The two most common word sizes are 32 and 64 bits. + +A computer memory is often called random access memory, or RAM. This simply means that the access time for any part of the memory is the same; in order to ex- amine location (say) 97, it is not necessary to first look through locations 1 to 96. It is possible to go directly to location 97. A slightly better term might have been access at random . The memory itself is highly ordered. + +# 2.1.3 Bus + +A bus is a set of connections between the CPU and other components. The bus is used for a variety of purposes. These include address signals, which tell the mem- ory which words are wanted next and data lines, which are used to transfer data to and from memory and to and from other parts of the computer system. This is typ- ical of many systems, but systems do vary considerably; so while the information above may not be true in specific cases, it provides a general model. + +# 2.2 Other components of a computer system + +So far the computer we have described is not sufficiently versatile. We have to add on other pieces of electronics to make it really useful. + +# 2.2.1 Disks + +These are devices for storing collections of bits , which are inevitably organised in reality into bytes and files. One advantage of adding these to our computer system is that we can switch the machine off, go away and come back at a later time and continue with what we were doing. + +Memory is expensive and fast, whereas disks are slower but cheaper. Most com- puter systems balance speed against cost, and have a small memory in relation to disk capacity. + +Many people will be familiar with the two main types of disks on early personal computers (PCs) or microcomputers: floppy disks and hard disks. Floppy disks now come in one main physical size, 3-1/2 inch, but smaller ones are also used. Hard disks are inside the system, and most people do not see them. + +Optical drives are an essential part of present day systems. They exist in a variety of flavours including simple read-only CD, rewritable CD and DVD forms. + +# 2.2.2 Others + +There are a large number of other input and output devices. These vary consider- ably from system to system, depending on the work being carried out. They include: + +• Network (ethernet or wifi) cards for access to local and wide area net- works. • Modems for access from home, mainly to the Internet. • Printers of a variety of types. • Colour plotters. • Phototypesetter s. • Pens. • Sound interfaces, both for speech recognition and sound production. • Scanners. • Digital cameras. • Joy sticks. • Zip drives. + +# Chapter 2 + +• Memory sticks. + +The most important I/O devices are the keyboard and the screen, whether you use a terminal, PC or workstation. This book has been written assuming that most of your work will be done at one of these devices. + +Terminals fall into two categories, character-based devices (and the DEC VT series is a very popular one) and graphical devices (the X-Windows terminals are the most popular). Terminal access to remote systems is often provided on PCs using terminal emulation software, e.g., Telnet, WinQVT and X-Windows access to UNIX systems via software like Vista Exceed. + +PCs provide the opportunity for cheap and powerful desktop computing facilities, where the processing is done locally. + +Workstations are more powerful than microcomputers but this division is becoming rather blurred with the recent generations of processors. Screens on these devices are graphically oriented. Access to these systems is via a graphical or windows in- terface. + +This means that the device we use looks rather like an ordinary typewriter key- board, although some of the keys are different. However, the location of the letters, numbers and common symbols is fairly standard. Don't panic if you have never met a keyboard before. You don't have to know much more than where the keys are. Few programmers, even professionals, advance beyond the stage of using two index fingers and a thumb for typing. You will find that speed in typing is rarely important; it's accuracy that counts. + +One thing that people unfamiliar with keyboards often fail to realise is that what you have typed in is not sent to the computer until you press the carriage return key. To achieve any sort of communication you must press that key; it will be somewhere on the right-hand side of the keyboard, and will be marked return, c/r, send, enter , or something similar. + +# 2.3 Software + +So far we have not mentioned software. Software is the name given to the pro- grams that run on the hardware. Programs are written in languages. Computer languages are frequently divided into two categories: high level and low level. A low-level language (e.g., assembler) is closer to the hardware, whereas a high-level language (e.g., Fortran) is closer to the problem statement. There is typically a one-to-one correspondence between an assembly language statement and the actual hardware instruction. With a high-level language there is a one-to-many correspon- dence; one high-level statement will generate many machine-level instructions. + +A certain amount of general purpose software will have been provided by the man- ufacturer. This software will typically include the basic operating system, one or more compilers , an assembler , an editor , and a loader or link editor . + +• A compiler translates high-level statements into machine instructions. • An assembler translates low-level or assembly language statements into machine instructions. • An editor makes changes to text files, e.g., program sources. • A loader or link editor takes the output from the compiler and completes the process of generating something that can be executed on the hardware. + +These programs will vary considerably in size and complexity. Certain programs that make up the operating system will be quite simple and small (like copying utilities), whereas certain others will be relatively large and complex (like a com- piler). + +In this book we concentrate on software or programs that you write for your re- search or course work. As the book progresses you will be introduced to ways of building on what other people have produced and how to take advantage of the vast amount of software already written, tested and documented. + +# 2.4 Problems + +1. Distinguish between a memory address and memory contents. + +2. What does RAM stand for? + + 3. What would a WOM (write only memory) do? How would you use it? + + 4. What does CPU stand for? What does it do? 5 What does a compiler do? 6 What does a linker do? + +# 2.5 Bibliography + +Baer J.L., Computer Systems Architecture , Computer Science Press. + +Extremely readable coverage of this whole area. The version could do with an update, but it is still a very impressive coverage. Highly recommended. Bhandarkar D.P., Alpha Implementation and Architecture: Complete Reference and Guide , Digital Press, 1996. Excellent source of information on the Alpha architecture. + +Intel currently make a lot of material available on their web site. Two useful URLs are: + +•http://www.intel.com/and • http://developer.intel.com/ Well worth a look. Many publications are available in Adobe Acrobat Portable Document Format — PDF. Reeves C.M., An Introduction to Logical Design of Digital Circuits , CUP, 1972. This book provides coverage of the construction of the very simple electronic building blocks from which most modern computer systems are made. Rela- tively theoretical. Tannenbaum A.S., Structured Computer Organisation , Prentice-Hall, 1976. Very good coverage looking at a computer system in terms of a hierarchy of levels. An easy read. + +# Introduction to Operating Systems + +“‘Where shall I begin your Majesty’ he asked. + + ‘Begin at the beginning,’ the King said, gravely ‘and go on till you come to the end then stop.’” + +Lewis Carroll, Alice's Adventures in Wonderland + +# Aims + +The aims of this chapter are: + +• To provide a brief history of operating system development. • To look briefly at some commonly used operating systems: • DOS and Windows. • UNIX and X-Windows. • Linux and X-Windows. • VMS and Open VMS. + +# 3 Introduction to Operating Systems + +A simple definition of an operating system is the suite of programs that make the hardware usable. Most computer systems provide one. They vary considerably from those available on early microcomputers, like CP/M, to DOS and the various versions of Microsoft Windows on PCs and UNIX with X-Windows and Linux with X-Windows on workstations and supercomputers. + +From the designer's point of view operating systems are mainly resource managers. They allow management of the CPU, disks and I/O devices. They have to provide a user interface for computer operators, professional programmers (whether sys- tems or applications), administrators of the system, and finally the casual end user. As can be imagined, these groups have different functional requirements. It is therefore useful to look at the development of operating systems, and see a shift from satisfying the requirements of the professional to satisfying the requirements of the casual end user. + +# 3.1 History of operating systems + +# 3.1.1 The 1940s + +Early computer systems had no operating systems in the modern sense of the word. An early commercially available machine was the IBM 604 which could undertake some 60 program steps before using punch cards as backing store. The end user had intimate knowledge of the machine and programmed at a very low level. + +# 3.1.2 The 1950s + +This era saw a rapid change in the capabilities of operating systems. They were de- signed to make efficient use of an expensive resource. Jobs were batched so that the time between jobs was minimised. The end user was now distanced from the machine. This era saw rapid development in program language design and a nota- ble end to the period was the design of Algol 60. + +# 3.1.3 The 1960s + +The next milestone was the introduction of multi programming. Probably initially seen as a way of making efficient use of hardware it heralded the idea of time sharing. A time-sharing system is characterised by the conversational nature of the interaction and the use of a keyboard. This had a tremendous impact on the range of uses that a computer system had and on the program development process. + +# 3.1.4 The 1960s and 1970s + +The realisation that computer systems could be used in a wide range of human ac- tivities saw the development of large, general purpose systems, and probably the most famous of these was the IBM 360 series. These systems were some of the most complex programming endeavours undertaken, and most projects were late and well over budget. These costly mistakes helped lead to the establishment of software engineering as a discipline. + +The contribution of the time-sharing system to program development was quickly realised to be considerable. A system that was developed during this period was UNIX — and this operating system has a very sharp set of tools to aid in program development. + +# 3.1.5 The 1970s, 1980s, and 1990s + +The 1970s represented a period of relative stability with newer and more complex versions of existing systems. + +During this period the importance of graphical interfaces emerged and, with drop- ping hardware costs, graphical interfaces started to dominate. + +The Apple Macintosh heralded a new era and became a popular choice for many people. At the same time graphical interfaces were being added to existing major operating systems, with X-Windows and associated higher-level systems hiding raw UNIX from many users and Microsoft Windows on the Intel family of proces- sors. + +Linux (a free UNIX variant) is popular in the scientific field, and a very good al- ternative to DOS and Windows on the Intel family of processors. + +# 3.2 Networking + +Networking simplistically is a way of connecting two or more computer systems. Networking computer systems is not new. One of the first was the SAGE military network, funded by the US DoD in the 1950s. Networking capability has undergone a massive increase during the computer age. Local networks of two or three systems through tens of systems in small research groups and organizations are now extremely common place. It is not unusual now to have in excess of a thousand network connected devices on one local area net- work. Wide-area networking is also quite common, and most major organisations now have networks spanning a country or even the whole world. One of the most widely used wide-area networks in the academic and scientific world is the Internet, and there are many millions of systems on the Internet at the time of writing this book. A number of books on networking are included in the bibliography. + +# Chapter 3 + +# 3.3 Problems + +1. What type of system do you use, i.e., is it a stand alone microcomputer, termi- nal, workstation, etc? + +2. Is it networked, and if so in what way? + +3. Is wide-area networking available? + +4. Is a graphical interface available? + +# 3.4 Bibliography + +Brooks F.P., The Mythical Man Month: Essays on Software Engineering, Addi- son-Wesley, 1982. + +A very telling coverage of the development of the operating system for the IBM 360 series of systems. A must for any one involved in the longer term in program development. Deitel H.M., An Introduction to Operating Systems , Addison-Wesley, 1984. One of the most accessible books on operating systems with coverage of pro- cess management, storage management, processor management, auxiliary storage management, performance, networks and security, with case studies of the major players including UNIX, VMS, CP/M, MVS, VM, DOS and Win- dows. Feit S., TCP/IP, Architecture, Protocols, and Implementation, McGraw-Hill. A more technical book than Kroll, well written with a wealth of information for the more inquisitive reader. Kroll E., The Whole Internet User's Guide and Catalog, O'Reilly, 1994. The Internet book. Written with very obvious enthusiasm by Mister Internet himself! + +# Introduction to Using a Computer System + +“Maybe one day we will be glad to remember even these things.” + +Virgil + +# Aims + +The aims of this chapter are to introduce some of the fundamentals of using a com- puter system, including: + +• Files. • Editors. • Systems access and networking. + +# Chapter 4 + +# 4 Introduction to Using a Computer System + +There are a number of concepts that underpin your use of any computing system. Sitting at a high-resolution colour screen with a myriad of icons this may not be immediately apparent, but developing an appreciation of it will help considerably in the long term and when you inevitably move from one system to another. + +# 4.1 Files + +A file is a collection of information that you refer to by name, e.g., if you were to use a word processor to prepare a letter then the letter would exist independently as a file on that system, generally on a disk. With graphical interfaces there will be a systematic iconic representation of files. + +There will be many ways of manipulating files on the operating system that you work on. You will use an editor to make changes to a file. This file might be the source of a program and you could then use a compiler to compile your program. The compiler will generate a number of files, some of interest to you, others for its own use. There will be commands in the operating system to make copies of files, back files up onto a variety of media, etc. + +# 4.2 Editors + +All general purpose computer systems have at least one editor so that you can modify programs and data. Screen editors are by far the easiest to use, with changes you make to the file being immediately visible on the screen in front of you. + +Some editors will have sophisticated command modes of operation with pattern matching allowing very powerful text-processing capabilities. These can automate many common tasks, taking away the manual, repetitive drudgery of screen-based editing. + +# 4.3 Single-user systems + +These are becoming increasingly common, in use both at work and in the home. The PC is a very popular choice in the scientific community. They offer ease of use and access to a considerable amount of raw processing power for computer-in- tensive applications. + +# 4.4 Networked systems + +It is quite common to interconnect the above to local and wide-area networks. This same network would also have file servers, printers, plotters, mail gateways, etc. Both authors have PCs with modems at home and have access via the telephone system to the Internet. + +Workstations are generally networked in an environment like the above, providing very powerful processing capability. + +# 4.5 Multiuser systems + +One step above microcomputers and workstations are multiuser systems. The di- viding lines between microcomputers, individual workstations and multiuser systems are rapidly becoming blurred. + +Multiuser systems, especially the larger ones, are very popular as they relieve the casual end users from much of the drudgery of the day-to-day tasks: backing up disks, installing new versions of the software, locating and fixing problems with software that doesn't work quite as it should, etc., are carried out by a system man- ager or operator. + +Here we find one person with the role of registering new users, backing up the file system, sorting out printer problems, networking problems, etc. This also means that not all of the users of the system have to remember rather arcane and some- times rather magical commands! They can get on with solving their actual problems. + +# 4.6 Other useful things to know + +You will soon need to know what files you are working with and there will be commands to do this. There will be a need to get rid of files and there will be commands to achieve this. + +There will be ways of getting on-line help, and help as a command is (for once!) used by a variety of operating systems. On UNIX systems the rather more unintel- ligible man command is available. + +There will be commands to print program listings and data files + +With networked and multiuser systems there will be commands to send and receive electronic mail to/from other users. It is easy to send and reply to mail from people across the world, often in hours and even minutes. Table 4.1 has examples of some common operating system commands in DOS, UNIX, Linux and VMS. + +![](images/692357db2ae1dc022acaba05b19c145813a48c83e888033c251aa1a9c1725a77.jpg) + +Table 4.1 Common Operating System Commands + +# 4.7 Common methods of using computer systems to develop Fortran programs + +The following are some of the ways in which you can use a computer system to develop Fortran programs: + +• PC running Windows and X-Windowing software to access a remote sys- tem with a Fortran compiler installed, GUI interface. • PC running Linux and X-Windows to access a remote system with a For- tran compiler installed, GUI interface. • PC running Windows and telnet or ssh to access a remote system with a Fortran compiler installed, terminal-style interface. • PC running Linux and telnet or ssh to access a remote system with a For- tran compiler installed, terminal-style interface. • PC running Windows, local Fortran compiler installed. • PC running Linux, local Fortran compiler installed. • Proprietary workstation, local compiler installed. • Proprietary workstation, accessing compiler on remote system. + +All will have one thing in common and that is that the following cycle is used: + +• Edit your program. • Compile the program. • Run the program. • Check the answers. • Go back and edit the program to correct the errors and repeat until the an- swers are what you expect! + +# 4.8 Bibliography + +The main sources here are the manuals and documentation provided by the sup- plier of whatever system you use. These are increasingly of a very high standard. However they are inevitably written to highlight the positive and downplay the negative aspects of the systems. The next sources are the many third-party books written and widely available throughout the world. These vary considerably in price from basic introductory coverages to very comprehensive reference works. These are a very good complement to the first. The following URL is a very good source of UNIX information. + +• http://unixhelp.ed.ac.uk/ + +Gilly D., UNIX in a Nutshell, O'Reilly. + +A very good quick reference guide. Assumes some familiarity with UNIX. Current edition (at the time of writing this book) was System V Release IV, with Solaris 2.0. Also provides coverage of the various shells, Bourne, Korn and C. Microsoft, Windows User's Guide , Microsoft Press. Good coverage of Windows and suitable for the beginner and intermediate level user. Sufficient for most users. A massive improvement over earlier ver- sions. + +# Introduction to Problem Solving + +“They constructed ladders to reach to the top of the enemy's wall, and they did this by calculating the height of the wall from the number of layers of bricks at a point which was facing in their direction and had not been plastered. The layers were counted by a lot of people at the same time, and though some were likely to get the figure wrong the majority would get it right... Thus, guessing what the thick- ness of a single brick was, they calculated how long their ladder would have to be.” + +Thucydides, The Peloponnesian War + +“‘When I use a word,’ Humpty Dumpty said, in a rather scornful tone, ‘it means just what I choose it to mean — neither more nor less.’ + +‘The question is,’ said Alice, ‘whether you can make words mean so many differ- ent things.’” + +Lewis Carroll, Through the Looking Glass and What Alice Found There + +# Aims + +The aims of this chapter are: + +• To examine some of the ideas and concepts involved in problem solving. • To introduce the concept of an algorithm. • To introduce two ways of approaching algorithmic problem solving. • To introduce the ideas involved with systems analysis and design, i.e., to show the need for pencil and paper study before using a computer system. + +Chapter 5 + +# 5 Introduction to Problem Solving + +It is informative to consider some of the dictionary definitions of problem: + +• A matter difficult of settlement or solution, Chambers . • A question or puzzle propounded for solution, Chambers . • A source of perplexity, Chambers . • Doubtful or difficult question, Oxford . • Proposition in which something has to be done, Oxford . • A question raised for enquiry, consideration, or solution, Webster's . • An intricate unsettled question, Webster's . + +A common thread seems to be a question that we would like answered or solved. So one of the first things to consider in problem solving is how to pose the prob- lem. This is often not as easy as is seems. Two of the most common methods to use here are: + +• In natural language. • In artificial or stylised language. + +Both methods have their advantages and disadvantages. + +# 5.1 Natural language + +Most people use natural language and are familiar with it, and the two most com- mon forms are the written and spoken word. Consider the following language usage: + +• The difference between a 3-year-old child and an adult describing the world. • The difference between the way an engineer and a physicist would ap- proach the design of a car engine. • The difference between a manager and a worker considering the implica- tions of the introduction of new technology. + +Great care must be taken when using natural language to define a problem and a solution. It is possible that people use the same language to mean completely dif- ferent things, and one must be aware of this when using natural language whilst problem solving. + +Natural language can also be ambiguous: Old men and women eat cheese. Are both the men and women old? + +# 5.2 Artificial language + +The two most common forms of artificial language are technical terminology and notations. Technical terminology generally includes both the use of new words and alternate use of existing words. Consider some of the concepts that are useful when examining the expansion of gases in both a theoretical and practical fashion: + +• Temperature. • Pressure. • Mass. • Isothermal expansion. • Adiabatic expansion. + +Now look at the following: + +• A chef using a pressure cooker. • A garage mechanic working on a car engine. • A doctor monitoring blood pressure. • An engineer designing a gas turbine. + +Each has a particular problem to solve, and all will approach their problem in their own way; thus they will each use the same terminology in slightly different ways. + +# 5.2.1 Notations + +Some examples of notations are: + +• Algebra. • Calculus. • Logic. + +All of the above have been used as notations for describing both problems and their solutions. + +# 5.3 Resumé + +We therefore have two ways of describing problems and they both have a learning phase until we achieve sufficient understanding to use them effectively. Having ar- rived at a satisfactory problem statement we next have to consider how we get the solution. It is here that the power of the algorithmic approach becomes useful. + +# Chapter 5 + +# 5.4 Algorithms + +An algorithm is a sequence of steps that will solve part or all of a problem. One of the most easily understood examples of an algorithm is a recipe. Most people have done some cooking, if only making toast and boiling an egg. + +A recipe is made up of two parts: + +• A check list of things you need. • The sequence or order of steps. + +Problems can occur at both stages, e.g., finding out halfway through the recipe that you do not have an ingredient or utensil; finding out that one stage will take an hour when the rest will be ready in ten minutes. Note that certain things can be done in any order — it may not make any difference if you prepare the potatoes before the carrots. + +There are two ways of approaching problem solving when using a computer. They both involve algorithms , but are very different from one another. They are called top-down and bottom-up. + +# 5.4.1 Top-down + +In a top-down approach the problem is first specified at a high or general level: prepare a meal. It is then refined until each step in the solution is explicit and in the correct sequence, e.g., peel and slice the onions, then brown in a frying pan be- fore adding the beef. One drawback to this approach is that it is very difficult to teach to beginners because they rarely have any idea of what primitive tools they have at their disposal. Another drawback is that they often get the sequencing wrong, e.g., now place in a moderately hot oven is frustrating because you may not have lit the oven (sequencing problem) and secondly because you may have no idea how hot moderately hot really is. However, as more and more problems are tackled, top-down becomes one of the most effective methods for programming. + +# 5.4.2 Bottom-up + +Bottom-up is the reverse to top-down! As before you start by defining the problem at a high level, e.g., prepare a meal. However, now there is an examination of what tools, etc. you have available to solve the problem. This method lends itself to teaching since a repertoire of tools can be built up and more complicated problems can be tackled. Thinking back to the recipe there is not much point in trying to cook a six course meal if the only thing that you can do is boil an egg and open a tin of beans. The bottom-up approach thus has advantages for the beginner. How- ever, there may be a problem when no suitable tool is available. A colleague and friend of the authors learned how to make Bechamel sauce, and was so pleased by his success that every other meal had a course with a Bechamel sauce. Try it on your eggs one morning. Here is a case of specifying a problem, prepare a meal , and using an inappropriate but plausible tool, Bechamel sauce. + +The effort involved in tackling a realistic problem, introducing the constructs as and when they are needed and solving it is considerable. This approach may not lead to a reasonably comprehensive coverage of the language, or be particularly useful from a teaching point of view. Case studies do have great value, but it helps if you know the elementary rules before you start on them. Imagine learning French by studying Balzac, before you even look at a French grammar book. You can learn this way but even when you have finished, you may not be able to speak to a Frenchman and be understood. A good example of the case study approach is given in the book Software Tools , by Kernighan and Plauger. + +In this book our aim is to gradually introduce more and more tools until you know enough to approach the problem using the top-down method, and also realise from time to time that it will be necessary to develop some new tools. + +# 5.4.3 Stepwise refinement + +Both of the above techniques can be combined with what is called stepwise refine- ment . The original ideas behind this approach are well expressed in a paper by Wirth, entitled “Program Development by Stepwise Refinement” , published in 1971. It means that you start with a global problem statement and break the prob- lem down in stages, into smaller and smaller subproblems that become more and more amenable to solution. When you first start programming the problems you can solve are quite simple, but as your experience grows you will find that you can handle more complex problems. + +When you think of the way that you solve problems you will probably realise that unless the problem is so simple that you can answer it straightaway some thinking and pencil and paper work are required. An example that some may be familiar with is in practical work in a scientific discipline, where coming unprepared to the situation can be very frustrating and unrewarding. It is therefore appropriate to look at ways of doing analysis and design before using a computer. + +# 5.4.4 Modular programming + +As the problems we try solving become more complex we need to look at ways of managing the construction of programs that comprise many parts. Modula 2 was one of the first languages to support this methodology and we will look at modular programming in more depth in a subsequent chapter. + +# 5.4.5 Object oriented programming + +There is a class of problems that are best solved by the treatment of the compo- nents of these problems as objects. We will look at the concepts involved in object oriented programming and object oriented languages in the next chapter. + +# Chapter 5 + +# 5.5 Systems analysis and design + +When one starts programming it is generally not apparent that one needs a method- ology to follow to become successful as a programmer. This is usually because the problems are reasonably simple, and it is not necessary to be explicit about all of the stages one has gone through in arriving at a solution. As the problems become more complex it is necessary to become more rigorous and thorough in one's ap- proach, to keep control in the face of the increasing complexity and to avoid making mistakes. It is then that the benefit of systems analysis and design becomes obvious. Broadly we have the following stages in systems analysis and design: + +• Problem definition. • Feasibility study and fact finding. • Analysis. • Initial system design. • Detailed design. • Implementation. • Evaluation. • Maintenance. + +and each problem we address will entail slightly different time spent in each of these stages. Let us look at each stage in more detail. + +# 5.5.1 Problem definition + +Here we are interested in defining what the problem really is. We should aim at providing some restriction on both the scope of the problem, and the objectives we set ourselves. We can use the methods mentioned earlier to help us out. It is essen- tial that the objectives are: + +• Clearly defined. • Understood and agreed to by all people concerned, when more than one person is involved. • Realistic. + +# 5.5.2 Feasibility study and fact finding + +Here we look to see if there is a feasible solution. We would try and estimate the cost of solving the problem and see if the investment was warranted by the bene- fits, i.e., cost-benefit analysis. + +# 5.5.3 Analysis + +Here we look at what must be done to solve the problem. Note that we are inter- ested in finding out what we need to do, but that we do not actually do it at this stage. + +# 5.5.4 Design + +Once the analysis is complete we know what must be done, and we can proceed to the design. We may find there are several alternatives, and we thus examine alter- nate ways in which the problem can be solved. It is here that we use the techniques of top-down and bottom-up problem solving, combined with stepwise refinement to generate an algorithm to solve the problem. We are now moving from the logi- cal to the physical side of the solution. This stage ends with a choice among several alternatives. Note that there is generally not one ideal solution, but several, each with its own advantages and disadvantages. + +# 5.5.5 Detailed design + +Here we move from the general to the specific, The end result of this stage should be a specification that is sufficiently tightly defined specification to generate actual program code. + +It is at this stage that it is useful to generate pseudocode . This means writing out in detail the actions we want carried out at each stage of our overall algorithm. We gradually expand each stage (stepwise refinement) until it becomes Fortran — or whatever language we want. + +# 5.5.6 Implementation + +It is at this stage that we actually use a computer system to create the program(s) that will solve the problem. It is here that we actually need to know enough about a programming language to use it effectively to solve our problem. This is only one stage in the overall process, and mistakes at any of the stages can create se- rious difficulties. + +# 5.5.7 Evaluation and testing + +Here we try to see if the program(s) we have produced will actually do what they are supposed to. We need to have data sets that enable us to say with confidence that the program really does work. This may not be an easy task, as quite often we only have numeric methods to solve the problem, which is why we are using the computer in the first place — hence we are relying on the computer to provide the proof; i.e., we have to use a computer to determine the veracity of the programs — and as Heller says, Catch 22. + +# Chapter 5 + +# 5.5.8 Maintenance + +It is rare that a program is run once and never used again. This means that there will be an ongoing task of maintaining the program, generally to make it work with different versions of the operating system or compiler, and to incorporate new features not included in the original design. It often seems odd when one starts programming that a program will need maintenance, as we are reluctant to regard a program in the same way as a mechanical object like a car that will eventually fall apart through use. Thus maintenance means keeping the program working at some tolerable level, often with a high level of investment in manpower and resources. Research in this area has shown that anything up to $80\%$ of the manpower invest- ment in a program can be in maintenance. + +# 5.6 Conclusions + +A drawback, inherent in all approaches to programming and to problem solving in general, is the assumption that a solution is indeed possible. There are problems which are simply insoluble — not only problems like balancing a national budget, weather forecasting for a year, or predicting which radioactive atom will decay, but also problems which are apparently computationally solvable. + +Knuth gives the example of a chess problem — determining whether the game is a forced victory for white. Although there is an algorithm to achieve this, it requires an inordinately long time to complete. For practical purposes it is unsolvable. + +Other problems can be shown mathematically to be undecidable. The work of Gödel in this area has been of enormous importance, and the bibliography contains a number of references for the more inquisitive and mathematically orientated reader. The Hofstader coverage is the easiest, and least mathematical. + +As far as possible we will restrict ourselves to solvable problems, like learning a programming language. + +Within the formal world of Computer Science our description of an algorithm would be considered a little lax. For our introductory needs it is sufficient, but a more rigorous approach is given by Hopcroft and Ullman in Introduction to Au- tomata Theory, Languages and Computation , and by Beckman in Mathematical Foundations of Programming. + +# 5.7 Problems + +1. What is an algorithm? + +2. What distinguishes top-down from bottom-up approaches to problem solving? Illustrate your answer with reference to the problem of a car, motor-cycle or bicy- cle having a flat tire. + +# 5.8 Bibliography + +Aho A.V., Hopcroft J.E., Ullman J.D., The Design and Analysis of Computer Algo- rithms , Addison-Wesley, 1982. Theoretical coverage of the design and analysis of computer algorithms. Beckman F.S., Mathematical Foundations of Programming , Addison-Wesley, 1981 Good clear coverage of the theoretical basis of computing. Bulloff J.J., Holyoke T.C., Hahn S.W., Foundations of Mathematics — Symposium Papers Commemorating the $60^{t h}$ Birthday of Kurt Gödel , Springer-Verlag, 1969. The comment by John von Neumann highlights the importance of Gödel's work, .. Kurt Gödel's achievement in modern logic is singular and monumen- tal — indeed it is more than a monument, it is a landmark which will remain visible far in space and time. Whether anything comparable to it has occurred in the logic of modern times may be debated. In any case, the conceivable proxima are very, very few. The subject of logic has certainly changed its na- ture and possibilities with Gödel's achievement. Dahl O.J., Dijkstra E.W., Hoare C.A.R., Structured Programming , Academic Press, 1972. This is the seminal book on structured programming. Davis M., Computability and Unsolvability , Dover, 1982. The book is an introduction to the theory of computability and non computability — the theory of recursive functions in mathematics. Not for the mathematically faint hearted! Davis W.S., Systems Analysis and Design , Addison-Wesley, 1983. Good introduction to systems analysis and design, with a variety of case stud- ies. Also looks at some of the tools available to the systems analyst. Fogelin R.J., Wittgenstein , Routledge and Kegan Paul, 1980. The book provides a gentle introduction to the work of the philosopher Wittgenstein, who examined some of the philosophical problems associated with logic and reason. Gödel K., On Formally Undecidable Propositions of Principia Mathematica and Related Systems , Oliver and Boyd, 1962. An English translation of Gödel's original paper by Meltzer, with quite a lengthy introduction by R.B. Braithwaite, then Knightbridge Professor of Moral Philosophy at Cambridge University, England, and classified under phi- losophy at the library at King's, rather than mathematics. Hofstadter D., The Eternal Golden Braid , Harvester Press, 1979. A very readable coverage of paradox and contradiction in art, music and logic, looking at the work of Escher, Bach and Gödel, respectively. + +Hopcroft J.E., Ullman J.D., Introduction to Automata Theory, Languages and + +Comprehensive coverage of the theoretical basis of computing. Kernighan B.W., Plauger P.J., Software Tools, Addison-Wesley, 1976. Interesting essays on the program development process, originally using a nonstandard variant of Fortran. Also available using Pascal. Knuth D.E., The Art of Computer Programming , Addison-Wesley, Vol 1. Fundamental Algorithms , 1974 Vol 2. Semi-numerical Algorithms , 1978 Vol 3. Sorting and Searching , 1972 Contains interesting insights into many aspects of algorithm design. Good source of specialist algorithms, and Knuth writes with obvious and infectious enthusiasm (and erudition). Millington D., Systems Analysis and Design for Computer Applications , Ellis Horwood, 1981. Short and readable introduction to systems analysis and design. Wirth N., Program Development by Stepwise Refinement , Communications of the ACM, April 1971, Volume 14, Number 4, pp. 221-227. Clear and simple exposition of the ideas of stepwise refinement. + +# Introduction to Programming Languages + +“We have to go to another language in order to think clearly about the problem.” Samuel R. Delany , Babel-17 + +# Aims + +The primary aim of this chapter is to provide a short history of program language development and give some idea as to the concepts that have had an impact on Fortran 95. It concentrates on some but not all of the major milestones of the last 40 years, in roughly chronological order. The secondary aim is to show the breadth of languages available. The chapter concludes with coverage of a small number of more specialised languages. + +# Chapter 6 + +# 6 Introduction to Programming Languages + +It is important to realise that programming languages are a recent invention. They have been developed over a relatively short period — 45 years — and are still un- dergoing improvement. Time spent gaining some historical perspective will help you understand and evaluate future changes. This chapter starts right at the begin- ning and takes you through some, but not all, of the developments during this 45 year span. The bulk of the chapter describes languages that are reasonably widely available commercially, and therefore ones that you are likely to meet. The chapter concludes with a coverage of some more specialised and/or recent developments. + +# 6.1 Some early theoretical work + +Some of the most important early theoretical work in computing was that of Turing and von Neumann. Turing's work provided the base from which it could be shown that it was possible to get a machine to solve problems. The work of von Neumann added the concept of storage and combined with Turing's work to provide the basis for most computers designed to this day. + +# 6.2 What is a programming language? + +For a large number of people a programming language provides the means of get- ting a digital computer to solve a problem. There is a wide range of problems and an equally wide range of programming languages, with particular languages being suited to a particular class of problems, all of which often appears bewildering to the beginner. + +# 6.3 Program language development and engineering + +There is much in common between the development of programming languages and the development of anything from the engineering world. Consider the car: old cars offer much of the same functionality as more modern ones, but most people prefer driving newer models. The same is true of programming languages, where you can achieve much with the older languages, but the newer ones are easier to use. + +# 6.4 The early days + +A concept that proves very useful when discussing programming languages is that of the level of a machine. By this is meant how close a language is to the underly- ing machine that the program runs on. In the early days of programming (up to 1954) there were only two broad categories: machine languages and assemblers. The language that digital machines use is that of 0 and 1, i.e., they are binary de- vices. Writing a program in terms of patterns of 0 and 1 was not particularly satisfactory and the capability of using more meaningful mnemonics was soon in- troduced. Thus it was realised quite quickly that one of the most important aspects of programming languages is that they have to be read and understood by both ma- chines and humans. + +# 6.4.1 Fortran — The Early Days + +The next stage was the development of higher-level languages. The first of these was Fortran and it was developed over a 3 year period from 1954 to 1957 by an IBM team led by John Backus. This group achieved considerable success, and helped to prove that the way forward lay with high-level languages for com- puter-based problem solving. Fortran stands for for mula tran slation and was used mainly by people with a scientific background for solving problems that had a sig- nificant arithmetic content. It was thus relatively easy, for the time, to express this kind of problem in Fortran. + +By 1966 and the first standard Fortran: + +• Was widely available. • Was easy to teach. • Had demonstrated the benefits of subroutines and independent compila- tion. • Was relatively machine independent. • Often had very efficient implementations. + +Possibly the single most important fact about Fortran was, and still is, its widepread usage in the scientific community. + +# 6.4.2 Fortran 77 + +The next standard in 1977 (actually 1978, and thus out by one — a very common programming error, more of this later!) added character handling, but little else in the way of major new features, really tidying up some of the deficiencies of the 1966 standard. + +# 6.4.3 Cobol + +The business world also realised that computers were useful and several languages were developed, including FLOWMATIC, AIMACO, Commercial Translator and FACT, leading eventually to Cobol — Co mmon B usiness O rientated L anguage. There is a need in commercial programming to describe data in a much more com- plex fashion than for scientific programming, and Cobol had far greater capability in this area than Fortran. The language was unique at the time in that a group of competitors worked together with the objective of developing a language that would be useful on machines used by other manufacturers. + +The contributions made by Cobol include: + +• Firstly the separation among: • The task to be undertaken. • The description of the data involved. • The working environment in which the task is carried out. • Secondly a data description mechanism that was largely machine inde- pendent. • Thirdly its effectiveness for handling large files. • Fourthly the benefit to be gained from a programming language that was easy to read. + +Modern developments in computing — of report generators, file-handling software, fourth-generation development tools, and especially the increasing availability of commercial relational database management systems — are gradually replacing the use of Cobol, except where high efficiency and/or tight control are required. + +# 6.4.4 Algol + +Another important development of the 1950s was Algol. It had a history of devel- opment from Algol 58, the original Algol language, through Algol 60 eventually to the Revised Algol 60 Report. Some of the design criteria for Algol 58 were: + +• The language should be as close as possible to standard mathematical no- tation and should be readable with little further explanation. • It should be possible to use it for the description of computing processes in publications. • The new language should be mechanically translatable into machine pro- grams. + +A sad feature of Algol 58 was the lack of any input/output facilities, and this meant that different implementations often had incompatible features in this area. + +The next important step for Algol occurred at a UNESCO-sponsored conference in June 1959. There was an open discussion on Algol and the outcome was Algol 60, and eventually the Revised Algol 60 Report. + +It was at this conference that John Backus gave his now famous paper on a method for defining the syntax of a language, called Backus Normal Form, or BNF. The full significance of the paper was not immediately recognised. However, BNF was to prove of enormous value in language definition, and helped provide an interface point with computational linguistics. + +The contributions of Algol to program language development include: + +• Block structure. • Scope rules for variables because of block structure. • The BNF definition by Backus — most languages now have a formal def- inition. • The support of recursion. • Its offspring. + +Thus Algol was to prove to make a contribution to programming languages that was never reflected in the use of Algol 60 itself, in that it has been the parent of one of the main strands of program language development. + +# 6.5 Chomsky and program language development + +Programming languages are of considerable linguistic interest, and the work of Chomsky in 1956 in this area was to prove of inestimable value. Chomsky's sys- tem of transformation al grammar was developed in order to give a precise mathematical description to certain aspects of language. Simplistically, Chomsky describes grammars, and these grammars in turn can be used to define or generate corresponding kinds of languages. It can be shown that for each type of grammar and language there is a corresponding type of machine. It was quickly realised that there was a link with the earlier work of Turing. + +This link helped provide a firm scientific base for programming language develop- ment, and modern compiler writing has come a long way from the early work of Backus and his team at IBM. It may seem unimportant when playing a video game at home or in an arcade, but for some it is very comforting that there is a firm the- oretical basis behind all that fun. + +# 6.6 Lisp + +There were also developments in very specialized areas. List processing was prov- ing to be of great interest in the 1950s and saw the development of IPLV between 1954 and 1958. This in turn led to the development of Lisp at the end of the 1950s. Lisp has proved to be of considerable use for programming in the areas of artificial intelligence, playing chess, automatic theorem proving and general prob- lem solving. It was one of the first languages to be interpreted rather than compiled. Whilst interpreted languages are invariably slower and less efficient in their use of the underlying computer systems than compiled languages, they do provide great opportunities for the user to explore and try out ideas whilst sitting at a terminal. The power that this gives to the computational problem solver is con- siderable. + +Possibly the greatest contribution to program language development made by Lisp was its functional notation. One of the major problems for the Lisp user has been the large number of Lisp flavours, and this has reduced the impact that the lan- guage has had and deserved. + +# 6.7 Snobol + +Snobol was developed to aid in string processing, which was seen as an important part of many computing tasks, e.g., parsing of a program. Probably the most im- portant thing that Snobol demonstrated was the power of pattern matching in a programming language, e.g., it is possible to define a pattern for a title that would include Mr, Mrs, Miss, Rev, etc., and search for this pattern in a text using Snobol. Like Lisp it is generally available as an interpreter rather than a compiler, but com- piled versions do exist, and are often called Spitbol. Pattern-matching capabilities are now to be found in many editors and this makes them very powerful and useful tools. It is in the area of text manipulation that Snobol's greatest contribution to program language development lies. + +# 6.8 Second-generation languages + +# 6.8.1 PL/1 and Algol 68 + +It is probably true that Fortran, Algol 60 and Cobol are the three main first-genera- tion high-level languages. The 1960s saw the emergence of PL/1 and Algol 68. PL/1 was a synthesis of features of Fortran, Algol 60 and Cobol. It was soon real- ised that whilst PL/1 had great richness and power of expression this was in some ways offset by the greater difficulties involved in language definition and use. + +These latter problems were also true of Algol 68. The report introduced its own syntactic and semantic conventions and thus forced another stage in the learning process on the prospective user. However, it has a small but very committed user population who like the very rich facilities provided by the language. + +# 6.8.2 Simula + +Another strand that makes up program language development is provided by Simula, a general purpose programming language developed by Dahl, Myhrhaug and Nygaard of the Norwegian Computing Centre. The most important contribu- tion that Simula makes is the provision of language constructs that aid the programming of complex, highly interactive problems. It is thus heavily used in the areas of simulation and modelling. It was effectively the first language to offer the opportunity of object orientated programming, and we will come back to this very important development in programming languages later in this chapter. + +# 6.8.3 Pascal + +The designer of Pascal, Niklaus Wirth, had participated in the early stages of the design of Algol 68 but considered that the generality and complexity of Algol 68 was a move in the wrong direction. Pascal (like Algol 68) had its roots in Algol 60 but aimed at providing expressive power through a small set of straightforward concepts. This set is relatively easy to learn and helps in producing readable and hence more comprehensible programs. + +It became the language of first choice within the field of computer science during the 1970s and 1980s, and the comment by Wirth sums up the language very well: “although Pascal had no support from industry, professional societies, or govern- ment agencies, it became widely used. The important reason for this success was that many people capable of recognising its potential actively engaged themselves in its promotion. As crucial as the existence of good implementations is the avail- ability of documentation. The conciseness of the original report made it attractive for many teachers to expand it into valuable textbooks. Innumerable books ap- peared between 1977 and 1985, effectively promoting Pascal to become the most widespread language used in introductory programming courses. Good course ma- terial and implementations are the indispensable prerequisites for such an evolution.” + +# 6.8.4 APL + +APL is another interesting language of the early 1960s. It was developed by Iverson early in the decade and was available by the mid to late 1960s. It is an in- terpretive vector- and matrix-based language with an extensive set of operators for the manipulation of vectors, arrays, etc., of whatever data type. As with Algol 68 it has a small but dedicated user population. A possibly unfair comment about APL programs is that you do not debug them, but rewrite them! + +# 6.8.5 Basic + +Basic stands for B eginners A ll Purpose S ymbolic I nstruction C ode, and was devel- oped by Kemeny and Kurtz at Dartmouth during the 1960s. Its name gives a clue to its audience and it is very easy to learn. It is generally interpreted, though com- piled versions do exist. It is probably the most heavily used language on micros and home computers. It has proved to be well suited to the rapid development of small programs. It is much criticised because it lacks features that encourage or force the adoption of sound programming techniques. + +# 6.8.6 C + +There is a requirement in computing to be able to access the underlying machine directly or at least efficiently. It is therefore not surprising that computer profes- sionals have developed high-level languages to do this. This may well seem a contradiction, but it can be done to quite a surprising degree. Some of the earliest published work was that of Martin Richards on the development of BCPL. + +This language directly influenced the work of Ken Thompson and can be clearly seen in the programming languages B and C. The UNIX operating system is al- most totally written in C and demonstrates very clearly the benefits of the use of high-level languages wherever possible. + +With the widespread use of UNIX within the academic world C gained consider- able ground during the 1970s and 1980s. UNIX systems also offered much to the professional software developer, and became widely used for large-scale software development and as Ritchie says: “ C is quirky, flawed, and an enormous success. While accidents of history surely helped, it evidently satisfied a need for a system implementation language efficient enough to displace assembly language, yet suffi- ciently abstract and fluent to describe algorithms and interactions in a wide variety of environments. ” + +# 6.9 Some other strands in language development + +There are many strands that make up program language development and some of them are introduced here. + +# 6.9.1 Abstraction, stepwise refinement and modules + +Abstraction has proved to be very important in programming. It enables a complex task to be broken down into smaller parts concentrating on what we want to hap- pen rather than how we want it to happen. This leads almost automatically to the ideas of stepwise refinement and modules, with collections of modules to perform specific tasks or steps. + +# 6.9.2 Structured programming + +Structured programming in its narrowest sense concerns itself with the develop- ment of programs using a small but sufficient set of statements and, in particular, control statements. It has had a great effect on program language design, and most languages now support the minimal set of control structures. + +In a broader sense structured programming subsumes other objectives, including simplicity, comprehensibility, verifiability, modifiability and maintenance of pro- grams. + +# 6.9.3 Standardisation + +The purposes of a standard are quite varied and include: + +Investment in people: by this we mean that the time spent in learning a standard language pays off in the long term, as what one learns is applica- + +ble on any hardware/software platform that has a standard conformant compiler. + +• Portability: one can take the code one has written for one hardware/soft- ware platform and move it to any hardware/software platform that has a standard conformant compiler. + +• Known reference point: when making comparisons one starts with refer- ence to the standard first, and then between the additional functionality of the various implementations + +These are some but not all of the reasons for the use of standards. Their impor- tance is summed up beautifully by Ronald G. Ross in his introduction to the Cannan and Otten book on the SQL standard: “ Anybody who has ever plugged in an electric cord into a wall outlet can readily appreciate the inestimable benefits of workable standards. Indeed, with respect to electrical power, the very fact that we seldom even think about such access (until something goes wrong) is a sure sign of just how fundamentally important a successful standard can be. ” + +Appendix A contains notes on what standards apply at this time for the languages covered. + +# 6.10 Ada + +Ada represents the culmination of many years of work in program language devel- opment. It was a collective effort and the main aim was to produce a language suitable for programming large-scale and real-time systems. Work started in 1974 with the formulation of a series of documents by the American Department of De- fence (DoD), which led to the Steelman documents. It is a modern algorithmic language with the usual control structures and facilities for the use of modules, and allows separate compilation with type checking across modules. + +Ada is a powerful and well-engineered language. Its widespread use is certain as it has the backing of the DoD. However, it is a large and complex language and con- sequently requires some effort to learn. It seems unlikely to be widely used except by a small number of computer professionals. + +# 6.11 Modula + +Modula was designed by Wirth during the 1970s at ETH, for the programming of embedded real-time systems. It has many of the features of Pascal, and can be taken for Pascal at a glance. The key new features that Modula introduced were those of processes and monitors. + +As with Pascal it is relatively easy to learn and this makes it much more attractive than Ada for most people, achieving much of the capability without the complex- ity. + +# 6.12 Modula 2 + +Wirth carried on developing his ideas about programming languages and the culmi- nation of this can be seen in Modula 2. In his words: “In 1977, a research project with the goal to design a computer system (hardware and software) in an inte- grated approach, was launched at the Institut fur Informatik of ETH Zurich. This system (later to be called Lilith) was to be programmed in a single high level lan- guage, which therefore had to satisfy requirements of high level system design as well as those of low level programming of parts that closely interact with the given hardware. Modula 2 emerged from careful design deliberations as a language that includes all aspects of Pascal and extends them with the important module concept and those of multi-programming. Since its syntax was more in line with Modula than Pascal's the chosen name was Modula 2.” + +The language's main additions with regard to Pascal are: + +• The module concept, and in particular the facility to split a module into a definition part and an implementation part. • A more systematic syntax which facilitates the learning process. In partic- ular, every structure starting with a keyword also ends with a keyword, i.e., is properly bracketed. • The concept of process as the key to multi programming facilities. • So-called low-level facilities, which make it possible to breach the rigid type consistency rules and allow one to map data with Modula 2 structure onto a store without inherent structure. • The procedure type, which allows procedures to be dynamically assigned to variables. + +A sad feature of Modula 2 has been the long time taken to arrive at a standard for the language. + +# 6.13 Other language developments + +The following is a small selection of language developments that the authors find interesting — they may well not be included in other people's coverage. + +# 6.13.1 Logo + +Logo is a language that was developed by Papert and colleagues at the Artificial Intelligence Laboratory at MIT. Papert is a professor of both mathematics and edu- cation, and has been much influenced by the psychologist Piaget. The language is used to create learning environments in which children can communicate with a computer. The language is primarily used to demonstrate and help children develop fundamental concepts of mathematics. Probably the turtle and turtle geometry are known by educationists outside of the context of Logo. Turtles have been incorpo- rated into the Smalltalk computer system developed at Xerox Palo Alto Research Centre — Xerox PARC. + +# 6.13.2 Postscript, TeX and LaTeX + +The 1980s saw a rapid spread in the use of computers for the production of printed material. The 3 languages are each used quite extensively in this area. + +Postscript is a low-level interpretive programming language with good graphics ca- pabilities. Its primary purpose is to enable the easy production of pages containing text, graphical shapes and images. It is rarely seen by most end users of modern desktop publishing systems, but underlies many of these systems. It is supported by an increasing number of laser printers and typesetters. + +TeX is a language designed for the production of mathematical texts, and was de- veloped by Donald Knuth. It linearises the production of mathematics using a standard computer keyboard. It is widely used in the scientific community for the production of documents involving mathematical equations. + +LaTex is Leslie Lamport's version of TeX, and is regarded by many as more friendly. It is basically a set of macros that hide raw TeX from the end user. The TeX/LaTeX ratio is probably 1 to 9 (or so I'm reliably informed by a TeXie). + +# 6.13.3 Prolog + +Prolog was originally developed at Marseille by a group led by Colmerauer in 1972/73. It has since been extended and developed by several people, including Pereira (L.M.), Pereira (F), Warren and Kowalski. Prolog is unusual in that it is a vehicle for logic programming. Most of the languages described here are basically algorithmic languages and require a specification of how you want something done. Logic programming concentrates on the what rather than the how. The lan- guage appears strange at first, but has been taught by Kowalski and others to 10-year-old children at schools in London. + +# 6.13.4SQL + +SQL stands for Structured Query Language, and was originally developed by peo- ple mainly working for IBM in the San Jose Research Laboratory. It is a relational database language, and enables programmers to define, manipulate and control data in a relational database. Simplistically, a relational database is seen by a user as a collection of tables, comprising rows and columns. It has become the most impor- tant language in the whole database field. + +# 6.13.5 ICON + +ICON is in the same family as Snobol, and is a high-level general purpose pro- gramming language that has most of the features necessary for efficient processing of nonnumeric data. Griswold (one of the original design team for Snobol) has learnt much since the design and implementation of Snobol, and the language is a joy to use in most areas of text manipulation. + +It is available for most systems via anonymous FTP from a number of sites on the Internet. + +# 6.14 Object orientated programming — OOP + +OOP represents a major advance in program language development. The concepts that this introduces include: + +• Classes. • Objects. • Messages. • Methods. + +These in turn draw on the ideas found in more conventional programming lan- guages and correspond to + +• Extensible data types. • Instances of a class. • Dynamically bound procedure calls. • Procedures of a class. + +Inheritance is a very powerful high-level concept introduced with OOP. It enables an existing data type with its range of valid operations to form the basis for a new class, with more data types added with corresponding operations, and the new type is compatible with the original. + +As was mentioned earlier, the first language to offer functionality in this area was Simula, and thus the ideas originated in the 1960s. The book Simula Begin by Birtwistle, Dahl, Myhrhaug and Nygaard is well worth a read as it represents one of the first books to introduce the concepts of OOP. + +# 6.14.1 Oberon and Oberon 2 + +As Wirth says: “The programming language Oberon is the result of a concen- trated effort to increase the power of Modula-2 and simultaneously to reduce its complexity. Several features were eliminated, and a few were added in order to in- crease the expressive power and flexibility of the language.” + +Oberon and Oberon 2 are thus developments beyond Modula 2. The main new concept added to Oberon was that of type extension. This enables the construction of new data types based on existing types and allows one to take advantage of what has already been done for that existing type. + +Language constructs removed included: + +• Variant records. • Opaque types. • Enumeration types. • Subrange types. • Local modules. • WITH statement. • Type transfer functions. • Concurrency. + +The short paper by Wirth provides a fuller coverage. It is available at ETH via anonymous FTP. + +# 6.14.2 Smalltalk + +# Language plus use of a computer system. + +Smalltalk has been under development by the Xerox PARC Learning Research Group since the 1970s. In their words: “Smalltalk is a graphical, interactive pro- gramming environment. As suggested by the personal computer vision, Smalltalk is designed so that every component in the system is accessible to the user and can be presented in a meaningful way for observation and manipulation. The user in- terface issues in Smalltalk revolve around the attempt to create a visual language for each object. The preferred hardware system for Smalltalk includes a high reso- lution graphical display screen and a pointing device such as a graphics pen or mouse. With these devices the user can select information viewed on the screen and invoke messages in order to interact with the information. ” Thus Smalltalk represents a very different strand in program language development. The ease of use of a system like this has long been appreciated and was first demonstrated commercially in the Macintosh microcomputers. + +Wirth has spent some time at Xerox PARC and has been influenced by their work. In his own words “ the most elating sensation was that after sixteen years of work- ing for computers the computer now seemed to work for me .” This influence can be seen in the design of the Lilith machine, the original Modula 2 engine, and in the development of Oberon as both a language and an operating system. + +# Chapter 6 + +# 6.14.3 $\mathbf{C++}$ + +Stroustrup did his Ph.D thesis at the Computing Laboratory, Cambridge University, England, and worked with Simula. He had previously worked with Simula at the University of Aarhus in Denmark. His comments are illuminating: “ but was pleas- antly surprised by the way the mechanisms of the Simula language became increasingly helpful as the size of the program increased. The class and co-routine mechanisms of Simula and the comprehensive type checking mechanisms ensured that problems and errors did not (as I - and I guess most people - would have ex- pected) grow linearly with the size of the program. Instead, the total program acted like a collection of very small (and therefore easy to write, comprehend and debug) programs rather than a single large program.” + +He designed $\mathrm{C++}$ to provide Simula's functionality within the framework of C's ef- ficiency, and he succeeded in this goal as $\mathrm{C++}$ is a widely used object oriented programming language. The major disadvantage now concerns the largely incom- patible class libraries that exist. It is hoped that the various standards bodies address this problem in the immediate future. + +# 6.14.4 Java + +Bill Joy (of Sun fame) had by the late 1980s decided that $\mathrm{C++}$ was too compli- cated and that an object oriented environment based upon $\mathrm{C++}$ would be of use. At around about the same time James Gosling (mister emacs) was starting to get frus- trated with the implementation of an SGML editor in $\mathrm{C++}$ . Oak was the outcome of Gosling's frustration. + +Sun over the next few years ended up developing Oak for a variety of projects. It wasn't until Sun developed their own web browser, Hotjava, that Java as a lan- guage hit the streets. And as the saying goes the rest is histor y. + +Java is a relatively simple object oriented language. Whilst it has its origins in $\mathrm{C++}$ it has dispensed with most of the dangerous features. It is OO throughout. Every- thing is a class. + +It is interpreted and the intermediate byte code will run on any machine that has a Java virtual machine for it. This is portability at the object code level, unlike porta- bility at the source code level — which is what we expect with most conventional languages. Some of the safe features of the language include: + +• Built in garbage collection. • No pointers — everything is passed by reference. + +It is multithreaded, which makes it a delight for many applications. It has an exten- sive windows toolkit, the so called AWT that was in the original release of the language and Swing that came in later. It achieves much of what Visual Basic 6 offers but within the framework of a far more powerful language. Development en- vironments are becoming widely available to aid in this task. + +A major drawback is the rapid development of the language and the large number of different versions, and further compounded by the different virtual machines available. + +# 6.14.5 Visual Basic + +Visual Basic (VB) has developed into one of the most widely used development platforms for Windows. Its main strength is the ability to quickly put a visual inter- face onto an a program. The following are some dates for the various versions that Microsoft have released:- + +• VB 1.0 May 1991 for Windows. • VB 1.0 for MS-DOS September 1992. • VB 2.0 November 1992. • VB 3.0 Summer 1993. • VB 4.0 August 1995. • VB 5 February 1997. • VB 6.0 Summer 1998. • VB .NET 2002. Two major drawbacks are: • It only runs under Microsoft Windows • It is a proprietary programming language and has been changed several times by Microsoft. The .net version is a backwards incompatible upgrade to previous versions. + +It is one of the most widely used programming languages on the Windows plat- forms. + +# 6.14.6 C# + +$\mathrm{C}\#$ is a new language from Microsoft and is a key part of their .net framework. It is a modern, well-engineered language in the same family of programming lan- guages in terms of syntax as C, $\mathrm{C++}$ and Java. If you have a knowledge of one of these languages it will look very familiar. + +One of the design goals was to produce a component oriented language, and to build on the work that Microsoft had done with OLE, ActiveX and COM: + +• ActiveX is a set of technologies that enables software components to in- teract with one another in a networked environment, regardless of the lan- guage in which they were created. ActiveX was built on the Component Object Model (COM). • COM is the object model on which ActiveX Controls and OLE are built. COM allows an object to expose its functionality to other components and to host applications. It defines both how the object exposes itself and how this exposure works across processes and networks. COM also defines the object's life cycle. • OLE is a mechanism that allows users to create and edit documents con- taining items or objects created by multiple applications. OLE was origi- nally an acronym for Object Linking and Embedding. However, it is now referred to simply as OLE. Parts of OLE not related to linking and embed- ding are now part of Active technology. + +Other design goals included creating a language: + +• Where everything is an object — C# also has a mechanism for going be- tween objects and fundamental types (integers, reals, etc.). • Which would enable the construction of robust and reliable software — it has garbage collection, exception handling and type safety. • Which would use a $\mathrm{C}/\mathrm{C++}/\mathrm{Java}$ syntax which is already widely known and thus help programmers converting from one of these languages to C#. + +Microsoft has submitted $\mathrm{C\#}$ to the ECMA for formal standardisation and it became an ISO standard in 2003 - ISO/IEC 23270. Visit + +• http://en.wikipedia.org/wiki/C Sharp programming language#Standard- ization + +for up-to-date information on the standardisation effort. + +# 6.15 Fortran 90 + +Almost as soon as the Fortran 77 standard was complete and published, work be- gan on the next version. The language drew on many of the ideas covered in this chapter and these help to make Fortran 90 a very promising language. Some of the new features included: + +• New source form, with blanks being significant and names being up to 31 characters. • Better control structures. • Control of the precision of numerical computation. + +• Array processing. • Pointers. • User defined data types and operators. • Procedures. • Modules. • Recursion. • Dynamic storage allocation. + +We will look into all of these in turn. + +# 6.16 Fortran 1995 + +Fortran was next standardised in 1996 — yet again out by one! Firstly we have a clear up of some of the areas in the standard that had emerged as requiring clarifi- cation. Secondly Fortran 95 added the following major concepts: + +• The FORALL construct. • PURE and ELEMENTAL procedures. • Implicit initialisation of derived-type objects. • Initial association status for pointers. + +The first two help considerably in parallelization of code. + +Minor features include amongst others: + +• Automatic deallocation of allocatable arrays. • Intrinsic SIGN function distinguishes between $-0$ and $+0$ . • Intrinsic function NULL returns disconnected pointer. • Intrinsic function CPU_TIME returns the processor time. • References to some pure functions are allowed in specification statements. • Nested WHERE constructs. • Masked ELSEWHERE construct. • Small changes to the CEILING, FLOOR, MAXLOC and MINLOC intrin- sic functions. + +Some of these were added to keep Fortran in line with High Performance Fortran (HPF). More details are given later. + +Part 2 of the standard (ISO/IEC 1539-2:1994) adds the functional specification for varying length character data type, and this extends the usefulness of Fortran for character applications very considerably. + +# 6.17 ISO technical reports TR15580 and TR15581 + +There are two additional reports that have been published on Fortran. TR 15580 specifies three modules that provide access to IEEE floating point arithmetic and TR15581 allows the use of the ALLOCATABLE attribute on dummy arguments, function results and structure components. + +# 6.18 Fortran 2003 + +The language is known as Fortran 2003 even though the language did not make it through the standardisation process until 2004. It is a major revision. + +• Derived-type enhancements: parameterised derived types (allows the kind, length, or shape of a derived type’s components to be chosen when the derived type is used), mixed component accessibility (allows different components to have different accessibility), public entities of private type, improved structure constructors, and finalisers. • Object oriented programming support: enhanced data abstraction (allows one type to extend the definition of another type), polymorphism (allows the type of a variable to vary at run time), dynamic type allocation, SELECT TYPE construct (allows a choice of execution flow depending upon the type a polymorphic object currently has), and type-bound proce- dures. • The ASSOCIATE construct (allows a complex expression or object to be denoted by a simple symbol). • Data manipulation enhancements: allocatable components, deferred-type parameters, VOLATILE attribute, explicit type specification in array con- structors, INTENT specification of pointer arguments, specified lower bounds of pointer assignment and pointer rank remapping, extended ini- tialisation expressions, MAX and MIN intrinsics for character type, and enhanced complex constants. • Input/output enhancements: asynchronous transfer operations (allow a pro- gram to continue to process data while an input/output transfer occurs), stream access (allows access to a file without reference to any record structure), user specified transfer operations for derived types, user speci- fied control of rounding during format conversions, the FLUSH statement, named constants for preconnected units, regularisation of input/output keywords, and access to input/output error messages. + +• Procedure pointers. • Scoping enhancements: the ability to rename defined operators (supports greater data abstraction) and control of host association into interface bod- ies. • Support for IEC 60559 (IEEE 754) exceptions and arithmetic (to the ex- tent a processor’s arithmetic supports the IEC standard). • Interoperability with the C programming language (allows portable access to many libraries and the low-level facilities provided by C and allows the portable use of Fortran libraries by programs written in C). • Support for international usage: (ISO 10646) and choice of decimal or comma in numeric formatted input/output. • Enhanced integration with the host operating system: access to command line arguments and environment variables and access to the processor’s error messages (improves the ability to handle exceptional conditions). + +It is not clear at this time when compilers will be available that fully conform to the Fortran 2003 standard. At the time of writing this book some compilers had started to implement some of the 2003 features. Up-to-date information can be found at + +• http://www.kcl.ac.uk/fortran + +where we make available a number of Fortran resources including details of com- pilers, books, code restructers, etc. + +# 6.19 DTR 19767 enhanced module facilities + +The module system in Fortran has a number of shortcomings and this DTR ad- dresses some of the issues. + +One of the major issues was the so-called recompilation cascade. Changes to one part of a module forced recompilation of all code that used the module. Modula 2 addressed this issue by distinguishing between the definition or interface and im- plementation. This can now be achieved in Fortran via submodules. + +If a module as specified by International Standard ISO/IEC 1539-1:2004 is used to package proprietary software, the source text of the module cannot be published as authoritative documentation of the interface of the module without either exposing trade secrets or requiring the expense of separating the implementation from the in- terface every time a revision is published. + +Using facilities specified in this technical report, one can easily publish the source text of the module as authoritative documentation of its interface, while witholding publication of the source text of the submodules that contain the implementation details and the trade secrets embodied within them. + +# 6.20 Internet resources + +The Internet provides access to a wealth of information regarding the Fortran fam- ily of languages. + +# 6.20.1 Standards information + +The official home of the standard is + +• http://www.nag.co.uk/sc22wg5/ + +We recommend visiting the site to keep up to date with Fortran developments. + +Their official ftp server can be found at + +•ftp://ftp.nag.co.uk/sc22wg5/ + +Copies of all working documents can be found there. + +Also have a look at + +• ftp://ftp.nag.co.uk/sc22wg5/N1601-N1650/Index.txt + +The following is a version of this file as of November 2004. + +ISO/IEC JTC1/SC22/WG5 N1650 + +INDEX OF DOCUMENTS 1601-1650 + +(Documents enclosed in square brackets are not yet available) + +1601 Draft International Standard for Fortran 2003 (Maine) 1602 Draft TR on Enhanced Module Facilities (Snyder) 1603 Response to the PDTR ballot (Reid) 1604 Report from Netherlands (van Waveren) 1605 Abstract for the TR on Enhanced Module Facilities (Snyder) 1606 Index of meetings (SD7) (Muxworthy) 1607 SC22 Project Information (SD3) (Reid) + +1608 WG5 Business Plan and Convener's Report to SC22 (Reid) 1609 Result of Enhanced Module Facilities DTR ballot (JTC 1) 1610 Result of Fortran 2003 DIS ballot (JTC 1) + +The documents in square brackets are not yet available. + +# 6.20.2 Fortran discussion lists + +The first to look at is the Fortran 90 list. Details can be found at • http://www.jiscmail.ac.uk/lists/COMP-FORTRAN-90.html + +If you subscribe you will have access to people involved in Fortran standardisation, language implementors for most of the hardware and software platforms, people using Fortran in many very specialised areas, people teaching Fortran, etc. + +There is also a comp.lang.fortran list available via USENET news. This provides access to people worldwide with enormous combined expertise in all aspects of Fortran. Invariably someone will have encountered your problem or one very much like it and have one or more solutions. + +There are many people on the Internet who will make the time to provide you with very valuable advice. As a point of network etiquette please do not waste band- width with questions that are answered in the FAQ. Please also spend some time developing an understanding of your problem and making some attempt to see if the answer lies in the documentation or manuals. In computing services and techni- cal support many user problems are labelled RTFM — read the fabulous manual. + +# 6.20.3 Other sources + +The following URLs are very useful: + +• Fortran 90 FAQ, maintained by Michel Olagnon • http://www.ifremer.fr/ditigo/molagnon/fortran90/engfaq.html • http://www.kcl.ac.uk/fortran • The Fortran Market, maintained by Walt Brainerd • http://www.fortran.com/fortran/market.html • Fortran90/95 Information, maintained by Mike Metcalf • http://www.kcl.ac.uk/fortran • Fortran FAQ, maintained by Keith Bierman, Sun + +# Chapter 6 + +• http://www.fortran.com/fortran/FAQ/cont.html • http://www.kcl.ac.uk/fortran + +# 6.21 Summary + +It is hoped that you now have some idea about the wide variety of uses that pro- gramming languages are put to. + +# 6.22 Bibliography + +Fortran 2003 Standard, ISO/IEC DIS 1539-1:2004(E) + +DTR 19767: Enhanced Module Facilities: ISO/IEC TR 19767:2004(E) + +The ISO home page is + +• http://www.iso.org/ + +The standard was published on $18^{\mathrm{th}}$ November 2004. + +The J3 home page is: • http://j3-fortran.org + +The WG5 home page is: + +• http://www.nag.co.uk/sc22wg5/ + +Both have copies of working documents. + +Adobe Systems Incorporated, Postscript Language: Tutorial and Cookbook , Addi- son-Wesley, 1985. + +Adobe Systems Incorporated, Postscript Language: Reference Manual, Addi- son-Wesley, 1985. + +Adobe System Incorporated, Postscript Language: Program Design, Addison-Wes- ley, 1985. + +The three books provide a comprehensive coverage of the facilities and capa- bilities of Postscript. + +ACM SIG PLAN, History of Programming Languages Conference — HOPL-II , ACM Press, 1993. + +One of the best sources of information on programming language develop- ments, from an historical perspective. There is coverage of Ada, Algol 68, C, $\mathrm{C++}$ , CLU, Concurrent Pascal, Formac, Forth, Icon, Lisp, Pascal, Prolog, Smalltalk and Simulation Languages by the people involved in the original de- sign and or implementation. Very highly recommended. This is the second in the HOPL series, and the first was edited by Wexelblat. Details are given later. + +Adams J.C., Brainerd W.S., Martin J.T., Smith B.T., Wagener J.L., Fortran 90 Handbook: Complete ANSI/ISO Reference, McGraw-Hill, 1992. A complete coverage of the language. As with the Metcalf and Reid book some of the authors were on the X3J3 committee. Originally expensive, but very thorough. Annals of the History of Computing, Special Issue: Fortran's 25 Anniversary, ACM, Article 6,1, 1984. Very interesting comments, some anecdotal, about the early work on Fortran. Barnes J., Programming in Ada 95, Addison-Wesley, 1996. One of the best Ada books. He was a member of the original design team Bergin T.J., Gibson R.G., History of Programming Languages , Addison-Wesley, 1996. This is a formal book publication of the Cenference Prceedings of HOPL II. The earlier work is based on preprints of the papers. Birtwistle G.M., Dahl O. J., Myhrhaug B., Nygaard K., SIMULA BEGIN , Chart- well-Bratt Ltd, 1979. A number of chapters in the book will be of interest to programmers unfamil- iar with some of the ideas involved in a variety of areas including systems and models, simulation, and co-routines. Also has some sound practical advice on problem solving. Brinch-Hansen P., The Programming Language Concurrent Pascal, IEEE Transac- tions on Software Engineering, June 1975, 199-207. Looks at the extensions to Pascal necessary to support concurrent processes. Cannan S., Otten G., SQL — The Standard Handbook , McGraw-Hill, 1993. Very thorough coverage of the SQL standard, ISO 9075:1992(E). Chivers I.D., Clark M.W., History and Future of Fortran, Data Processing, vol. 27 no 1, January/February 1985. Short article on an early draft of the standard, around version 90. Chivers Ian, Essential C# Fast, Springer, ISBN 1-85233-562-9 A quick introduction to the C# programming language. Chivers I.D., A Practical Introduction to Standard Pascal , Ellis Horwood, 1986. A short introduction to Pascal. Date C.J., A Guide to the SQL Standard , Addison-Wesley, 1997. Date has written extensively on the whole database field, and this book looks at the SQL language itself. As with many of Date's works quite easy to read. Appendix F provides a useful SQL bibliography. + +Deitel H.M., Deitel P.J., Java: How to Program , Prentice-Hall, 1999. A very good introduction to Java. Deitel H.M., Deitel P.J., Nieto T.R., Simply Visual Basic .Net , Prentice-Hall, 2003. Good practical introduction to VB .NET. Eckstein R., Loy M., Wood D., Java Swing , O'Reilly, 1998. Comprehensive coverage of the visual interface features available in Java. Flanagan D., Java in a Nutshell , O'Reilly, 1996. Just what you would expect from this series. Very useful reference text. Geissman L.B., Separate Compilation in Modula 2 and the Structure of the Mod- ula 2 Compiler on the Personal Computer Lilith , Dissertation 7286, ETH Zurich Harbison S.P., Steele G.L., A C Reference Manual , Prentice-Hall, 2002. Very good coverage of the various flavours of C, including K&R C, Standard C 1989, Standard C 1995, Standard C 1999 and Standard $\mathrm{C++}$ Jacobi C., Code Generation and the Lilith Architecture, Dissertation 7195, ETH Zurich Fascinating background reading concerning Modula 2 and the Lilith architec- ture. Goldberg A., Robson D., Smalltalk 80: The Language and its Implementation , Ad- dison-Wesley, 1983. Written by some of the Xerox PARC people who have been involved with the development of Smalltalk. Provides a good introduction (if that is possible with the written word) of the capabilities of Smalltalk. Goos G., Hartmanis J. (Eds), The Programming Language Ada — Reference Man- ual, Springer Verlag, 1981. The definition of the language. Griswold R.E., Poage J.F., Polonsky I.P., The Snobol4 Programming Language , Prentice-Hall, 1971. The original book on the language. Also provides some short historical mate- rial on the language. Griswold R.E., Griswold M.T., The Icon Programming Language , Prentice-Hall, 1983. The definition of the language with a lot of good examples. Also contains in- formation on how to obtain public domain versions of the language for a variety of machines and operating systems. + +Hoare C.A.R., Hints on Programming Language Design, SIGACT/SIGPLAN Symposium on Principles of Programming Languages, October 1973. The first sentence of the introduction sums it up beautifully: “I would like in this paper to present a philosophy of the design and evaluation of program- ming languages which I have adopted and developed over a number of years, namely that the primary purpose of a programming language is to help the programmer in the practice of his art. ” Jenson K., Wirth N., Pascal: User Manual and Report , Springer-Verlag, 1975. The original definition of the Pascal language. Understandably dated when one looks at more recent expositions on programming in Pascal. Kemeny J.G., Kurtz T.E., Basic Programming , Wiley, 1971. The original book on Basic by its designers. Kernighan B.W., Ritchie D.M., The $C$ Programming Language, Prentice-Hall, 1978 The original work on the C language, and thus essential for serious work with C. Kowalski R., Logic Programming in the Fifth Generation , The Knowledge Engi- neering Review, The BCS Specialist Group on Expert Systems. A short paper providing a good background to Prolog and logic programming, with an extensive bibliography. Knuth D. E., The TeXbook , Addison-Wesley, 1986. Knuth writes with an tremendous enthusiasm and perhaps this is understand- able as he did design TeX. Has to be read from cover to cover for a full understanding of the capability of TeX. Lyons J., Chomsky , Fontana/Collins, 1982. A good introduction to the work of Chomsky, with the added benefit that Chomsky himself read and commented on it for Lyons. Very readable. Malpas J., Prolog: A Relational Language and its Applications , Prentice-Hall, 1987. A good introduction to Prolog for people with some programming background. Good bibliography. Looks at a variety of versions of Prolog. Marcus C., Prolog Programming: Applications for Database Systems, Expert Sys- tems and Natural Language Systems, Addison-Wesley. Coverage of the use of Prolog in the above areas. As with the previous book aimed mainly at programmers, and hence not suitable as an introduction to Prolog as only two chapters are devoted to introducing Prolog. + +Metcalf M. and Reid J., Fortran 90 Explained , Oxford Science Publications, 1992. A clear compact coverage of the main features of Fortran 8x. Reid was secre- tary of the X3J3 committee. Mossenbeck H., Object-Orientated Programming in Oberon-2, Springer-Verlag, 1995. One of the best introductions to OOP. Uses Oberon-2 as the implementation language. Highly recommended. Papert S., Mindstorms - Children, Computers and Powerful Ideas , Harvester Press, 1980.Very personal vision of the uses of computers by children. It challenges many conventional ideas in this area. Sammet J., Programming Languages: History and Fundamentals , Prentice-Hall, 1969. Possibly the most comprehensive introduction to the history of program lan- guage development — ends unfortunately before the 1980s. Sethi R., Programming Languages: Concepts and Constructs , Addison-Wesley, 1989.The annotated bibliographic notes at the end of each chapter and the extensive bibliography make it a useful book. Reiser M., Wirth N., Programming in Oberon — Steps Beyond Pascal and Mod- ula, Addison-Wesley, 1992. Good introduction to Oberon. Revealing history of the developments behind Oberon. Reiser M., The Oberon System: User Guide and Programmer's Manual, Addision-Wesley, 1991. How to use the Oberon system, rather than the language. Stroustroup B., The $C++$ Programming Language, $3^{r d}$ Edition , Addison-Wesley, 1997. The $\mathrm{C++}$ book. Written by the designer of the language. Massive improve- ment over the earlier editions. Young S. J., An Introduction to Ada, $2^{n d}$ Edition , Ellis Horwood, 1984. A readable introduction to Ada. Greater clarity than the first edition. Wexelblat, History of Programming Languages, HOPL I , ACM Monograph Series, Academic Press, 1978. Very thorough coverage of the development of programming languages up to June 1978. Sessions on Fortran, Algol, Lisp, Cobol, APT, Jovial, GPSS, Simula, JOSS, Basic, PL/I, Snobol and APL, with speakers involved in the original languages. Very highly recommended. + +Wirth N., An Assessment of the Programming Language Pascal , IEEE Transac- tions on Software Engineering, June 1975, 192-198. Wirth N., History and Goals of Modula 2 , Byte, August 1984, 145-152. Straight from the horse's mouth! Wirth N., On the Design of Programming Languages , Proc. IFIP Congress 74, 386-393, North-Holland. Wirth N., The Programming Language Pascal, Acta Informatica 1, 35-63, 1971. Wirth N., Modula: a language for modular multi programming, Software Practice and Experience , 7, 3–35, 1977. Wirth N., Programming in Modula 2 , Springer-Verlag, 1983. The original definition of the language. Essential reading for anyone consider- ing programming in Modula 2 on a long term basis. Wirth N. Type Extensions, ACM Trans. on Prog. Languages and Systems , 10, 2 (April 1988), 2004-214 Wirth N. From Modula 2 to Oberon, Software — Practice and Experience , 18,7 (July 1988), 661–670 Wirth N., Gutknecht J., Project Oberon: The Design of an Operating System and Compiler, Addison-Wesley, 1992. Fascinating background to the development of Oberon. Highly recommended for anyone involved in large scale program development, not only in the areas of programming languages and operating systems, but more generally. + +# Introduction to Programming + +“Though this be madness, yet there is method in't” Shakespeare + +“‘Plenty of practice’ he went on repeating, all the time that Alice was getting him on his feet again. ‘plenty of practice.’” + +The White Knight, Through the Looking Glass and What Alice Found There, Lewis Carroll + +# Aims + +The aims of the chapter are: + +• To introduce the idea that there is a wide class of problems that can be solved with a computer and, further, that there is a relationship between the kind of problem to be solved and the choice of programming language that is used. • To give some of the reasons for the choice of Fortran. • To introduce the fundamental components or kinds of statements to be found in a general purpose programming language. • To introduce the three concepts of name, type and value. • To illustrate the above with sample programs based on three of the five intrinsic data types: • character, integer and real • To introduce some of the formal syntactical rules of Fortran. + +# Chapter 7 + +# 7 Introduction to Programming + +We have seen that an algorithm is a sequence of steps that will solve a part or the whole of a problem. A program is the realisation of an algorithm in a programming language, and there are at first sight a surprisingly large number of programming languages. The reason for this is that there is a wide range of problems that are solved using a computer, e.g., the telephone company generating itemised bills or the meteorological centre producing a weather forecast. These two problems make different demands on a programming language, and it is unlikely that the same lan- guage would be used to solve both. + +The range of problems that you want to solve will therefore strongly influence your choice of programming language. FORTRAN stands for FORmula TRANsla- tion, which gives a hint of the expected range of problems for which it is suitable. + +# 7.1 Language strengths and weaknesses + +Some of the reasons for choosing Fortran are: + +• It is a modern and expressive language, with much of the power of Ada without the complexity. • The language is now suitable for a very wide class of both numeric and nonnumeric problems. • The language is widely available in both the educational and scientific sectors. • A lot of software already exists, written in either Fortran 77 or its prede- cessor, Fortran 66, also known as Fortran IV. This code can be recompiled with standard conforming Fortran 90, 95 and 2003 compilers which protects any major investment in existing code. Some $15\%$ of code worldwide is estimated to be in Fortran. + +There are a few warts, however. Given that there has to be backwards compatibil- ity with Fortran 77 some of the syntax is clumsy to say the least. However, a considerable range of problems can now be addressed quite cleanly, if one sticks to a subset of the language and adopts a consistent style. + +# 7.2 Elements of a programming language + +As with ordinary (so-called natural ) languages, e.g., English, French, Gaelic, Ger- man, etc., programming languages have rules of syntax, grammar and spelling. The application of these rules in a programming language is more strict. A program has to be unambiguous, since it is a precise statement of the actions to be taken. Many everyday activities are rather vaguely defined — Buy some bread on your way home — but we are generally sufficiently adaptable to cope with the variations which occur as a result. If, in a program to calculate wages, we had an instruction Deduct some money for tax and insurance we could have an awkward problem when the program calculated completely different wages for the same person for the same amount of work every time it was run. One of the implications of the strict syntax of a programming language for the novice is that apparently silly error messages will appear when one first starts writing programs. As with many other new subjects you will have to learn some of the jargon to understand these mes- sages. + +Programming languages are made up of statements. We will look at the various kinds of statements briefly below. + +# 7.2.1 Data description statements + +These are necessary to describe the kinds of data that are to be processed. In the wages program, for example, there is obviously a difference between people's names and the amount of money they earn, i.e., these two things are not the same, and it would not make any sense adding your name to your wages. The technical term for this is data type — a wage would be of a different data type (a number) to a surname (a sequence of characters). + +# 7.2.2 Control structures + +A program can be regarded as a sequence of statements to solve a particular prob- lem, and it is common to find that this sequence needs to be varied in practice. Consider again the wages program. It will need to select among a variety of cir- cumstances (say married or single, paid weekly or monthly, etc,), and also to repeat the program for everybody employed. So there is the need in a program- ming language for statements to vary and/or repeat a sequence of statements. + +# 7.2.3 Data-processing statements + +It is necessary in a programming language to be able to process data. The kind of processing required will depend on the kind or type of data. In the wages program, for example, you will need to distinguish between names and wages. Therefore there must be different kinds of statements to manipulate the different types of data, i.e., wages and names . + +# 7.2.4 Input and output (I/O) statements + +For flexibility, programs are generally written so that the data that they work on exist outside the program. In the wages example the details for each person em- ployed would exist in a file somewhere, and there would be a record for each person in this file. This means that the program would not have to be modified each time a person left, was ill, etc., although the individual records might be up- dated. It is easier to modify data than to modify a program, and it is less likely to produce unexpected results. To be able to vary the action there must be some mechanism in a programming language for getting the data into and out of the pro- gram. This is done using input and output statements, sometimes shortened to I/O statements. + +Let us now consider a simple program which will read in somebody's first name and print it out: + +PROGRAM ch0701 + + ! + + ! This program reads in and prints out a name + + ! IMPLICIT NONE CHARACTER $^{\star}20$ :: First_Name + + ! PRINT \*,' Type in your first name.' PRINT \*,' up to 20 characters' READ \*,First_Name PRINT \*,First_Name + + ! + +END PROGRAM ch0701 + +There are several very important points to be covered here, and they will be taken in turn: + +• Each line is a statement. • There is a sequence to the statements. The statements will be processed in the order that they are presented, so in this example the sequence is print, read, print. • The first statement names the program. It makes sense to choose a name that conveys something about the purpose of the program. • The next three lines are comment statements. They are identified by a !. Comments are inserted in a program to explain the purpose of the pro- gram. They should be regarded as an integral part of all programs. It is es- sential to get into the habit of inserting comments into your programs straightaway. • The IMPLICIT NONE statement means that there has to be explicit typ- ing of each and every data item used in the program. It is good program- ming practice to include this statement in every program that you write, as it will trap many errors, some often very subtle in their effect. Using an analogy with a play, where there is always a list of the persona involved + +before the main text of the play we can say that this statement serves the + +• The CHARACTER $^{*}20$ statement is a type declaration. It was mentioned earlier that there are different kinds of data. There must be some way of telling the programming language that these data are of a certain type, and that therefore certain kinds of operations are allowed and others are banned or just plain stupid! It would not make sense to add a name to a number, e.g., what does Fred $+~10$ mean? So this statement defines that the variable First_Name is to be of type CHARACTER and only charac- ter operations are permitted. The concept of a variable is covered in the next section. Character variables of this type can hold up to 20 characters. • The PRINT statements print out an informative message to the terminal — in this case a guide as to what to type in. The use of informative mes- sages like this throughout your programs is strongly recommended. • The READ statement is one of the I/O statements. It is an instruction to read from the terminal or keyboard; whatever is typed in from the termi- nal will end up being associated with the variable First_Name. Input/out- put statements will be explained in greater detail in later sections. • The PRINT statement is another I/O statement. This statement will print out what is associated with the variable First_Name and, in this case, what you typed in. • The END PROGRAM statement terminates this program. It can be thought of as being similar to a full stop in natural language, in that it fin- ishes the program in the same way that a period ( . ) ends a sentence. Note the use of the name given in the PROGRAM statement at the start of the program. • Note also the use of the asterisk in three different contexts. • Indentation has been used to make the structure of the program easier to determine. Programs have to be read by human beings and we will look at this in more depth later. • Lastly, when you do run this program, character input will terminate with the first blank character. + +The above program illustrates the use of some of the statements in the Fortran lan- guage. Let us consider the action of the READ \* statement in more detail — in particular, what is meant by a variable and a value. + +# 7.3 Variables — name, type and value + +The idea of a variable is one that you are likely to have met before, probably in a mathematical context. Consider the following: + +circumference $=2\;\pi$ r + +This is an equation for the calculation of the circumference of a circle. The follow- ing represents a translation of this into Fortran: + +circumference $=2\mathrm{~}^{*}\mathrm{\pi}^{\mathrm{~*~}}$ radius + +There are a number of things to note about this equation: + +• Each of the variables on the right-hand side of the equals sign (pi and ra- dius) will have a value , which will allow the evaluation of the expression. • When the expression is fully evaluated the value is assigned to the vari- able on the left-hand side of the equals sign. • In mathemetics the multiplication is implied in Fortran we have to use the \* operator to indicate that we want to multiply 2 by pi by the radius. • We do not have access to mathematical symbols like $\pi$ in Fortran but have to use variable names based on letters from the Roman alphabet. + +The whole line is an example of an arithmetic assignment statement in Fortran. + +The following arithmetic assignment statement illustrates clearly the concepts of name and value, and the difference in the equals sign in mathematics and comput- ing: + +$$ +i=i+1 +$$ + +In Fortran this reads as take the current value of the variable $i$ and add one to it, store the new value back into the variable $i.$ , i.e., $i$ takes the value $_{i+1}$ . Alge- braically, + +$$ +i=i+1 +$$ + +does not make any sense. + +![Variables can be of different types. Table 7.1 shows some of those available in Fortran. ](images/20c325640f2ff90ccd03cc64435f47a33940479171e0c2996f029c406335adb2.jpg) + +Table 7.1 Variable, Type and Value + +Note the use of capitalisation and underscores to make the variable names easier to read. + +The concept of data type seems a little strange at first, especially as we commonly think of integers and reals as numbers. However, the benefits to be gained from this distinction are considerable. This will become apparent after you have written several programs. + +Let us now consider another program, one that reads in three numbers, adds them up and prints out both the total and the average: + +PROGRAM ch0702 + + ! + + ! This program reads in three numbers and sums + + ! and averages them + +IMPLICIT NONE REAL :: N1,N2,N3,Average $\begin{array}{r l}{\mathbf{\sigma}=}&{{}0\cdot0}\end{array}.$ , Total = 0.0 INTEGER :: $\mathrm{~\textit~{~N~}~}=\mathrm{~\textit~{~3}~}$ PRINT \*,' Type in three numbers.' PRINT \*,' Separated by spaces or commas' READ \*,N1,N2,N3 $\mathtt{T o t a l e}=\mathtt{N1}\;\;+\;\;\mathtt{N2}\;\;+\;\;\mathtt{N3}$ Average $=$ Total/N PRINT \*,'Total of numbers is ',Total PRINT \*,'Average of the numbers is ',Average END PROGRAM ch0702 + +# Chapter 7 + +# 7.4 Notes + +The program has been given a name that means something. + +There are comments at the start of the program describing what it does. + +The IMPLICIT NONE statement ensures that all data items introduced have to oc- cur in a type declaration. + +The next two statements are type declarations. They define the variables to be of real or integer type. Remember integers are whole numbers, whereas real numbers are those which have a decimal point. For example, 2 is an integer and 2.7, 2.00000001, and 2.0 are all real numbers. One of the fundamental distinctions in Fortran is between integers and reals. Type declarations must always come at the start of a program, before any processing is done. Note that the variables have been given sensible names to aid in making the program easier to understand. + +The variables Average, Total and $\mathbf{N}$ are also given initial values within the type declaration. Variables are initially undefined in Fortran, so the variables N1, N2, N3 fall into this category, as they have not been given values at the time that they are declared. + +The first PRINT statement makes a text message (in this case what is between the apostrophes) appear at the terminal. As was noted earlier, it is good practice to put out a message like this so that you have some idea of what you are supposed to type in. + +The READ statement looks at the input from the keyboard (i.e., what you type) and in this instance associates these values with the three variables. These values can be separated by commas (,), spaces ( ), or even by pressing the carriage return key, i.e., they can appear on separate lines. + +The next statement actually does some data processing. It adds up the values of the three variables (N1, N2, and N3) and assigns the result to the variable Total. This statement is called an arithmetic assignment statement , and is covered more fully in the next chapter. + +The next statement is another data-processing statement. It calculates the average of the numbers entered and assigns the result to Average. We could have actually used the value 3 here instead, i.e., written Average $\fallingdotseq$ Total/3 and have exactly the same effect. This would also have avoided the type declaration for N. However, the original example follows established programming practice of declaring all variables and establishing their meaning unambiguously. We will see further exam- ples of this type throughout the book. + +Indentation has been used to make the structure of the program easier to determine. + +The sum and average are printed out with suitable captions or headings. Do not write programs without putting captions on the results. It is too easy to make mis- takes when you do this, or even to forget what each number means. + +Finally we have the end of the program and again we have the use of the name in the PROGRAM statement. + +# 7.5 Some more Fortran rules + +There are certain things to learn about Fortran which have little immediate mean- ing and some which have no logical justification at all, other than historical precedence. Why is a cat called a cat? At the end of several chapters there will be a brief summary of these rules or regulations when necessary. Here are a few: + +• Source is free format. • Lowercase letters are permitted, but not required to be recognised. • Multiple statements may appear on one line and are separated by the semicolon character. • There is an order to the statements in Fortran. Within the context of what you have covered so far, the order is: • PROGRAM statement. • Type declarations, e.g., IMPLICIT, INTEGER, REAL or CHARAC- TER. • Processing and I/O statements. • END PROGRAM statement. • Comments may appear anywhere in the program, after PROGRAM and before END; they are introduced with a ! character, and can be in line. • Names may be up to 31 characters in length and include the underscore character. • Lines may be up to 132 characters. • Up to 39 continuation lines are allowed (using the ampersand $(\&)$ as the continuation character). • The syntax of the READ and PRINT statement introduced in these exam- ples is • READ format, input-item-list . • PRINT format, output-item-list . + + where format is \* in the examples and called list directed format- ting.  and input-item-list is a list of variable names separated by com- mas.  and output-item-list is a list of variable names and/or a sequence of characters enclosed in either ' or " , again separated by com- mas. + +• If the IMPLICIT NONE statement is not used, variables that are not ex- plicitly declared will default to REAL if the first letter of the variable name is A-H or O-Z, and to INTEGER if the first letter of the variable name is I-N. + +# 7.6 Fortran character set + +The following summarises the Fortran character set: + +A-Z Letters 0-9 Digits _ Underscore Blank $=$ Equal + Plus - Minus \* Asterisk / Slash or oblique \ Backslash ( Left parenthesis ) Right parenthesis [ Left square bracket ] Right square bracket {Left curly bracket}Right curly bracket, Comma . Period or decimal point : Colon ; Semicolon ! Exclamation mark “ Quotation mark $\%$ Percent&Ampersand\~ Tilde $@$ Commercial at $<$ Less than $>$ Greater than ? Question mark ' Apostrophe \` Grave accent $\wedge$ Circumflex accent | Vertical bar or line $\S$ Currency symbol # Number sign + +# 7.7 Good programming guidelines + +The following are guidelines, and do not form part of the Fortran language defini- tion: + +• Use comments to clarify the purpose of both sections of the program and the whole program. • Choose meaningful names in your programs. • Use indentation to highlight the structure of the program. Remember that the program has to be read and understood by both humans and a com- puter. • Use IMPLICIT NONE in all programs you write to minimise errors. + +Do not rely on the rules for explicit typing, as this is a major source of errors in programming. + +# 7.8 Compilers + +A number of hardware platforms, operating systems and compilers have been used when writing this book and the two earlier books on Fortran 95 and Fortran 90: + +• DEC VAX under VMS and later OPEN VMS with the NAG Fortran 90 compiler. • DEC Alpha under OPEN VMS using the DEC Fortran 90 compiler. • Sun Ultra Sparc under Solaris: • NAGACE F90 compiler. • NAGWare F95 compiler. • Sun (Release 1.x) F90 compiler. • Sun (Release 2.x) F90 compiler. • PCs under DOS and Windows: • DEC/Compaq Fortran 90 and Fortran 95 compilers. •Intel Compiler (7.x, 8.x).• Lahey Futitsu Fortran 95 (5.7). • NAG Fortran 95 Compiler. • NAG Salford Fortran 90 Compiler. • Salford Fortran 95 Compiler. + +# Chapter 7 + +• PCs under Linux: • Intel Compiler. • Lahey Fujitsu Fortran 95 Pro (6.1). •NAG Fortran 95 (4.x, 5.x). + +It is very illuminating to use more than one compiler whilst developing programs. + +# 7.9 Program development + +A number of ways of developing programs have been used, including: + +• Using DEC terminals to log into the DEC VAX and DEC Alpha systems. • Using PCs running terminal emulation software to log into the DEC VAX and DEC Alpha systems. • Using terminal emulation software to log into the SUN Ultra Sparc. • Using X-Windows software to log into the SUN Ultra Sparc systems. • Using a DOS box and simple command line prompt. • Using an integrated development environment, e.g., Microsoft Developer Studio. + +It is likely that you will end up doing at least one of the above and probably more. The key stages involved are: + +• Creating and making changes to the Fortran program source. • Saving the file. • Compiling the program: • If there are errors you must go back to the Fortran source and make the changes indicated by the compiler error messages. • Linking if successful to generate an executable: • Automatic link. This happens behind the scenes and the executable is generated for you immediately. • Manual link. You explicitly invoke the linker to generate the execut- able. • Running the program. • Determining whether the program actually works and gives the results ex- pected. + +These steps must be taken regardless of the hardware platform, operating system and compiler you use. Some people like working at the operating system prompt (e.g., DOS or UNIX), and others prefer working within a development environ- ment. Both have their strengths and weaknesses. + +# 7.10 Problems + +1. Compile and run example 1 in this chapter. Experiment with the following types of input. + +Ian + +Ian Chivers + +"Jane Margaret Sleightholme" + +2. Compile and run example 2 in this chapter. + +Think about the following points: + +• Is there a difference between separating the input by spaces or commas? • Do you need the decimal point? • What happens when you type in too many data? • What happens when you type in too few data? + +If you have access to more than one compiler repeat the above and compare the re- sults. + +3. Write a program that will read in your name and address and print them out in reverse order. + +Think about the following points: + +• How many lines are there in your name and address? • What is the maximum number of characters in the longest line in your name and address? • What happens at the first blank character of each input line? • Which characters can be used in Fortran to enclose each line of text typed in and hence not stop at the first blank character? • If you use one of the two special characters to enclose text what happens if you start on one line and then press the return key before terminating the text? + +The action here will vary between Fortran implementations. + +# Arithmetic + +“Taking Three as the subject to reason about — A convenient number to state — We add Seven, and Ten, and then multiply out By One Thousand diminished by Eight. The result we proceed to divide, as you see, By Nine Hundred and Ninety and Two: Then subtract Seventeen, and the answer must be Exactly and perfectly true.” + +Lewis Carroll, The Hunting of the Snark + +“Round numbers are always false.” + +Samuel Johnson + +# Aims + +The aims of this chapter are to introduce: + +• The rules for the evaluation of arithmetic expressions to ensure that they are evaluated as you intend. • The idea of truncation and rounding applied to reals. • The use of the PARAMETER statement to define or set up constants. • The concepts and ideas involved in numerical computation, including: • Specifying data types using kind-type parameters. • The concept of numeric models and positional number systems for in- teger and real arithmetic and their implementation on binary devices. • Testing the numerical representation of different kind types on a sys- tem. + +# Chapter 8 + +# 8 Arithmetic + +Most problems in the academic and scientific communities require arithmetic eval- uation as part of the algorithm. As the rules for the evaluation of arithmetic in Fortran may differ from those that you are probably familiar with, you need to learn the Fortran rules thoroughly. In the previous chapter, we introduced the arith- metic assignment statement, emphasising the concepts of name, type and value. Here we will consider the way that arithmetic expressions are evaluated in Fortran. + +Table 8.1 lists the five arithmetic operators available in Fortran. + +![](images/bc1f19043c2e7534663c219257ffb7573a9baf49d416090e5f27f95eaa7845d9.jpg) + +Table 8.1 Fortran Operators + +Exponentiation is raising to a power. Note that the exponentiation operator is the \* character twice . + +The following are some examples of valid arithmetic assignment statements in For- tran: + +Taxable_Income $=$ Gross_Wage - Personal allowance Cost $=$ Bill $^+$ Vat $^+$ Service Delta $=$ Deltax/Deltay Area $=$ Pi \* Radius \* Radius Cube $=$ Big \*\* 3 + +The above expressions are all simple, and there are no problems when it comes to evaluating them. However, now consider the following: + +Tax $=$ Gross_Wage - Personal Allowance \* Tax_Rate + +This is a poorly written arithmetic expression. There is a choice of doing the sub- traction before or after the multiplication. Our everyday experience says that the subtraction should take place before the multiplication. However, if this expression were evaluated in Fortran the multiplication would be done before the subtraction. A complete program to show the correct form in Fortran is as follow: + +IMPLICIT NONE + + ! + + ! Example of a Fortran program to calculate net pay + + ! given an employee's gross pay + + ! REAL :: Gross_wage, Net_wage, Tax REAL:: Tax_rate $=$ 0.25INTEGER :: Personal allowance $=$ 4800 CHARACTER\*60 :: Their_Name PRINT \*,'Input employees name' READ \*,Their_Name PRINT \*,'Input Gross wage' READ \*,Gross_wage Tax $=$ (Gross_wage - Personal allowance) \* Tax_rate Net_wage $=$ Gross_wage - Tax PRINT \*,'Employee: ',Their_Name PRINT \*,'Gross Pay: ', Gross_wage PRINT \*,'Tax: ',Tax PRINT \*,'Net Pay:',Net_wage END PROGRAM ch0801 + +We need to look at three areas here: + +• The rules for forming expressions — the syntax. • The rules for interpreting expressions — the semantics. • The rules for evaluating expressions — optimisation. + +The syntax rules determine which expressions are valid. The semantics determine a valid interpretation, and once this has been done the compiler can replace the ex- pression with any other one that is mathematically equivalent, generally in the interests of optimisation. + +The rules for the evaluation of expressions in Fortran are as follows: • Brackets are used to define priority in the evaluation of an expression. + +• Operators have a hierarchy of priority — a precedence. The hierarchy of operators is: + +• Exponentiation: when the expression has multiple exponentiation, the evaluation is from right to left. For example, + +$$ +\mathrm{L}=\mathrm{I}^{\mathrm{\Omega}^{**}}\mathrm{\boldmath~J~}^{**}\mathrm{\boldmath~K~} +$$ + +is evaluated by first raising J to the power K, and then using this result as the exponent for I; more explicitly, + +$$ +\mathrm{L}=\mathrm{I}^{\mathrm{\Omega}^{**}}\left(\mathrm{J}^{\mathrm{\Omega}^{**}}\mathrm{K}\right) +$$ + +Although this is similar to the way in which we might expect an algebraic expression to be evaluated, it is not consistent with the rules for multiplication and division, and may lead to some con- fusion. When in doubt, use brackets. + +• Multiplication and division: within successive multiplications and divisions, the rules regarding any mathematically equivalent expres- sion means that you must use brackets to ensure the evaluation you want For example, with + +$$ +\mathrm{A=B*C/D*E} +$$ + +for noninteger numeric types the compiler does not necessarily evaluate in a left to right manner, i.e., evaluate B times C, then divide the result by D and finally take that result and multiply by E. + +• Addition and subtraction: as for multiplication and division the rules regarding any equivalent expression apply. However, it is sel- dom that the order of addition and subtraction is important, unless other operators are involved. + +The following are all examples of valid arithmetic expressions in Fortran: + +Slope $=$ (Y1-Y2)/(X1-X2) $\begin{array}{r l}{\mathrm{X}\beth}&{{}=}\end{array}$ ( $^{-\mathrm{B}+}$ ((B\*B-4\*A\*C)\*\*0.5))/(2\*A) Q = Mass_D/2\*(Mass_A\*Veloc_A/Mass_D)\*\*2 + & ((Mass_A $\star$ Veloc_A) $\star\star_{2}$ )/2 + +Note that brackets have been used to make the order of evaluation more obvious. It is often possible to write involved expressions without brackets, but, for the sake of clarity, it is often best to leave the brackets in, even to the extent of inserting a few extra ones to ensure that the expression is evaluated correctly. The expression will be evaluated just as quickly with the brackets as without. Also note that none of the expressions is particularly complex. The last one is about as complex as you should try: with more complexity than this it is easy to make a mistake. + +The rule regarding any equivalent expression means if A, B and C are numeric then the following are true: + +$$ +\begin{array}{l}{\mathrm{A+B=B+A}}\\ {\mathrm{-A+B=B-A}}\\ {\mathrm{A+B+C=A+(B+C)}}\end{array} +$$ + +The last is nominally evaluated left to right, as the additions are of equal prece- dence: + +$$ +\begin{array}{r l}&{\mathrm{\boldmath~A~}^{*}\mathrm{\boldmath~B=B~}^{*}\mathrm{\boldmath~A~}}\\ &{\mathrm{\boldmath~A~}^{*}\mathrm{\boldmath~B~}^{*}\mathrm{\boldmath~C=A~}^{*}\left(\mathrm{\boldmath~B~}^{*}\mathrm{\boldmath~C}\right)}\end{array} +$$ + +and again the last is nominally evaluated left to right, as the multiplications are of equal precedence: + +$$ +\begin{array}{r l}&{\mathrm{~A~}^{*}\mathrm{~B~}-\mathrm{~A~}^{*}\mathrm{~C=A~}^{*}\mathrm{~(B~}-\mathrm{C)~}}\\ &{\mathrm{~A~/~B~/~C=A~/~(B~}^{*}\mathrm{~C)~}}\end{array} +$$ + +The last is true for noninteger numeric types only. + +Problems arise when the value that a faulty expression yields lies within the range of expected values and the error may well go undetected. This may appear strange at first, but a computer does exactly what it is instructed to do. If, through a misun- derstanding on the part of a programmer, the program is syntactically correct but logically wrong from the point of view of the problem definition, then this will not be spotted by the compiler. If an expression is complex, break it down into succes- sive statements with elements of the expression on each line, e.g., + +$$ +\begin{array}{r c l}{{\mathrm{Temp}}}&{{=}}&{{\tt B}}\end{array}\star\begin{array}{c c l}{{\tt B}}&{{-}}&{{4}}\end{array}\star\begin{array}{c c l}{{\tt A}}&{{\star}}&{{\tt C}}\\ {{\mathrm{Temp}}}&{{\star}}&{{0.5}}\end{array}\bigr)\begin{array}{c c l}{{}}&{{}/}&{{(\mathrm{~\boldmath~\it~2~\star~\bar{~}~})}}\end{array} +$$ + +and + +$$ +\begin{array}{r c l}{{\mathrm{Moment}\mathrm{~=~\Mas\Sigma_A~\ast~\nabla~V e l o c\_A~}}}&{{}}&{{}}\\ {{Q\mathrm{~=~\Mas\Sigma_D~\!~~/~2~\ast~\Sigma_(~\mathsf{~M o m e n t}~\!~/~\!~M a s s\_D~\!~~)~\!~~\ast\!\ast\!~2~\!~\!~+~\!~\&~\!~}}}\\ {{\mathrm{~\!\!~~\!(~\!~M o m e n t~\!~\ast\!\ast\!~2\!~)~\!~~/~\!~2~}}}&{{}}&{{}}\end{array} +$$ + +# 8.1 Rounding and truncation + +When arithmetic calculations are performed one of the following can occur: + +• Truncation. This operation involves throwing away part of the number, e.g., with 14.6 truncating the number to two figures leaves 14. • Rounding . Consider 14.6 again. This is rounded to 15. Basically, the number is changed to the nearest whole number. It is still a real number. What do you think will happen with 14.5; will this be rounded up or down? + +You must be aware of these two operations. They may occasionally cause prob- lems in division and in expressions with more than one data type. + +To see some of the problems that can occur consider the examples below: + +PROGRAM ch0802 IMPLICIT NONE REAL :: A,B,C INTEGER :: I $\mathrm{~\textit~{~A~}~}=\mathrm{~\1~}.5$ $\begin{array}{c c c}{\tt B}&{=}&{2\cdot0}\end{array}$ C = A / B $\mathrm{~\small~\mathbb~{~T~}~}=\mathrm{~\small~\mathbb~{~A~}~}$ / B PRINT \*,A,B PRINT $^{\star}\,,\,\mathrm{C}$ PRINT \*,I END PROGRAM ch0802 + +After executing these statements C has the value 0.75, and I has the value zero! This is an example of type conversion across the $=$ sign. The variables on the right are all real, but the last variable on the left is an integer. The value is therefore made into an integer by truncation. In this example, 0.75 is real, so I becomes zero when truncation takes place. + +Consider now an example where we assign into a real variable (so that no trunca- tion due to the assignment will take place), but where part of the expression on the righthand side involves integer division: + +PROGRAM ch0803 IMPLICIT NONE INTEGER :: I,J,K REAL :: Answer $\begin{array}{l l l}{{{\mathbb I}}}&{{=}}&{{{\displaystyle5}}}\\ {{{\mathbb J}}}&{{=}}&{{{\displaystyle2}}}\\ {{{\mathbb K}}}&{{=}}&{{{\displaystyle4}}}\end{array}$ Answer $=\quad\mathbb{T}$ / J \* K PRINT \*,I + +PRINT \*,J PRINT \*,K PRINT \*,Answer END PROGRAM ch0803 + +The value of ANSWER is 8, because the I/J term involves integer division. The expected answer of 10 is not that different from the actual one of 8, and it is cases like this that cause problems for the unwary, i.e., where the calculated result may be close to the actual one. In complicated expressions it would be easy to miss something like this. + +To recap, truncation takes place in Fortran: + +• Across ${\mathrm{an}}={\mathrm{sign}}$ , when a real is assigned to an integer. • In integer division. + +It is very important to be careful when attempting mixed mode arithmetic — that is, when mixing reals and integers. If a real and an integer are together in a divi- sion or multiplication, the result of that operation will be real; when addition or subtraction takes place in a similar situation, the result will also be real. The prob- lem arises when some parts of an expression are calculated using integer arithmetic and other parts with real arithmetic: + +$$ +\!\!\!\!\!\!\!\!\!\!\ensuremath{\texttt{C}}\!\!\!\!\!\!\!\!=\!\!\!\!\!\!\!\!\!\mathbb{A}\!\!\!\!\!\!\!\!\!\!^{\!}\!\!\!\!\!\!\!\!\!+\!\!\!\!\!\!\!\!\!\!\!\mathbb{B}\!\!\!\!\!\!\!\!\!\!^{\!}\!\!\!\!\!\!\!\!\!-\!\!\!\!\!\!\!\!\!\!\mathbb{I}\!\!\!\!\!\!\!\!\!\!\!^{\!}\!\!\!\!\!\!\!\!\!\!+\!\!\!\!\!\!\!\!\!\!\!\mathbb{B}\!\!\!\!\!\!\!\!\!\!\!^{\!}\!\!\!\!\!\!\!\!\!\!+\!\!\!\!\!\!\!\!\!\!\!\mathbb{A}\!\!\!\!\!\!\!\!\!\!\!^{\!}\!\!\!\!\!\!\!\!\!\!+\!\!\!\!\!\!\!\!\!\!\!\mathbb{B}\!\!\!\!\!\!\!\!\!\!\!^{\!}\!\!\!\!\!\!\!\!\!\!+\!\!\!\!\!\!\!\!\!\!\!\mathbb{B}\!\!\!\!\!\!\!\!\!\!\!^{\!}\!\!\!\!\!\!\!\!\!\!+\!\!\!\!\!\!\!\!\!\!\!\mathbb{A}\!\!\!\!\!\!\!\!\!\!\! +$$ + +The integer division is carried out before the addition and subtraction; hence the result of $\mathrm{I/J}$ is integer, although all the other parts of the expression will be carried out with real arithmetic. + +# 8.2 Time taken for light to travel from the Sun to Earth + +How long does it take for light to reach the Earth from the Sun? Light travels 9.46 $10^{12}\,\mathrm{km}$ in 1 year. We can take a year as being equivalent to 365.25 days. (As all school children know, the astronomical year is 365 days, 5 hours, 48 minutes and 45.9747 seconds — hardly worth the extra effort.) The distance between the Earth and Sun is about $150{,}000{,}000{\mathrm{~km}}$ . There is obviously a bit of imprecision involved in these figures, not least since the Earth moves in an elliptical orbit, not a circular one. One last point to note before presenting the program is that the elapsed time will be given in minutes and seconds. Few people readily grasp fractional parts of a year: + +PROGRAM ch0804 IMPLICIT NONE REAL :: Light_Minute, Distance, Elapse INTEGER :: Minute, Second + +REAL , PARAMETER :: Light_Year $\scriptstyle{\dot{=}9}$ . $46^{\star}10^{\star\star}12 + +$ ! Light_year : Distance travelled by light + + ! in one year in km + + ! Light_minute : Distance travelled by light + + ! in one minute in km + + ! Distance : Distance from sun to earth in km + + ! Elapse : Time taken to travel a + + ! distance (Distance) in minutes + + ! Minute : Integer number part of elapse + + ! Second : Integer number of seconds + + ! equivalent to fractional part of elapse + + ! Light_minute $=$ Light_Year/(365.25 \* 24.0 \* 60.0) Distance $=$ 150.0 \* 10 \*\* 6 Elapse $=$ Distance / Light_minute Minute $=$ Elapse Second $=$ (Elapse - Minute) \* 60 Print \*,' Light takes ' , Minute,' Minutes' Print $^{\star}\,,\,\,^{\bullet}$ ' , Second,' Seconds' Print \*,' To reach the earth from the sun' END PROGRAM ch0804 + +The calculation is straightforward; first we calculate the distance travelled by light in 1 minute, and then use this value to find out how many minutes it takes for light to travel a set distance. Separating the time taken in minutes into whole-number minutes and seconds is accomplished by exploiting the way in which Fortran will truncate a real number to an integer on type conversion. The difference between these two values is the part of a minute which needs to be converted to seconds. Given the inaccuracies already inherent in the exercise, there seems little point in giving decimal parts of a second. + +It is worth noting that some structure has been attempted by using comment lines to separate parts of the program into fairly distinct chunks. Note also that the com- ment lines describe the variables used in the program. + +Can you see any problems with this example? + +# 8.3 The PARAMETER statement + +This statement is used to provide a way of associating a meaningful name with a constant in a program. Consider a program where $\pi$ was going to be used a lot. It would be silly to have to type in 3.14159265358, etc., every time. There would be a lot to type and it is likely that a mistake could be made typing in the correct value. It therefore makes sense to set up $\pi$ once and then refer to it by name. How- ever, if PI was just a variable then it would be possible to do the following: + +REAL :: li,pi . pi=3.14159265358 . pi=4\*alpha/beta . + +The $\mathrm{\pi}{=}4^{*}$ alpha/beta statement should have been $\scriptstyle{\mathrm{li}=4^{*}}$ alpha/beta. What has hap- pened is that, through a typing mistake (p and l are close together on a keyboard), an error has crept into the program. It will not be spotted by the compiler. Fortran provides a way of helping here with the PARAMETER statement, which should be preceded with a type declaration. The following are correct examples of the PA- RAMETER statement: + +and + +REAL , PARAMETER :: Charge $\mathbf{\omega}=\mathbb{1}$ .6021917 + +The advantage of the PARAMETER statement is that you could not then assign another value to pi, C or Charge. If you tried to do this, the compiler would gener- ate an error message. + +A PARAMETER statement may contain an arithmetic expression, so that some rel- atively simple arithmetic may be performed in setting up these constants. The evaluation must be confined to addition, subtraction, multiplication, division and integer exponentiation. The following examples help to demonstrate the possibili- ties: + +# 8.4 Range, precision and size of numbers + +The range on integer numbers and the precision and the size of floating point num- bers in computing are directly related to the number of bits allocated to their internal representation. Tables 8.2 and 8.3 summarise this information for the two most common bit sizes in use for integers and reals — 32 bits and 64 bits. + +Table 8.2 looks at integer numbers. + +![Table 8.2 Word Size and Integer Numbers ](images/f0a3d8d9260859284fb0e1fbbb62afeda2ce33bc64b44360d99460b8694c56d3.jpg) + +![Table 8.3 is a corresponding table for real numbers. ](images/260ad58281e53412fe805ec55283258ec6ba64c534188971beb95851682ac2ca.jpg) + +Table 8.3 Word Size and Real Numbers + +Note that access to what the hardware supports is dependent on the operating sys- tem and compiler as well. + +Precision is not the same as accuracy. In this age of digital timekeeping, it is easy to provide an extremely precise answer to the question What time is it ? This an- swer need not be accurate, even though it is reported to tenths (or even hundredths!) of a second. Do not be fooled into believing that an answer reported to ten places of decimals must be accurate to ten places of decimals. The computer can only retain a limited precision. When calculations are performed, this limita- tion will tend to generate inaccuracies in the result. The estimation of such inaccuracies is the domain of the branch of mathematics known as Numerical Analysis. + +To give some idea of the problems, consider an imaginary decimal computer which retains two significant digits in its calculations. For example, 1.2, 12.0, 120.0 and 0.12 are all given to two-digit precision. Note therefore that 1234.5 would be rep- resented as 1200.0 in this device. When any arithmetic operation is carried out, the result (including any intermediate calculations) will have two significant digits. Thus: + +$$ +130+12=140\;(\mathrm{rounding\;down\;from\;}142) +$$ + +and similarly: + +$$ +17~/~3=5.7~(\mathrm{rounding~up~from}~5.666666...) +$$ + +and: + +$$ +16\ ^{*}\ 16=260 +$$ + +Where there are more involved calculations, the results can become even less at- tractive. Assume we wish to evaluate + +$$ +(16*16)\,/\,0.14 +$$ + +We would like an answer in the region of 1828.5718, or, to two significant digits, 1800.0. If we evaluate the terms within the brackets first, the answer is 260/0.14, or 1857.1428; 1900.0 on the two-digit machine. Thinking that we could do better, we could rewrite the fraction as + +Which gives a result of 1800.0. + +Algebra shows that all these evaluations are equivalent if unlimited precision is available. + +Care should also be taken when is one is near the numerical limits of the machine. Consider the following: + +$$ +\mathrm{~\boldmath~{~Z~}~}=\mathrm{~\boldmath~{~B~}~}^{\star}\mathrm{~\boldmath~{~C~}~}/\mathrm{~\boldmath~{~D~}~} +$$ + +where B, C and $\mathrm{D}$ are all ${10}^{30}$ and we are using 32-bit floating point numbers where the maximum real is approximately ${10}^{38}$ . Here the product $\textrm{B}^{*}\textrm{C}$ generates a number of ${10}^{60}$ — beyond the limits of the machine. This is called overflow as the number is too large. Note that we could avoid this problem by retyping this as + +$$ +\mathrm{~\sf~Z~}\mathrm{~\ensuremath~{~=~}~}\mathrm{~\sf~B~}\mathrm{~\star~}\mathrm{~\sf~(~C~\!~\!~/~}\mathrm{~\!~D~\!~)~} +$$ + +where the intermediate result would now be $10^{30}/10^{30}$ , i.e., 1. + +There is an inverse called underflow when the number is too small, which is illus- trated below: + +$$ +\mathrm{~\textsf~{~Z~}~}=\mathrm{~\mathrm{~\mathrm{~X}}}\mathbb{1}\mathrm{~\textsf~{~*~}~}\mathrm{~\mathrm{~Y}}\mathbb{1}\mathrm{~\textsf~{~*~}~}\mathrm{~\mathrm{~Z}}\mathbb{1} +$$ + +where X1 and Y1 are ${{10}^{-20}}$ and Z1 is $10^{20}$ . The intermediate result of $_{\mathrm{X1}}*_{\mathrm{Y1}}$ is ${{10}^{-40}}$ — again beyond the limits of the machine. This problem could have been overcome by retyping as + +$$ +\mathrm{~\sf~Z~}~=~\mathrm{~\sf~X1~}^{~\star~}~~(\mathrm{Y1~}^{~\star~}~~\mathrm{Z1}) +$$ + +This is a particular problem for many scientists and engineers with all machines that use 32-bit arithmetic for integer and real calculations. This is because there are a number of physical constants (Plank constant, elementary charge, Bohr magneton etc.,) that will cause arithmetic problems due to their size. This is rarely a problem with machines with hardware support for 64-bit arithmetic. + +How we get around this problem and how we move our programs from one plat- form to another making sure that we are working with the same precision and same range of numbers are covered in detail in the next section. + +# 8.5 Health warning: optional reading, beginners are advised to leave until later + +It is very important in scientific programming to know the range and precision of data on the hardware platform on which we are working. The facilities provided in Fortran 95 now allow programmers to specify the range and precision they wish to use and the compiler will choose an appropriate type. + +If it is not possible to offer the precision and range requested the compiler returns an error code. To avoid this happening the programmer needs to query the com- puter first for details of its data representations before trying to run a program which specifies range and precision. + +In order to do this we use the KIND intrinsic function, (intrinsic functions are cov- ered in depth in Chapter 14 and Appendix D), e.g.: + +REAL :: X PRINT \*,'Kind number for X = ',KIND(X) + +This will print out the kind number used by your system to represent default REAL variables. These kind numbers are arbitrary and there is usually no meaning at- tached to them. + +Consider the following program, which demonstrates the use of the KIND func- tion: + +PROGRAM ch0805 IMPLICIT NONE + +INTEGER :: i REAL :: r CHARACTER\*1 :: c LOGICAL :: l COMPLEX :: cp PRINT \*,' Integer ',KIND(i) PRINT \*,' Real ',KIND(r) PRINT \*,' Char ',KIND(c) PRINT \*,' Logical ',KIND(l) PRINT \*,' Complex ',KIND(cp) END PROGRAM ch0805 + +It is worthwhile actually typing this program in and seeing what answers you get for the system you are working on. Output from a PC compiler is given below: + +[FTN90 Version 1.12 Copyright (c)SALFORD SOFTWARE LTD 1992 & ] [ (c)THE NUMERICAL ALGORITHMS GROUP 1991,1992] NO ERRORS [FTN90] Program entered Integer 3 Real 1 Char 1 Logical 2 Complex 1 + +The following is the output from the DEC Fortran 90 compiler under OPEN VMS on an Alpha 2100: + +Program entered Integer 4 Real 4 Char 1 Logical 4 Complex 4 + +Thus it is up to each compiler implementation to decide what kind numbers are as- sociated with each type and kind variation. Thus the kind value on its own should not be used across platforms to try to achieve portability. + +In fact, specifying a kind number actually is not what is intended by the Fortran standard, so two intrinsic functions + +SELECTED IN T KIND + +and + +SELECTED_REAL_KIND + +are available instead. They are used to specify the range of numbers for integers and the range and precision of numbers for reals, and the compiler will return the appropriate kind numbers that it has assigned to such representations. These kind numbers can be assigned to parameters called kind type parameters, which can be used with REAL and INTEGER type declarations. Let's consider the two main nu- meric types to see how this works. + +# 8.5.1 Selecting different INTEGER kind types + +The Fortran standard specifies that only one integer kind needs to be available, but often a machine's architecture or compiler implementation will offer more. Most compiler implementations will offer the following: + +• 8-bit or one-byte integers. • 16-bit or two-byte integers. • 32-bit or four-byte integers. + +and 64-bit or eight byte integers will be available on certain platforms and imple- mentations. The most common reason for choosing 8-bit or 16-bit integers is to reduce the memory requirements of your program and the most common reason for choosing 64-bit integers is to solve specialised problems in mathematics requiring large integer numbers. + +To choose an integer kind other than the default, you specify the range of the num- bers you require it to lie in, in terms of a power of 10; e.g., + +INTEGER, PARAMETER :: First $=$ SELECTED IN T KIND (2) INTEGER (First) :: I,J + +selects an integer kind parameter, First, with representation which includes all inte- gers between $\widetilde{-}10^{2}$ and $10^{2}$ , i.e., numbers in the range $-100$ to 100. The integer kind parameter can be used in brackets after the integer type statement to specify variables of this integer kind, e.g., I and J. + +If there is no integer kind representation for the range specified, the SE- LECTED_INT_KIND function returns $^{-1}$ . Unfortunately it is not possible to then test for $^{-1}$ in a type statement, i.e., you will get a compile time error message. We suggest that you run the program in Section 8.5.9 to find the limits of your ma- chine's architecture before trying to specify a kind parameter that it can't support. + +# 8.5.2 Selecting different REAL kind types + +The Fortran standard specifies that there must be at least two representations of the real type, the default plus one other. Often there are more, depending on what the underlying hardware can support. When working with real data there are two things to specify — range and precision. The precision is the minimum number of significant digits (all floating point numbers are normalised) to which real numbers are stored, and the range is the power of 10 of the largest number to be repre- sented. So, for example, to specify that a variable R has a kind type that supports 15 significant figures and a range $10^{\pm307}$ we define a real kind parameter, Long, and then use this with the REAL type declaration for R as follows: + +INTEGER, PARAMETER :: Long $=$ SELECTED_REAL_KIND(15,307)REAL (Long) :: R + +The only problem is if the underlying hardware can't support this specification, in which case the function will return $^{-1}$ if the requested precision is unavailable, $^{-2}$ if the range is unavailable, and $^{-3}$ if both are unavailable. As we mentioned earlier with integer kinds, it is not possible to test for negative values in a type declara- tion, so before trying to use different kind types, or even just the default types, you need to know what kind types your machine supports and their range and preci- sion. + +# 8.5.3 Specifying kind types for literal integer and real constants + +A literal constant is a data object whose value cannot change. An integer constant 1 is of default integer kind and a real constant 10.3 is a default real constant. If in a program you have chosen a real kind type, other than the default, then to be con- sistent and also to make sure that all real arithmetic is done to the precision specified, you need to declare all real constants to be of this kind type. This is done by giving the literal constant followed by an underscore and a kind number or kind type parameter, e.g. + +constant_kind + +For the earlier example with a kind type parameter Long, a real literal constant of this type would be given as + +-22.36_Long + +It is not recommended to use the actual kind number because, as we have seen, these are not portable across machines. + +The convention we use throughout this book if we require a numeric kind type other than the defaults is to specify a kind type parameter, e.g., + +INTEGER, PARAMETER :: Long $=$ SELECTED_REAL_KIND(15,307) + +and then use it with REAL type declarations, e.g., + +REAL (Long) :: R + +This still doesn't make programs completely portable across different hardware platforms, so you will firstly need to run a program which tests the range of data representations. Before doing this we need to know a bit more about the underly- ing representation of numerical data on computer systems. + +# 8.5.4 Positional number systems + +Most people take arithmetic completely for granted and rarely think much about the subject. It is necessary to look at it in a bit more depth if we are to understand what the computer is doing in this area. + +Our way of working with numbers is essentially a positional one. When we look at the number 1024, for example, we rarely think of it in terms of $1\mathrm{~}^{*}\mathrm{~}1000+0\mathrm{~}^{*}\mathrm{~}100$ ${}+2\ ^{*}\ 10+4\ ^{*}\ 1.$ . Thus the normal decimal system we use in everyday life is a po- sitional one, with a base of 10. + +We are probably aware that we can use other number bases, and 2, 8 and 16 are fairly common alternate number bases. As the computer is a binary device it uses base 2. + +We are also reasonably familiar with a mantissa exponent or floating point combi- nation when the numbers get very large or very small, e.g., a parsec is commonly expressed as $3.08^{\mathrm{~*~}}10^{\mathrm{~**~}}16.$ , and here the mantissa is 3.08, and the exponent is 10 \*\* 16. + +The above information will help in understanding the way in which integers and reals are represented on computer systems. + +# 8.5.5 Bit data type and representation model + +The model is only defined for positive integers (or cardinal numbers), where they are represented as a sequence of binary digits, and is based on the model: + +$$ +i=\sum_{k\mathop{=}0}^{n-1}b_{k}~2^{k} +$$ + +where $i$ is the integer value, $n$ is the number of bits, and $b_{k}$ is a bit value of 0 or 1, with bit numbering starting at 0, and reading right to left. Thus the integer 43 and bit pattern 101011 is given by: + +$$ +43=(1\ ^{*}\ 32)+(0\ ^{*}\ 16)+(1\ ^{*}\ 8)+(0\ ^{*}\ 4)+(1\ ^{*}\ 2)+(1\ ^{*}\ 1) +$$ + +or + +$$ +43=(1\ ^{*}2^{5})+(0\ ^{*}2^{4})+(1\ ^{*}2^{3})+(0\ ^{*}2^{2})+(1\ ^{*}2^{1})+(1\ ^{*}2^{0}) +$$ + +# 8.5.6 Integer data type and representation model + +The integer data type is based on the model + +$$ +i=S\,\sum_{k=1}^{q}\,\,l_{k}\,\,r^{\,k-1} +$$ + +where $i$ is the integer value, $s$ is the sign, $q$ is the number of digits (always posi- tive), $r$ is the radix or base (integer greater than 1), and $l_{k}$ is a positive integer (less than r). + +A base of 2 is typical so 1023 is + +$$ +\begin{array}{l}{{1023=(1\,*\,2^{9})+(1\,*\,2^{8})+(1\,*\,2^{7})+(1\,*\,2^{6})+(1\,*\,2^{5})+(1\,*\,2^{4})+(1\,*\,2^{3})}}\\ {{\qquad+\;(1\,*\,2^{2})+(\,1\,*\,2^{1})+(1\,*\,2^{0})}}\end{array} +$$ + +# 8.5.7 Real data type and representation model + +The real data type is based on the model + +$$ +x=s\ b^{\ e}\ \sum_{k=1}^{m}\ f_{k}\ b^{\,-k} +$$ + +where $x$ is the real number, $s$ is the sign, $b$ is the radix or base (greater than 1), m is the number of bits in the mantissa, $e$ is an integer in the range $e_{m i n}$ to $e_{m a x}$ , and $f_{k}$ is a positive number less than $b$ . + +This means that with, for example, a 32-bit real there would be 8 bits allocated to the exponent and 24 to the mantissa. One of the bits in each part would be used to represent the sign and is called the sign bit. This reduces the number of bits that can actually be used to represent the mantissa and exponent to 31 and 7, respec- tively. There is also the concept of normalisation, where the exponent is adjusted so that the most significant bit is in position $22\mathrm{~-~}$ bits are typically numbered 0–22, rather than 1–23. This form of representation is not new, and is first docu- mented around $1750\,\mathrm{\BC}$ , when Babylonian mathematicians used a sexagesimal (radix 60) positional notation. It is interesting that the form they used omitted the exponent! + +This is the theoretical basis of the representation of these three data types in For- tran. Remember from Chapter 2 that the computer is essentially a binary device, and works at the lowest level with sequences of zeros and ones. + +This information together with the following provide a good basis for writing por- table code across a range of hardware. + +# Chapter 8 + +# 8.5.8 IEEE 754 + +The first standard IEEE 754: 1985 covered binary floating point arithmetic. The later IEEE 754: 1987 standard added decimal arithmetic. + +A considerable amount of hardware now offers support for the IEEE 754 standard. The standard can be purchased from + +• http://standards.ieee.org + +Work is under way on the next version and you can find out details of the current state of play at + +•http://grouper.ieee.org/groups/754/There are quite a lot of good links from this site. + +# 8.5.9 Testing the numerical representation of different kind types on a system + +You are now ready to write or adapt a program to run on your system in order to test the range of integer kind types and the range and precision of real kind types. + +The following program selects several integer and real kind types and by calling the intrinsic functions KIND, HUGE, PRECISION and EPSILON produces most of the information you need to know about for these kind types. Table 8.4 provides details of what these functions do. + +![](images/37ba36a21bf4384c63acdac070a1760db350172d6e3f2e9ad2df46d5bb04d635.jpg) + +Table 8.4 Numeric Query Functions + +A complete program using the above is as follows: + +PROGRAM ch0806 + +! + + ! Examples of the use of the kind + +! function and the numeric inquiry functions + + ! + + ! Integer arithmetic + + ! + + ! 32 bits is a common word size, + + ! and this leads quite cleanly + + ! to the following + + ! 8 bit integers + + ! -128 to 127 $\it{10^{\star\star}2} + +$ ! 16 bit integers + + ! -32768 to 32767 10\*\*4 + + ! 32 bit integers + + ! -2147483648 to 2147483647 10\*\*9 + + ! + + ! 64 bit integers are increasingly available. + + ! This leads to + + ! -9223372036854775808 to + + ! 9223372036854775807 10\*\*19 + + ! + + ! You may need to comment out some of the following + + ! depending on the hardware platform and compiler + + ! that you use. INTEGER :: I INTEGER ( SELECTED_INT_KIND( 2)) :: I1 INTEGER ( SELECTED_INT_KIND( 4)) :: I2 INTEGER ( SELECTED_INT_KIND( 9)) :: I3 INTEGER ( SELECTED IN T KIND(10)) :: I4 + + ! Real arithmetic + + ! + + ! 32 and 64 bit reals are normally available. + + ! + + ! 32 bit reals 8 bit exponent, 24 bit mantissa + + ! + + ! 64 bit reals 11 bit exponent 53 bit mantissa + + ! REAL :: R REAL ( SELECTED_REAL_KIND( 6, 37)) :: R1 REAL ( SELECTED REAL KIND(15,307)) :: R2 REAL ( SELECTED REAL KIND(15,310)) :: R3 + +PRINT \*,' ' PRINT \*,' Integer values' PRINT \*,' Kind Huge' PRINT \*,' ' PRINT \*,' ',KIND(I ),' ',HUGE(I ) PRINT \*,' ' PRINT \*,' ',KIND(I1 ),' ',HUGE(I1 ) PRINT \*,' ',KIND(I2 ),' ',HUGE(I2 ) PRINT \*,' ',KIND(I3 ),' ',HUGE(I3 ) PRINT \*,' ',KIND(I4 ),' ',HUGE(I4 ) PRINT \*,' ' PRINT \*,' Real values' PRINT \*,' Kind Huge ',& 'Precision epsilon' PRINT \*,' ' PRINT \*,' ',KIND(R ),' ',HUGE(R ),& '',PRECISION(R),' ',EPSILON(R)PRINT \*,' ' PRINT \*,' ',KIND(R1 ),' ',HUGE(R1 ),& '',PRECISION(R1),' ',EPSILON(R1)PRINT \*,' ',KIND(R2 ),' ',HUGE(R2 ),& ' ',PRECISION(R2),' ',EPSILON(R2) PRINT \*,' ',KIND(R3 ),' ',HUGE(R3 ),& ' ',PRECISION(R3),' ',EPSILON(R3) + +END PROGRAM ch0806 + +The output from the Intel compiler under Windows is: + +Integer values Kind Huge + +4 2147483647 1 127 2 32767 4 2147483647 8 9223372036854775807 + +Real values Kind Huge + +# 4 3.4028235E+38 + +6 1.1920929E-07 + +4 3.4028235E+38 + +6 1.1920929E-07 8 1.797693134862316 E+308 15 2.220446049250313 E-016 16 1.189731495357231765085759326628007 E+4932 33 1.925929944387235853055977942584927E-0034 + +Integer values Kind Huge + +4 2147483647 1 127 2 32767 4 2147483647 8 9223372036854775807 + +Real values Kind Huge Precision epsilon + +43.40282347E+3861.19209290E-0743.40282347E+3861.19209290E-078 1.797693134862316 E+308 15 + +2.220446049250313 E-16 + +16 1.1897314953572317650857593266280070 E+4932 33 + + 1.9259299443872358530559779425849273E-0034 + +The output from the Salford compiler under Windows is: + +Integer values Kind Huge + +![](images/48af6a5fc2bdee28742056f1e3751ea31a68dffeac5fc1510319c6692826d7cc.jpg) + +Run this program on whatever system you have access to and compare the output with the above examples. + +# 8.5.10 Binary representation of different integer kind type numbers + +For those who wish to look at the internal binary representation of integer numbers with a variety of kinds, we have included the following program + +SELECTED IN T KIND( 2) means provide at least an integer representation with numbers between $-10^{2}$ and $+10^{2}$ . + +SELECTED IN T KIND( 4) means provide at least an integer representation with numbers between $^-10^{4}$ and $+10^{4}$ . + +SELECTED IN T KIND( 9) means provide at least an integer representation with numbers between $^-10^{9}$ and $+10^{9}$ . + +We use the INT function to convert from one integer representation to another. + +We use the logical function BTEST to determine whether the binary value at that position within the number is a zero or a one, i.e., if the bit is set. + +I_in_Bits is a character string that holds a direct mapping from the internal binary form of the integer and a text string that prints as a sequence of zeros or ones: + +PROGRAM ch0807 + + ! + + ! Use the bit functions in Fortran to write out a + + ! 32 bit integer number as a sequence of + + ! zeros and ones + + ! INTEGER :: J INTEGER :: I INTEGER ( SELECTED_INT_KIND( 2)) :: I1 INTEGER ( SELECTED_INT_KIND( 4)) :: I2 INTEGER ( SELECTED_INT_KIND( 9)) :: I3 CHARACTER (LEN $=\!32$ ) :: I_in_Bits PRINT \*,' Type in an integer ' READ \* , I $\scriptstyle\mathbb{T}\,\mathbb{1}=\mathbb{T}\mathrm{NNT}$ (I,KIND(2)) $\scriptstyle\mathbb{T}\,2\,=\,\mathbb{T}\mathrm{NT}$ (I,KIND(4)) $\uptau\textgreater$ INT(I,KIND(9)) I_in_Bits $=$ ' ' DO $\uptau\!\!=\!0$ ,7 IF (BTEST(I1,J)) THEN I_in_Bits( ${\u{'}}\ 8\,{-}\,{\u{J}}:8\,{-}\,{\u{J}}\ )={'}\ \u{1}$ ' ELSE I_in_Bits(8-J:8-J)='0' END IF END DO PRINT \*,' 1 2 3' PRINT \*,'1234567890123456789012345678901234567890' PRINT \*,I1 PRINT \*,I_in_Bits DO $\uptau\!\!=\!0$ ,15 IF (BTEST(I2,J)) THEN I_in_Bits $(\,\mathbb{1}\,6\!-\!\!\mathbb{J}:\!\mathbb{1}\,6\!-\!\!\mathbb{J}\,)=\,^{\!\!\!\mathsf{\nu}}\,\mathbb{1}$ ' ELSE I_in_Bits( $\left(\mathbb{1}6\mathbb{-}\mathbb{J}:\mathbb{1}6\mathbb{-}\mathbb{J}\right)={}^{\mathsf{t}}~0$ ' END IF END DO PRINT \*,I2 PRINT \*,I_in_Bits DO $\uptau\!\!=\!0$ ,31 IF (BTEST(I3,J)) THEN I_in_Bits( $\left.32\mathrm{-}\mathsf{J}:32\mathrm{-}\mathsf{J}\right)={}^{\mathsf{t}}\,\mathbb{1}$ ' ELSE + +# Chapter 8 + +I_in_Bits(32-J:32-J)='0' END IF END DO PRINT \*,I3 PRINT \*,I_in_Bits END PROGRAM ch0807 + +The DO loop indices follow the convention of an 8-bit quantity starting at bit 0 and ending at bit 7, 16-bit quantities starting at 0 and ending at 15, etc. + +The numbers written out follow the conventional mathematical notation of having the least significant quantity at the right-hand end of the digit sequence, i.e., with 127 in decimal we have $^{1\mathrm{~*~}}100,\,2\mathrm{~*~}10$ and $^\textrm{\scriptsize7*1}$ , so 00100001 in binary means 1 $^*\ 32\ +\ 1\ ^{*}\ 1$ decimal. + +Try running this program on the system you are using. Does it produce the results you expect? Experiment with a variety of numbers. Try at least the following 0, $+1$ , –1, –128, 127, 128, –32768, 32767, 32768. + +# 8.5.11 Binary representation of a real number + +The following program is a simple variant of the previous one, but we now look at a floating point number: + +PROGRAM ch0808 + +! + + ! Use the bit functions in Fortran to write out a + + ! 32 bit integer number equivalenced to a real + + ! using the transfer intrinsic as a sequence of + + ! zeros and ones + + ! IMPLICIT NONE INTEGER :: I,J CHARACTER ( $\mathtt{L E N}{=}32$ ) :: I_in_Bits $=$ " " REAL :: $\scriptstyle{\mathrm{X}}=-1$ .0 PRINT \*,' 1 2 3' PRINT \*,'1234567890123456789012345678901234567890' PRINT \*,I_in_Bits $\mathbb{T}=$ TRANSFER(x,I) DO $\uptau\!\!=\!0$ ,31 IF (BTEST(i,J)) THEN I_in_Bits( $\left.32\mathrm{-}\mathsf{J}:32\mathrm{-}\mathsf{J}\right)={}^{\mathsf{t}}\,\mathbb{1}$ ' ELSE I_in_Bits(32-J:32-J)='0' END IF + +END DO PRINT \*,x PRINT \*,I_in_Bits END PROGRAM ch0808 + +We use the intrinsic function transfer to help out here. The BTEST intrinsic takes an integer argument, so we need to copy the bit pattern of the real number into an integer variable. + +# 8.5.12 Summary of how to select the appropriate kind type + +To write programs that will perform arithmetically in a similar fashion on a variety of hardware requires an understanding of: + +• The integer data representation model and in practice the word size of the various integer kind types. • The real data representation model and in practice the word size of the various real kind types and the number of bits in both the mantissa and exponent. + +Armed with this information we can then choose a kind type that will ensure mini- mal problems when moving from one platform to another. End of health warning! + +# 8.6 Variable status + +Fortran has two concepts regarding the status of a variable: defined and undefined. If a program does not provide an initial value (in a type statement) for a variable then its status is said to be undefined. Consider the following code segment taken from the earlier example that calculated the sum and average of three numbers: + +REAL :: N1, N2, N3, Average $\mathbf{\xi}=\left(\begin{array}{l l}\end{array}\right.$ 0.0, Total=0.0 INTEGER :: $\mathtt{N}\!\!=\!\!3$ + +In the above the variables Average, Total and $\mathbf{N}$ all have a defined status. How- ever, N1, N2 and N3 are said to be undefined. The use of undefined values is implementation dependent and therefore not portable. Care must be taken when writing programs to ensure that your variables have a defined status wherever pos- sible. We will look at this area again in subsequent chapters. + +# 8.7 Summary + +The following are some practical rules and guidelines: • Learn the rules for the evaluation of arithmetic expressions. + +• Break expressions down where necessary to ensure that the expressions are evaluated in the way you want. • Take care with truncation owing to integer division in an expression. Note that this will only be a problem where both parts of the division are IN- TEGER. • Take care with truncation owing to the assignment statement when there is an integer on the left-hand side of the statement, i.e., assigning a real into an integer variable. • When you want to set up constants which will remain unchanged through- out the program, use the PARAMETER statement. • Do not confuse precision and accuracy. • Learn what the default KINDs are for the numeric types you work with, what the maximum and minimum values and precision are for REAL data, and what the maximum and minimum are for INTEGER data. • You have been introduced to the use of the functions DIGITS, HUGE and PRECISION, and some of the concepts involved in their use. We will look at functions in much greater depth later on. + +# 8.8 Problems + +1. Compile and run examples 1 through 3 in this chapter. + +2. Have another look at example 4. Compile and run it. It will generate an error on some systems. Can you see where the error is? Appendix D contains sample output from several compilers. + +2.. Write a program to calculate the period of a pendulum. This is given mathemat- ically as + +$$ +t\,{=}\,2\pi\sqrt{\frac{l e n g t h}{9.81}} +$$ + +Use the following Fortran arithmetic assignment statement: + +$$ +\begin{array}{r c c c c c c c c l}{\mathrm{T}}&{=}&{2}&{\star}&{\tt P I}&{\star}&{(\tt L E N G T H}&{/}&{9\cdot81)}&{\star\star}&{.5}\end{array} +$$ + +The length (LENGTH) is in metres, and the time (T) in seconds. $\pi$ was given a value earlier in this chapter. + +Repeat the above using two other methods. Try a hand-held calculator and a spreadsheet. Do you get the same answers? + +3. Base conversion. + +In this chapter you have seen a brief coverage of base conversion. The following program illustrates some of the problems that can occur when going from base 10 to base 2 and back again. Which numbers will convert without loss? + +program base_conversion real :: ${\tt x1}\!=\!1\;.\;0$ real :: $_{{\bf x}2\,=\,0\;.\;1}$ real :: $\scriptstyle{\mathrm{x}}3\,=\,0\;.\;0\,\perp$ real :: $\tt{x4\!=\!0.001}$ real :: $\mathbf{x}5\!=\!0\;.\;0\,0\,0\,1$ print \*,' ',x1 print \*,' ',x2 print \*,' ',x3 print \*,' ',x4 print \*,' ',x5 end program base_conversion + +Which do you think will provide the same number as originally entered? + +4. Simple subtraction. In this chapter we looked at representing floating point num- bers in a finite number of bits. + +Try the following program: + +program subtract real :: $\mathsf{a}\!=\!\mathsf{1}\,.\,0\,0\,0\,2$ real :: $\mathsf{b}\!\!=\!\!1\,.\,0\,0\,0\,\bot$ real :: c $\mathsf{C}\!=\!\mathsf{a}\!-\!\mathsf{b}$ print \*,a print $^{\star}\,,\,\flat$ print $^{\star}\,,\,\mathsf{C}$ end program subtract + +5. Expression equivalence. We introduced some of the rules that apply in Fortran for expression evaluation. In mathematics the following is true: + +$$ +(x^{2}–y^{2})=(x^{*}x–y^{*}y)=(x–y)^{*}(x+y) +$$ + +Try the following program: + +program expression equivalence + + ! + + ! simple evaluation of $\scriptstyle\mathrm{X}^{\star}\mathrm{X-Y}^{\star}\mathrm{Y} + +$ ! when x and y are similar + +! + + ! we will evaluate in three ways. + + ! real :: $\scriptstyle\times=\perp$ .002 real :: ${\tt y}{=}1\cdot0\,0\,1$ real :: t1,t2,t3,t4,t5 $\mathsf{t}\,\mathsf{1}\!=\!\mathsf{x}\!-\!\mathsf{y}$ $\scriptstyle\tan\in2=\mathrm{x}+\mathrm{y}$ print \*,t1 print \*,t2 $\scriptstyle\tan3\,=\,\tan^{\star}\tan2$ $\scriptstyle\mathrm{t}\,4=\mathrm{x}^{\star\star}\,2-\mathrm{y}^{\star\star}\,2$ $\scriptstyle{\mathsf{t}}\,5=\!\mathrm{x}^{\star}\mathrm{x}-\mathrm{y}^{\star}\mathrm{y}$ print \*,t3 print \*,t4 print \*,t5 end program expression equivalence + +Solve the problem with pencil and paper, calculator and Excel. + +The last three examples show that you must be careful when using a computer to solve problems. + +6. The following is a simple variant of ch0804. In this case we initialise light year in an assignment statement. Do you think you will get the same results as from running the earlier example? Appendix E contains the output from several compil- ers. + +PROGRAM ch0804p IMPLICIT NONE REAL :: Light_Minute, Distance, Elapse INTEGER :: Minute, Second REAL :: Light_Year + + ! Light_year : Distance travelled by light + + ! in one year in km + + ! Light_minute : Distance travelled by light + + ! in one minute in km + + ! Distance : Distance from sun to earth in km + + ! Elapse : Time taken to travel a + + ! distance (Distance) in minutes + + ! Minute : Integer number part of elapse + + ! Second : Integer number of seconds + + ! equivalent to fractional part of elapse Light_Year $=\!9$ .46\*10\*\*12 Light_minute $=$ Light_Year/(365.25 \* 24.0 \* 60.0) Distance $=$ 150.0 \* 10 \*\* 6 Elapse $=$ Distance / Light_minute Minute $=$ Elapse Second $=$ (Elapse - Minute) \* 60 Print \*,' Light takes ' , Minute,' Minutes' Print $^{\star}\,,\,\,^{\bullet}$ ' , Second,' Seconds' Print \*,' To reach the earth from sun' + +END PROGRAM ${\tt c h o804p}$ + +# 8.9 Bibliography + +Some understanding of numerical analysis is essential for successful use of Fortran when programming. As Froberg says “ numerical analysis is a science — computa- tion is an art .” The following are some of the more accessible books available. + +Froberg C.E., Introduction to Numerical Analysis , Addison-Wesley, 1969. + +The short chapter on numerical computation is well worth a read; it covers some of the problems of conversion between number bases and some of the errors that are introduced when we compute numerically. The Samuel Johnson quote owes its inclusion to Froberg! IEEE, IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std 754-1985 , Institute of Electrical and Electronic Engineers Inc. The formal definition of IEEE 754. Knuth D., Seminumerical Algorithms , Addison-Wesley, 1969. A more thorough and mathematical coverage than Wakerly. The chapter on positional number systems provides a very comprehensive historical coverage of the subject. As Knuth points out the floating point representation for num- bers is very old, and is first documented around 1750 B.C. by Babylonian mathematicians. Very interesting and worthwhile reading. Sun, Numerical Computation Guide , SunPro, 1993. Very good coverage of the numeric formats for IEEE Standard 754 for Binary Floating-Point Arithmetic. All SunPro compiler products support the features of the IEEE 754 standard. Wakerly J.F., Microcomputer Architecture and Programming , Wiley, 1981. The chapter on number systems and arithmetic is surprisingly easy. There is a coverage of positional number systems, octal and hexadecimal number system conversions, addition and subtraction of nondecimal numbers, representation of negative numbers, two's complement addition and subtraction, one's com- plement addition and subtraction, binary multiplication, binary division, bcd or binary coded decimal representation and fixed and floating point representa- tions. There is also coverage of a number of specific hardware platforms, including DEC PDP-11, Motorola 68000, Zilog Z8000, TI 9900, Motorola 6809 and Intel 8086. A little old but quite interesting nevertheless. + +# Arrays 1 Some Fundamentals + +“Thy gifts, thy tables, are within my brain Full charactered with lasting memory.” + +William Shakespeare, The Sonnets + +“Here, take this book, and peruse it well: The iterating of these lines brings gold.” + +Christopher Marlowe, The Tragical History of Doctor Faustus + +# Aims + +The aims of the chapter are to introduce the fundamental concepts of arrays and DO loops, in particular: + +• To introduce the idea of tables of data and some of the formal terms used to describe them: • Array. • Vector. • List and linear list. • To discuss the array as a random access structure where any element can be accessed as readily as any other and to note that the data in an array are all of the same type. • To introduce the twin concepts of data structure and corresponding control structure. • To introduce the statements necessary in Fortran to support and manipu- late these data structures. + +Chapter 9 + +# 9 Arrays 1: Some Fundamentals + +9.1 Tables of data + +Consider the examples below. + +# 9.1.1 Telephone directory + +A telephone directory consists of the following kinds of entries: + +![](images/6aad02dd2cd668aa2e5ec0d425004c4876f80ecaf062eb9a30233e816f1055d1.jpg) + +This structure can be considered in a variety of ways, but perhaps the most com- mon is to regard it as a table of data, where there are three columns and as many rows as there are entries in the telephone directory. + +Consider now the way we extract information from this table. We would scan the name column looking for the name we are interested in, and then read along the row looking for either the address or telephone number, i.e., we are using the name to look up the item of interest. + +# 9.1.2 Book catalogue + +A catalogue could contain: + +![](images/15ec3d719ff1cff3b0874120c1cc0bdb5b021680700accc004465b3adc5900ef.jpg) + +Again, this can be regarded as a table of data, having three columns and many rows. We would follow the same procedure as with the telephone directory to ex- tract the information. We would use the Name to look up what books are available. + +![9.1.3 Examination marks or results This could consist of: ](images/505e50fe0762e6aa29a4b40800ad373dbfdfe8728ce213a5d1532a8289d9c056.jpg) + +This can again be regarded as a table of data. This example has seven columns and five rows. We would again look up information by using the Name. + +# 9.1.4 Monthly rainfall + +The following data are the monthly average rainfall for London: + +![](images/bd5a3489fde3da7664cbe1d2d128492fbe197fd17d5d751c24c9ce421e270546.jpg) + +In this table there are two columns and twelve rows. To find out what the rainfall was in July, we scan the table for July in the Month column and locate the value in the same row, i.e., the rainfall figure for July. + +These are just some of the many examples of problems where the data that are be- ing considered have a tabular structure. Most general purpose languages therefore have mechanisms for dealing with this kind of structure. Some of the special names given to these structures include: + +• Linear list. • List. • Vector. • Array. + +The term used most often here, and in the majority of books on Fortran program- ming, is array. + +# 9.2 Arrays in Fortran + +There are three key things to consider here: + +• The ability to refer to a set or group of items by a single name. • The ability to refer to individual items or members of this set, i.e., look them up. • The choice of a control structure that allows easy manipulation of this set or array. + +# 9.3 The DIMENSION attribute + +The DIMENSION attribute defines a variable to be an array. This satisfies the first requirement of being able to refer to a set of items by a single name. Some exam- ples are given below: + +REAL , DIMENSION(1:100) :: Wages INTEGER , DIMENSION(1:10000) :: Sample + +For the variable Wages it is of type REAL and an array of dimension or size 100, i.e., the variable array Wages can hold up to 100 real items. + +For the variable Sample it is of type INTEGER and an array of dimension or size 10,000, i.e., the variable Sample can hold up to 10,000 integer items. + +# 9.4 An index + +An index enables you to refer to or select individual elements of the array. In the telephone directory, book catalogue, exam marks table and monthly rainfall exam- ples we used the name to index or look up the items of interest. We will give concrete Fortran code for this in the example of monthly rain fall. + +# 9.5 Control structure + +The statement that is generally used to manipulate the elements of an array is the DO statement. It is typical to have several statements controlled by the DO state- ment, and the block of repeated statements is often called a $D O$ loop . Let us look at two complete programs that highlight the above. + +# 9.6 Monthly rainfall + +![Let us look at this earlier example in more depth now. Consider the following: ](images/8aeced57f8bd3afdfd892fdb9175d261e6fea0c3b62f9999c20ec6715f56afd0.jpg) + +Most of you should be familiar with the idea of the use of an integer as an alter- nate way of representing a month, e.g., in a date expressed as 1/3/2000, for $1^{\mathrm{st}}$ March 2000 (anglicised style) or January $3^{\mathrm{rd}}$ (americanised style). Fortran, in com- mon with other programming languages, only allows the use of integers as an index into an array. Thus when we write a program to use arrays we have to map between whatever construct we use in everyday life as our index (names in our ex- amples of telephone directory, book catalogue, and exam marks) to an integer representation in Fortran. The following is an example of an assignment statement showing the use of an index: + +RainFall(1)=3.1 + +We saw earlier that we could use the DIMENSION attribute to indicate that a vari- able was an array. In the above example Fortran statement our array is called RainFall. In this statement we are assigning the value 10.4 to the first element of the array; i.e., the rainfall for the month of January is 10.4. We use the index 1 to represent the first month. Consider the following statement: + +SummerAverage $=$ (RainFall(6) $^+$ RainFall(7) + & RainFall(8))/3 + +This statement says take the values of the rainfall for June, July and August, add them up and then divide by 3, and assign the result to the variable SummerAverage, thus providing us with the rainfall average for the three summer months — Northern Hemisphere of course. + +# 9.6.1 Example 1: Rainfall + +The following program reads in the 12 monthly values from the terminal, computes the sum and average for the year, and prints the average out. + +PROGRAM ch0901 IMPLICIT NONE REAL :: Total $=\!0$ .0, Average $=\!0$ .0 REAL , DIMENSION(1:12) :: RainFall INTEGER :: Month PRINT \*,' Type in the rainfall values' PRINT \*,' one per line' DO Month $\mathbf{\Psi}\!=\!\!1$ ,12 READ \*, RainFall(Month) ENDDO DO Month $\mathbf{\Psi}\!=\!\!1$ ,12 Total $=$ Total $^+$ RainFall(Month) ENDDO Average $=$ Total / 12 PRINT \*,' Average monthly rainfall was' PRINT \*, Average END PROGRAM ch0901 + +RainFall is the array name. The variable Month in brackets is the index. It takes on values from 1 to 12 inclusive, and is used to pick out or select elements of the array. The index is thus a variable and this permits dynamic manipulation of the array at run time. The general form of the DO statement is + +DO Counter $=$ Start, End, Increment + +The block of statements that form the loop is contained between the DO statement, which marks the beginning of the block or loop, and the ENDDO statement, which marks the end of the block or loop. + +In this program, the DO loops take the form: + +The body of the loop in the program above has been indented. This is not required by Fortran. However it is good practice and will make programs easier to follow. + +The number of times that the DO loop is executed is governed by the last part of the DO statement, i.e., by the + +Counter $=$ Start, End, Increment + +Start as it implies, is the initial value which the counter (or index, or control vari- able) takes. Each time the loop is executed, the value of the counter will be increased by the value of increment, until the value of end is reached. If increment is omitted, it is assumed to be 1. No other element of the DO statement may be omitted. In order to execute the statements within the loop (the body ) it must be possible to reach end from start. Thus zero is an illegal value of increment . In the event that it is not possible to reach end, the loop will not be executed and control will pass to the statement after the end of the loop. + +In the example above, both loops would be executed 12 times. In both cases, the first time around the loop the variable MONTH would have the value 1, the second time around the loop the variable MONTH would have the value 2, etc., and the last time around the loop MONTH would have the value 12. + +# 9.7 People's weights + +In the table below we have ten people, with their names as shown. We associate each name with a number — in this case we have ordered the names alphabeti- cally, and the numbers therefore reflect their ordering. WEIGHT is the array name . The number in brackets is called the index and it is used to pick out or select ele- ments of the array. The table is read as the first element of the array WEIGHT has the value 85, the second element has the value 76 , etc. + +![](images/40771b2cfd5f0fe9a21241af214bb842788ce33a325bac3dd5cd0cf9158c0548.jpg) + +# 9.7.1 Example 2: Setting array size with a parameter + +In the first example we so-called hard coded the number 12, which is the number of months, into the program. It occurred four times. Modifying the program to work with a different number of months would obviously be tedious and potentially error prone. + +In this example we parameterise the size of the array and reduce the effort in- volved in modifying the program to work with a different number of people: + +PROGRAM ch0902 + + ! The program reads up to number of people weights + + ! into the array Weight + + ! Variables used + + ! Weight, holds the weight of the people + + ! Person, an index into the array + + ! Total, total weight + + ! Average, average weight of the people + + ! Parameters used + + ! NumberOfPeople ,10 in this case. + + ! The weights are written out so that + + ! they can be checked + + ! + +IMPLICIT NONE INTEGER , PARAMETER :: Number Of People = 10 REAL :: Total $=$ 0.0, Average $\begin{array}{r l}{\mathbf{\sigma}=}&{{}0\mathbf{\sigma}.\;0}\end{array}$ + +INTEGER :: Person REAL , DIMENSION(1:Number of People) :: Weight DO Person $\mathbf{\Psi}^{=}\mathbb{1}$ ,Number Of People PRINT \*, ' Type in the weight for person ',Person READ \*,Weight(Person) Total $=$ Total $^+$ Weight(Person) ENDDO Average $=$ Total / Number Of People PRINT \*,' The total of the weights is ',Total PRINT \*,' Average Weight is ',Average PRINT \*,' ',Number of People,' Weights were ' DO Person $\mathbf{\Psi}^{=}\mathbb{1}$ ,Number Of People PRINT \*,Weight(Person) ENDDO + +# 9.8 Summary + +The DIMENSION attribute declares a variable to be an array, and must come at the start of a program unit, with other declarative statements. It has two forms and examples of both of them are given below. In the first case we explicitly specify the upper and lower limts: + +REAL , DIMENSION(1:Number of People) :: Weight + +In the second case the lower limit defaults to 1 + +REAL , DIMENSION(Number of People) :: Weight + +The latter form will be seen in legacy code, especially Fortran 77 code suites. + +The PARAMETER attribute declares a variable to have a fixed value that cannot be changed during the execution of a program. In our example above note that this statement occurs before the other declarative statements that depend on it. To recap the statements covered so far, the order is summarised below. + +![](images/2397999c263c89584045d91f4ec2c2c55af2a92a7fcc105350a007977fc4b6c5.jpg) + +We choose individual members using an index, and these are always of integer type in Fortran. + +The DO loop is a very convenient control structure for manipulating arrays, and we use indentation to clearly identify loops. + +# 9.9 Problems + +1. Compile and run examples 1 and 2 from this chapter. + +2. Using a DO loop and an array rewrite the program which calculated the average of five numbers (Question 3 in Chapter 8) and increase the number of values read in from five to ten. + +3.1 Modify the program that calculates the total and average of people's weights to additionally read in their heights and calculate the total and average of their heights. Use the data given below, which have been taken from a group of first year undergraduates: + +Height Weight 1.85 85 1.80 76 1.85 85 1.70 90 + +1.75 69 1.67 83 1.55 64 1.63 57 1.79 65 1.78 76 + +3.2 Your body mass index is given by your weight (in kilos) divided by your height (in metres) squared. Calculate and print out the BMI for each person. + +Grades of obesity according to Garrow as follows: + +Grade 0 (desirable) 20–24.9 Grade 1 (overweight) 25–29.9 Grade 2 (obese) 30–40 Grad 3 (morbidly obese) ${>}40$ Ideal BMI range, Men, Range $20.1{-}25~\mathrm{kg}/\mathrm{m}^{2}$ Women, Range $18.7{-}23.8~\mathrm{kg}/\mathrm{m}^{2}$ + +3.3 When working on either a UNIX system or a PC in a DOS box it is possible to use the following characters to enable you to read data from a file or write output to a file when running your program: + +Character Meaning < read from file > write to file + +On a typical UNIX system we could use + +a.out $<$ data.dat $>$ results.txt + +to read the data from the file called data.dat and write the output to a file called results.txt. + +On a PC in a DOS box the equivalent would be + +program.exe $<$ data.dat $>$ results.txt + +This is a quick and dirty way of developing programs that do simple I/O; we don't have to keep typing in the data and we also have a record of the behaviour of the program. Rerun the program that prints out the BMI values to write the output to a file called results.txt. Examine this file in an editor. + +4. Modify the program that read in your name to read in ten names. Use an array and a DO loop. When you have read the names into the array write them out in re- verse order on separate lines. + +Hint : Look at the formal syntax of the DO statement. + +5. Modify the rainfall program (which assumes that the measurement is in inches) to convert the values to centimetres. One inch equals 2.54 centimetres. Print out the two sets of values as a table. + +Hint: Use a second array to hold the metric measurements. + +6. Combine the programs that read in and calculate the average weight with the one that reads in peoples names. The program should read the weights into one ar- ray and the names into another. Allow 20 characters for the length of a name. Print out a table linking names and weights. + +7. In an earlier chapter we used the following formula to calculate the period of a pendulum: + +$$ +\mathrm{T}=2\ ^{*}\ \mathrm{PI}\ ^{*}\ (\mathrm{LENGTH}\ /\ 9.81)\ ^{**}\ .5 +$$ + +Write a program that uses a DO loop to make the length go from 1 to 10 metres in 1-metre increments. + +Produce a table with two columns, the first of lengths and the second of periods. + +# Arrays 2 Further Examples + +“Sir, In your otherwise beautiful poem ( The Vision of Sin ) there is a verse which reads + +Every moment dies a man, every moment one is born. Obviously this cannot be true and I suggest that in the next edition you have it read Every moment dies a man, every moment 1 1/16 is born. Even this value is slightly in error but should be sufficiently accurate for poetry.” + +Charles Babbage in a letter to Lord Tennyson + +# Aims + +The aims of the chapter are to extend the concepts introduced in the previous chap- ter and in particular: + +• To set an array size at run time - ALLOCATABLE arrays. • To introduce the idea of an array with more than one dimension and the corresponding control structure to permit easy manipulation of higher-dimensioned arrays. • To introduce an extended form of the DIMENSION attribute declaration, and the corresponding alternative form to the DO statement, to manipulate the array in this new form. • To introduce the DO loop as a mechanism for the control of repetition in general, not just for manipulating arrays. • To formally define the block DO syntax. + +# Chapter 10 + +# 10 Arrays 2: Further Examples + +10.1 Varying the array size at run time + +The earlier examples set the array size in the following two ways: + +• Explicitly using a numeric constant • Implicitly using a parameterised variable + +In both cases we knew the size of the array at the time we compiled the program. We may not know the size of the array at compile time and Fortran provides the ALLOCATABLE attribute to accommodate this kind of problem. Consider the fol- lowing example. + +# PROGRAM ch1001 + +! + + ! This program is a simple variant of ch0902. + + ! The array is now allocatable + + ! and the user is prompted for the + + ! number of people at run time. + +IMPLICIT NONE INTEGER :: Number Of People REAL :: Total $\begin{array}{r l}{\mathbf{\sigma}=}&{{}0\mathbf{\sigma}.\;0}\end{array}$ , Average $=$ 0.0 INTEGER :: Person REAL , DIMENSION(:) , ALLOCATABLE :: Weight PRINT \*,' How many people?' READ \*,Number Of People ALLOCATE(Weight(1:Number Of People)) DO Person $\mathbf{\Psi}^{=}\mathbb{1}$ ,Number Of People PRINT \*, ' Type in the weight for person ',Person READ \*,Weight(Person) Total $=$ Total $^+$ Weight(Person) ENDDO Average $=$ Total / Number Of People PRINT \*,' The total of the weights is ',Total PRINT \*,' Average Weight is ',Average PRINT \*,' ',Number of People,' Weights were ' DO Person $\mathbf{\Psi}^{=}\mathbb{1}$ ,Number Of People PRINT \*,Weight(Person) ENDDO + +END PROGRAM ch1001 + +The first statement of interest is the type declaration with the dimension and allocatable attributes, e.g., + +REAL , DIMENSION(:) , ALLOCATABLE :: Weight + +The second is the ALLOCATE statement where the value of the variable Num- ber_of_people is not known until run time, e.g., + +ALLOCATE(Weight(1:Number Of People)) We will look more formally at these statements in Chapter 11. + +# 10.2 Higher-dimension arrays + +There are many instances where it is necessary to have arrays with more than one dimension. Consider the examples below. + +# 10.2.1 A map + +Consider the representation of the height of an area of land expressed as a two-di- mensional table of numbers e.g., we may have some information represented in a simple table as follows: + +![](images/39cd1c36ee37a175ea9e890998b73bd3c46f78a3af5aac2675c0492011619f48.jpg) + +The values in the array are the heights above sea level. The example is obviously artificial, but it does highlight the concepts involved. For those who have forgotten their geography, lines of latitude run east–west (the equator is a line of latitude) and lines of longitude run north–south (they go through the poles and are all of the same length). In the above table therefore the latitude values are ordered by row and the longitude values are ordered by column. + +A program to manipulate this data structure would involve something like the fol- lowing: + +# Chapter 10 + +PROGRAM C1002 + + ! Variables used + + ! Height - used to hold the heights above sea level + + ! Long - used to represent the longitude + + ! Lat - used to represent the latitude + + ! both restricted to integer values. + + ! Correct - holds the correction factor IMPLICIT NONE INTEGER , PARAMETER :: Size = 3 INTEGER :: Lat , Long REAL , DIMENSION(1:Size,1:Size) :: Height REAL , PARAMETER :: Correct $=$ 10.0 DO Lat $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size DO Long $\begin{array}{r l}{\mathbf{\Sigma}=}&{{}\ \mathbb{1}}\end{array}$ ,Size PRINT \*,' Type in value at ',Lat,' ',Long READ \* , Height(Lat,Long) ENDDO ENDDO DO Lat $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size DO Long $\begin{array}{r l}{\mathbf{\Sigma}=}&{{}\ \mathbb{1}}\end{array}$ ,Size Height(Lat,Long) $=$ Height(Lat,Long) $^+$ Correct ENDDO ENDDO PRINT \* , ' Corrected data is ' DO Lat $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size DO Long $\begin{array}{r l}{\mathbf{\Sigma}=}&{{}\ \mathbb{1}}\end{array}$ ,Size PRINT $\star$ , Height(Lat,Long) ENDDO ENDDO END PROGRAM C1002 + +Note the way in which indentation has been used to highlight the structure in this example. Note also the use of a textual prompt to highlight which data value is ex- pected. Running the program highlights some of the problems with the simple I/O used in the example above. We will address this issue in the next example. + +The inner loop is said to be nested within the outer one. It is very common to en- counter problems where nesting is a natural way to express the solution. Nesting is permitted to any depth. Here is an example of a valid nested DO loop: + +DO ! Start of outer loop DO ! Start of inner loop + +ENDDO ! End of inner loop ENDDO ! End of outer loop + +This example introduces the concept of two indices, and can be thought of as a row and column data structure. + +# 10.2.2 Example 3: Sensible tabular output + +The first example had the values printed in a format that wasn't very easy to work with. In this example we introduce a so-called implied DO loop, which enables us to produce neat and humanly comprehensible output: + +PROGRAM C1003 + + ! Variables used + + ! Height - used to hold the heights above sea level + + ! Long - used to represent the longitude + + ! Lat - used to represent the latitude + + ! both restricted to integer values. IMPLICIT NONE INTEGER , PARAMETER :: Size $=$ 3 INTEGER :: Lat , Long REAL , DIMENSION(1:Size,1:Size) :: Height REAL , PARAMETER :: Correct $=$ 10.0 DO Lat $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size DO Long $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size READ \* , Height(Lat,Long) Height(Lat,Long) $=$ Height(Lat,Long) $^+$ Correct ENDDO ENDDO DO Lat $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size PRINT \* , (Height(Lat,Long),Long $\backprime\!=\!1$ ,3) ENDDO END PROGRAM C1003 + +The key statement in this example is + +PRINT \* , (Height(Lat,Long),Long $\backprime\!=\!1$ ,3) + +This is called an implied DO loop, as the longitude variable takes on values from 1 through 3 and will write out all three values on one line. + +We will see other examples of this statement as we go on. + +# Chapter 10 + +# 10.2.3 Example 4: Average of three sets of values + +This example extends the previous one. Now we have three sets of measurements and we are interested in calculating the average of these three sets. The two new data sets are: + +9.5 39.5 69.5 19.5 49.5 79.5 29.5 59.5 89.5 + +and + +10.5 40.5 70.5 20.5 50.5 80.5 30.5 60.5 90.5 + +and we have chosen the values to enable us to quickly check that the calculations for the averages are correct. + +This program also uses implied DO loops to read the data, as data in files are gen- erally tabular: + +PROGRAM C1004 + + ! Variables used + + ! H1,H2,H3 - used to hold the heights above sea level + + ! H4 - used to hold the average of the above + + ! Long - used to represent the longitude + + ! Lat - used to represent the latitude + + ! both restricted to integer values. IMPLICIT NONE INTEGER , PARAMETER :: Size = 3 INTEGER :: Lat , Long REAL , DIMENSION(1:Size,1:Size) :: H1,H2,H3,H4 DO Lat $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size READ \* , (H1(Lat,Long), Long $\scriptstyle{\dot{\mathbf{\tau}}}={\boldsymbol{\perp}}$ ,Size) ENDDO DO Lat $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size READ \* , (H2(Lat,Long), Long $\scriptstyle{\dot{\mathbf{\tau}}}={\boldsymbol{\perp}}$ ,Size) ENDDO DO Lat $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size READ \* , (H3(Lat,Long), Long $\scriptstyle{\dot{\mathbf{\tau}}}={\boldsymbol{\perp}}$ ,Size) ENDDO DO Lat $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size DO Long $\begin{array}{r l}{\mathbf{\Sigma}=}&{{}\ \mathbb{1}}\end{array}$ ,Size H4(Lat,Long) $=$ ( H1(Lat,Long) $^+$ H2(Lat,Long) + & + +H3(Lat,Long) ) / Size ENDDO ENDDO DO Lat $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ ,Size PRINT \* , (H4(Lat,Long),Long $\scriptstyle{\cdot=1}$ ,3) ENDDO END PROGRAM C1004 + +The original data was accurate to three significant figures. The output from the above has spurious additional accuracy. We will look at how to correct this in the later chapter on output. + +# 10.2.4 Example 5: Booking arrangements in a theatre or cinema + +A theatre or cinema consists of rows and columns of seats. In a large cinema or a typical theatre there would also be more than one level or storey. Thus, a program to represent and manipulate this structure would probably have a two-d or three-d array. Consider the following program extract: + +PROGRAM ch1005 IMPLICIT NONE INTEGER , PARAMETER :: $\tt N R{=}5$ INTEGER , PARAMETER :: $\scriptstyle{\mathrm{NC}}=\perp0$ INTEGER , PARAMETER :: $\tt N F\!=\!3$ INTEGER :: Row,Column,Floor CHARACTER\*1 , DIMENSION(1:NR,1:NC,1:NF) :: Seats $="$ ' DO Floor $\mathbf{\omega}=\mathbb{1}$ ,NF DO Row $\scriptstyle\cdot=\perp$ ,NR READ \*,(Seats(Row,Column,Floor),Column $\v{=}\,\v{1}$ ,NC) ENDDO ENDDO PRINT \*,' Seat plan is' DO Floor $\mathbf{\omega}=\mathbb{1}$ ,NF PRINT \*,' Floor $=$ ',Floor DO Row $\scriptstyle\cdot=\perp$ ,NR PRINT \*,(Seats(Row,Column,Floor),Column $\mathbf{\Psi}^{=}\mathbb{1}$ ,NC) ENDDO ENDDO END PROGRAM ch1005 + +Note here the use of the term PARAMETER in conjunction with the INTEGER declaration. This is called an entity orientated declaration. An alternative to this is an attribute-orientated declaration, e.g., + +INTEGER :: NR,NC,NF PARAMETER :: NR ${\tt=}{\tt5}$ , $\scriptstyle{\mathrm{NC}}=\perp0$ , $\tt N F\!=\!3$ + +and we will be using the entity-orientated declaration method throughout the rest of the book. This is our recommended method as you only have to look in one place to determine everything that you need to know about an entity. + +# 10.3 Additional forms of the DIMENSION attribute and DO loop statement + +10.3.1 Example 6: Voltage from $\mathbf{-20}$ to $\mathbf{+20}$ volts + +Consider the problem of an experiment where the independent variable voltage varies from $-20$ to $+20$ volts and the current is measured at 1-volt intervals. For- tran has a mechanism for handling this type of problem: + +PROGRAM C1006 IMPLICIT NONE REAL , DIMENSION(-20:20) :: Current REAL :: Resistance INTEGER :: Voltage PRINT \*,' Type in the resistance' READ \*, Resistance DO Voltage $=$ -20,20 Current(Voltage) $=$ Voltage/Resistance PRINT \*, Voltage, ' ', Current(Voltage) ENDDO END PROGRAM C1006 + +We appreciate that, due to experimental error, the voltage will not have exact inte- ger values. However, we are interested in representing and manipulating a set of values, and thus from the point of view of the problem solution and the program this is a reasonable assumption. There are several things to note. + +This form of the DIMENSION attribute + +DIMENSION(First:Last) + +is of considerable use when the problem has an effective index which does not start at 1. + +There is a corresponding form of the DO statement which allows processing of problems of this nature. This is shown in the above program. The general form of the DO statement is therefore: + +DO counter ${}^{\ast}=$ start, end, increment + +where start, end and increment can be positive or negative. Note that zero is a le- gitimate value of the dimension limits and of a DO loop index. + +# 10.3.2 Example 7: Longitude from $\mathbf{-180}$ to $\mathbf{+180}$ + +Consider the problem of the production of a table linking time difference with lon- gitude. The values of longitude will vary from $-180$ to $+180$ degrees, and the time will vary from $+12$ hours to $-12$ hours. A possible program segment is: + +PROGRAM ch1007 IMPLICIT NONE REAL , DIMENSION(-180:180) :: Time $=\!0$ INTEGER :: Degree,Strip REAL :: Value DO Degree $=$ -180,165,15 Value $=$ Degree/15. DO Strip $=\!0$ ,14 Time(Degree $^+$ Strip) $=$ Value ENDDO ENDDO DO Degree $=$ -180,180 PRINT \*,Degree,' ',Time(Degree) END DO END PROGRAM ch1007 + +# 10.3.3 Notes + +The values of the time are not being calculated at every degree interval. + +The variable Time is a real variable. It would be possible to arrange for the time to be an integer by expressing it in either minutes or seconds. + +This example takes no account of all the wiggly bits separating time zones or of British Summer Time. + +What changes would you make to the program to accommodate $+180?$ What is the time at $-180$ and $+180?$ + +# 10.4 The DO loop and straight repetition + +# 10.4.1 Example 8: Table of temperatures + +Consider the production of a table of liquid measurements. The independent vari- able is the litre value; the gallon and US gallon are the dependent variables. Strictly speaking, a program to do this does not have to have an array, i.e., the DO loop can be used to control the repetition of a set of statements that make no refer- ence to an array. The following shows a complete but simple conversion program: + +PROGRAM ch1008 IMPLICIT NONE + +! + + ! 1 us gallon = 3.7854118 litres + + ! 1 uk gallon $=$ 4.545 litres + + ! INTEGER :: Litre REAL :: Gallon,USGallon DO Litre $\begin{array}{r l}{\mathbf{\Sigma}=}&{{}\mathbb{1}\,,\,\mathbb{1}\,0}\end{array}$ Gallon $=$ Litre \* 0.2641925 USGallon $=$ Litre $\star$ 0.220022 PRINT \*,Litre, ' ',Gallon,' ',USGallon END DO END PROGRAM ch1008 + +Note here that the DO statement has been used only to control the repetition of a block of statements — there are no arrays at all in this program. + +This is the other use of the DO statement. The DO loop thus has two functions — its use with arrays as a control structure and its use solely for the repetition of a block of statements. + +# 10.4.2 Example 9: Means and standard deviations + +In the calculation of the mean and standard deviation of a list of numbers, we can use the following formulae. It is not actually necessary to store the values, nor to accumulate the sum of the values and their squares. In the first case, we would possibly require a large array, whereas in the second, it is conceivable that the ac- cumulated values (especially of the squares) might be too large for the machine. The following example uses an updating technique which avoids these problems, but is still accurate. The DO loop is simply a control structure to ensure that all the values are read in, with the index being used in the calculation of the updates: + +# PROGRAM ch1009 + +! Variables used are + + ! Mean - for the running mean + + ! SSQ - The running corrected sum of squares + + ! X - Input values for which + + ! mean and sd required + + ! W - Local work variable + + ! SD - Standard Deviation + + ! R - Another work variable + +IMPLICIT NONE REAL :: Mean $=\!0$ .0,SSQ $=\!0$ .0,X,W,SD,R INTEGER :: I,N PRINT \*,' ENTER THE NUMBER OF READINGS' READ\*,N PRINT\*,' ENTER THE ',N,' VALUES, ONE PER LINE' DO $\mathbb{T}\!=\!1$ ,N READ\*,X $\scriptstyle{\bar{W}}=X$ -Mean $\scriptstyle\mathrm{R}=\mathbb{T}-\mathbb{1}$ Mean $\b=$ (R\*Mean+X)/I $\scriptstyle S S Q=S S Q+W^{\star}W^{\star}\mathrm{R}/\mathrm{\,}\mathrm{L}$ ENDDO SD=(SSQ/R)\*\*0.5 PRINT \*,' Mean is ',Mean PRINT \*,' Standard deviation is ',SD END PROGRAM ch1009 + +# 10.5 Summary + +Arrays can have up to seven dimensions. + +DO loops may be nested, but they must not overlap. + +The DIMENSION attribute allows limits to be specified for a block of information which is to be treated in a common way. The limits must be integer, and the sec- ond limit must exceed the first, e.g., + +REAL , DIMENSION(-123:-10) :: List REAL , DIMENSION(0:100,0:100) :: Surface REAL , DIMENSION(1:100) :: Value + +The last example could equally be written REAL , DIMENSION(100) :: Value + +where the first limit is omitted and is given the default value 1. The array LIST would contain 114 values, while Surface would contain 10201. + +A DO statement and its corresponding ENDDO statement define a loop. The DO statement provides a starting value, terminal value, and optionally, an increment for its index or counter. + +The increment may be negative, but should never be zero. If it is not present, the default value is 1. It must be possible for the terminating value to be reached from the starting value. + +# Chapter 10 + +The counter in a DO loop is ideally suited for indexing an array, but it may be used anywhere that repetition is needed, and of course the index or counter need not be used explicitly. + +The formal syntax of the block DO construct is + +[ do-construct-name : ] DO [label] [ loop-control ] [execution-part-construct ] [ label ] end-do + +where the forms of the loop control are + +[ , ] scalar-variable-name $=$ scalar-numeric-expression , scalar-numeric-expression [ , scalar-numeric-expression ] + +and the forms of the end-do are + +END DO [ do-construct-name ] CONTINUE + +and $[]$ identify optional components of the block DO construct. This statement is looked at in much greater depth in Chapter 16. + +# 10.6 Problems + +1. Compile and run all the examples in this chapter, except example 5. This is cov- ered separately later. + +2. Modify the first example to convert the height in feet to height in metres. The conversion factor is one 1 equals 0.305 metres. + +Hint: You can either overwrite the height array or introduce a second array. + +3. The following are two equations for temperature conversion + +$$ +\begin{array}{c}{{\tt c=\mathrm{~5~/9~*~}(t{\mathrm{-}}32)}}\\ {{\tt f=32+\mathrm{~9~/5~*~}t}}\end{array} +$$ + +Write a complete program where t is an integer DO loop variable and loop from $-50$ to 250. Print out the values of c, t and f on one line. What do you notice about the c and f values? + +4. Write a program to print out the 12 times table. Typical output would be of the form: + +1 \* 12 = 12 + +2 \* 12 = 24 3 \* 12 = 36 + +etc. + +Hint: You don't need to use an array here. + +5. Write a program to read the following data into a two-dimensional array: + +1 2 3 4 5 6 7 8 9 + +Calculate totals for each row and column and produce an output similar to that below: + +1 2 3 6 4 5 6 15 7 8 9 24 12 15 18 + +Hint 1: Example ch0902 shows how to sum over a loop. + +Hint 2 : You need to introduce two one-dimensional arrays to hold the row and column totals. You need to index over the rows to get the column totals and over the columns to get the row totals. + +6. Modify the above to produce averages for each row and column as well as the totals. + +9. Using the following data from Problem 2 in Chapter 9: + +1.85 85 1.80 76 1.85 85 1.70 90 1.75 69 1.67 83 1.55 64 1.63 57 1.79 65 1.78 76 + +Use the program that evaluated the mean and standard deviation to do so for these heights and weights. + +# Chapter 10 + +In the first case use the program as is and run it twice, first with the heights then with the weights. + +What changes would you need to make to the program to read a height and a weight in a pair? + +Hint : You could introduce separate scalar variables for the heights and weights. + +10. Example 5 looked at seat bookings in a cinema or theatre. Here is an example of a sample data file for this program + +The key for this is as follows: + +$\mathrm{C}=$ Confirmed Booking + +$\mathrm{P}=$ Provisional Booking + +$\mathrm{E}=$ Seat Empty + +Compile and run the program. The output would benefit from adding row and col- umn numbers to the information displayed. We will come back to this issue in a subsequent chapter on output formatting. + +The data are in a file on the web and the address is given below. • http://www.kcl.ac.uk/fortran + +Problem 3.3 in the last chapter shows how to read data from a file. + +# Whole Array and Additional Array Features + +“A good notation has a subtlety and suggestiveness which at times make it seem almost like a live teacher.” + +Bertrand Russell + +# Aims + +The aims of the chapter are: + +• To look more formally at the terminology required to precisely describe arrays. • To introduce ways in which we can manipulate whole arrays and parts of arrays (sections). • ALLOCATABLE arrays — ways in which the size of an array can be de- ferred until execution time. • To introduce the concept of array element ordering and physical and vir- tual memory. • To introduce ways in which we can initialise arrays using array construc- tors. • To introduce the WHERE statement and array masking. • To introduce the FORALL statement and construct. + +# Chapter 11 + +# 11 Whole Array and Additional Array Features + +# 11.1 Terminology + +Fortran supports an abundance of array handling features. In order to make the de- scription of these features more precise a number of additional terms have to be covered and these are introduced and explained below. + +# 11.1.1 Rank + +The number of dimensions of an array is called its rank. A one-dimensional array has rank 1, a two-dimensional array has rank 2 and so on. + +# 11.1.2 Bounds + +An array's bounds are the upper and lower limits of the index in each dimension. + +# 11.1.3 Extent + +The number of elements along a dimension of an array is called the extent. + +INTEGER, DIMENSION(-10:15):: Current + +has bounds $-10$ and 15 and an extent of 26. + +# 11.1.4 Size + +The total number of elements in an array is its size. + +# 11.1.5 Shape + +The shape of an array is determined by its rank and its extents in each dimension. + +# 11.1.6 Conformable + +Two arrays are said to be conformable if they have the same shape, that is, they have the same rank and the same extent in each dimension. + +# 11.1.7 Array element ordering + +Array element ordering states that the elements of an array, regardless of rank, form a linear sequence. The sequence is such that the subscripts along the first di- mension vary most rapidly, and those along the last dimension vary most slowly. This is best illustrated by considering, for example, a rank 2 array A defined by + +REAL , DIMENSION(1:4,1:2) :: A + +A has 8 real elements whose array element order is A(1,1), A(2,1), A(3,1), A(4,1), A(1,2), A(2,2), A(3,2), A(4,2) i.e., mathematically by column and not row. + +# 11.2 Whole array manipulation + +The examples of arrays so far have shown operations on arrays via array elements. One of the significant features of Fortran is its ability to manipulate arrays as whole objects. This allows arrays to be referenced not just as single elements but also as groups of elements. Along with this ability comes a whole host of intrinsic procedures for array processing. These procedures are mentioned in Chapter 14, and listed in alphabetical order with examples in Appendix D. + +# 11.2.1 Assignment + +An array name without any indices can appear on both sides of assignment and in- put and output statements. For example, values can be assigned to all the elements of an array in one statement: + +REAL, DIMENSION(1:12):: Rainfall Rainfall $=\!0$ .0 + +The elements of one array can be assigned to another: + +INTEGER, DIMENSION(1:50) :: A,B + + ... + +$\tt A\!\!=\!\!\tt B$ + +Arrays A and B must be conformable in order to do this. + +The following example is illegal since X is rank 1 and extent 20, whilst $Z$ is rank 1 and extent 41. + +REAL, DIMENSION(1:20) :: X REAL, DIMENSION(1:41) :: Z + + $\mathtt{X}\!\!=\!\!5\,0$ .0 + + $\scriptstyle\mathrm{Z}\,=\!\mathrm{X}$ + +But the following is legal because both arrays are now conformable, i.e., they are both of rank 1 and extent 41: + +REAL , DIMENSION (-20:20) :: X REAL , DIMENSION (1:41) :: Y + + ${\mathrm{X}}{=}\,5\,0\ .\ 0 + +$ $\tt Y\!=\!\tt X$ + +# 11.2.2 Expressions + +All the arithmetic operators available to scalars are available to arrays, but care must be taken because mathematically they may not make sense. + +REAL , DIMENSION (1:50) :: A,B,C,D,E $\mathrm{C=\mathbb{A}+\mathbb{B}}$ + +adds each element of A to the corresponding element of B and assigns the result to C. + +$\scriptstyle\mathbb{E}=\mathbb{C}^{\star}\mathbb{D}$ + +multiplies each element of C by the corresponding element of D. This is not vector multiplication. To perform a vector dot product there is an intrinsic procedure DOT_PRODUCT, and an example of this is given in a subsequent section on array constructors. + +For higher dimensions + +REAL ,DIMENSION (1:10,1:10) :: F,G,H $\mathrm{F{=}F^{\star\star}\,0\,.\,5}$ + +takes the square root of every element of F. + +$$ +\scriptstyle{\mathrm{H}=\mathrm{F}+\mathrm{G}} +$$ + +adds each element of $\mathrm{F}$ to the corresponding element of G. + +$$ +\scriptstyle\mathrm{H}=\mathrm{F}^{\star}\mathrm{G} +$$ + +multiplies each element of F by the corresponding element of G. The last statement is not matrix multiplication. An intrinsic procedure MATMUL performs matrix multiplication; further details are given in Appendix D. + +Consider the following example, which is a solution to a problem set earlier, but is now addressed using some of the whole array features of Fortran + +PROGRAM ch1101 IMPLICIT NONE INTEGER , PARAMETER :: $\scriptstyle{\mathrm{N}}=\pm2$ REAL , DIMENSION(1:N) :: RainFall_ins $=\!0$ .0 REAL , DIMENSION(1:N) :: RainFall_cms $=\!0$ .0 INTEGER :: Month PRINT \*, ' Input the rainfall values in inches' READ \*, RainFall_ins RainFall_cms $=$ RainFall_ins \* 2.54 DO Month $\mathbf{\Psi}\!=\!\!1$ ,N PRINT \* , ' ', Month , ' ' , & RainFall_ins(Month) , ' ' , & RainFall_cms(Month) + +# Whole Array and Additional Array Features + +END DO END PROGRAM ch1101 + +The statements + +REAL , DIMENSION(1:N) :: RainFall_ins $=\!0$ .0 REAL , DIMENSION(1:N) :: RainFall_cms $=\!0$ .0 + +are examples of whole array initialisation. Each element of the arrays is set to 0.0. The statement + +READ \*, RainFall_ins + +is an example of whole array I/O, where we no longer have to use a DO loop to read each element in. + +Finally, we have the statement + +RainFall_cms $=$ RainFall_ins \* 2.54 + +which is an example of whole array arithmetic and assignment. + +Here is a two-dimensional example: + +PROGRAM ch1102 + + ! This program reads in a grid of temperatures + + ! (degrees Fahrenheit) at 25 grid references + + ! and converts them to degrees Celsius IMPLICIT NONE REAL, DIMENSION (1:5,1:5) :: Fahrenheit, Celsius INTEGER :: Long, Lat + + ! + + ! Read in the temperatures + +DO Lat $\mathbf{\omega}=\mathbb{1}$ ,5 PRINT \*, ' For Latitude $=$ ',Lat DO Long $\backprime\!=\!1$ ,5 PRINT \*, ' For Longitude', Long READ \*,Fahrenheit( Long, Lat) END DO END DO + + ! + + ! Conversion applied to all values + + ! + +Celsius $=$ 5.0/9.0 \* (Fahrenheit - 32.0) + +# Chapter 11 + +PRINT \* , Celsius PRINT $\star$ , Fahrenheit END PROGRAM ch1102 + +Note the use of whole arrays in the print statements. The output does look rather messy though, and also illustrates array element ordering. + +# 11.3 Array sections + +Often it is necessary to access part of an array rather than the whole, and this is possible with Fortran's powerful array manipulation features. + +# 11.3.1 Rank 1 array example + +Consider the following: + +program ch1103 implicit none integer , dimension(-5:5) :: x integer :: i x(-5:-1) = -1 x(0) = 0 x(1:5)=1do i=-5,5 print \*,' ',i,' ',x(i)end do end program ch1103 + +The statement + +$$ +\begin{array}{r l r}{{\sf x}\left(\,-\,5:-\,1\,\right)}&{{}=}&{-\,1}\end{array} +$$ + +is working with a section of an array. It assigns the value $^{-1}$ to elements $\mathtt{X}(-5)$ through $\mathbf{\deltaX}(-1)$ . + +The statement + +$$ +\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ +$$ + +is also working with an array section. It assignes the value 1 to elements x(1) through $\mathbf{\delta}_{\mathbf{X}}(5)$ . + +# 11.3.2 Rank 2 array example + +In Chapter 9 we gave an example of a table of examination marks, and this is given again below: + +![](images/ad9b1fd7562ca5ddca4287afa8b3a3aeb2f76a84847b835a0675baf4f96291af.jpg) + +The following program reads the data in, scales column 3 by 2.5 as the Biology marks were out of 40 (the rest are out of 100), calculates the averages for each subject and for each person and prints out the results. + +real , dimension(1:nrow) & :: People_average = 0.0 real , dimension(1:ncol) &:: Subject_Average = 0.0 integer :: r,c do $\scriptstyle\mathtt{T}=1$ ,nrow read \*, exam_results(r,1:ncol)end do Exam_Results(1:nrow,3) = 2.5 \* Exam_Results(1:nrow,3) do $\scriptstyle\mathtt{T}=1$ ,nrow do $\scriptstyle C={\underline{{1}}}$ ,ncol people_average(r) $=$ people_average(r) + & exam_results(r,c) end do end do people_average $=$ people_average / ncol do $\scriptstyle C={\underline{{1}}}$ ,ncol do $\scriptstyle\mathtt{T}=1$ ,nrow subject_average(c) $=$ subject_average(c) + & exam_results(r,c) end do + +# Chapter 11 + +end do subject_average $=$ subject_average / nrow print \*,' People averages' print \*, people_average print \*, ' Subject averages' print \*, subject_average end program ch1104 + +The statement + +read \*, exam_results(r,1:ncol) + +uses sections to replace the implied DO loop in the earlier example. + +The statement + +Exam_Results(1:nrow,3) = 2.5 \* Exam_Results(1:nrow,3) + +uses array sections in the arithmetic and the assignment. + +# 11.4 Array constructors + +Arrays can be given intial values in Fortran using array constructors. Some exam- ples are given below. + +# 11.4.1 Rank 1 array example — explicit values + +PROGRAM ch1105 IMPLICIT NONE integer :: $\mathtt{n}\!\!=\!\!\bot2$ REAL :: Tota $\mathsf{L}\!=\!0\;.\;0$ , Average $=\!0$ .0 REAL , DIMENSION(1:n) :: RainFall = & (/3.1,2.0,2.4,2.1,2.2,2.2,1.8,2.2,2.7,2.9,3.1,3.1/) INTEGER :: Month DO Month $\mathbf{\Psi}\!=\!\!1$ ,n Total $=$ Total $^+$ RainFall(Month) ENDDO Average $=$ Total / n PRINT \*,' Average monthly rainfall was' PRINT \*, Average END PROGRAM ch1105 + +The statement + +REAL , DIMENSION(1:n) :: RainFall = & (/3.1,2.0,2.4,2.1,2.2,2.2,1.8,2.2,2.7,2.9,3.1,3.1/) + +provides initial values to the elements of the array Rainfall. + +# 11.4.1.1 Rank 1 array example and implied DO loop + +The next example uses a simple variant: + +PROGRAM ch1106 IMPLICIT NONE + + ! + + ! 1 us gallon $=$ 3.7854118 litres + + ! 1 uk gallon $=$ 4.545 litres + + ! integer , parameter :: $\mathtt{n}\!=\!\mathtt{10}$ integer :: i INTEGER , dimension( $\cdot\,\mathtt{{1}}:\mathtt{{n}})$ ) :: Litre $=$ (/(i,i=1,n)/) REAL, dimension(1:n) :: Gallon,USGallonGallon $=$ Litre $\star$ 0.2641925 USGallon $=$ Litre $\star$ 0.220022 DO $\begin{array}{r c l}{\dot{\mathrm{~\textemdash~}}}&{=}&{\mathbb{1}\,,\mathfrak{n}}\end{array}$ PRINT \*,Litre(i), ' ',Gallon(i),' ',USGallon(i) END DO END PROGRAM ch1106 + +The statement + +INTEGER , dimension( $\mathtt{L}:\mathtt{n}\,.$ ) :: Litre $=$ (/(i,i=1,n)/) + +initialises the 10 elements of the Litre array to the values 1,2,3,4,5,6,7,8,9,10 respectively. + +# 11.4.1.2 Rank 1 array example and the DOT_PRODUCT intrinsic + +The following example uses an array constructor and the intrinsic procedure DOT_PRODUCT: + +INTEGER , DIMENSION(1:3) :: X,Y INTEGER :: Result + + ${\mathrm{X}}{=}\left(\,/\,1\,,\,3\,,\,5\,/\,\right)$ ) + + $\mathbb{Y}{=}\left(\,/\,2\,,\,4\,,\,6\,/\,\right)$ Result $=$ DOT_PRODUCT(X,Y) + +and Result has the value 44, which is obtained by the normal mathematical dot product operation, $1^{*}2+3^{*}4+5^{*}6.$ . + +The general form of the array constructor is ( $/$ a list of expressions/) where each expression is of the same type. + +To construct arrays of higher rank than one the intrinsic function RESHAPE must be used. An introduction to intrinsic functions is given in Chapter 14, and an al- phabetic list with a full explanation of each function is given in Appendix D. To use it in its simplest form: + +Matrix $=$ RESHAPE ( Source, Shape) + +where Source is a rank 1 array containing the values of the elements required in the new array, Matrix, and Shape is a rank 1 array containing the shape of the new array Matrix. + +We consider the rank 1 array $\mathbf{B}{=}$ (1,3,5,7,9,11), and we wish to store these values in a rank 2 array A, such that A is the matrix: + +$$ +A=\left(\begin{array}{l l}{1}&{7}\\ {3}&{9}\\ {5}&{11}\end{array}\right) +$$ + +The following code extract is needed: + +INTEGER, DIMENSION(1:6) :: B INTEGER, DIMENSION(1:3, 1:2) :: A B = (/1,3,5,7,9,11/) A $=$ RESHAPE(B,(/3,2/)) + +Note that the elements of the source array B must be stored in the array element order of the required array A. + +The following example illustrates the additional forms of the RESHAPE function that are used when the number of elements in the source array is less than the number of elements in the destination. The complete form is + +RESHAPE(Source, Shape, Pad, Order) + +Pad and Order are optional. See Appendix D for a complete explanation of Pad and Order : + +program ch1107 implicit none integer , dimension(1:2,1:4) :: x integer , dimension(1:8) :: $\mathtt{y}{=}\,(\,/\,1\,,\,2\,,\,3\,,\,4\,,\,5\,,\,6\,,\,7\,,\,8\,/\,)$ integer , dimension(1:6) :: $z\!=(\,/\,1\,,\,2\,,\,3\,,\,4\,,\,5\,,\,6\,/\,)$ integer :: r,c print \*,' Source array y' print \*,y print \*,' Source array z' + +print \*,z print \*,' Simple reshape sizes match' $\times{}=$ reshape(y,(/2,4/))do $\scriptstyle\mathtt{T}=1$ ,2 print \*,(x(r,c), $_{\mathbf{C}}{=}1$ ,4) end do print \*,' Source 2 elements smaller pad with 0' $\times{}=$ reshape(z,(/2,4/),(/0,0/)) do $\scriptstyle\mathtt{T}=1$ ,2 print \*,(x(r,c), $_{\mathbf{C}}{=}1$ ,4) end do print \*,' As previous now specify order as $_{1}\star_{2}$ ' $\times{}=$ reshape(z,(/2,4/),(/0,0/),(/1,2/)) do $\scriptstyle\mathtt{T}=1$ ,2 print \*,(x(r,c), $_{\mathbf{C}}{=}1$ ,4) end do print \*,' As previous now specify order as $_2\!\star\!_{\perp}$ ' $\times{}=$ reshape(z,(/2,4/),(/0,0/),(/2,1/)) do $\scriptstyle\mathtt{T}=1$ ,2 print \*,(x(r,c), $_{\mathbf{C}}{=}1$ ,4) end do end program ch1107 + +# 11.4.2 Rank 1 example with step size of 2 in implied DO loop + +Consider the following example: + +program ch1108 implicit none integer :: i integer , dimension(1:10) :: $\mathbf{x}{=}\left(\mathbf{\beta}/\left(\dot{\mathbf{\alpha}}\mathbf{l}\right.,\dot{\mathbf{\alpha}}{=}\mathbf{1}\mathbf{\beta},\mathbf{1}0\mathbf{\beta}\right)\mathbf{\beta}/\mathbf{\gamma}\right)$ ) integer , dimension(1:5) :: ${\mathsf{o d d}}\!=\!\left(\,\prime\,(\,\dot{\bot}\,,\,\dot{\bot}\!=\!\!1\,,\,\underline{{{1}}}\,0\,,\,2\,)\,\right/\,\right)$ integer , dimension(1:5) :: even even $=\!\!\mathbf{x}$ (2:10:2) print \*,' x' print \*,x print \*,' odd' print \*,odd print \*,' even' print \*,even end program ch1108 + +The statement + +steps through the array 2 at a time. + +The statement + +$$ +\mathsf{e v e n}\!\!=\!\!\operatorname{x}\left(\,2:\!10:\!2\,\right) +$$ + +shows an array section where we go from elements two through ten in steps of two. The 2:10:2 is an example of a subscript triplet in Fortran, and the first 2 is the lower bound, the 10 is the upper bound, and the last 2 is the increment. Fortran uses the term stride to mean the increment in a subscript triplet. + +# 11.4.3 Rank 1 array and the SUM intrinsic function + +The following example is based on ch1105. It uses the SUM intrinsic to calculate the sum of all the values in the Rainfall array. + +PROGRAM ch1109 IMPLICIT NONE REAL :: Total $=\!0$ .0, Average $=\!0$ .0 REAL , DIMENSION(12) :: RainFall $=\quad\&$ (/3.1,2.0,2.4,2.1,2.2,2.2,1.8,2.2,2.7,2.9,3.1,3.1/) INTEGER :: Month Total $=$ SUM(RainFall)Average $=$ Total / 12 PRINT \*,' Average monthly rainfall was' PRINT \*, Average END PROGRAM ch1109 + +The statement + +Total $=$ SUM(RainFall) + +replaces the statements below from the earlier example + +DO Month $\mathbf{\Psi}\!=\!\!1$ ,n Total $=$ Total $^+$ RainFall(Month) ENDDO + +In this example SUM adds up all of the elements of the array Rainfall. + +So we have three ways of processing arrays: + +• Element by element. + +• Using sections. + +• On a whole array basis. + +The ability to use sections and whole arrays when programming is a major advance of the element by element processing supported by Fortran 77. + +# 11.4.4 Rank 2 arrays and the SUM intrinsic function + +This example is based on the earlier exam results program: + +program ch1110 implicit none integer , parameter :: nrow $=\!5$ integer , parameter :: ncol $=\!6$ real , dimension(1:nrow\*ncol) :: results = & (/50 , 47 , 28 , 89 , 30 , 46 , & 37 , 67 , 34 , 65 , 68 , 98 , & 25 , 45 , 26 , 48 , 10 , 36 , & 89 , 56 , 33 , 45 , 30 , 65 , & 68 , 78 , 38 , 76 , 98 , 65/) REAL , DIMENSION(1:nrow,1:ncol) :: Exam_Results & $\begin{array}{r l}{\mathbf{\sigma}=}&{{}\,0\cdot0}\end{array}.$ real , dimension(1:nrow) :: People_average & $\begin{array}{r l}{\mathbf{\sigma}=}&{{}\,0\cdot0}\end{array}.$ real , dimension(1:ncol) :: Subject_Average & $\begin{array}{r l}{\mathbf{\sigma}=}&{{}\,0\cdot0}\end{array}.$ integer :: r,c exam_results $=\quad\&$ reshape(results,(/nrow,ncol/),(/0.0,0.0/),(/2,1/)) Exam_Results(1:nrow,3) $=$ 2.5 \* Exam_Results(1:nrow,3) subject_average $=$ sum(exam_results,dim $\mathbf{\Psi}_{:=1}$ ) people_average $=$ sum(exam_results,dim $_{1}{=}2$ ) people_average $=$ people_average / ncol subject_average $=$ subject_average / nrow print \*,' People averages' print \*, people_average print \*, ' Subject averages' print \*, subject_average end program ch1110 + +This example has several interesting array features: + +• We initialise a rank 1 array with the values we want in our exam marks array. The data are laid out in the program as they would be in an external file in rows and colums. • We use RESHAPE to initialise our exam marks array. We use the fourth parameter (/2,1/) to populate the rank 2 array with the data in row order. • We use SUM with a DIM of 1 to compute the sums for the subjects. • We use SUM with a DIM of 2 to compute the sums for the people. + +# 11.5 Masked array assignment and the WHERE statement + +Fortran has array assignment both on an element by element basis and on a whole array basis. There is an additional form of assignment based on the concept of a logical mask. + +Consider the example of time zones given in Chapter 10. The Time array will have values that are both negative and positive. We can then associate the positive values with the concept of east of the Greenwich meridian, and the negative values with the concept of west of the Greenwich meridian e.g.: + +PROGRAM ch1111 IMPLICIT NONE REAL , DIMENSION(-180:180) :: Time $=\!0$ INTEGER :: Degree,Strip REAL :: Value CHARACTER (LEN $\scriptstyle{\stackrel{\cdot}{=}}1$ ) , DIMENSION(-180:180) & :: Direction $=$ ' ' DO Degree $=$ -180,165,15 Value $=$ Degree/15. DO Strip $=\!0$ ,14 Time(Degree $^+$ Strip) $=$ Value ENDDO ENDDO DO Degree $=$ -180,180 PRINT \*,Degree,' ',Time(Degree) END DO WHERE (Time $>$ 0.0) Direction $\cdot^{-}$ 'E'ELSEWHERE (Time $<$ 0.0) Direction $=$ 'W' ENDWHEREPRINT \*,direction END PROGRAM ch1111 + +# 11.5.1 Notes + +The arrays must be conformable, i.e., in our example Time and Direction are the same shape. + +The selective assignment is achieved through the WHERE statement. + +Both the WHERE and ELSEWHERE blocks can be executed. + +The formal syntax is: + +WHERE (array logical assignment) array assignment block ELSEWHEREarray assignment block END WHERE + +The first array assignment is executed where Time is positive and the is executed where Time is negative. For further coverage of logical expressions see Chapters 15 and 18. + +# 11.6 The FORALL statement and FORALL construct + +The FORALL statement and FORALL construct were introduced into Fortran to keep it inline with High Performance Fortran — HPF. They indicate to the com- piler that the code can be optimised on a parallel processor. Consider the following example where a value is subtracted from the diagonal elements of a square matrix A: + +FORALL ( $\scriptstyle\mathbb{T}=1$ :N) A(I,I) $=$ A(I,I) - Lamda END FORALL + +The FORALL construct allows the calculations to be carried out simultaneously in a multiprocessor environment. + +# 11.6.1 Syntax + +FORALL ( triplet [ , triplet ] ... [ , mask ] ) variable $=$ expression FORALL ( triplet [ , triplet ] ... [ , mask ] ) pointer $=>$ target + +The triplet specifies a value set for an index variable. It has the following syntax: + +index $=$ first : last [ : stride ] + +First, last and stride are scalar integer expressions. + +Mask is a scalar logical expression: + +Name is an optional name, which identifies the FORALL construct. + +# 11.6.2 Array element ordering and physical and virtual memory + +Fortran compilers will store arrays in memory according to the array element or- dering scheme. Whilst the standard says nothing about how this is implemented it generally means in contiguous memory locations. + +There will be a limit to the amount of physical memory available on any computer system. To enable problems that require more than the amount of physical memory available to be solved, most implementations will provide access to virtual mem- ory, which in reality means access to a portion of a physical disk. + +Access to virtual memory is commonly provided by a paging mechanism of some description. Paging is a technique whereby fixed-sized blocks of data are swapped between real memory and disk as required. + +In order to minimise paging (and hence reduce execution time) array operations should be performed according to the array element order. + +Some common page sizes are: + +• Sun UltraSparc – 4Kb, 8Kb. • DEC Alpha – 8Kb, 16Kb, 32Kb., 64Kb. • Intel 80x86 – 4Kb. • Intel Pentium PIII – 4Kb. + +The Intel PIII also supports large pages (2Mb and 4Mb) — see the reference at the end of the chapter for more details. + +# 11.7 Summary + +We can now perform operations on whole arrays and partial arrays (array sections) without having to refer to individual elements. This shortens program development time and greatly clarifies the meaning of programs. + +Array constructors can be used to assign values to rank 1 arrays within a program unit. The RESHAPE function allows us to assign values to a two or higher rank ar- ray when used in conjunction with an array constructor. + +We have introduced the concept of a deferred-shape array. Arrays do not need to have their shape specified at compile time, only their rank. Their actual shape is deferred until runtime. We achieve this by the combined use of the ALLOCATABLE attribute on the variable declaration and the ALLOCATE state- ment, which makes Fortran a very flexible language for array manipulation. + +# 11.8 Problems + +1. Give the rank, bounds, extent and size of the following arrays: + +REAL , DIMENSION(1:15) :: A INTEGER , DIMENSION(1:3,0:4) :: B REAL , DIMENSION(-2:2,0:1,1:4) :: C INTEGER , DIMENSION(0:2,1:5) :: D + +Which two of these arrays are conformable? + +2. Use the SUM intrinsic function (see Appendix D) to calculate the total rainfall in the rainfall program example in Chapter 9. + +3. Write a program to read in five rank 1 arrays, A, B, C, D, E and then store them as five columns in a rank 2 array TABLE. + +4. Take the first part of Problem 2 in Chapter 10 and rewrite it using the SUM in- trinsic function. + +# 11.9 Bibliography + +Bhandarkar D.P., Alpha Implementation and Architecture: Complete Reference and Guide, Digital Press. + +Intel, Intel Architecture Software Developer's Manual Volume 3: System Pro- gramming + +This is available as a PDF file from Intel. Try: • http://developer.intel.com/design/PentiumIII/manuals/ + +# Output of Results + +“Why, sometimes I've believed as many as six impossible things before breakfast.” Lewis Carroll, Through the Looking-Glass and What Alice Found There + +“All the persons in this book are real and none is fictitious even in part.” Flann O'Brien, The Hard Life + +# Aims + +The aims here are to introduce the facilities for producing neat output and to show how to write results to a file, rather than to the terminal. In particular: + +• The A, I, E, F, and X layout or edit descriptors. • The OPEN, WRITE, and CLOSE statements. + +# Chapter 12 + +# 12 Output of Results + +When you have used PRINT \* a few times it becomes apparent that it is not al- ways as useful as it might be. The data are written out in a way which makes some sense, but may not be especially easy to read. Real numbers are written out with all their significant places, which is very often rather too many, and it is often dif- ficult to line up the columns for data which are notionally tabular. It is possible to be much more precise in describing the way in which information is presented by the program. To do this, we use FORMAT statements. Through the use of the FORMAT we can: + +• Specify how many columns a number should take up. • Specify where a decimal point should lie. • Specify where there should be white space. • Specify titles. + +The FORMAT statement has a label associated with it; through this label, the PRINT statement associates the data to be written with the form in which to write them. + +# 12.1 Integers — I format or edit descriptor + +Integer format is reasonably straightforward, and offers clues for formats used in describing other numbers. I3 is an integer taking three columns. The number is right justified, a bit of jargon meaning that it is written as far to the right as it will go, so that there are no trailing or following blanks. Consider the following exam- ple: + +PROGRAM ch1201 INTEGER :: T PRINT \*,' ' PRINT \*,' Twelve times table' PRINT \*,' ' DO $\scriptstyle{\mathrm{T}}={\boldsymbol{\perp}}$ ,12 PRINT 100, T, $\mathbb{T}^{\star}\bar{\bot}^{2}$ 100 FORMAT(' ',I3,' \* 12 = ',I3) END DO END PROGRAM ch1201 + +The first statement of interest is PRINT 100, T, $\mathbb{T}^{\star}\bar{\bot}^{2}$ + +The 100 is a statement label. There must be a format statement with this label in the program. The variables to be written out are $\mathrm{T}$ and $12^{*}\mathrm{T}$ . + +The second statement of interest is + +100 FORMAT(' ',I3,' \* 12 = ',I3) + +Inside the brackets we have + +' ' Print out what occurs between the quote marks, in this case one space. , The comma separates items in the FORMAT statement. I3 Print out the first variable in the PRINT statement right justified in three columns , Item separator. ' \* 12 = ' Print out what occurs between the quote characters. , Item separator I3 Print out the second variable (in this case an expression) right justified in three colums. + +All of the output will appear on one line. + +Now consider the following example: + +program ch1202 implicit none integer :: big=10 integer :: i do $\dot{\bot}\!=\!\bot$ ,40 print 100,i,big 100 format(1x,i3,2x,i12) big ${}^{\prime}{=}$ big\*10 end do end program ch1202 + +The new feature in the format statement is the $1\mathrm{x}$ and $2\mathrm{x}$ edit descriptor. This is another way of getting white space into the output, and in this case one space and two spaces, respectively. + +This program will loop and the variable big will overflow, i.e., go beyond the range of valid values for a 32-bit integer. Does the program crash or generate a run time error? This is the output from the NAG f95 compiler and the Intel Fortran 95 compiler. + +1 10 2 100 3 1000 4 10000 5 100000 6 1000000 7 10000000 8 100000000 9 1000000000 10 1410065408 11 1215752192 12 -727379968 13 1316134912 14 276447232 15 -1530494976 16 1874919424 17 1569325056 18 -1486618624 19 -1981284352 20 1661992960 21 -559939584 22 -1304428544 23 -159383552 24 -1593835520 25 1241513984 26 -469762048 27 -402653184 28 268435456 29 -1610612736 30 1073741824 31 -2147483648 32 0 33 0 34 0 35 0 36 0 37 0 38 0 39 0 40 0 + +Is there a compiler switch to trap this kind of error? + +# Output of Results + +# 12.2 Reals — F format or edit descriptor + +The F format can be seen as an extension of the integer format, but here we have to deal with the decimal point. The form of the F format specifies where the deci- mal point will occur, and how many digits follow it. Thus, F7.4 means: + +• There is a total width of seven. • There is a decimal point • There are four digits after the decimal point. + +This means that since the decimal point is also written out, there may be up to two digits before the decimal point. As in the case of the integer, any minus sign is part of the number, and would take up one column. Thus, the format F7.4 may be used for numbers in the range + +–9.9999 to 99.9999 + +Let us look at the last example more closely. When a number is written out, it is rounded; that is to say, if we write out 99.99999 in an F7.4 format, the program will try to write out 100.0000! This is bad news, since we have not left enough room for all those digits before the decimal point. What happens? Asterisks will be printed. In the example above, a number out of range of the format's capabilities would be printed as: + +\*\*\*\*\*\*\* + +What would a format of $\mathrm{F7.0}$ do? Again, seven columns have been set aside to ac- commodate the number and its decimal point, but this time no digits follow the point. + +99. -21375. + +are examples of numbers written in this format. With an F format, there is no way of getting rid of the decimal point. + +The numbers making up the parts of the descriptors must all be positive integers. The definition of a real format is therefore F followed by two integer numbers, separated by a decimal point. The first integer must exceed the second, and the second must be greater than or equal to zero. The following are valid examples: + +F4.0 F6.2 F12.2 F16.8 + +# Chapter 12 + +but these are not valid: + +F4.4 F6.8 F-3.0 F6 F.2 + +The program in Section 12.2.1 illustrates the use of both I format and F format. + +# 12.2.1 Metric and imperial conversion + +program ch1203 implicit none integer :: fluid real :: litres real :: pints do fluid $\mathbf{\Psi}_{\cdot}\!=\!\!\mathbf{1}$ ,10 litres $=$ fluid / 1.75 pints $=$ fluid \* 1.75 print 100 , pints,fluid,litres 100 format(' ',F7.3,' ',I3,' ',F7.3) end do end program ch1203 + +Pints will be printed out in F7.3 format, fluid will be printed out in I3 format and litres will be printed out in F7.3 format. + +# 12.2.2 Overflow and underflow + +Consider the following program: + +program ch1204 implicit none integer :: i real :: small = 1.0 real :: big $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}\cdot0}\end{array}$ do $\dot{\bot}\!=\!\bot$ ,50 print 100,i,small,big 100 format(' ',i3,' ',f7.3,' ',f7.3) small $=$ small/10.0 big ${}^{\prime}{=}$ big\*10.0 end do end program ch1204 + +In this program the variable small will underflow and big will overflow. The out- put from the Intel compiler is: + +1 1.000 1.000 2 0.100 10.000 3 0.010 100.000 4 0.001 \*\*\*\*\*\*\* 5 0.000 \*\*\*\*\*\*\* 6 0.000 \*\*\*\*\*\*\* 7 0.000 \*\*\*\*\*\*\* 8 0.000 \*\*\*\*\*\*\* 9 0.000 \*\*\*\*\*\*\* 10 0.000 \*\*\*\*\*\*\* 11 0.000 \*\*\*\*\*\*\* 12 0.000 \*\*\*\*\*\*\* 13 0.000 \*\*\*\*\*\*\* 14 0.000 \*\*\*\*\*\*\* 15 0.000 \*\*\*\*\*\*\* 16 0.000 \*\*\*\*\*\*\* 17 0.000 \*\*\*\*\*\*\* 18 0.000 \*\*\*\*\*\*\* 19 0.000 \*\*\*\*\*\*\* 20 0.000 \*\*\*\*\*\*\* 21 0.000 \*\*\*\*\*\*\* 22 0.000 \*\*\*\*\*\*\* 23 0.000 \*\*\*\*\*\*\* 24 0.000 \*\*\*\*\*\*\* 25 0.000 \*\*\*\*\*\*\* 26 0.000 \*\*\*\*\*\*\* 27 0.000 \*\*\*\*\*\*\* 28 0.000 \*\*\*\*\*\*\* 29 0.000 \*\*\*\*\*\*\* 30 0.000 \*\*\*\*\*\*\* 31 0.000 \*\*\*\*\*\*\* 32 0.000 \*\*\*\*\*\*\* 33 0.000 \*\*\*\*\*\*\* 34 0.000 \*\*\*\*\*\*\* 35 0.000 \*\*\*\*\*\*\* 36 0.000 \*\*\*\*\*\*\* 37 0.000 \*\*\*\*\*\*\* 38 0.000 \*\*\*\*\*\*\* 39 0.000 \*\*\*\*\*\*\* 40 0.000 Infini + +41 0.000 Infini 42 0.000 Infini 43 0.000 Infini 44 0.000 Infini 45 0.000 Infini 46 0.000 Infini 47 0.000 Infini 48 0.000 Infini 49 0.000 Infini 50 0.000 Infini + +When the number is too small for the format, the printout is what you would prob- ably expect. When the number is too large, you get asterisks. When the number actually overflows the Intel compiler tells you that the number is too big and has overflowed. However the program ran to completion and did not generate a run time error. + +# 12.3 Reals — E format or edit descriptor + +The exponential or scientific notation is useful in cases where we need to provide a format which may encompass a wide range of values. If likely results lie in a very wide range, we can ensure that the most significant part is given. It is possible to give a very large F format, but alternatively, the E format may be used. This takes a form such as + +E10.4 + +which looks something like the F, and may be interpreted in a similar way. The 10 gives the total width of the number to be printed out, that is, the number of col- umns it will take. The number after the decimal point indicates the number of positions to be written after the decimal point. Since all exponent format numbers are written so that the number is between 0.1 and 0.9999..., with the exponent tak- ing care of scale shifts, this implies that the first four significant digits are to be printed out. + +Taking a concrete example, 1000 may be written as $10^{**}3$ , or as $0.1^{\mathrm{~*~}}10^{**}4$ . This gives us the two parts: 0.1 gives the significant digits (in this case only one signifi- cant digit), while the $10^{**}4$ gives the exponent, namely 4 or $+4$ . In a form that looks more like Fortran, this would be written $1\mathrm{E}{+}04$ , where the $_{\textrm{E+04}}$ means $10^{**}4$ . + +There is a minimum size for an exponential format. Because of all the extra bits and pieces it requires: + +• The decimal point. + +• The sign of the entire number. • The sign of the exponent. • The magnitude of the exponent. • The E. + +The width of the number less the number of significant places should not be less than 6. In the example given above, E10.4 meets this requirement. When the expo- nent is in the range 0 to 99, the E will be printed as part of the number; when the exponent is greater, the E is dropped, and its place is taken by a larger value; how- ever, the sign of the exponent is always given, whether it is positive or negative. The sign of the whole number will usually only be given when it is negative. This means that if the numbers are always positive, the rule of six given above can be modified to a rule of five . It is safer to allow six places over, since, if the format is insufficient, all you will get are asterisks. + +The most common mistake with an E format is to make the edit descriptor too small, so that there is insufficient room for all the padding to be printed. Formats like E8.4 just don't work (on output anyway). The following four are valid E for- mats on output: + +E9.3 E11.2 E18.7 E10.4 + +but the next five would not be acceptable as output formats, for a variety of rea- sons: + +E11.7 E6.3 E4.0 E10 E7.3 + +# 12.3.1 Simple E format example + +This is the same as ch1204 except that we have replaced the F formatting with E formatting: + +program ch1205 implicit none integer :: i real :: small = 1.0 real :: big $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}\cdot0}\end{array}$ + +# Chapter 12 + +do $\dot{\bot}\!=\!\bot$ ,50 print 100,i,small,big 100 format(' ',i3,' ',e10.4,' ',e10.4) small $=$ small/10.0 big ${}^{\prime}{=}$ big\*10.0 end do end program ch1205 + +We now have three ways to print out floating point numbers and each has its use. The PRINT \* is very useful when developing programs. + +# 12.4 Spaces + +You have seen two ways of generating spaces on output. The first is to use ' char- acters to enclose blanks in the format statement. The second is to use the X edit descriptor. Consider the following. + +PRINT 100, ALPHA,BETA 100 FORMAT(1X,F10.4,10X,F10.3) + +The 10X is read rather like any of the other format elements — logically it should have been X10, to correspond to I10 or F10.4, but that would be allowing intuition to run away with you. Clearly the X3J3 committee felt it important that Fortran should have inconsistencies, just like a natural language. + +Remember that these blanks are in addition to any generated as a result of the lead- ing blanks on numbers (if any are present). If you wish to leave a single space, you must still precede the X by a number (in this case, 1); simply writing X is illegal. The general form is therefore a positive integer followed by X. + +# 12.5 Characters — A format or edit descriptor + +This is perhaps the simplest output of all. Since you will already have declared the length of a character variable in your declarations, + +CHARACTER (10) :: B + +when you come to write out B, the length is known — thus you need only specify that a character string is to be output: + +PRINT 100,B 100 FORMAT(1X,A) + +If you feel you need a little extra control, you can append an integer value to the A, like A10 (A9 or A1), and so on. If you do this, only the first 10 (9 or 1) char- acters are written out; the remainder are ignored. Do note that 10A1 and A10 are not the same thing. 10A1 would be used to print out the first character of ten char- acter variables, while A10 would write out the first 10 characters of a single character variable. The general form is therefore just A, but if more control is re- quired, this may be followed by a positive integer. + +The following program is a simple rewrite of a program from Chapter 7. + +PROGRAM ch1206 + + ! + + ! This program reads in and prints out + + ! your first name + + ! IMPLICIT NONE CHARACTER (20) :: First_name + + ! PRINT \*,' Type in your first name.' PRINT \*,' up to 20 characters' READ \*,First_Name PRINT 100,First_Name 100 FORMAT(1x,A) + +! END PROGRAM ch1206 + +# 12.5.1 Headings + +A simple heading is given in the program below: + +program ch1207 implicit none integer :: fluid real :: litres real :: pints PRINT \*,' Pints Litres' do fluid $\mathbf{\omega}=\mathbb{1}$ ,10 litres $=$ fluid / 1.75 pints $=$ fluid \* 1.75 print 100 , pints,fluid,litres 100 format(' ',f7.3,' ',i3,' ',f7.3) end do end program ch1207 + +# Chapter 12 + +# 12.6 Mixed type output in a FORMAT statement + +The following example shows how to mix and match character, integer and real output in one FORMAT statement: + +PROGRAM ch1208 IMPLICIT NONE CHARACTER (LEN $\dot{=}\bar{\perp}5$ ) :: Firstname INTEGER :: age REAL :: weight CHARACTER (LEN $\mathbf{\omega}=\mathbb{1}$ ) :: sex PRINT \*,' Type in your first name ' READ \*,Firstname PRINT \*,' type in your age in years' READ \*,age PRINT \*,' type in your weight in kilos' READ \*,weight PRINT \*,' type in your sex ( $\mathrm{f/m})$ )' READ \*,sex PRINT \*,' your personal details are' PRINT \* PRINT 100 100 FORMAT(4x,'first name', 4x , 'age' , 1x , & 'weight' , $_{2\times}$ , 'sex') PRINT 200 , firstname, age , weight , sex 200 FORMAT(1x , a , 2x , i3 , 2x , f5.2 , 2x, a)END PROGRAM ch1208 + +Take care to match up the variables with the appropriate edit descriptors. You also need to count the number of characters and spaces when lining up the heading. + +# 12.7 Common mistakes + +It must be stressed that an integer can only be printed out with an I format, and a real with an F (or E) format. You cannot use integer variables or expressions with F or E edit descriptors or real variables and expressions with I edit descriptors. If you do, unpredictable results will follow. There are (at least) two other sorts of er- rors you might make in writing out a value. You might try to write out something which has never actually been assigned a value; this is termed an indefinite value. You might find that the letter I is written out. In passing, note that many loaders and link editors will preset all values to zero — i.e., unset (indefinite) values are actually set to zero. On better systems there is generally some way of turning this facility off, so that undefined is really indefinite. More often than not, indefinite values are the result of mistyping rather than of never setting values. It is not un- common to type O for 0, or 1 for either I or l. The other likely error is to try to print out a value greater than the machine can calculate — out of range values. Some machines will print out such values as R, but some will actually print out something which looks right, and such overflow and underflow conditions can go unnoticed. Be wary. + +# 12.8 OPEN (and CLOSE) + +One of the particularly powerful features of Fortran is the way it allows you to ma- nipulate files. Up to now, most of the discussion has centred on reading from and writing to the terminal. It is also possible to read and write to one or more files. This is achieved using the OPEN, WRITE, READ and CLOSE statements. In a later chapter we will consider reading from files but here we will concentrate on writing . + +# 12.8.1 The OPEN statement + +This statement sets up a file for either reading or writing. A typical form is + +OPEN (UNIT $\vDash\vDash$ ,FILE $=$ 'DATA') + +The file will be known to the operating system as DATA (or will have DATA as the first part of its name), and can be written to by using the UNIT number. This statement should come before you first read from or write to the file DATA. + +It is not possible to write to the file DATA directly; it must be referenced through its unit number. Within the Fortran program you write to this file using a statement such as + +WRITE(UNIT $\mathbf{\omega}=\mathbb{1}$ ,FMT $\backprime\!=\!100$ ) XVAL,YVAL + +or + +WRITE(1,100) XVAL,YVAL + +These two statements are equivalent. Besides opening a file, we really ought to CLOSE it when we have finished writing to it: + +CLOSE(UNIT $\scriptstyle{1=1}$ ) + +In fact, on many systems it is not obligatory to OPEN and CLOSE all your files. Almost certainly, the terminal will not require this, since INPUT and OUTPUT units will be there by default. At the end of the job, the system will CLOSE all your files. Nevertheless, explicit OPEN and CLOSE cannot hurt, and the added clarity generally assists in understanding the program. + +The following program contains all of the above statements: + +# Chapter 12 + +program ch1209 implicit none integer :: fluid real :: litres real :: pints open (unit $\mathbf{\omega}=\mathbb{1}$ ,file $=$ 'ch1209.txt') write(unit $\mathbf{\omega}=\mathbb{1}$ ,fmt $=\!2\,0\,0$ ) 200 format(' Pints Litres') do fluid $\mathbf{\Psi}_{\cdot}\!=\!\!\mathbf{1}$ ,10 litres $=$ fluid / 1.75 pints $=$ fluid $\star$ 1.75 write(unit $\mathbf{\omega}=\mathbb{1}$ ,fmt $=\!\perp0\,0$ ) , pints,fluid,litres 100 format(' ',f7.3,' ',i3,' ',f7.3) end do close(1) end program ch1209 + +# 12.8.2 Writing + +PRINT is always directed to the file OUTPUT; in the case of interactive working, this is the terminal. This is not a very flexible arrangement. WRITE allows us to direct output to any file, including OUTPUT. The basic form of the WRITE is + +WRITE(6,100) X,Y,Z + +or + +WRITE(UNIT $=\!6$ ,FMT=100) X,Y,Z + +The latter form is more explicit, but the former is probably the one most widely used. We have an example here of the use of positionally dependent parameters in the first case and equated keywords in the second. With the exceptions of the PRINT statement and the READ \* form of the READ, all of the input/output state- ments allow the unit number and the format labels to be specified either by an equated keyword (or specifier) or in a positionally dependent form. If you use the explicit UNIT $\leftleftarrows$ and $\mathrm{FT}\!=$ it does not matter what order the elements are placed in, but if you omit these keywords, the unit number must come first, followed by the format label. + +UNIT $\mathit{\dot{\Pi}}_{=6}$ means that the output will be written to the file given the unit number 6. In the next chapter we will cover the way in which you may associate file names and unit numbers, but, for the moment, we will assume that the default is being used. The name of the file, as defined by the system, will depend on the particular system you use; a likely name is something like DATA06, TAPE6, or FILE0006. One easy way to find out (apart from asking someone) is to create such a file from a program and then look at the names of your files after the program has finished. A great many of computing's minor complexities can be clarified by simple experi- mentation. + +FMT $\mathord{=}100$ simply gives the label of the format to be used. The overworked asterisk may be used, either for the unit or for the format: + +UNIT $\mathbf{\dot{\rho}}=^{*}\mathbf{\rho}$ will write to OUTPUT (the terminal) FMT $\mathbf{\dot{\omega}}=^{*}$ will produce output controlled by the list of variables, often called list directed output. + +The following three statements are therefore equivalent: + +WRITE(UNIT $=\star$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) X,Y,Z WRITE(\*,\*) X,Y,Z PRINT\*,X,Y,Z + +There are other controls possible on the WRITE, which will be elaborated later. + +# 12.9 Repetition + +Often we need to print more than one number on a line and want to use the same layout descriptor. Consider the following: + +PRINT 100,A,B,C,D + +If each number can be written with the same layout descriptor, we can abbreviate the FORMAT statement to take account of the pattern: + +100 FORMAT(1X,4F8.2) + +is equivalent to + +as you might anticipate. If the pattern is more complex, we can extend this ap- proach: + +PRINT 100,I,A,J,B,K,C 100 FORMAT(1X,3(I3,F8.2)) + +Bracketing the description ensures that we repeat the whole entity: + +is equivalent to + +100 FORMAT(1X,I3,F8.2, I3,F8.2, I3,F8.2) + +Repetition with brackets can be rather more complex. In order to give some over- view of formatted Fortran output, it is helpful to delve a little into the history of the language. Many of the attributes of Fortran can be traced back to the days of single-user mainframes (with often a fraction of the power of many contemporary microcomputers and workstations). These would generally take input from punched cards (the traditional 80-column Hollerith card), and would generate output on a line printer. In this sort of environment, the individual punched card had a signifi- cance which lines in a file do not have today. Each card could be seen as a single entity — a physical record unit. The record was seen as an element of a subdivi- sion within a file. Even then, there was some confusion between the notion of physical records and files split into logically distinct subunits, since these subunits might also be termed records. The present Fortran standard merely says that a re- cord does not necessarily correspond to a physical entity , although a punched card is usually considered to be a record. This leaves us sitting at our terminals in a be- mused state, especially since we may have no idea what a punched card looks like (an ideal state of affairs!). + +It is important to have some notion of a record, since most of the formal defini- tions dealing with output (and input) are couched in terms of records. Every time an input or output statement is executed your nominal position in the file changes. If we think in terms of individual records (which may be cards), the notions of cur- rent, preceding and next record seem fairly straightforward. The current record is simply the one we have just read or written, and the other definitions follow natu- rally. + +The situation becomes less clear when we realise that a single output statement may generate many lines of output: + +WRITE(UNIT $=6$ , $\scriptstyle{\mathrm{FMT}=101}$ ) A,B,C 101 FORMAT(1X,F10.4) + +writes out three separate lines. Looking at the output alone, there is no way to dis- tinguish this from the output generated by + +WRITE(UNIT $^{1\!}{=}\,6$ , $\scriptstyle{\mathrm{FMT}=101}$ ) A WRITE(UNIT $=6$ , $\scriptstyle{\mathrm{FMT}=101}$ ) B WRITE(UNIT $^{1\!}{=}\,6$ , $\scriptstyle{\mathrm{FMT}=101}$ ) C 101 FORMAT(1X,F10.4) + +In the latter case we would probably be happy to consider each line a record, al- though in the previous example we might swither between considering all three lines (generated by a single statement) a single record or three records. Consider the first of these two examples more closely; each time the format is exhausted — that is to say, each time we run out of format description, we start again on a new line (a new record). A new record is begun as each F10.4 is begun. The correct in- terpretation is therefore that three records have been written. + +The same sort of thing happens in more complex FORMAT statements: + +WRITE(UNIT $=\!6$ ,FM $\scriptstyle{\mathrm{T}=105}$ ) X,I,Y 105 FORMAT(1X,F8.4,I3,(F8.4)) + +would write out a single record containing a real, an integer and a real. Using the same format statement with WRITE ( UNIT $\mathord{:=}6$ , FMT=105 ) X,I,Y,Z would write out two records. The first containing the values of X, I and Y and the second con- taining only Z. If there were still more values + +WRITE(UNIT $=\!6$ , $\mathrm{FMT}{=}105$ ) X,I,Y,Z,A + +would print out three records. The group in brackets — the (F8.4) — is repeated until we run out of items. + +# 12.10 Some more examples + +Since it is the last open bracket which determines the position at which the format is repeated, simply writing + +WRITE(UNIT $=\!6$ ,FMT $\scriptstyle1=100$ ) A,I,B,C,J 100 FORMAT(1X,F8.4,I3,F8.2) + +would imply that A, I and B would be written on one line then, returning to the last open brackets (in this case the only open brackets), a new record (or line) is begun to write out C and J. A statement like + +would return to the (F8.4) group, and then continue to the I3 and F8.2 before re- peating again (if necessary). The same thing happens if the (F8.4) had no brackets around it. On the other hand + +contains superfluous brackets around the F8.4, since the repeat statement will never return to that group. Are you confused yet? This all seems very esoteric, and re- ally, we have only hinted at the complexity which is possible. It is seldom that you have to create complex FORMAT statements, and clarity is far more important than brevity. + +When patterned or repeated output is used, we may want to stop when there are no more numbers to write out. Take the following example: + +WRITE(UNIT $\mathbf{\omega}=\mathbb{1}$ , $\scriptstyle{\mathrm{FMT}=100}$ ) A,B,C,D 100 FORMAT(1X,4(F6.1,',')) + +This will give output which looks like + +37.4, 29.4, 14.2, -9.1, + +The last comma should not be there. We can suppress these unwanted elements by using the colon: + +which would then give us + +Since we run out of data at the fourth item, D, the output following is not written out. It is a small point, but it does look a lot tidier. There are other ways of achiev- ing the same thing. + +This helps to illustrate another point, namely that you may have formats which are more extensive than the lists which reference them: + +WRITE(UNIT $\mathbf{\omega}=\mathbb{1}$ ,F $\scriptstyle{\mathrm{MT}=100}$ ) A,B,C WRITE(UNIT $\mathbf{\omega}=\mathbb{1}$ , $\scriptstyle{\mathrm{FMT}=100}$ ) X,Y 100 FORMAT(1X,6F8.2) + +Both WRITE statements use the format provided, although they write out different numbers of data, and neither uses up the whole format. + +# 12.11 Implied DO loops and array sections for array output + +The following program shows how to use both implied DO loops and array sec- tions to output an array in a neat fashion: + +program ch1210 implicit none integer , parameter :: nrow ${\tt=}5$ integer , parameter :: ncol $=\!6$ real , dimension(1:nrow\*ncol) :: results $=\quad\&$ (/50 , 47 , 28 , 89 , 30 , 46 , & 37 , 67 , 34 , 65 , 68 , 98 , & 25 , 45 , 26 , 48 , 10 , 36 , & 89 , 56 , 33 , 45 , 30 , 65 , & 68 , 78 , 38 , 76 , 98 , 65/) + +REAL , DIMENSION(1:nrow,1:ncol) :: Exam_Results=0.0 real , dimension(1:nrow) :: People_average = 0.0 real , dimension(1:ncol) :: Subject_Average $=$ 0.0 integer :: r,c exam_results $=\quad\&$ reshape(results,(/nrow,ncol/),(/0.0,0.0/),(/2,1/)) Exam_Results(1:nrow,3) $=$ 2.5 \* Exam_Results(1:nrow,3) subject_average $=$ sum(exam_results,dim $\mathbf{\Psi}_{:=1}$ ) people_average $=$ sum(exam_results,dim $_{1}{=}2$ ) people_average $=$ people_average / ncol subject_average $=$ subject_average / nrow do $\scriptstyle\mathtt{T}=1$ ,nrow print 100 , (exam_results(r,c), $\scriptstyle C=\perp$ ,ncol) ,& people_average(r) 100 format(1x,6( $\updownarrow\mathrm{x}$ ,f5.1),' = ',f6.2) end do print \*,' ==== ==== ==== ==== ==== ====' print 110, subject_average(1:ncol) 110 format(1x,6(1x,f5.1))end program ch1210 + +The print 100 uses an implied DO loop and the print 110 uses an array section. Take care when using whole arrays. Consider the following program: + +PROGRAM ch1211 REAL , DIMENSION(10,10) :: Y INTEGER :: $\mathrm{\ensuremath{NR O W S}}\!=\!6$ INTEGER :: NCOLS $=7$ INTEGER :: I,J INTEGER :: $\mathtt{K}\!=\!0$ + +DO $\mathbb{T}\!=\!1$ ,NROWS DO $\uptau\!\!=\!\!1$ ,NCOLS $\mathbb{K}{=}\mathbb{K}{+}\mathbb{1}$ $\mathtt{Y}\left(\,\mathtt{I}\,,\,\mathtt{J}\,\right)=\mathtt{K}$ END DO END DO WRITE(UNIT $=\star$ ,FMT=100)Y + +# Chapter 12 + +100 FORMAT(1X,10F10.4) + +END PROGRAM ch1211 + +There are several points to note with this example. Firstly, this is a whole array reference, and so the entire contents of the array will be written; there is no scope for fine control. Secondly, the order in which the array elements are written is ac- cording to Fortran's array element ordering, i.e., the first subscript varying 1 to 10 (the array bound), with the second subscript as 1, then 1 to 10 with the second sub- script as 2 and so on; the sequence is + +![](images/6bd565925b04f1520079a91af02fc37f70de77e89c1e84e2342875be80ce18be.jpg) + +Thirdly we have defined values for part of the array. This program behaves differ- ently with the following compilers: + +• Sun Fortran 90. • NagWare F95. • Compaq F95. + +If you have access to more than one compiler then try out this example. + +# 12.12 Formatting for a line printer + +There is one extension to format specifications which is relevant to line printers. Fortran defines four special characters which have an effect on standard line print- ers when they occur in the first character position of a line. This means that a lineprinter which is not under your immediate control can be used to produce neat output by sending a file to be printed on it. This has a variety of names including, spooling , queueing and routing depending on the system. You should check with your local system for the exact mechanism to achieve this. + +The special characters are $+,\;0,\;1$ and blank. To be used, they must be the first character of the output in each line — as if they were to be printed in column 1. In fact, a standard line printer never prints a character that occurs in column 1 at all. + +Whenever a WRITE statement is begun, the printer advances to a new record; i.e., a new line is begun before any data are transferred. If the first character is a spe- cial character, then this will be interpreted by the line printer. If the first character to be printed is a blank, the printer continues printing on that line. The first charac- ter is also known as the carriage control character. + +The blank is a do nothing special control. It signifies that the line is to be printed as it is. + +The zero indicates that you wish to leave an extra line; this is often useful in spac- ing out results to make the output more readable. + +The 1 makes the output skip down to the top of the next page. This is clearly use- ful for separating logically distinct chunks of output. If you obtain a line printer listing of your compiled program, each segment will start at the top of a new page. + +The plus is a no advance or overprint character. It suppresses the effect of the line advance which a WRITE generates. No new line is begun and the previous line is overprinted with the new. Overprinting can be useful especially when you wish to print out grey scale maps but its use is rather restricted. In particular, it can be a dangerous control character. If you have a format starting with a plus in a loop, you can make the printer overprint again and again and again . . . and again and again, until it has hammered itself into a pulp. This is not a good idea. + +Similarly, accidental use of the 1 as a control character in a loop will give you lots of blank pages. It is just a bit embarrassing to be presented with a 6 inch stack of paper which is (almost) blank, because you had a 1 repeatedly in column 1. + +# 12.12.1 Mechanics of carriage control + +The following are all quite reasonable ways of generating the blank in column 1: + +WRITE(UNIT $=\!6$ ,FMT $\scriptstyle1\,=\,\pm\,0\,0$ )A 100 FORMAT(' ',F10.4) + +or WRITE(UNIT $=\!6$ , $\scriptstyle\mp\mathrm{{MT}}=\pm\,0\,0$ )A 100 FORMAT(1X,F10.4) or WRITE(UNIT $=\!6$ ,FMT $\backprime{=}100$ )A 100 FORMAT(' THE ANSWER IS ',F10.4) Note, however, that WRITE(UNIT $=\!6$ , $\scriptstyle{\mathrm{FMT}=100}$ )A 100 FORMAT(F8.4) + +could result in problems. If A contained the value 100.2934, the result on a line printer would be + +00.2934 + +printed at the top of a new page. The 1 is taken as carriage control, and the rest of the line then printed. + +Accidentally printing zeros in column 1 is a little more difficult, but + +WRITE(UNIT $=\!6$ , $\mathrm{FMT}{=}100$ )I 100 FORMAT(I1) + +might just do it. Don't. + +Remember that this only applies to line printer output, and not to the terminal. Since Fortran only defines four characters as carriage control, you will find that anything else in column 1 will give unpredictable results. On some systems, a fair number of alternatives may be defined by the installation, and they may do some- thing useful. On other systems, they may do something, but they may also fail to print the rest of the line. This can be very perplexing. Beware. + +# 12.12.2 Generating a new line on both line printers and terminals + +There are several ways of generating new lines, other than with a 0 in column 1 of your line printer output. A more general approach, which works on both terminals and line printers, is through the oblique or slash, /. Each time this is encountered in a FORMAT statement, a new line is begun. + +PRINT 101,A,B 101 FORMAT(1X,F10.4/1X,F10.4) + +would give output like + +100.2317 -4.0021 + +This is the same as (F10.4) would have given, but clearly it opens up lots of possi- bilities for formatting output more tidily: + +PRINT 102,NVAL,XMAX,XMIN 102 FORMAT(' NUMBER OF VALUES READ IN WAS: ',I10/ & ' MAXIMUM VALUE IS: ',F10.4/ & ' MINIMUM VALUE IS: ',F10.4) + +may be easier to read than using only one line, and it is certainly more compact to write than using three separate print statements. It is not necessary to separate / by commas, although if you do nothing catastrophic will happen. + +You may also begin a format description with a /, in order to generate an extra line or even generate lots of lines with lots of slashes; e.g., + +WRITE(UNIT $=\!6$ ,FMT=103)A,B 103 FORMAT(//1X,F10.4,4(/),1X,F10.4) + +will leave two lines before printing A, and then will generate four new lines before writing B (i.e., there will be three lines between A and B — the fourth new line will contain B). While a slash by itself, or with another slash, does not have to be separated by commas from other groups, a more complex grouping, $4(/)$ , does have to have commas and brackets to delimit it. + +# 12.13 Timing of writing formatted files + +The following example looks at the amount of time spent in different sections of a program with the main emphasis on formatted output: + +program ch1212 implicit none integer , parameter :: n=1000000 integer , dimension(1:n) :: x real , dimension(1:n) :: y integer :: i real :: t,t1,t2,t3,t4,t5 character $^{\star}\!\perp\!0$ :: comment open(unit $=\!\perp0$ ,file $=$ 'ch1212.txt') call cpu_time(t) $\scriptstyle{\mathbb{C}}\,\mathtt{I}\,=\,\mathtt{C}$ comment $="$ Intial ' print 100,comment,t1 do $\dot{\bot}\!=\!\bot$ ,n $\textsf{x}(\dot{\bot})=\dot{\bot}$ end do call cpu_time(t) $\scriptstyle\pm2=\pm-\pm1$ comment $\c=\qquad^{!}$ Integer ' print 100,comment,t2 y=real(x) call cpu_time(t) t3=t-t1-t2 + +comment $=$ ' real ' print 100,comment,t2 do $\dot{\bot}\!=\!\bot$ ,n write(10,200) x(i) 200 format(1x,i10)end do call cpu_time(t) $\mathsf{t}\,\mathsf{4}\!=\!\mathsf{t}\!-\!\mathsf{t}\,\mathsf{1}\!-\!\mathsf{t}\,\mathsf{2}\!-\!\mathsf{t}\,\mathsf{3}$ comment $=$ ' i write ' print 100,comment,t4 do $\scriptstyle{\dot{\bot}}\,=\,\!1$ ,n write(10,300) y(i)300 format(1x,f10.0)end do call cpu_time(t) $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\$ comment $=$ ' r write ' print 100,comment,t5 100 format(1x,a, $_{2\times}$ ,f7.3)end program ch1212 + +There is a call to the built-in intrinsic cpu_time to obtain timing information. Timing details for a number of compilers follow: + +Intel Nag Salford Intial 0.063 0.046 0.094 Integer 0.031 0.031 0.016 real 0.031 0.031 0.016 i write 10.109 16.968 2.453 r write 12.281 101.860 3.453 + +Formatted output takes up a lot of time, as we are converting from an internal bi- nary representation to an external decimal form. + +# 12.14 Timing of writing unformatted files + +The following program is a variant of the above but now the output is in unformat- ted or binary form: + +program ch1213 implicit none integer , parameter :: n=1000000 integer , dimension(1:n) :: x real , dimension(1:n) :: y + +integer :: i real :: t,t1,t2,t3,t4,t5 character $^{\star}\!\perp\!0$ :: comment open(unit $=\!\perp0$ ,file $=$ 'ch1213.txt',form $\b=$ 'unformatted') call cpu_time(t) $\scriptstyle{\mathbb{C}}\,\mathtt{I}\,=\,\mathtt{C}$ comment $="$ Intial ' print 100,comment,t1 do $\dot{\bot}\!=\!\bot$ ,n $\textsf{x}(\dot{\bot})=\dot{\bot}$ end do call cpu_time(t) $\scriptstyle\pm2=\pm-\pm1$ comment $=$ ' Integer ' print 100,comment,t2 $\mathtt{Y}^{=}$ real $(\mathbf{\nabla}\mathbf{x})$ call cpu_time(t) t3=t-t1-t2 comment $=$ ' real ' print 100,comment,t2 write(10) x call cpu_time(t) t4=t-t1-t2-t3 comment $=$ ' i write ' print 100,comment,t4 write(10) y call cpu_time(t) t5=t-t1-t2-t3-t4 comment $=$ ' r write ' print 100,comment,t5 100 format(1x,a, $_{2\times}$ ,f7.3)end program ch1213 + +Timing details for a number of compilers follows: + +Intel Nag Salford Intial 0.063 0.062 0.172 Integer 0.031 0.030 0.031 real 0.031 0.030 0.031 i write 0.031 0.064 0.063 r write 0.031 0.062 0.063 + +# Chapter 12 + +Unformatted is very efficient in terms of time. It also has the benefit for real or floating point numbers of no information loss. + +In Chapter 13 we will look at timing information reading in formatted and unfor- matted files. + +# 12.15 Summary + +You have been introduced in this chapter to the use of format or layout descriptors which will give you greater control over output. + +The main features are: + +• The I format for integer variables. • The E and F formats for real numbers. • The A format for characters. • The X, which allows insertion of spaces. + +Output can be directed to files as well as to the terminal through the WRITE state- ment. + +The WRITE, together with the OPEN and CLOSE statements, also introduces the class of Fortran statements which use equated keywords, as well as positionally de- pendent parameters. + +The FORMAT statement and its associated layout or edit descriptor are powerful and allow repetition of patterns of output (both explicitly and implicitly). + +When output is to be directed to a line printer, the following four characters: + +• + • 0 • 1 • (blank) + +allow reasonable control over the layout. Care must to be taken with these charac- ters, since it is possible to decimate forests with little effort. + +# 12.16 Problems + +1. Rewrite the temperature conversion program which was Problem 8 in Chapter 10 to actually produce the output shown. + +2. Write a litres and pints conversion program to produce a similar kind of output to problem one above. Start at 0 and make the central column go up to 50. One pint is 0.568 litres. + +3. Information on car fuel consumption is usually given in miles per gallon in Brit- ain and the United States and in litres per 100 kilometres in Europe. Just to add an extra problem US gallons are 0.8 imperial gallons. + +Prepare a table which allows conversion from either US or imperial fuel consump- tion figures to the metric equivalent. Use the PARAMETER statement where appropriate: + +1 imperial gallon $=4.54596$ litres 1 mile = 1.60934 kilometres + +4. The two most commonly used operating systems for Fortran programming are UNIX and DOS. It is possible to use the operating system file redirection symbols $<$ and $>$ to read from a file and write to a file, respectively. Rerun the program in problem 1 to write to a file. Examine the file using an editor. + +5. Modify any of the above to write to a file rather than the terminal. What changes are required to produce a general output which will be suitable for both the terminal and a line printer? Is this degree of generality worthwhile? + +6. To demonstrate your familiarity with formats, reformat problems 1, 2 or 3 to use E formats, rather than F (or vice versa). + +7. Modify the temperature conversion program to produce output suitable for a line printer. Use the local operating system commands to send the file to be printed. + +8. Repeat for the litres and pints program. + +9. What features of Fortran reveal its evolution from punched card input? + +10. Try to create a real number greater than the maximum possible on your com- puter — write it out. Try to repeat this for an integer. You may have to exercise some ingenuity. + +11. Check what a number too large for the output format will be printed as on your local system — is it all asterisks? + +12. Write a program which stores litres and corresponding pints in arrays. You should now be able to control the output of the table (excluding headings — al- though this could be done too) in a single WRITE or PRINT statement. If you don't like litres and pints, try some other conversion (£ sterling to US dollars, leagues to fathoms, Scots miles to Betelgeusian pfnings). The principle remains the same. + +13. Fortran is an old programming language and the text formatting functionality discussed in this chapter assumes very dumb printing devices. + +The primary assumption is that we are dealing with so-called monospace fonts, i.e., that digits, alphabetic characters, punctuation, etc., all have the same width. + +If you are using a PC try using: + +• Notepad + +and + +• Word + +to open your programs and some of the files created in this chapter. What happens to the layout? + +If you are using Notepad look at the Word wrap and set Font options under the edit menu. + +What fonts are available? What happens to the layout when you choose another font? + +If you are using Word what fonts are available? What happens when you make changes to your file and exit Word? Is it sensible to save a Fortran source file as a Word document? + +# Reading in Data + +“Winnie-the-Pooh read the two notices very carefully, first from left to right, and afterwards, in case he had missed some of it, from right to left.” + +A A Milne, Winnie-the-Pooh “For Madmen Only” + +Hermann Hesse, Steppenwolf + +# Aims + +The aims of this chapter are to introduce some of the ideas involved in reading data into a program. In particular, using the following: + +• Reading from fixed fields. • Integers, reals and characters. • Blanks — nulls or zeros? • READ — extensions. • error handling on input. • OPEN — associating unit numbers and file names. • CLOSE • REWIND • BACKSPACE + +# Chapter 13 + +# 13 Reading in Data + +# 13.1 Reading from the terminal or keyboard versus reading from files + +It is unlikely that you would use fixed formats when reading numeric input from the terminal or keyboard; they are more likely to be used when reading data from a file. However the examples that follow do it. We look at reading from files later in this chapter. + +# 13.2 Fixed fields on input + +All the formats described earlier are available, and again they are limited to partic- ular types. Integers may only be input by the I format, reals with F and E, and character (alphanumeric) with A. + +# 13.2.1 Integers and the I format + +Integers are read in with the I edit descriptor. Whereas, on output, integers appear right justified, on input they may appear anywhere in the field you have delimited. Blanks (by default) are considered not to exist for the purpose of the value read, al- though they do contribute to the field width. Apart from the digits 0 to 9, the only other characters which may appear in an integer field are – and $^+$ . + +Consider the following 12 times table: + +![](images/aca14be0a64a0fc792255ddac07bef80ff1f9391ffce47576cf4f02766a62f13.jpg) + +The following is a program to read the first and last columns of integer data: + +program ch1301 implicit none integer , parameter :: $\mathtt{n}\!\!=\!\!12$ integer :: i + +integer , dimension(1:n) :: x integer , dimension(1:n) :: y do $\dot{\bot}\!=\!\bot$ ,n read 100,x(i),y(i) 100 format( $_{2\times}$ ,i2,9x,i3) print 200,x(i),y(i)200 format(1x,i3,2x,i3) end do end program ch1301 + +The + +read 100,x(i),y(i) + +will try reading values into $\mathrm{x(i)}$ and y(i) using format statement + +100 format(2x,i2,9x,i3) + +which will skip the first two characters on the line or record, read the first value from the next two columns, skip the next nine characters and read the last value from the next three characters. + +We recommend that when working with formatted files you to use a text editor that displays the column and line details + +Notepad under Windows has a status bar option under the View menu. Gvim under Windows has line and column information available. Under Redhat, vim and gedit both display line and column information. User SuSe, kedit and vim display line and column information. There should be an editor available on your system that has this option. + +# 13.2.2 Reals and the F format + +Real numbers may be input using a variety of formats and we will look at the F fomat in this example. Consider the following BMI data: + +1.85 85 + + 1.80 76 + + 1.85 85 + + 1.70 90 + + 1.75 69 + + 1.67 83 + + 1.55 64 + + 1.63 57 + + 1.79 65 + + 1.78 76 + +# Chapter 13 + +The following program will read in the data: + +program ch1302 implicit none integer , parameter :: $\mathtt{n}\!=\!\mathtt{10}$ real , dimension(1:n) :: h real , dimension(1:n) :: w real , dimension(1:n) :: bmi integer :: i do $\dot{\bot}\!=\!\bot$ ,n read 100, h(i),w(i) 100 format(f4.2, $_{2\times}$ ,f3.0) end do bmi $\mathtt{=}\mathtt{W}$ / $(\mathtt{h}^{\star}\mathtt{h})$ ) do $\dot{\bot}\!=\!\bot$ ,n print 200,bmi(i) 200 format( $_{2\times}$ ,f5.0) end do end program ch1302 + +To read in the heights we need a total width of four columns with two after the decimal point. We then skip two spaces and read in the weights. The data in the file do not have a decimal point! + +# 13.2.3 Reals and the E Format + +An exponential format number (which may be read in $\mathrm{F}$ or E formats) can take a number of different forms. The most obvious is the explicit form + +-1.2E-4 + +where all the components of the value are present — the significant digits to the left of the E, the E itself, and the exponent to the right. We can drop (almost) any two of these three components, so: + +-1.2 + + -1.2E + + -1.2-4 + + -4 + +are all valid values. Only the first two are interpreted as the same numerical value, and just giving the exponent part would be interpreted by the format as giving only the significant digits. If the exponent is to be given, there must be some significant digits as well. It is not even enough to give the E and assume that the program will interpret this as 10 to the power exponent . + +E-4 + +is not an acceptable exponential format value, although + +1E-4 + +would be. + +There are opportunities for confusion with E formats. + +READ(UNIT $^{\ast}{=}^{\ast}$ ,FMT $\vDash Q$ ) X,Y 102 FORMAT(2E10.3) + +with: + +10.23 -2 + +would be interpreted as X taking the value 10.23E-2 and Y taking the value 0.0, while with + +102 FORMAT(2F8.3) + +X would be 10.23, and Y would be $-2.0$ . + +Although the decimal point may also be dropped, this might generate confusion as well. While + +4E3 45 45E-4 45-4 + +are all valid forms, if an E format is used, a special conversion takes place. A for- mat like E10.8, when used with integral significant digits (no decimal point), uses the 8 as a negative power of 10 scaling e.g.' + +3267E05 + +converts to + +$$ +3267^{\star}\underline{{1}}0^{\star\star}8^{\star}\underline{{1}}0^{\star\star}5 +$$ + +or + +3267\*10\*\*3 + +or + +![3.267 Therefore, the interpretation of, say, 136, read in E format, would depend on the format used: ](images/effe13f58d43c66ca6511d2c29529c7c95fa6cf18d0d6e614d68dbe0b4a725ac.jpg) + +One implication of all this is that the format you use to input a variable may not be suitable to output that same variable. So given the data: + +136 136 136 136 + + 136. + + 136. + + 136. + + 136. + +and the program + +program ch1303 implicit none real :: x read 100,x 100 format(e10.0) print \*,x read 200,x 200 format(e10.4) print \*,x read 300,x 300 format(e10.10) print \*,x read \*,x + +print \*,x read 100,x print \*,x read 200,x print \*,x read 300,x print \*,x read \*,x print \*,x end program ch1303 + +We get the following output when the program is compiled with the Intel compiler: + +136.0000 1.3600000E-02 1.3600000E-08 136.0000 136.0000 136.0000 136.0000 136.0000 + +Other compilers may give slightly different formatting of the output. + +# 13.3 Blanks, nulls and zeros + +You can control how Fortran treats blanks in input through two special format in- structions, BN and BZ. BN is a shorthand form of blanks become null, that is, a blank is treated as if it were not there at all. BZ is therefore blanks become zeros. + +As we have already seen, 1 4 (i.e., the two digits separated by a blank) read in I3 format would be read as 14; similarly, 14 (one-four-blank) is also 14 when the BN format is in operation. All of the blanks are ignored for the purposes of interpreting the number. They help to create the width of the number, but otherwise contribute nothing. This is the default, which will be in operation unless you specify other- wise. + +The BZ descriptor turns blanks into zeros. Thus, 1 4 (one-blank-four) read in I3 format is 104, and 14 (one-four-blank) is 140. + +There is one place where we must be very careful with the use of the BZ format + + — when using exponent format input. Consider + +5.321E+02 + +# Chapter 13 + +read in (BZ,E10.3) format. We have specified a field which is ten characters wide; therefore the blank in column 10, which follows the $_{\mathrm{E+02}}$ , is read as a zero, mak- ing this $_{\mathrm{E+020}}$ . This is probably not what was required. + +# 13.4 Characters + +When characters are read in, it is sufficient to use the A format, with no explicit mention of the size of the character string, since this size (or length) is determined in the program by the CHARACTER declaration. This implies that any extra char- acters would not be read in. You may however read in less: + +CHARACTER (10) :: LIST . . READ(UNIT ${\tt I}\!=\!5$ ,FMT $\scriptstyle1\equiv100$ )LIST 100 FORMAT(A1) + +would read only the first character of the input. The remaining nine characters of LIST would be set to blank. + +The notion of blanks as nulls or zeros has no meaning for characters. The blank is a legitimate character and is treated as meaningful, completely distinct from the notion of a null or a zero. + +A simple variant on ch1301 which uses the character variable temp to hold the text between the two numbers appears below: + +program ch1304 implicit none integer , parameter :: $\mathtt{n}\!\!=\!\!12$ integer :: i integer , dimension(1:n) :: x integer , dimension(1:n) :: y character $^{\star\,9}$ :: temp do $\dot{\bot}\!=\!\bot$ ,n read 100,x(i),temp,y(i) 100 format( $_{2\times}$ ,i2,a,i3) print 200,x(i),y(i)200 format(1x,i3, $_{2\times}$ ,i3) end do end program ch1304 + +Note that in the format statement we just use the A edit descriptor and the number of characters to read is picked up from the variable declaration. + +# Reading in Data + +# 13.5 Skipping spaces and lines + +The X format is also useful for input. There may be fields in your data which you do not wish to read. These are easily omitted by the X format: + +READ(UNIT $^{\ast}{=}^{\ast}$ ,FMT $\scriptstyle1\equiv100$ ) A,B 100 FORMAT(F10.4,10X,F8.3) + +Similarly, you can jump over or ignore entire records by using the oblique. Do note, however, that + +READ(UNIT $^{\ast}{=}^{\ast}$ ,FMT $=$ 100) A,B 100 FORMAT(F10.4/F10.4) + +would read A from one line (or record) and B from the next. To omit a record be- tween A and B, the format would need to be + +100 FORMAT(F10.4//F10.4) + +Another way to skip over a record is + +READ(UNIT $=\star$ , $\scriptstyle{\mathrm{FMT}=100}$ ) 100 FORMAT() + +with no variable name at all. + +# 13.6 Reading + +As you have already seen, reading, or the input of information, is accomplished through the READ statement. We have used + +READ \*,X,Y + +for list directed input from the terminal, and + +READ(UNIT $^{\ast}{=}^{\ast}$ ,FMT $\acute{=}$ 100) X,Y + +for formatted input from the terminal. These forms may be expanded to + +READ(UNIT $^{\ast}{=}^{\ast}$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) X,Y + +READ(UNIT $^{\ast}{=}^{\ast}$ ,FMT=100) X,Y + +for input from the terminal, or to + +# Chapter 13 + +READ(UNIT ${\tt I}\!=\!5$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) X,Y + +or + +READ(UNIT ${\tt I}\!=\!5$ ,FMT=100) X,Y + +when we wish to associate the READ statement with a particular unit number (or format label, for formatted input). As with the WRITE statement, these last two READ statements may be abbreviated to + +READ(5,\*) X,Y + +and + +READ(5,100) X,Y + +# 13.7 File manipulation again + +The OPEN and CLOSE statements are also relevant to files which are used as in- put, and they may be used in the same ways. Besides introducing the notion of manipulating lots of files, the OPEN statement allows you to change the default for the treatment of blanks. The default is to treat blanks as null, but the statement BLANK $=^{1}$ 'ZERO' changes the default to treat blanks as zeros. There are other pa- rameters on the OPEN, which are considered elsewhere. + +Once you have OPENed a file, you may not issue another OPEN for the same file until it has been CLOSEd, except in the case of the BLANK $=$ parameter. You may change the default back again with + +OPEN(UNIT $\mathrm{\Sigma=}\beth0$ ,FILE $=$ 'Example.dat') READ(UNIT $\mathrm{\Sigma=}\beth0$ ,FMT $\scriptstyle1=100$ ) A,B ... OPEN(UNIT $\mathrm{\Sigma=}\beth0$ ,FILE $\mathbf{\varepsilon}=\mathbf{\varepsilon}^{\mathsf{I}}$ 'Example.dat',BLANK $=$ 'ZERO') READ(UNIT $\mathrm{\Sigma=}\beth0$ ,FMT $\scriptstyle1=100$ ) A,B + +This implies that, within the same input file, you may treat some records as blank for null, and some as blank for zero. This sounds very dangerous, and is better done by manipulating individual formats if it has to be done at all. + +Given that you may write a file, you may also rewind it, in order to get back to the beginning. The syntax is similar to the other commands: + +REWIND(UNIT $\vDash\vDash$ ) + +This often comes in useful as a way of providing backing storage, where interme- diate data can be stored on file and then used later in the processing. + +The notion of records in Fortran input and output has been introduced. If you are confident in your understanding of this ambiguous and nebulous concept, you can backspace through a file, using the statement + +BACKSPACE(UNIT $\vDash\vDash$ ) + +which moves back over a single record on the designated file. There is no point in trying to BACKSPACE or REWIND if the input is from the keyboard or terminal. + +# 13.8 Reading using array sections + +Consider the following output: + +50.0 47.0 70.0 89.0 30.0 46.0 = 55.33 37.0 67.0 85.0 65.0 68.0 98.0 = 70.00 25.0 45.0 65.0 48.0 10.0 36.0 = 38.17 89.0 56.0 82.5 45.0 30.0 65.0 = 61.25 68.0 78.0 95.0 76.0 98.0 65.0 = 80.00 ==== ==== ==== ==== ==== ==== 53.8 58.6 79.5 64.6 47.2 62.0 + +A program to read this file using array sections is as follows: + +program ch1305 implicit none integer , parameter :: nrow ${\tt=}5$ integer , parameter :: ncol $=\!6$ REAL , DIMENSION(1:nrow,1:ncol) :: Exam_Results=0.0 real , dimension(1:nrow) :: People_average = 0.0 real , dimension(1:ncol) :: Subject_Average $=$ 0.0 integer :: r,c do $\scriptstyle\mathtt{T}=1$ ,nrow read 100,(exam_results(r,1:ncol)),people_average(r) 100 format(1x,6(1x,f5.1),4x,f6.2)end do read \* read 110, subject_average(1:ncol) 110 format(1x,6(1x,f5.1))do $\scriptstyle\mathtt{T}=1$ ,nrow print 200,(exam_results(r,c), $_{C}{=}1$ ,ncol),people_average(r) + +# Chapter 13 + +200 format(1x,6(1x,f5.1),' = ',f6.2) end do print \*,' ==== ==== ==== ==== ==== ====' print 210, subject_average(1:ncol) 210 format(1x,6(1x,f5.1))end program ch1305 + +Note also the use of + +read \* + +to skip a line. + +If you are on a UNIX or Linux system use diff to compare the input and output files. They should be the same. + +# 13.9 Timing of reading formatted files + +A program to read a formatted file is shown below: + +program ch1306 implicit none integer , parameter :: n=1000000 integer , dimension(1:n) :: x real , dimension(1:n) :: y integer :: i real :: t,t1,t2,t3,t4,t5 character $^{\star}\!\perp\!0$ :: comment open(unit $=\!\perp0$ ,file $=$ 'ch1306.txt') call cpu_time(t) $\scriptstyle{\mathbb{C}}\,\mathtt{I}\,=\,\mathtt{C}$ comment $="$ ' Intial ' print 100,comment,t1 do $\dot{\bot}\!=\!\bot$ ,n read(10,200) x(i)200 format( $_{1\times}$ ,i10) end do call cpu_time(t) $\scriptstyle\pm2=\pm-\pm1$ comment $\begin{array}{r l r}{\mathbf{\Sigma}}&{{}=}&{\mathbf{\Sigma}^{\prime}\quad\dot{\mathbf{\Sigma}}}\end{array}$ read ' print 100,comment,t2 do $\scriptstyle{\dot{\bot}}\,=\,\!1$ ,n read(10,300) y(i)300 format(1x,f10.0)end do + +call cpu_time(t) $\mathsf{t}\,3\!=\!\mathsf{t}-\mathsf{t}\,2-\mathsf{t}\,2$ comment $=$ ' r read ' print 100,comment,t3 100 format(1x,a, $_{2\times}$ ,f7.3)do $\scriptstyle{\dot{\bot}}\,=\,\!1$ ,10 print \*,x(i), ' ' , y(i)end do end program ch1306 + +Some timing data from the Intel compiler follows: + +Intial 0.063 i read 1.922 r read 1.828 1 1.000000 2 2.000000 3 3.000000 4 4.000000 5 5.000000 6 6.000000 7 7.000000 8 8.000000 9 9.000000 10 10.00000 + +# 13.10 Timing of reading unformatted files + +The following is a program to read from an unformatted file: + +program ch1307 implicit none integer , parameter :: n=1000000 integer , dimension( $\perp\,:\!\mathrm{n}$ ) :: x real , dimension(1:n) :: y integer :: i real :: t,t1,t2,t3,t4,t5 character $^{\star}\!\perp\!0$ :: comment open(unit $=\!\perp0$ ,file $=$ 'ch1307.txt',form $\b=$ 'unformatted') call cpu_time(t) $\scriptstyle{\mathbb{C}}\,\mathtt{I}\,=\,\mathtt{C}$ comment $="$ Intial ' print 100,comment,t1 read(10) x + +# Chapter 13 + +call cpu_time(t) $\scriptstyle\pm2=\pm-\pm1$ comment $=$ ' i read ' print 100,comment,t2 read (10) y call cpu_time(t) $\mathsf{t}\,3\!=\!\mathsf{t}-\mathsf{t}\,2\!-\!\mathsf{t}\,2$ comment $=$ ' r read ' print 100,comment,t3 100 format( $_{1\times}$ ,a, $_{2\times}$ ,f7.3) do $\scriptstyle{\dot{\bot}}\,=\,\!1$ ,10 print \*,x(i), ' ' , y(i)end do end program ch1307 + +Some timing data from the Intel compiler follows. + +Intial 0.047 i read 0.063 r read 0.063 1 1.000000 2 2.000000 3 3.000000 4 4.000000 5 5.000000 6 6.000000 7 7.000000 8 8.000000 9 9.000000 10 10.00000 + +# 13.11 Errors when reading + +In discussing some aspects of input, it has been pointed out that errors may be made. Where such errors are noticed, in the sense that something illegal is being attempted, there are two options: + +• Print a diagnostic message, and allow correction of the mistake. • Print a diagnostic message, and terminate the program. + +The only time that the first makes sense is when you are interacting with a pro- gram at a terminal. Some Fortran implementations provide correction facilities in a case like this, but most do not. + +Chapter 21 looks at how we handle errors in input data, together with a more in-depth coverage of file I/O. + +# 13.12 Summary + +Values may be read in from the keyboard, terminal or from another file through fixed formats. + +Much of the structure of input format statements is very similar to that of the out- put formats. Broadly speaking, data written out in a particular format may be read in by the same format. However, there is greater flexibility, and quite a variety of forms can be accepted on input. + +A key distinction to make is the interpretation of blanks, as either nulls or zeros; alternative interpretations can radically alter the structure of the input data. + +Fortran allows file names to be associated with unit numbers through the OPEN statement. This statement allows control of the interpretation of blanks, although this can also be done through the BN and BZ formats. + +Files can also be manipulated through REWIND and BACKSPACE. + +# 13.13 Problems + +1. Write a program that will read in two reals and one integer, using + +FORMAT(F7.3,I4,F4.1) + +and that, in one instance treats blanks as zeros and in the second treats them as nulls. Use PRINT \* to print the numbers out immediately after reading them in. What do you notice? Can you think of instances where it is necessary to use one rather than the other? + +2. Write a program to read in and write out a real number using + +FORMAT(F7.2) + +What is the largest number that you can read in and write out with this format? What is the largest negative number that you can read in and write out with this format? What is the smallest number, other than zero, that can be read in and writ- ten out? + +3. Rewrite two of the earlier programs that used READ,\* and PRINT,\* to use FORMAT statements. + +4. Write a program to read the file created by either the temperature conversion program or the litres and pints conversion program. Make sure that the programs ignore the line printer control characters and any header and title information. This kind of problem is very common in programming (writing a program to read and possibly manipulate data created by another program). 5. Use the OPEN, REWIND, READ and WRITE statements to input a value (or values) as a character string, write this to a file, rewind the file, read in the values again, this time as real variables with blanks treated as null, and then repeat with blanks as zeros. 6. Demonstrate that input and output formats are not symmetric — i.e., what goes in does not necessarily come out. + + 7. Can you suggest why Fortran treats blanks as null rather than zero? + + 8. What happens at your terminal when you enter faulty data, inappropriate for the formats specified? We will look at how we address this problem in Chapter 21. + +“It is a capital mistake to theorise before one has data.” + +Sir Arthur Conan Doyle + +# Aims + +The aims of this chapter are: + +• To review the process of file creation at a terminal. • To introduce more formally the idea of the file as a fundamental entity. • To show how files can be declared explicitly by the OPEN and CLOSE statements. • To introduce the arguments for the OPEN and CLOSE statements. • To demonstrate the interaction between the READ/WRITE statements and the OPEN/CLOSE statements. + +# 14 Files + +When you work interactively on a terminal, you are working with files, files that contain programs, files that contain data, and perhaps files that are libraries. The file is fundamental to most modern operating systems, and almost all operations are carried out on files. + +In this chapter we are going to extend some of your ideas about files. Let us con- sider what kinds of files you have met so far: + +1. Text files. These are the source of your programs, compilation listings, etc. They can be examined by printing them. They can also be transmitted around a computer system fairly easily. A file sent to a printer is a text file. Mail mes- sages are generally plain text files. Note that when mail messages arrive in your mail box they will then typically contain additional nonprintable informa- tion. + +2. Data files. These exist in two main forms: firstly those prepared by using an editor, (hence a text file) and those prepared using a package or program, in a computer readable form, but not directly readable by a human. 3. Binary, object or relocatable files, e.g., output from the compiler, satellite data. They cannot be printed. To examine files like these you need to use spe- cial utilities, provided by most operating systems. + +The above categories account for the majority of files that you have met so far. + +If you use a word processor then you will also have met files that are textual with additional nonprintable information. + +Let us now consider how we can manipulate files using Fortran. They will gener- ally be data files, and will thus be text files. They can therefore be listed, etc., using standard operating system commands. + +# 14.1 Data files in Fortran + +These allow us to associate a logical unit number with any arbitrary file name dur- ing the running of the program; e.g., + +OPEN(UNIT $\vDash\vDash$ ,FILE $=$ 'DATA') + +would associate the name DATA and the logical unit 1, so that + +READ(UNIT $\vDash\vDash$ ,FMT=100) X + +would read from DATA. Note that for this to work on some operating systems the file DATA must be local to the session; we specify the name as a character vari- able. If we then wanted to use a subsequent data file, we could have another OPEN statement, but if we want to use the same logical unit number, we must first CLOSE the file + +CLOSE(UNIT $\mathbf{\omega}=\mathbb{1}$ ,FILE $=$ 'DATA') + +before we + +OPEN(UNIT $\vDash\vDash$ ,FILE $=$ 'DATA2') + +In this way we can keep referring to logical unit 1, but change the file associated with it. This can be useful in interactive programs where we wish to analyse differ- ent sets of data, e.g.: + +PROGRAM ch1401 IMPLICIT NONE REAL :: X CHARACTER (7) :: WHICH + +DO WRITE(UNIT $^{1\!}{=}\,6$ , $\mathrm{{FMT}=`}$ ('' DATA SET NAME, OR END'')')READ(UNIT ${\tt=}5$ ,FMT $="$ (A)') WHICH IF(WHICH $==$ 'END') EXIT OPEN(UNIT $\vDash\vDash$ ,FILE $=$ WHICH) READ(UNIT $\vDash\vDash$ ,FMT $\scriptstyle1\equiv100$ ) X ... CLOSE(UNIT $\scriptstyle{1=1}$ ,FILE $=$ WHICH)END DO END PROGRAM ch1401 + +One useful feature of the OPEN statement is that there are other parameters. What would happen, for example, if the file is not there? To take care of this you can use the IOSTAT and STATUS keywords, e.g., + +OPEN(UNIT $\vDash\vDash$ ,FILE $=$ 'DATA',IOSTAT $=$ FileStat,STATUS $=$ 'OLD') STATUS can be equated to one of four values: + +STATUS $=$ 'OLD' STATUS $=$ 'NEW' STATUS $=$ 'SCRATCH' STATUS $=$ 'UNKNOWN' + +If we say STATUS $\risingdotseq$ NEW', we are creating a new file and it should not matter whether a file of the same name is present; 'SCRATCH' does not concern us, while 'UNKNOWN' implies that if a file of the correct name is present use it, but if not create a 'NEW’ one. If you omit the STATUS $=$ keyword altogether, the value 'UN- KNOWN’ will be assumed. If we use STATUS $\risingdotseq$ ’OLD’ and the file is not present, this will cause an error which will be reflected in the value associated with the variable Open File Status . Consider the following example: + +OPEN(UNIT $\vDash\vDash$ ,FILE $=$ 'DATA',IOSTAT $^{\ast}{=}$ FileStat,STATUS $=$ 'OLD') IF (FileStat $>\quad0$ ) THEN PRINT \*,' Error opening file, please check' STOP END IF READ(UNIT $\vDash\vDash$ ,FMT $=\!.$ 100) X ... + +The program will terminate after printing an appropriate error message. The stan- dard defines that if an error occurs then IOSTAT will return a positive integer value. A value of zero is returned if there is no error. + +# 14.2 Summary of options on OPEN + +UNIT: The unit number of the file to be opened. + +IOSTAT: The I/O status specifier designates a variable to store a value indicating the status of a data transfer operation. It takes the following form: + +IOSTAT $\mathbf{\dot{\omega}}=\!i\mathbf{\dot{\omega}}$ -var + +$i$ -var + +is a scalar integer variable. When a data transfer statement is executed, $i.$ -var is set to one of the following values: + +• A positive integer indicating that an error condition occurred. • A negative integer indicating that an end-of-file or end-of-record condition occurred. The actual values vary between compilers. • Zero indicating no error, end-of-file, or end-of-record condition oc- curred. + +Execution continues with the statement following the data transfer statement or the statement identified by a branch specifier (if any). + +An end-of-file condition occurs only during execution of a sequential READ state- ment; an end-of-record condition occurs only during execution of a nonadvancing READ statement. + +FILE: Character expression specifying the file name. + +STATUS: Character expression specifying the file status. It can be one of 'OLD' , 'NEW', 'SCRATCH' or 'UNKNOWN'. + +ACCESS : Character expression specifying whether the file is to be used in a se- quential or random fashion. Valid values are SEQUENTIAL (the default) or DIRECT. + +The two most common access mechanisms for files are sequential and direct. Con- sider a file with 1000 records. To get at record 789 in a sequential file means reading or processing the first 788 records. To get at record 789 in a direct access file means using a record number to immediately locate record 789. + +FORM: Character expression specifying + +FORMATTED if the file is opened for formatted I/O + +or UNFORMATTED if the file is opened for unformatted I/O + +The default is formatted for sequential access files and unformatted for direct ac- cess files. If the file exists, FORM must be consistent with its present characteristics. + +As noted earlier data are maintained internally in a binary format, not immediately comprehensible by humans. When we wish to look at the data we must write it in a formatted fashion, i.e., as a sequence of printable ASCII characters — text, or the written word. This formatting will carry with it an overhead in terms of the time required to do it. It will also carry with it the penalty of conversion from one num- ber base (internally binary) to another and also loss of significance due to rounding with whatever edit descriptors are used, e.g., writing out as F7.4. + +If we are interested in reusing data on the same system and compiler then we can use the unformatted option and avoid both the time overhead (as there is no con- version between the internal and external formats) and the loss of significance associated with formatted data. + +Please note that unformatted files are rarely portable between different computer systems, and sometimes even between different compilers on the same system. + +We will look again at the use of unformatted files in Chapter 28 when we deal with efficiency and the space-time trade-off. + +RECL: Integer variable or constant specifying the record length for a direct access file. It is specified in characters for a formatted file and words for an unformatted file. + +BLANK : Character expression having one of the following values: + +'NULL' if blanks are to be ignored on reading. Note that a field of all blanks is treated as 0! + +# Chapter 14 + +'ZER $O^{\prime}$ if blanks are to be treated as zeros. + +# 14.3 More foolproof I/O + +Fortran provides a way of writing more foolproof programs involving I/O. This is done via the IOSTAT keyword on the READ statement. Consider the following: + +PROGRAM ch1402 IMPLICIT NONE INTEGER :: IO_Stat_Number $\scriptstyle{\underline{{\mathbf{\zeta}}}}=-1$ INTEGER :: I DO READ (UNIT $=\star$ , $\scriptstyle{\mathrm{FMT}=10}$ ,& IOSTAT $^{\ast}{=}$ IO_Stat_Number) I 10 FORMAT(I3) PRINT \*,' iostat $\mathbf{\mu}=\mathbf{\sigma}^{\mathsf{I}}$ ,IO_Stat_Number PRINT \*,I IF (IO_Stat_Number $==0$ ) EXIT END DO END PROGRAM ch1402 + +The following data input should be tried and the values of IO_Stat_Number should be examined + +• A valid three-digit number $^+$ [RETURN] key • A three-digit number with an embedded blank, e.g., $^\textrm{\scriptsize12+}$ [RETURN] key • [RETURN] key only • $\mathrm{[TTRL]+Z}$ • Any other nonnumeric character on the keyboard • $100200300\;+$ [RETURN] key • [CTRL] + C + +This will then enable you to write programs that handle common I/O errors. + +Consider the following: + +PROGRAM ch1403 INTEGER , DIMENSION(10) :: A $=\!\delta\!c$ $(\,/-\mathtt{L}\,,\,-\mathtt{L}\,,\,-\mathtt{L}\,,\,-\mathtt{L}\,,\,-\mathtt{L}\,,\,-\mathtt{L}\,,\,-\mathtt{L}\,,\,-\mathtt{L}\,,\,-\mathtt{L}\,,\,-\mathtt{L}\,,\,-\mathtt{L}\,/\,)$ INTEGER :: IO_Stat_Number $=\!0$ INTEGER :: I OPEN(UNIT $\vDash\vDash$ ,FILE $=$ 'DATA.DAT',STATUS $=$ 'OLD') + +DO $\mathbb{T}\!=\!1$ ,10 READ (UNIT $\vDash\vDash$ ,FMT $\vDash\vDash$ 0,IOSTAT $=$ IO_Stat_Number) A(I) 10 FORMAT(I3) IF (IO_Stat_Number $==\quad0$ ) THEN CYCLE ELSEIF (IO_Stat_Number $\begin{array}{r l}{\mathrm{{==}}}&{{}\end{array}$ ) THEN PRINT \*,' End of file detected at line ',I PRINT \*,' Please check data file' EXIT ELSEIF (IO_Stat_Number > 0 ) THEN PRINT \*,' Non numeric data at line ',I PRINT \*,' Please correct data file' EXIT ENDIF END DO DO $\mathbb{T}\!=\!1$ ,10 PRINT \* , ' I = ',I,' A(I) = ',A(I) ENDDO END PROGRAM ch1403 + +The above program is system specific but interestingly the following compilers re- turn the same value for end of file. They return different values for nonnumeric data: + +• NAG/Salford compiler. • DEC Alpha OPENVMS compiler. • NagAce Fortran 90 under Solaris. • Sun F90 compiler (release 2.x). • Nag F95 compiler under Solaris. • Compaq/Dec F95, 6.01A. + +What happens with a completely blank line? + +Note that in the above example the testing for the various conditions only exits the DO loop for reading data from the file. This means that execution would continue with the statement immediately after the END DO statement. This may not be what we want in all cases, and the EXIT may be replaced with a STOP statement to terminate execution immediately. + +# 14.4 Summary + +The file is a fundamental entity within the operating system. + +# Chapter 14 + +A file may be manipulated in Fortran by associating its name with a unit number. All subsequent communication within the program is through the unit number. When a file is opened there are a large number of equatable keywords which may be employed to establish its characteristics. + +The default file type used in Fortran is sequential formatted, but several other eso- teric types may be used. + +# 14.5 Problems + +1. Write a program to write the first 500 integers to a file using formatted I/O. Put 10 values on a line, with a blank as the first character of the line, and eight col- umns allowed for each integer, with two spaces between integer fields. + +Now write a program to read this file into an array, and write the numbers in re- verse order over the original data, i.e., the data file now contains the first 500 numbers in descending order. + +Now modify the first program to add the next 500 integers to the same file, so that the file now comprises the first 500 numbers in descending order, and the next 500 numbers in ascending order. + +2. To write and maintain a crude database of student details, we might do the fol- lowing: create separate files for each year — CLAS1, CLAS2, CLAS3, or COF84, COF85, COF86, and so on. In either case there is an unchanging prefix, CLAS or COF, and a variable suffix, which identifies membership within the overall group. In each of the files we may wish to record details like name, date of birth, address, courses taken, etc. Such files will require updating as details change or as errors are noted. Write (or sketch out) a program which would select and maintain such records and would allow corrected files to be printed out. While you might feel that the most appropriate tool for this job is an editor, you might find it too power- ful a tool. An editor can leave files in a sorry state. Naturally, any program like this should be helpful (so called ' user friendly '). Is this sort of information sensitive enough to require security checks and passwords? + +# Functions + +“I can call spirits from the vasty deep. Why so can I, or so can any man; but will they come when you do call for them?” + +William Shakespeare, King Henry IV, part 1 + +# Aims + +The aims of this chapter are: + +• To consider some of the reasons for the inclusion of functions in a programming language. • To introduce, with examples, some of the predefined functions available in Fortran 95. • To introduce a classification of intrinsic functions, generic, elemental, transformation al. • To introduce the concept of a user defined function. • To introduce the concept of a recursive function. • To introduce the concept of user defined elemental and pure functions. • To briefly look at scope rules in Fortran 95 for variables and functions. • To look at internal user defined functions. + +# 15 Functions + +The role of functions in a programming language and in the problem-solving pro- cess is considerable and includes: + +• Allowing us to refer to an action using a meaningful name, e.g., SINE(X) a very concrete use of abstraction. • Providing a mechanism that allows us to break a problem down into parts, giving us the opportunity to structure our problem solution. • Providing us with the ability to concentrate on one part of a problem at a time and ignore the others. • Allowing us to avoid the replication of the same or very similar sections of code when solving the same or a similar subproblem which has the sec- ondary effect of reducing the memory requirements of the final program. • Allowing us to build up a library of functions or modules for solving par- ticular subproblems, both saving considerable development time and in- creasing our effectiveness and productivity. + +Some of the underlying attributes of functions are: + +• They take parameters or arguments. • The parameter can be an expression. • A function will normally return a value and the value returned is normally dependent on the parameter(s). • They can sometimes take arguments of a variety of types. + +Most languages provide both a range of predefined functions and the facility to de- fine our own. We will look at the predefined functions first. + +# 15.1 An introduction to predefined functions and their use + +Fortran provides over a hundred intrinsic functions and subroutines. For the pur- poses of this chapter a subroutine can be regarded as a variation on a function. Subroutines are covered in more depth in a later chapter. They are used in a straightforward way. If we take the common trigonometric functions, sine, cosine and tangent, the appropriate values can be calculated quite simply by: + +X=SIN(Y) Z=COS(Y) A=TAN(Y) + +This is in rather the same way that we might say that X is a function of Y, or $\mathrm{X}$ is sine Y. Note that the argument, Y, is in radians not degrees . + +# 15.1.1 Example 1: Simple function usage + +A complete example is given below: + +PROGRAM ch1501 REAL :: X PRINT \*,' Type in an angle (in radians)' READ \*,X PRINT \*,' Sine of ', X ,' = ',SIN(X) END PROGRAM ch1501 + +These functions are called intrinsic functions. A selection is follows: + +![](images/177a37525a01f9cff4890f234664dad3c97d23c24cd7faf4d6af037bf997fc10.jpg) +A complete list is given in Appendix D. + +# 15.2 Generic functions + +All but four of the intrinsic functions and procedures are generic, i.e., they can be called with arguments of one of a number of kind types. + +# Chapter 15 + +# 15.2.1 Example 2: The ABS generic function + +The following short program illustrates this with the ABS intrinsic function: + +PROGRAM ch1502 IMPLICIT NONE COMPLEX :: $\mathsf{C}\!=(\,\mathbb{1}\,,\,\mathbb{1}\,)$ )REAL :: $\scriptstyle\mathrm{~R=}10\,.\,9$ INTEGER :: $\mathbb{T}\!=\!-27$ PRINT \*,ABS(C) PRINT \*,ABS(R)PRINT \*,ABS(I) END PROGRAM ch1502 + +The four nongeneric functions are LGE, LGT, LLE and LLT — the lexical charac- ter comparison functions. + +Type this program in and run it on the system you use. + +It is now possible with Fortran 95 for the arguments to the intrinsic functions to be arrays. It is convenient to categorise the functions into either elemental or transformation al, depending on the action performed on the array elements. + +# 15.3 Elemental functions + +These functions work with both scalar and array arguments, i.e., with arguments that are either single or multiple valued. + +# 15.3.1 Example 3: Elemental function use + +Taking the earlier example with the evaluation of sine as a basis, we have: + +PROGRAM ch1503 REAL , DIMENSION(5) :: X = (/1.0,2.0,3.0,4.0,5.0/) PRINT \*,' Sine of ', X , $\mathrm{~\ensuremath~{~\vert~\vert~}~}\mathrm{~\ensuremath~{~\vert~\vert~}~}$ ',SIN(X) END PROGRAM ch1503 + +In the above example the sine function of each element of the array X is calculated and printed. + +# 15.4 Transformation al functions + +Transformation al functions are those whose arguments are arrays, and work on these arrays to transform them in some way. + +# 15.4.1 Example 4: Simple transformation al use + +To highlight the difference between an element-by-element function and a transformation al function consider the following examples: + +PROGRAM ch1504 IMPLICIT NONE REAL , DIMENSION(5) :: X = (/1.0,2.0,3.0,4.0,5.0/) + + ! Elemental function PRINT \*,' Sine of ', X ,' = ',SIN(X) + + ! Transformation al function PRINT \*,' Sum of ', X ,' = ',SUM(X) END PROGRAM ch1504 + +The SUM function adds each element of the array and returns the SUM as a scalar, i.e., the result is single valued and not an array. + +# 15.4.2 Example 5: Intrinsic DOT_PRODUCT use + +The following program uses the transformation al function DOT_PRODUCT: + +PROGRAM ch1505 IMPLICIT NONE REAL , DIMENSION(5) :: X = (/1.0,2.0,3.0,4.0,5.0/) PRINT \*,' Dot product of X with X is' PRINT \*,' ',DOT_PRODUCT(X,X) END PROGRAM ch1505 + +Try typing these examples in and running them to highlight the differences be- tween elemental and transformation al functions. + +# 15.5 Notes on function usage + +You should not use variables which have the same name as the intrinsic functions; e.g., what does SIN(X) mean when you have declared SIN to be a real array? + +When a function has multiple arguments care must be taken to ensure that the ar- guments are in the correct position and of the appropriate kind type. + +You may also replace arguments for functions by expressions, e.g., + +$$ +\mathrm{~\boldmath~X~}=\mathrm{~\sf~L~O~G~}(\,2\cdot0\,) +$$ + +or + +X = LOG(ABS(Y)) + +# Chapter 15 + +or + +X = LOG(ABS(Y)+Z/2.0) + +# 15.6 Example 6: Easter + +This example uses only one function, the MOD (or modulus). It is used several times, helping to emphasise the usefulness of a convenient, easily referenced func- tion. The program calculates the date of Easter for a given year. It is derived from an algorithm by Knuth, who also gives a fuller discussion of the importance of its algorithm. He concludes that the calculation of Easter was a key factor in keeping arithmetic alive during the Middle Ages in Europe. Note that determination of the Eastern churches' Easter requires a different algorithm: + +PROGRAM ch1506 IMPLICIT NONE INTEGER :: Year, Metcyc, Century, Error1, Error2, Day INTEGER :: Epact, Luna, Temp + + ! A program to calculate the date of Easter PRINT \*,' Input the year for which Easter' PRINT \*,' is to be calculated' PRINT \*,' enter the whole year, e.g. 1978 ' READ \*,Year + + ! calculating the year in the 19 year + + ! metonic cycle using variable metcyc Metcyc $=$ MOD(Year,19) $^{+1}$ IF(Year $<=~\,\,\mathbb{1582}$ )THEN Day $=$ (5\*Year)/4 Epact $=$ MOD(11\*Metcyc-4,30)+1 ELSE + + ! calculating the Century-century Century $=$ (Year/100) $+1 + +$ ! accounting for arithmetic inaccuracies + + ! ignores leap years etc. Error1 $=$ (3\*Century/4)-12 Error2 $=$ ((8\*Century ${+5}$ )/25)-5 + + ! locating Sunday Day $=$ (5\*Year/4)-Error1-10 + + ! locating the epact(full moon) Temp $\begin{array}{r l}{\mathbf{\Sigma}}&{{}=\quad\mathbb{1}\mathbb{1}}\end{array}$ \* Metcyc $^+$ 20 $^+$ Error2 - Error1 Epact $=$ MOD(Temp,30) IF(Epact $<=~\mathrm{~\textmu~}0$ ) THEN Epact $=$ 30 $^+$ Epact + +ENDIF IF((Epact $==\ \ 25$ .AND. Metcyc $>$ 11) & .or. Epact $==\quad24$ )THEN Epact $=$ Epact $+1$ ENDIF ENDIF + + ! finding the full moon Luna $=$ 44 - epact IF (Luna $<\ \ 2\,\beth$ ) THEN Luna $=$ Luna $+30$ ENDIF + + ! locating Easter Sunday Luna $=$ Luna $+7-$ (MOD(Day $^{\cdot+}$ Luna,7)) + + ! locating the correct month IF(Luna $>$ 31)THEN Luna $=$ Luna - 31 PRINT \*,' for the year ',YEAR PRINT \*,' Easter falls on April ',Luna ELSE PRINT \*,' for the year ',YEAR PRINT \*,' Easter falls on march ',Luna ENDIF END PROGRAM ch1506 + +We have introduced a new statement here, the IF THEN ENDIF, and a variant the IF THEN ELSE ENDIF. A more complete coverage is given in the chapter on con- trol structures. The main point of interest is that the normal sequential flow from top to bottom can be varied. In the following case, + +IF (expression) THEN block of statements + +ENDIF + +if the expression is true the block of statements between the IF THEN and the ENDIF is executed. If the expression is false then this block is skipped, and execu- tion proceeds with the statements immediately after the ENDIF. + +In the following case, + +IF (expression) THEN + +block 1 + +ELSE + +# Chapter 15 + +block 2 + +ENDIF + +if the expression is true block 1 is executed and block 2 is skipped. If the expres- sion is false then block 2 is executed and block 1 is skipped. Execution then proceeds normally with the statement immediately after the ENDIF. + +As well as noting the use of the MOD generic function in this program, it is also worth noting the structure of the decisions. They are nested , rather like the nested DO loops we met earlier. + +# 15.7 Complete list of predefined functions + +Due to the large number of predefined functions it is useful to classify them, and the following is one classification. + +# 15.7.1 Inquiry functions + +These functions return information about their arguments. They can be further subclassified into BIT, CHARACTER, NUMERIC, ARRAY, POINTER, ARGU- MENT PRESENCE: + +Bit BIT_SIZE + +Character LEN + +Numeric DIGITS, EPSILON, EXPONENT, FRACTION, HUGE, KIND, MAXEXPONENT, MINEXPONENT, NEAREST, PRECISION, RADIX, RANGE, RRSPACING, SCALE, SET_EXPONENT, SELECTED IN T KIND, SELECTED_REAL_KIND, SPACING, TINY Array ALLOCATED, LBOUND, SHAPE, SIZE, UBOUND, Pointer ASSOCIATED, NULL Argument PRESENT + +Presence + +# 15.7.2 Transfer and conversion functions + +These functions convert data from one type and kind type to another type and kind type. Most of them are by necessity generic. + +Transfer ACHAR, AIMAG, AINT, ANINT, CHAR, CMPLX, CONJG, and DBLE, IACHAR, IBITS, ICHAR, INT, LOGICAL, NINT, Conversion REAL, TRANSFER 15.7.3 Computational functions These functions actually carry out a computation of some sort and return the result of that computation: Numeric ABS, ACOS, ASIN, ATAN, ATAN2, CEILING, COS, COSH, DIM, DOT_PRODUCT, DPROD, EXP, FLOOR, LOG, LOG10, MATMUL, MAX, MIN, MOD, MODULO, SIGN, SIN, SINH, SQRT, TAN, TANH Character ADJUSTL, ADJUSTR, INDEX, LEN_TRIM, LGE, LGT, LLE, LLT, REPEAT, SCAN, TRIM, VERIFY Bit BTEST, IAND, IBCLR, IBSET, IEOR, IOR, ISHFT, ISHFTC, NOT 15.7.4 Array functions Reduction ALL, ANY, COUNT, MAXVAL, MINVAL, PRODUCT, SUM Construction MERGE, PACK, SPREAD, UNPACK Reshape RESHAPE Manipulation CSHIFT, EOSHIFT, TRANSPOSE Location MAXLOC, MINLOC 15.7.5 Predefined subroutines Date and CPU_TIME, DATE_AND_TIME, SYSTEM_CLOCK Time Random RANDOM_NUMBER, RANDOM_SEED Number Other MVBITS + +# Chapter 15 + +An alphabetical list of all intrinsic functions and procedures is given in Appendix D. This list provides the following information: + +• Function name. • Description. • Argument name and type. • Result type. • Classification. • Examples of use. + +Appendix D should be consulted for a more complete and thorough understanding of intrinsic functions and their use in Fortran 95. + +# 15.8 Supplying your own functions + +There are two stages here: firstly, to define the function and, secondly, to reference or use it. Consider the calculation of the greatest common divisor of two integers. + +# 15.8.1 Example 7: Simple user defined function + +The following defines a function to achieve this: + +INTEGER FUNCTION GCD(A,B) IMPLICIT NONE INTEGER , INTENT(IN) :: A,B INTEGER :: Temp IF ( $\mathrm{~\AA~}<\mathrm{~\textmu~}$ ) THEN Temp $\mathrm{=A}$ ELSE Temp $\mathtt{=B}$ ENDIF DO WHILE ((MOD(A,Temp) /= 0) .OR. (MOD(B,Temp) / $=\!0$ )) Temp $=$ Temp-1 END DO GCD ${}^{1=}$ Temp END FUNCTION GCD + +To use this function, you reference or call it with a form like: + +PROGRAM ch1507 IMPLICIT NONE INTEGER :: I,J,Result INTEGER :: GCD + +PRINT \*,' Type in two integers' READ \*,I,J Result $=$ GCD(I,J) PRINT \*,' GCD is ',Result END PROGRAM ch1507 + +The first line of the function + +INTEGER FUNCTION GCD(A,B) + +has a number of items of interest: + +• Firstly the function has a type, and in this case the function is of type IN- TEGER, i.e., it will return an integer value. • The function has a name, in this case GGD. • The function takes arguments or parameters, in this case A and B. + +The structure of the rest of the function is the same as that of a program, i.e., we have declarations, followed by the executable part. This is because both a program and a function can be regarded as a so-called program unit . We will look into this more fully in later chapters. + +In the declaration we also have a new attribute for the INTEGER declaration. The two parameters A and B are of type integer, and the INTENT(IN) attribute means that these parameters will NOT be altered by the function. + +The value calculated is returned through the function name somewhere in the body of the executable part of the function. In this case GCD appears on the left-hand side of an arithmetic assignment statement at the bottom of the function. The end of the function is signified in the same way as the end of a program: + +END FUNCTION GCD + +We then have the program which actually uses the function GCD. In the program the function is called or invoked with I and J as arguments. The variables are called A and B in the function, and references to A and B in the function will use the values that I and J have respectively in the main program. We will look into the whole area of argument association in much greater depth in later chapters. + +Note also a new control statement, the DO WHILE ENDDO. In the following case, + +DO WHILE (expression) block of statements + +# Chapter 15 + +# ENDDO + +the block of statements between the DO WHILE and the ENDDO is executed whilst the expression is true. There is a more complete coverage in Chapter 16. + +We have two options here regarding compilation. Firstly, to make the function and the program into one file, and invoke the compiler once. Secondly, to make the function and program into separate files, and invoke the compiler twice, once for each file. With large programs comprising one program and several functions it is probably worthwhile to keep the component parts in different files and compile in- dividually, whereas if it consists of a simple program and one function then keeping things together in one file makes sense. + +Try this program out on the system you work with. + +# 15.9 An introduction to the scope of variables and local variables + +One of the major strengths of Fortran is the ability to work on parts of a problem at a time. This is achieved by the use of program units (a main program, one or more functions and one or more subroutines) to solve discrete subproblems. Inter- action between them is limited and can be isolated, for example, to the arguments of the function. Thus variables in the main program can have the same name as variables in the function and they are completely separate variables, even though they have the same name. Thus we have the concept of a local variable in a pro- gram unit. We will look into this area again after a coverage of recursion and very thoroughly after the coverage of subroutines and modules. + +In the example above I, J, Result, are local to the main program. The declaration of GCD is to tell the compiler that it is an integer, and in this case it is an external function. + +A and B in the function GCD do not exist in any real sense; rather they will be re- placed by the actual variable values from the calling routine, in this case by whatever values I and J have. Temp is local to GCD. + +# 15.10 Recursive functions + +There is an additional form of the function header that must be used when the function is recursive. Recursion means the breaking down of a problem into a sim- pler but identical subproblem. The concept is best explained with reference to an actual example. Consider the evaluation of a factorial, e.g., 5!. From simple mathe- matics we know that the following is true: + +$5!\!=\!\!5^{*}4!$ + +$4!\!\!=\!\!4^{*}3!$ + +$3!\!\!=\!\!3^{*}2!$ $2!\!=\!\!2^{*}1!$ 1!=1 and thus $5!=5^{*}4^{*}3^{*}2^{*}1$ or 120. + +# 15.10.1 Example 8: Recursive factorial evaluation + +Let us look at a program with recursive function to solve the evaluation of factori- als. + +PROGRAM ch1508 IMPLICIT NONE INTEGER :: I, F, Factorial PRINT \*,' Type in the number, integer only' READ \*,I DO WHILE( $\mathbb{T}\!<\!0$ ) PRINT \*,' Factorial only defined for ' PRINT \*,' positive integers: Re-input' READ \*,I END DO $\mathrm{F}\!=$ Factorial(I) PRINT \*,' Answer is', F END PROGRAM ch1508 + +RECURSIVE INTEGER FUNCTION Factorial(I) RESULT(Answer) IMPLICIT NONE INTEGER , INTENT(IN):: I IF ( $\mathbb{T}\!=\!=\!0$ ) THEN Answer $\mathbf{\Psi}\!=\!\!1$ ELSE Answer ${}^{\ast}=$ I\*Factorial(I-1) END IF END FUNCTION Factorial + +What additional information is there? Firstly, we have an additional attribute on the function header that declares the function to be recursive. Secondly, we must re- turn the result in a variable, in this case Answer. Let us look now at what happens when we compile and run the whole program (both function and main program). If we type in the number 5 the following will happen: + +• The function is first invoked with argument 5. The ELSE block is then taken and the function is invoked again. + +• The function now exists a second time with argument 4. The ELSE block is then taken and the function is invoked again. • The function now exists a third time with argument 3. The ELSE block is then taken and the function is invoked again. • The function now exists a fourth time with argument 2. The ELSE block is then taken and the function is invoked again. • The function now exists a fifth time with argument 1. The ELSE block is then taken and the function is invoked again. • The function now exists a sixth time with argument 0. The IF BLOCK is executed and Answer $=\!1$ . This invocation ends and we return to the previ- ous level, with Answer $\scriptstyle{\stackrel{\ldots}{=}1^{*}1}$ . • We return to the previous invocation and now Answer ${\stackrel{}{=}}2^{*}1$ . • We return to the previous invocation and now Answe ${\stackrel{}{=}}3^{*}2$ . • We return to the previous invocation and now Answer ${=}4^{*}6$ . • We return to the previous invocation and now Answe $\scriptstyle=5^{*}24$ . + +The function now terminates and we return to the main program or calling routine. The answer 120 is the printed out. + +Add a PRINT $^{*}\mathrm{,I}$ statement to the function after the last declaration and type the program in and run it. Try it out with 5 as the input value to verify the above state- ments. + +Recursion is a very powerful tool in programming, and remarkably simple solu- tions to quite complex problems are possible using recursive techniques. We will look at recursion in much more depth in the later chapters on dynamic data types, and subroutines and modules. + +# 15.11 Example 9: Recursive version of GCD + +The following is another example of the earlier GCD function but with the algo- rithm in the function replaced with an alternate recursive solution: + +PROGRAM ch1509 IMPLICIT NONE INTEGER :: I,J,Result INTEGER :: GCD PRINT \*,' Type in two integers' READ \*,I,J Result $=$ GCD(I,J) PRINT \*,' GCD is ',Result + +END PROGRAM ch1509 + +RECURSIVE INTEGER FUNCTION GCD(I,J) RESULT(Answer) IMPLICIT NONE INTEGER , INTENT(IN) :: I,J IF ( $\scriptstyle{\overline{{\cup}}}\,=\,0$ ) THEN Answer $:=\mathbb{T}$ ELSE Answer ${}^{\ast}=$ GCD(J,MOD(I,J)) ENDIF END FUNCTION GCD + +Try this program out on the system you work with, look at the timing information provided, and compare the timing with the previous example. The algorithm is a much more efficient algorithm than in the original example, and hence should be much faster. On one system there was a twentyfold decrease in execution time be- tween the two versions. + +Recursion is sometimes said to be inefficient, and the following example looks at a nonrecursive version of the second algorithm. + +# 15.12 Example 10: After removing recursion + +The following is a variant of the above, with the same algorithm, but with the re- cursion removed: + +PROGRAM ch1510 IMPLICIT NONE INTEGER :: I,J,Result INTEGER :: GCD PRINT \*,' Type in two integers' READ \*,I,J Result $=$ GCD(I,J) PRINT \*,' GCD is ',Result END PROGRAM ch1510 + +INTEGER FUNCTION GCD(I,J)IMPLICIT NONE INTEGER , INTENT(INOUT) :: I,J INTEGER :: Temp DO WHILE ( $\upsigma/\!=\!0$ ) Temp $=$ MOD(I,J) $\scriptstyle{\mathbb{T}}={\mathbb{T}}$ $\scriptstyle{\overline{{\cup}}}=$ Temp + +# Chapter 15 + +END DO $\mathrm{GCD}{=}\mathbb{T}$ END FUNCTION GCD + +# 15.13 Pure functions + +Within the world of mathematics there is the concept of a pure function. This means that the function only returns a value, and has no effect on the arguments. Fortran 95 introduced the ability to write user defined pure functions. We will pro- vide examples in Chapter 26, when we have covered the additional syntax that is required. + +# 15.14 Elemental functions + +Fortran 77 introduced the concept of generic intrinsic functions. Fortran 90 added elemental intrinsic functions and the ability to write generic user defined functions. Fortran 95 squares the circle and enables us to write elemental user defined func- tions. We will show how this can be done in Chapter 26 when we have covered the additional syntax that is required. + +# 15.15 Internal functions + +An internal function is a more restricted and hidden form of the normal function definition. + +Since the internal function is specified within a program segment, it may only be used within that segment and cannot be referenced from any other functions or subroutines, unlike the intrinsic or other user defined functions. + +# 15.15.1 Example 11: Stirling's approximation + +In this example we use Stirling's approximation for large $n$ , + +$$ +n!={\sqrt{2\pi n}}{\left({\frac{n}{e}}\right)}^{n} +$$ + +and a complete program to use this internal function is given below: + +PROGRAM ch1511 IMPLICIT NONE REAL :: Result,N,R PRINT \*,' Type in N and R' READ \*,N,R + + ! NUMBER OF POSSIBLE COMBINATIONS THAT CAN + + ! BE FORMED WHEN + + ! R OBJECTS ARE SELECTED OUT OF A GROUP OF N + +! N!/R!(N-R)! Result $=$ Stirling(N)/(Stirling(R)\*Stirling(N-R))PRINT \*,Result PRINT \*,N,R CONTAINS REAL FUNCTION Stirling (X) REAL , INTENT(IN) :: X REAL , PARAMETER :: PI $^{=3}$ .1415927, E $^{=2}$ .7182828 Stirling ${}^{\circ}\underline{{=}}$ SQRT(2 ${}^{\star}\,\mathtt{P}\,\mathtt{T}\,{}^{\star}\,\mathtt{X}$ ) \* (X/E)\*\*X END FUNCTION Stirling END PROGRAM ch1511 + +The difference between this example and the earlier ones lies in the CONTAINS statement. The function is now an integral part of the program and could not, for example, be used elsewhere in another function. This provides us with a very pow- erful way of information hiding and making the construction of larger programs more secure and bug free. + +# 15.16 Resumé + +There are a large number of Fortran supplied functions and subroutines (intrinsic functions) which extend the power and scope of the language. Some of these func- tions are of generic type, and can take several different types of arguments. Others are restricted to a particular type of argument. Appendix D should be consulted for a fuller coverage concerning the rules that govern the use of the intrinsic functions and procedures. + +When the intrinsic functions are inadequate, it is possible to write user defined functions. Besides expanding the scope of computation, such functions aid in prob- lem visualisation and logical subdivision, may reduce duplication, and generally help in avoiding programming errors. + +In addition to separately defined user functions, internal functions may be em- ployed. These are functions which are used within a program segment. + +Although the normal exit from a user defined function is through the END, other, abnormal, exits may be defined through the RETURN statement. + +Communication with nonrecursive functions is through the function name and the function arguments. The function must contain a reference to the function name on the left-hand side of an assignment. Results may also be returned through the argu- ment list. + +We have also covered briefly the concept of scope for a variable, local variables, and argument association. This area warrants a much fuller coverage and we will do this after we have covered subroutines and modules. + +# Chapter 15 + +# 15.17 Function syntax + +The syntax of a function is: + + [function prefix] function statement & + + [RESULT (Result_name) ] + + [specification part] + + [execution_part] + + [internal sub program part] END [FUNCTION [function name]] and prefix is: [type specification] RECURSIVE or [RECURSIVE] type specification and the function statement is: FUNCTION function_name ([dummy argument name list]) [ ] represent optional parts to the specification. + +# 15.18 Rules and restrictions + +The type of the function must only be specified once, either in the function state- ment or in a type declaration. The names must match between the function header and END FUNCTION func- tion name statement. If there is a RESULT clause, that name must be used as the result variable, so all references to the function name are recursive calls. The function name must be used to return a result when there is no RESULT clause. + +We will look at additional rules and restrictions in later chapters. + +# 15.19 Problems + +1. Find out the action of the MOD function when one of the arguments is negative. Write your own modulus function to return only a positive remainder. Don't call it MOD! + +2. Create a table which gives the sines, cosines and tangents for $^{-1}$ to 91 degrees in 1 degree intervals. Remember that the arguments have to be in radians. What value will you give $\pi?$ One possibility is $\pi{=}4^{*}$ atan(1.0). Pay particular attention to the following angle ranges: + +• $^{-1,0,+1}$ • 29,30,31 • 44,45,46 • 59,60,61 • 89,90,91 + +What do you notice about sine and cosine at 0 and 90 degrees? What do you notice about the tangent of 90 degrees? Why do you think this is? + +Use a calculator to evaluate the sine, cosine at 0 and 90 degrees. Do the same for the tangent at 90 degrees. Does this surprise you? + +Repeat using a spreadsheet, e.g., Excel. + +Are you surprised? + +Repeat the Fortran program using one or more real kind types. + +3. Write a program that will read in the lengths $a$ and $b$ of a right-angled triangle and calculate the hypotenuse $c$ . Use the Fortran SQRT intrinsic. + +4. Write a program that will read in the lengths $a$ and $b$ of two sides of a triangle and the angle between them (in degrees). Calculate the length of the third side $c$ using the cosine rule: + +$$ +c^{2}=a^{2}+b^{2}-2a b c o s(\Theta) +$$ + +5. Write a function to convert an integer to a binary character representation. It should take an integer argument and return a character string that is a sequence of zeros and ones. Use the program in Chapter 8 as a basis for the solution. + +# 15.20 Bibliography + +Abramowitz M., Stegun I., Handbook of Mathematical Functions , Dover, 1968. + +• This book contains a fairly comprehensive collection of numerical algo- rithms for many mathematical functions of varying degrees of obscurity. It is a widely used source. + +Association of Computing Machinery (ACM) • Collected Algorithms , 1960–1974 + +• Transactions on Mathematical Software , 1975 — A good source of more specialied algorithms. Early algorithms tended to be in Algol, Fortran now predominates. + +# 15.20.1 Recursion and problem solving + +The following are a number of books that look at the role of recursion in problem solving and algorithms. + +Hofstader D. R., Gödel, Escher, Bach — an Eternal Golden Braid , Harvester Press. + +• The book provides a stimulating coverage of the problems of paradox and contradiction in art, music and mathematics using the works of Escher, Bach and Gödel, and hence the title. There is a whole chapter on recursive structures and processes. The book also covers the work of Church and Turing, both of whom have made significant contributions to the theory of computing. + +Kruse R.L., Data Structures and Program Design , Prentice-Hall, 1994. + +• Quite a gentle introduction to the use of recursion and its role in problem solving. Good choice of case studies with explanations of solutions. Pascal is used. + +Sedgewick R., Algorithms in Modula 3 , Addison-Wesley, 1993. + +• Good source of algorithms. Well written. The GCD algorithm was taken from this source. + +Vowels R.A., Algorithms and Data Structures in $F$ and Fortran, Unicomp, 1998. + +• The only book currently that uses Fortran 90/95 and F. Visit the Fortran web site for more details. They are the publishers. http://www.fortran.com/fortran/market.html + +Wirth N., Algorithms $^+$ Data Structures $=$ Programs , Prentice-Hall, 1976. + +• In the context of this chapter the section on recursive algorithms is a very worthwhile investment in time. + +Wood D., Paradigms and Programming in Pascal , Computer Science Press. + +• Contains a number of examples of the use of recursion in problem solv- ing. Also provides a number of useful case studies in problem solving. + +# 16 + +# Control Structures + +“Summarizing: as a slow-witted human being I have a very small head and I had better learn to live with it and to respect my limitations and give them full credit, rather than try to ignore them, for the latter vain effort will be punished by fail- ure.” + +Edsger W. Dijkstra, Structured Programming + +# Aims + +The aims of this chapter are to introduce: + +• Selection among various courses of action as part of the algorithm. • The concepts and statements in Fortran needed to support the above: Logical expressions and logical operators. One or more blocks of statements. • The IF THEN ENDIF construct. • The IF THEN ELSE IF ENDIF construct. • To introduce the CASE statement with examples. • To introduce the DO loop, in three forms with examples, in particular: The iterative DO loop. The DO WHILE form. The DO ... IF THEN EXIT END DO or repeat until form. The CYCLE statement. The EXIT statement. + +# Chapter 16 + +# 16 Control Structures + +When we look at this area it is useful to gain some historical perspective concern- ing the control structures that are available in a programming language. + +At the time of the development of Fortran in the 1950s there was little theoretical work around and the control structures provided were very primitive and closely related to the capability of the hardware. + +At the time of the first standard in 1966 there was still little published work re- garding structured programming and control structures. The seminal work by Dahl, Dijkstra and Hoare was not published until 1972. + +By the time of the second standard there was a major controversy regarding lan- guages with poor control structures like Fortran which essentially were limited to the GOTO statement. The facilities in the language had led to the development and continued existence of major code suites that were unintelligible, and the pejorative term spaghetti was applied to these programs. Developing an understanding of what a program did became an almost impossible task in many cases. + +Fortran missed out in 1977 on incorporating some of the more modern and intelli- gible control structures that had emerged as being of major use in making code easier to understand and modify. + +It was not until the 1990 standard that a reasonable set of control structures had emerged and became an accepted part of the language. The more inquisitive reader is urged to read at least the work by Dahl, Dijkstra and Hoare to develop some un- derstanding of the importance of control structures and the role of structured programming. The paper by Knuth is also highly recommended as it provides a very balanced coverage of the controversy of earlier times over the GOTO state- ment. + +# 16.1 Selection among courses of action + +In most problems you need to choose among various courses of action, e.g., + +• If overdrawn, then do not draw money out of the bank. • If Monday, Tuesday, Wednesday, Thursday or Friday, then go to work. • If Saturday, then go to watch Queens Park Rangers. • If Sunday, then lie in bed for another two hours. + +As most problems involve selection between two or more courses of action it is necessary to have the concepts to support this in a programming language. Fortran has a variety of selection mechanisms, some of which are introduced below. + +# 16.1.1 The BLOCK IF statement + +The following short example illustrates the main ideas: + +. wake up . . check the date and time IF (Today $==$ Sunday) THEN . . lie in bed for another two hours . ENDIF . . get up . make breakfast + +If today is Sunday then the block of statements between the IF and the ENDIF is executed. After this block has been executed the program continues with the state- ments after the ENDIF. If today is not Sunday the program continues with the statements after the ENDIF immediately. This means that the statements after the ENDIF are executed whether or not the expression is true. The general form is: + +IF (Logical expression) THEN . Block of statements . ENDIF + +The logical expression is an expression that will be either true or false; hence its name. Some examples of logical expressions are given below: + +$(\mathtt{A l p h a}\ >=\ 10\cdot1)$ Test if Alpha is greater than or equal to 10.1 (Balance <= 0.0) Test if overdrawn (( Today $==$ Saturday).OR.( Today $=$ Sunday)) Test if today is Saturday or Sunday ((Actual - Calculated) <= 1.0E-6) Test if Actual minus Calculated is less than or equal to 1.0E-6 + +![Fortran has the following relational and logical operators: ](images/f77e0dadea4ffb868e9ef6f8e0b650341b0e60ba15592486c69d1d33ad36a61a.jpg) + +The first six should be self-explanatory. They enable expressions or variables to be compared and tested. The last three enable the construction of quite complex com- parisons, involving more than one test; in the example given earlier there was a test to see whether today was Saturday or Sunday. + +Use of logical expressions and logical variables (something not mentioned so far) is covered again in a later chapter on logical data types. + +The 'IF expression THEN statements ENDIF' is called a BLOCK IF construct. There is a simple extension to this provided by the ELSE statement. Consider the following example: + +IF (Balance $>0.0$ ) THEN . draw money out of the bank ELSE . borrow money from a friend ENDIF Buy a round of drinks. + +In this instance, one or other of the blocks will be executed. Then execution will continue with the statements after the ENDIF statement (in this case buy a round ). + +There is yet another extension to the BLOCK IF which allows an ELSEIF state- ment. Consider the following example: + +IF (Today $==$ Monday) THEN . ELSEIF (Today $==$ Tuesday) THEN + +ELSEIF (Today $=$ Wednesday) THEN . ELSEIF (Today $=$ Thursday) THEN . ELSEIF (Today $=$ Friday) THEN . ELSEIF (Today $=$ Saturday) THEN . ELSEIF (Today $=$ Sunday) THEN . ELSE there has been an error. The variable Today has taken on an illegal value. ENDIF + +Note that as soon as one of the logical expressions is true, the rest of the test is skipped, and execution continues with the statements after the ENDIF. This implies that a construction like + +IF( $\mathrm{~\ensuremath~{~\mathbb~{~T~}~}~}<\mathrm{~\ensuremath~{~2~}~}$ )THEN ... ELSEIF( $,\mathrm{~\textperthousand~}$ )THEN ... ELSE ... ENDIF + +is inappropriate. If I is less than 2, the latter condition will never be tested. The ELSE statement has been used here to aid in trapping errors or exceptions. This is recommended practice. A very common error in programming is to assume that the data are in certain well-specified ranges. The program then fails when the data go outside this range. It makes no sense to have a day other than Monday, Tuesday, Wednesday, Thursday, Friday, Saturday or Sunday. + +# 16.1.2 Example 1: Quadratic roots + +A quadratic equation is: + +$$ +\textit{a x}^{2}+\textit{b x}+\textit{c}\!=\!0 +$$ + +This program is straightforward, with a simple structure. The roots of the quadratic are either real, equal and real, or complex depending on the magnitude of the term $\mathrm{B}\mathrm{~}^{**}\mathrm{~}2\mathrm{~-~}4\mathrm{~}^{*}\mathrm{~A~}^{*}\mathrm{~C~}$ . The program tests for this term being greater than or less than + +# Chapter 16 + +zero: it assumes that the only other case is equality to zero (from the mechanics of a computer, floating point equality is rare, but we are safe in this instance): + +PROGRAM ch1601 IMPLICIT NONE REAL :: A , B , C , Term , A2 , Root1 , Root2 + + ! + + ! a b and c are the coefficients of the terms + + ! $\mathsf{a}^{\star}\mathsf{x}^{\star\star}\mathsf{2}\!+\!\mathsf{b}^{\star}\mathsf{x}\!+\!\mathsf{C} + +$ ! find the roots of the quadratic, root1 and root2 + + ! PRINT\*,' GIVE THE COEFFICIENTS A, B AND C' READ\*,A,B,C $\mathrm{Tem}\;\;=\;\;{\mathbb{B}}^{\star}{\mathbb{B}}\;\;\;-\;\;\;4\;.\;^{\star}{\mathbb{A}}^{\star}{\mathbb{C}}$ $\begin{array}{r l r}{\mathbb{A}2}&{{}=}&{\mathbb{A}^{\star}2}\end{array}$ . + + ! if term $<\quad0$ , roots are complex + + ! if term $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\$ , roots are equal + + ! if term $>\quad0$ , roots are real and different IF(Term $<\quad0\cdot0$ )THEN PRINT\*,' ROOTS ARE COMPLEX' ELSEIF(Term $>\ \ \ 0\cdot0$ )THEN Term $=$ SQRT(Term) Root1 $=$ (-B+Term)/A2 Root2 $=$ (-B-Term)/A2 PRINT\*,' ROOTS ARE ',Root1,' AND ',Root2 ELSE Root1 $=$ -B/A2 PRINT\*,' ROOTS ARE EQUAL, AT ',Root1 ENDIF + +END PROGRAM ch1601 + +# 16.1.3 Note + +Given the understanding you now have about real arithmetic and finite precision will the ELSE block above ever be executed? + +# 16.1.4 Example 2: Date calculation + +This next example is also straightforward. It demonstrates that, even if the condi- tions on the IF statement are involved, the overall structure is easy to determine. The comments and the names given to variables should make the program self-explanatory. Note the use of integer division to identify leap years: + +PROGRAM ch1602 IMPLICIT NONE + +INTEGER :: Year , N , Month , Day , T + + ! + + ! calculates day and month from year and + + ! day-within-year + + ! t is an offset to account for leap years. + + ! Note that the first criteria is division by 4 + + ! but that centuries are only + + ! leap years if divisible by 400 + + ! not 100 (4 \* 25) alone. + + ! PRINT\*,' year, followed by day within year' READ\*,Year,N + + ! checking for leap years IF ((Year/4)\*4 == Year ) THEN $\scriptstyle{\mathrm{T}}={\underline{{1}}}$ IF ((Year/400)\*400 == Year ) THEN $\scriptstyle{\mathrm{T}}={\underline{{1}}}$ ELSEIF ((Year/100) $^{\star}\!\perp\!0\,0$ == Year) THEN $\scriptstyle{\mathrm{T}}=0$ ENDIF ELSE $\scriptstyle{\mathrm{T}}=0$ ENDIF + + ! accounting for February IF(N $>$ ( $5\,9\,{+}\,\mathrm{T}$ ))THEN $\scriptstyle\mathrm{Day=N}+2\,-\mathrm{T}$ ELSE Day $\tt\ddot{=}N$ ENDIF Month $\b=$ (Day+91)\*100/3055 Day $\leftleftarrows$ (Day+91)-(Month\*3055)/100 Month $\risingdotseq$ Month-2 PRINT\*,' CALENDAR DATE IS ', Day , Month , Year END PROGRAM ch1602 + +# 16.1.5 The CASE statement + +The CASE statement provides a very clear and expressive selection mechanism be- tween two or more courses of action. Strictly speaking it could be constructed from the IF THE ELSE IF ENDIF statement, but with considerable loss of clarity. Re- member that programs have to be read and understood by both humans and compilers! + +# Chapter 16 + +# 16.1.6 Example 3: Simple calculator + +PROGRAM ch1603 IMPLICIT NONE + + ! + + ! Simple case statement example + + ! INTEGER :: I,J,K CHARACTER :: Operator DO PRINT \*,' Type in two integers' READ \*, I,J PRINT \*,' Type in operator' READ '(A)',Operator Calculator : & SELECT CASE (Operator) CASE $(\ "+"\ )$ Calculator $\scriptstyle\mathrm{K=}\mathrm{T}+\mathrm{J}$ PRINT \*,' Sum of numbers is ',K CASE ('-') Calculator $\mathtt{K}{=}\mathbb{T}{-}\mathbb{T}$ PRINT \*,' Difference is ',K CASE ('/') Calculator $\scriptstyle{\mathrm{K}}=\mathbb{T}/\mathbb{T}$ PRINT \*,' Division is ',K CASE ('\*') Calculator $\scriptstyle\mathrm{K}=\mathbb{T}^{\star}\mathbb{T}$ PRINT \*,' Multiplication is ',K CASE DEFAULT Calculator EXIT END SELECT Calculator END DO END PROGRAM ch1603 + +The user is prompted to type in two integers and the operation that they would like carried out on those two integers. The CASE statement then ensures that the appro- priate arithmetic operation is carried out. The program terminates when the user types in any character other than $+,\,-,^{*}$ or /. + +The CASE DEFAULT option introduces the EXIT statement. This statement is used in conjunction with the DO statement. When this statement is executed con- trol passes to the statement immediately after the matching END DO statement. In the example above the program terminates, as there are no executable statements after the END DO. + +# 16.1.7 Example 4: Counting vowels, consonants, etc. + +This example is more complex, but again is quite easy to understand. The user types in a line of text and the program produces a summary of the frequency of the characters typed in: + +PROGRAM ch1604 IMPLICIT NONE + +! + + ! Simple counting of vowels, consonants, + + ! digits, blanks and the rest + + ! INTEGER :: Vowels $=\!0$ , Consonants $=\!0$ , Digits $=\!0$ INTEGER :: Blank $=\!0$ , Other $=\!0$ , I CHARACTER :: Letter CHARACTER (LEN $\mathtt{=}\mathtt{80}$ ) :: Line READ '(A)', Line DO $\mathbb{T}\!=\!1$ ,80 Letter ${}^{\ast}=$ Line(I:I) ! the above extracts one character at position I SELECT CASE (Letter) CASE ('A','E','I','O','U', & 'a','e','i','o','u') Vowels $=$ Vowels $^+$ 1 CASE ('B','C','D','F','G','H', & 'J','K','L','M','N','P', & 'Q','R','S','T','V','W', & 'X','Y','Z', & 'b','c','d','f','g','h', & 'j','k','l','m','n','p', & 'q','r','s','t','v','w', & 'x','y','z') Consonants $=$ Consonants + 1 CASE ('1','2','3','4','5','6','7','8','9','0') Digits $=$ Digits $^+$ 1 CASE (' ') Blank $\b=$ Blank $^+$ 1 CASE DEFAULT Other $=$ Other $+1$ END SELECT END DO + +# Chapter 16 + +PRINT \*, ' Vowels = ', Vowels PRINT \*, ' Consonants $=$ ', Consonants PRINT \*, ' Digits $\c=\qquad\qquad$ , Digits PRINT \*, ' Blanks $=$ ',Blank PRINT \*, ' Other characters $\c=\qquad\qquad$ , Other END PROGRAM ch1604 + +# 16.2 The three forms of the DO statement + +You have already been introduced in the chapters on arrays to the iterative form of the DO loop, i.e., + +DO Variable $=$ Start, End, Increment block of statements + +END DO + +A complete coverage of this form is given in the three chapters on arrays. + +There are two additional forms of the block DO that complete our requirements: + +DO WHILE (Logical Expression) block of statements + +END DO + +The first form is often called a WHILE loop as the block of statements executes whilst the logical expression is true, and the second form is often called a RE- PEAT UNTIL loop as the block of statements executes until the statement is true. + +Note that the WHILE block of statements may never be executed, and the RE- PEAT UNTIL block will always be executed at least once. + +# 16.2.1 Example 5: Sentinel usage + +The following example shows a complete program using this construct: + +PROGRAM ch1605 IMPLICIT NONE ! this program picks up the first occurrence + +! of a number in a list. + + ! a sentinel is used, and the array is 1 more + + ! than the max size of the list. INTEGER , ALLOCATABLE , DIMENSION(:) :: A INTEGER :: Mark INTEGER :: I,Howmany OPEN (UNIT $\vDash\vDash$ ,FILE $=$ 'DATA') PRINT \*,' What number are you looking for?' READ \*, Mark PRINT \*,' How many numbers to search?' READ \*,Howmany ALLOCATE(A(1:Howmany $+1$ )) READ(UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) (A(i), $\scriptstyle\mathbb{T}=1$ ,Howmany) $\scriptstyle{\mathbb{T}}=1$ A(Howmany $+1$ ) $=$ Mark DO WHILE(Mark / $=$ A(I)) $\mathbb{T}\!=\!\mathbb{T}\!+\!\mathbb{1}$ END DO IF( $\begin{array}{r l}{\mathbb{T}}&{{}==}\end{array}$ (Howmany $+1$ )) THEN PRINT\*,' ITEM NOT IN LIST' ELSE PRINT\*,' ITEM IS AT POSITION ',I ENDIF END PROGRAM ch1605 + +The repeat until construct is written in Fortran as: + +DO ... ... IF (Logical Expression) EXIT END DO + +There are problems in most disciplines that require a numerical solution. The two main reasons for this are either that the problem can only be solved numerically or that an analytic solution involves too much work. Solutions to this type of problem often require the use of the repeat until construct. The problem will typically re- quire the repetition of a calculation until the answers from successive evaluations differ by some small amount, decided generally by the nature of the problem. A program extract to illustrate this follows: + +REAL , PARAMETER :: TOL $\vartriangleleft$ .0E-6 + +DO ... CHANGE $_{\parallel}=$ ... IF (CHANGE $<=$ TOL) EXIT END DO + +Here the value of the tolerance is set to 1.0E–6. Note again the use of the EXIT statement. The DO END DO block is terminated and control passes to the state- ment immediately after the matching END DO. + +# 16.2.2 CYCLE and EXIT + +These two statements are used in conjunction with the block DO statement. You have seen examples above of the use of the EXIT statement to terminate the block DO, and pass control to the statement immediately after the corresponding END DO statement. + +The CYCLE statement can appear anywhere in a block DO and will immediately pass control to the start of the block DO. Examples of CYCLE and EXIT are given in later chapters. + +# 16.2.3 Example 6: $\mathbf{e}^{**}\!x$ evaluation + +The function etox illustrates one use of the repeat until construct. The function evaluates $\mathrm{e}^{**}x.$ . This may be written as + +$$ +1+x/1!+x^{2}/2!+x^{3}/3!\ldots +$$ + +or + +$$ +1+\sum_{n=1}^{\infty}\ {\frac{x^{\ n-1}}{(n-1)!}}{\frac{x}{n}} +$$ + +Every succeeding term is just the previous term multiplied by $x/n$ . At some point the term $x/n$ becomes very small, so that it is not sensibly different from zero, and successive terms add little to the value. The function therefore repeats the loop un- til $x/n$ is smaller than the tolerance. The number of evaluations is not known beforehand, since this is dependent on $x$ : + +REAL FUNCTION etox(X) IMPLICIT NONE REAL :: Term REAL , INTENT(IN) :: X INTEGER :: Nterm REAL , PARAMETER ::Tol $=$ 1.0E-6 etox $\mathbf{\Psi}\!=\!\!1$ .0 + +Term $\mathbf{\Psi}\!=\!\!1$ .0 Nterm $\scriptstyle{.=0}$ DO Nterm $=$ Nterm $+1$ Term $=$ ( X / Nterm) \* Term etox $=$ etox $^+$ Term IF(ABS(Term) $<=$ Tol)EXIT END DO END FUNCTION etox + +program ch1606 implicit none real :: etox real , parameter :: $\scriptstyle\times=\perp$ .0 real :: y print \*,' Fortran intrinsic ',exp(x) y=etox(x) print \*,' User defined etox ',y end program ch1606 + +The whole program compares the user defined function with the Fortran intrinsic exp function. + +# 16.2.4 Example 7: Wave breaking on an offshore reef + +This example is drawn from a situation where a wave breaks on an offshore reef or sand bar, and then reforms in the near-shore zone before breaking again on the coast. It is easier to observe the heights of the reformed waves reaching the coast than those incident to the terrace edge. + +Both types of loops are combined in this example. The algorithm employed here finds the zero of a function. Essentially, it finds an interval in which the zero must lie; the evaluations on either side are of different signs. The while loop ensures that the evaluations are of different signs, by exploiting the knowledge that the in- cident wave height must be greater than the reformed wave height (to give the lower bound). The upper bound is found by experiment, making the interval bigger and bigger. Once the interval is found, its mean is used as a new potential bound. The zero must lie on one side or the other; in this fashion, the interval containing the zero becomes smaller and smaller, until it lies within some tolerance. This ap- proach is rather plodding and unexciting, but is suitable for a wide range of problems + +Here is the program: + +# Chapter 16 + +PROGRAM Break + +IMPLICIT NONE + +REAL :: Hi , Hr , Hlow , High , Half , Xl REAL :: Xh , Xm , D REAL , PARAMETER :: Tol $\mathbf{\omega}=\mathbb{1}$ .0E-6 + + ! problem - find hi from expression given + + ! in function f + + ! $\mathrm{{F}{=}\bar{A}^{\star}}$ (1.0-0.8\*EXP(-0.6\*C/A))-B + + ! HI IS INCIDENT WAVE HEIGHT (C) + + ! HR IS REFORMED WAVE HEIGHT (B) + + ! D IS WATER DEPTH AT TERRACE EDGE (A) PRINT\*,' Give reformed wave height, and water depth' READ\*,Hr,d + + ! + + ! for Hlow- let Hlow ${}^{\ast}{=}$ hr + + ! for high- let high $\mathbf{\xi}.\overline{{\overline{{\mathbf{\xi}}}}}$ Hlow\*2.0 + + ! + + ! check that signs of function results are different + + ! Hlow $=$ Hr High $=$ Hlow\*2.0 Xl = F( Hlow, Hr, D) Xh $=$ F( High, Hr, D) + + ! DO WHILE ((XL\*XH) >= 0.0) HIGH $=$ $\mathrm{H}\mathbb{I}\mathrm{G}\mathrm{H}^{\star}2\cdot0$ $\begin{array}{r l r}{\mathrm{XH}}&{{}=}&{\mathrm{F}\left(\mathrm{H}\mathbb{I}\mathrm{G}\mathrm{H}\,,\mathrm{HR}\,,\mathbb{D}\right)}\end{array}$ XH = F(HIGH,HR,D) END DO + + ! DO HALF $=$ (HLOW+HIGH)\*0.5 $\scriptstyle{\mathrm{XM}}=\operatorname{F}$ (HALF,HR,D) IF((XL\*XM) < 0.0)THEN $\scriptstyle\mathrm{XH=XM}$ HIGH $=$ HALF ELSE $\scriptstyle{\mathrm{KL}}=\!{\mathrm{XM}}$ HLOW $=$ HALF ENDIF IF(ABS(HIGH-HLOW) $<=$ TOL)EXIT END DO PRINT\*,' Incident Wave Height Lies Between' PRINT\*,Hlow,' and ',High,' metres' + +CONTAINS REAL FUNCTION F(A,B,C) IMPLICIT NONE REAL , INTENT (IN) :: A REAL , INTENT (IN) :: B REAL , INTENT (IN) :: C $\mathrm{{F}{=}\mathrm{{A}}^{*}}$ (1.0-0.8\*EXP(-0.6\*C/A))-B END FUNCTION F END PROGRAM Break + +# 16.3 Summary + +You have been introduced in this chapter to several control structures and these in- clude: + +• The block if . • The if then else if . • The case construct. • The block do in three forms: • The iterative do or do variable $=$ start,end,increment ... end do . • The while construct, or do while ... end do . • The repeat until construct, or do ... if then exit end do . • The cycle and exit statements, which can be used with do statement in all three forms: • The do variable $=$ start,end,increment ... end do . • The while construct, or do while ... end do . • The repeat until construct, or do ... if then exit end do. + +These constructs are sufficient for solving a wide class of problems. There are other control statements available in Fortran, especially those inherited from For- tran 66 and Fortran 77, but those covered here are the ones preferred. We will look in Chapter 28 at one more control statement, the so-called GOTO statement, with recommendations as to where its use is appropriate. + +# 16.3.1 Control structure formal syntax + +CASE + +# Chapter 16 + +SELECT CASE ( case variable ) [ CASE case selector [executable construct ] ... ] ... [ CASE DEFAULT [executable construct ] END SELECT + +# DO + +DO [ label ] [executable construct ] ... + +do termination + +DO [ label ] [ , ] loop variable $=$ initial value , final value , [ increment ] [executable construct ] ... do termination + +DO [ label ] [ , ] WHILE (scalar logical expression ) [executable construct ] ... + +do termination + +# IF + +IF ( scalar logical expression ) THEN [executable construct ] ... + + [ ELSE IF ( scalar logical expression THEN [executable construct ] ... ] ...] + + [ ELSE [executable construct ] ...] END IF + +# 16.4 Problems + +1. Rewrite the program for the period of a pendulum. The new program should print out the length of the pendulum and period, for pendulum lengths from 0 to $100~\mathrm{cm}$ in steps of $0.5~\mathrm{cm}$ . The program should incorporate a function for the eval- uation of the period. + +2. Write a program to read an integer that must be positive. + +Hint . Use a DO WHILE to make the user re-enter the value. + +3. Using functions, do the following: • Evaluate $n$ ! from $n=0$ to $n=10$ . • Calculate 76! • Now calculate $(x^{**}n)/n!$ , with $x=13.2$ and $n=20$ . • Now do it another way. + +4. The program BREAK is taken from a real example. In the particular problem, the reformed wave height was 1 metre, and the water depth at the reef edge was 2 metres. What was the incident wave height? Rather than using an absolute value for the tolerance, it might be more realistic to use some value related to the re- formed wave height. These heights are unlikely to be reported to better than about $5\%$ accuracy. Wave energy may be taken as proportional to wave height squared for this example. What is the reduction in wave energy as a result of breaking on the reef or bar for this particular case. + +5. What is the effect of using INT on negative real numbers? Write a program to demonstrate this. + +6. How would you find the nearest integer to a real number? Now do it another way. Write a program to illustrate both methods. Make sure you test it for negative as well as positive values. + +7. The function etox has been given in this chapter. The standard Fortran function EXP does the same job. Do they give the same answers? Curiously the Fortran standard does not specify how a standard function should be evaluated, or even how accurate it should be. + +The physical world has many examples in which processes require that some threshold be overcome before they begin operation: critical mass in nuclear reac- tions, a given slope to be exceeded before friction is overcome, and so on. Unfortunately, most of these sorts of calculations become rather complex and not really appropriate here. The following problem tries to restrict the range of calcula- tion, whilst illustrating the possibilities of decision making. + +8. If a cubic equation is expressed as + +$$ +z^{3}+a_{\mathrm{~\,2~}}z^{2}+a_{\mathrm{~\,1~}}z+a_{\mathrm{~\,0~}}\!=\!0 +$$ + +and we let + +$$ +q\!=\!\frac{a_{_1}}{3}-\frac{(\;a_{_2}\;^{*}\;a_{_2}\;)}{9} +$$ + +and + +$$ +r\,{=}\,\frac{(\ a_{_1}\ ^{*}\ a_{_2}\ -3\ ^{*}\ a_{_0}\ )}{6}\,{-}\,\frac{(\ a_{_2}\ ^{*}\ a_{_2}\ ^{*}\ a_{_2}\ )}{27} +$$ + +# Chapter 16 + +we can determine the nature of the roots as follows: + +$q_{\circ}^{3}+r_{\circ}^{2}>0$ ; one real root and a pair of complex ${\overset{*}{q}}^{3}+r^{2}=0$ ; all roots real, and at least two equal $\overset{\cdot}{q}^{3}+\,r^{2}<0$ ; all roots real + +Incorporate this into a suitable program, to determine the nature of the roots of a cubic from suitable input. + +9. The form of breaking waves on beaches is a continuum, but for convenience we commonly recognise three major types: surging, plunging and spilling. These may be classified empirically by reference to the wave period, $T$ (seconds), the breaker wave height, $H_{b}$ (metres), and the beach slope, $m$ . These three variables are com- bined into a single parameter, $B$ , where + +$$ +B=H\!b/(g m T^{2}) +$$ + +$g$ is the gravitational constant $(981~\mathrm{cm~s}^{-2})$ ). If $B$ is less than 0.003, the breakers are surging; if $B$ is greater than 0.068, they are spilling, and between these values, plunging breakers are observed. + +(i) On the east coast of New Zealand, the normal pattern is swell waves, with wave heights of 1 to 2 metres and wave periods of 10 to 15 seconds. During storms, the wave period is generally shorter, say 6 to 8 seconds, and the wave heights higher, 3 to 5 metres. The beach slope may be taken as about 0.1. What changes occur in breaker characteristics as a storm builds up? + +(ii) Similarly, many beaches have a concave profile. The lower beach generally has a very low slope, say less than 1 degree $m\,=\,0.018)$ ), but towards the high-tide mark, the slope increases dramatically, to say 10 degrees or more $m=0.18)$ ). What changes in wave type will be observed as the tide comes in? + +# 16.5 Bibliography + +Dahl O.J., Dijkstra E.W., Hoare C.A.R., Structured Programming , Academic Press, 1972. + +• This is the original text, and a must. The quote at the start of the chapter by Dijkstra summarises beautifully our limitations when programming and the discipline we must have to master programming successfully. + +Knuth D.E., Structured Programming with GOTO Statements, in Current Trends in Programming Methodology, Volume 1, Prentice-Hall, 1977. + +• The chapter by Knuth provides a very succinct coverage of the arguments for the adoption of structured programming, and dispels many of the myths concerning the use of the GOTO statement. Highly recommended. + +# 17 + +Characters + +“These metaphysics of magicians, And necromantic books are heavenly; Lines, circles, letters and characters.” + +Christopher Marlowe, The Tragical History of Doctor Faustus + +# Aims + +The aims of this chapter are: + +• To extend the ideas about characters introduced in earlier chapters. • To demonstrate that this enables us to solve a whole new range of prob- lems in a satisfactory way. + +# Chapter 17 + +# 17 Characters + +For each type in a programming language there are the following concepts: + +• Values are drawn from a finite domain. • There are a restricted number of operations defined for each type. + +For the numeric types we have already met, integers and reals: + +• The values are either drawn from the domain of integer numbers or the domain of real numbers. • The valid operations are addition, subtraction, multiplication, division and exponentiation. + +For the character data type the basic unit is an individual character — any charac- ter which is available on your keyboard normally. To ensure portability we should restrict ourselves to the Fortran character set, that is: + +• the alphabetic characters A through Z, the digits or numeric characters 0 through 9, and the underscore character _ + +which may be used in variable names, and the complete Fortran character set is given in section 7.6 in Chapter 7. + +This provides us with 58 printing characters and omits many commonly used char- acters, e.g., lower case letters. However, if one does work with this set then one can ensure that programs are portable. + +As the most common current internal representation for the character data type uses 8 bits this should provide access to 256 $(2^{8}$ ) characters. However, there is lit- tle agreement over the encoding of these 256 possible characters, and the best you can normally assume is access to the ASCII character set, which is given in Ap- pendix B. One of the problems at the end of this chapter looks at determining what characters one has available. + +The only operations defined are concatenation (joining character strings together) and comparison. + +We will look into the area of character sets in more depth later in this chapter. + +We can declare our character variables: + +CHARACTER :: A, String, Line + +Note that there is no default typing of the character variable (unlike integer and real data types), and we can use any convenient name within the normal Fortran conventions. In the declaration above, each character variable would have been permitted to store one character. This is limiting, and, to allow character strings which are several units long, we have to add one item of information: + +CHARACTER (10) :: A CHARACTER (16) :: String CHARACTER (80) :: Line + +This indicates that A holds 10 characters, STRING holds 16, and LINE holds 80. If all the character variables in a single declaration contain the same number of characters, we can abbreviate the declaration to + +CHARACTER(80) :: LIST, STRING, LINE + +But we cannot mix both forms in the one declaration. We can now assign data to these variables, as follows: + +$\mathrm{\tt{A}}={}^{\mathrm{\tt{I}}}$ FIRST ONE 'STRING $\mathit{\Pi}^{\prime}\mathrm{\Delta}$ LONGER ONE ' LINE $=$ 'THE QUICK BROWN FOX JUMPS OVER THE LAZY DOG' + +The delimiter apostrophe (') or quotation mark (“) is needed to indicate that this is a character string (otherwise the assignments would have looked like invalid vari- able names). + +# 17.1 Character input + +In an earlier chapter we saw how we could use the READ \* and PRINT \* state- ments to do both numeric and character input and output or I/O. When we use this form of the statement we have to include any characters we type within delimiters (either the apostrophe ' or the quotation mark “). This is a little restricting and there is a slightly more complex form of the READ statement that allows one to just type the string on its own. The following two programs illustrate the differ- ences: + +PROGRAM ch1701 + + ! + + ! Simple character i/o + + ! CHARACTER (80) :: Line READ \*, Line PRINT \*, Line END PROGRAM ch1701 + +This form requires enclosing the string with delimiters. Consider the next form: + +# Chapter 17 + +PROGRAM ch1702 + +! + + ! Simple character i/o + + ! + +CHARACTER (80) :: Line READ '(A)' , Line PRINT \*,Line END PROGRAM ch1702 + +With this form one can just type the string in and input terminates with the car- riage return key. The additional syntax involves '(A)' where '(A)' is a character edit descriptor. The simple examples we have used so far have used implied format specifiers and edit descriptors. For each data type we have one or more edit descriptors to choose from. For the character data type only the A edit descriptor is available. + +# 17.2 Character operators + +The first manipulator is a new operator — the concatenation operator //. With this operator we can join two character variables to form a third, as in + +CHARACTER (5) :: FIRST, SECOND CHARACTER (10) :: THIRD FIRST $=$ 'THREE' SECOND $=$ 'BLIND' ... THIRD $\mathrm{=}$ FIRST//SECOND . THIRD $\mathrm{=}$ FIRST//'MICE' + +Where there is a discrepancy between the created length of the concatenated string and the declared lengths of the character strings, truncation will occur. For exam- ple, + +THIRD $\mathrm{=}$ FIRST//' BLIND MICE' + +will only append the first five characters of the string ' BLIND MICE' i.e., ' BLIN', and THIRD will therefore contain 'THREE BLIN'. + +What would happen if we assigned a character variable of length 'n' a string which was shorter than n? For example, + +CHARACTER (4) :: C2 $\mathsf{C}2\!=\!\mathsf{\dot{\Delta}A B}^{\dagger}$ + +The remaining two characters are considered to be blank, that is, it is equivalent to saying + +C2='AB + +However, while the strings 'AB' and 'AB ' are equivalent, 'AB' and ' AB' are not. In the jargon, the character strings are always left justified, and the unset char- acters are trailing blanks. + +If we concatenate strings which have 'trailing blanks', the blanks, or spaces, are considered to be legitimate characters, and the concatenation begins after the end of the first string. Thus + +CHARACTER (4) :: C2,C3 CHARACTER (8) :: JJ $\mathsf{C}2\!={}^{!}\,\mathrm{A}$ ' $\mathsf{C}3\!={}^{\!\mathsf{\iota}}\,\mathtt{M A N}$ ' $\scriptstyle{\overline{{\cup}}}{\overline{{=}}}{\mathsf{C}}2/{\mathsf{C}}3$ PRINT\*, 'THE CONCATENATION OF ',C2,' AND ',C3,' IS' PRINT\*,JJ + +would appear as + +THE CONCATENATION OF A AND MAN GIVES + +A MAN + +at the terminal. + +Sometimes we need to be able to extract parts of character variables — substrings. The actual notation for doing this is a little strange at first, but it is very powerful. To extract a substring we must reference two items: + +• The position in the string at which the substring begins. + +and + +• The position at which it ends. + +e.g., STRING $=$ 'SHARE AND ENJOY' + +# 17.3 Character substrings + +We may extract parts of this string: + +BIT $^{\ast}{=}$ STRING(3:5) + +would place the characters 'ARE' into the variable BIT. This may be manipulated further: + +BIT1 $=$ STRING(2:4)//STRING(9:9) BIT2 $=$ STRING(5:5) // & STRING(3:3)//STRING(1:1)//STRING(15:15) + +Note that to extract a single character we reference its beginning position and its end (i.e., repeat the same position), so that + +STRING(3:3) + +gives the single character 'A'. The substring reference can cut out either one of the two numerical arguments. If the first is omitted, the characters up to and including the reference are selected, so that + +SUB $,=$ STRING(:5) + +would result in SUB containing the characters 'SHARE'. When the second argu- ment is omitted, the characters from the reference are selected, so that + +SUB $,=$ STRING(11:) + +would place the characters 'ENJOY' in the variable SUB. In these examples it would also be necessary to declare STRING, SUB, BIT, BIT1 and BIT2 to be of CHARACTER type, of some appropriate length. + +Character variables may also form arrays: CHARACTER (10) , DIMENSION(20) :: A + +sets up a character array of twenty elements, where each element contains ten char- acters. In order to extract substrings from these array elements, we need to know where the array reference and the substring reference are placed. The array refer- ence comes first, so that + +DO $\mathbb{T}\!=\!1$ ,20 FIRST $\mathrm{\Lambda}^{!}{=}\mathrm{A}$ (I)(1:1)ENDO + +places the first character of each element of the array into the variable FIRST. The syntax is therefore 'position in array, followed by position within string'. + +Any argument can be replaced by a variable: + +STRING(I:J) + +This offers interesting possibilities, since we can, for example, strip blanks out of a string: + +PROGRAM ch1703 IMPLICIT NONE CHARACTER(80) :: String, Strip INTEGER :: IPOS,I,Length $\scriptstyle{\boldsymbol{\cdot}}=8\,0$ $\scriptstyle\mathbb{T P O S}\,=\,0$ PRINT \*,' Type in a string' READ '(A)',String DO $\mathbb{T}\!=\!1$ ,Length IF(String(I:I) /= ' ') THEN IPOS $=$ IPOS+1 Strip(IPOS:IPOS) $=$ String(I:I) ENDIF END DO PRINT\*,String PRINT\*,Strip END PROGRAM ch1703 + +# 17.4 Character functions + +There are special functions available for use with character variables: INDEX will give the starting position of a string within another string. If, for example, we were looking for all occurrences of the string 'Geology' in a file, we could construct something like: + +PROGRAM ch1704 IMPLICIT NONE CHARACTER (80) :: Line INTEGER :: I DO READ '(A)', Line $\mathbb{T}=$ INDEX(Line,'Geology') IF (I $/=~\,0$ ) THEN PRINT \*, ' String Geology found at position ',I PRINT \*, ' in line ', Line EXIT ENDIF ENDDO END PROGRAM ch1704 + +There are two things to note about this program. Firstly the INDEX function will only report the first occurrence of the string in the line; any later occurrences in any particular line will go unnoticed, unless you account for them in some way. Secondly, if the string does not occur, the result of the INDEX function is zero, and given the infinite loop (DO ENDDO) the program will crash at run time with an end of file error message. This isn't good programming practice. + +LEN provides the length of a character string. This function is not immediately useful, since you really ought to know how many characters there are in the string. However, as later examples will show, there are some cases where it can be useful. Remember that trailing blanks do count as part of the character string, and contrib- ute to the length. + +The following example illustrates the use of both LEN and LEN_TRIM: + +PROGRAM ch1705 IMPLICIT NONE CHARACTER (LEN $\scriptstyle{\dot{=}20}$ ) :: Name INTEGER :: Name_Length PRINT \*,' Type in your name' READ '(A)',Name + +! show LEN first + + ! + +Name_length $\mathbf{\xi}.\overline{{\overline{{\mathbf{\xi}}}}}$ LEN(Name) PRINT \*,' Name length is ',Name_length PRINT \*,' ',Name(1:Name_length),'<-end is here' Name_length $\b=$ LEN_TRIM(Name) PRINT \*,' Name length is ',Name_length PRINT \*,' ',Name(1:Name_Length),'<-end is here' END PROGRAM ch1705 + +# 17.5 Collating sequence + +The next group of functions need to be considered together. They revolve around the concept of a collating sequence. In other words, each character used in Fortran is ordered as a list and given a corresponding weight . No two weights are equal. Although Fortran has only 58 defined characters, the machine you use will gener- ally have more; 95 printing characters is a typical minimum number. On this type of machine the weights would vary from 0 to 94. There is a defined collating se- quence, the ASCII sequence, which is likely to be the default. The parts of the collating sequence which are of most interest are fairly standard throughout all col- lating sequences. + +In general, we are interested in the numerals (0–9), the alphabetics (A–Z) and a few odds and ends like the arithmetic operators $(+\mathrm{~-~}/\mathrm{~}^{*})$ , some punctuation (. and ,) and perhaps the prime ('). As you might expect, 0–9 carry successively higher weights (though not the weights 0 to 9), as do A to Z. The other odds and ends are a little more problematic, but we can find out the weights through the function ICHAR. This function takes a single character as argument and returns an integer value. The ASCII weights for the alphanumerics are as follows: + +0–9 48–57 A–Z 65–90 + +One of the exercises is to determine the weights for other characters. The reverse of this procedure is to determine the character from its weighting, which can be achieved through the function CHAR. CHAR takes an integer argument and re- turns a single character. Using the ASCII collating sequence, the alphabet would be generated from + +DO I=65,90 PRINT\*,CHAR(I) ENDDO + +![This idea of a weighting can then be used in four other functions: ](images/38d56528360cb0640a5ccbeeecb36cc6dfa4e755fd0b1c8acde14f0aa0e71220.jpg) + +In the sequence we have seen before, A is lexically less than B, i.e., its weight is less. Clearly, we can use ICHAR and get the same result. For example, + +is equivalent to + +but these functions can take character string arguments of any length. They are not restricted to single characters. + +These functions provide very powerful tools for the manipulation of characters, and open up wide areas of nonnumerical computing through Fortran. Text format- ting and word processing applications may now be tackled (conveniently ignoring the fact that lower-case characters may not be available). + +There are many problems that require the use of character variables. These range from the ability to provide simple titles on reports, or graphical output, to the pro- vision of a natural language interface to one of your programs, i.e., the provision of an English-like command language. Software Tools by Kernighan and Plauger contains many interesting uses of characters in Fortran. + +# 17.6 Summary + +Characters represent a different data type to any other in Fortran, and as a conse- quence there is a restricted range of operations which may be carried out on them. + +A character variable has a length which must be assigned in a CHARACTER dec- laration statement. + +Character strings are delimited by apostrophes (') or quotation marks (“). Within a character string, the blank is a significant character. + +Character strings may be joined together (concatenated) with the // operator. + +Substrings occurring within character strings may be also be manipulated. There are a number of functions especially for use with characters: + +•ACHAR• ADJUSTL • ADJUSTR • CHAR • IACHAR • INDEX • LEN • LEN_TRIM • LLE • LGE • LGT • LLT • REPEAT • SCAN + +• TRIM • VERIFY + +# 17.7 Problems + +1. Suggest some circumstances where PRIME $\risingdotseq$ ''' might be useful. What other al- ternative is there and why do you think we use that instead? + +2. Write a program to write out the weights for the Fortran character set. Modify this program to print out the weights of the complete implementation defined char- acter set for your version of Fortran 90. Is it ASCII? If not, how does it differ? + +3. Use the INDEX function in order to find the location of all the strings 'IS' in the following data: + +IF A PROGRAMMER IS FOUND TO BE INDISPENSABLE, THE BEST THING TO DO IS TO GET RID OF HIM AS QUICKLY AS POSSIBLE. + +4. Find the 'middle' character in the following strings. Do you include blanks as characters? What about punctuation? + +PRACTICE IS THE BEST OF ALL INSTRUCTORS. EXPERIENCE IS A DEAR TEACHER, BUT FOOLS WILL LEARN AT NO OTHER. + +5. In English, the order of occurrence of the letters, from most frequent to least is + +E, T, A, O, N, R, I, S, H, D, L, F, C, M, U, G, Y, P, W, B, V, K, X, J, Q, Z + +Use this information to examine the two files given in Appendix D (one is a trans- lation of the other) to see if this is true for these two extracts of text. The second text is in medieval Latin (c. 1320). Note that a fair amount of compression has been achieved by expressing the passage in Latin rather than modern English. Does this provide a possible model for information compression? + +6. A very common cypher is the substitution cypher, where, for example, every let- ter A is replaced by (say) an M, every B is replaced by (say) a Y, and so on. These encyphered messages can be broken by reference to the frequency of occurrence of the letters (given in the previous question). + +Since we know that (in English) E is the most commonly occurring letter, we can assume that the most commonly occurring letter in the encyphered message repre- sents an E; we then repeat the process for the next most common and so on. Of course, these correspondences may not be exact, since the message may not be long enough to develop the frequencies fully. + +However, it may provide sufficient information to break the cypher. + +# Chapter 17 + +The file given in Appendix E contains an encoded message. Break it. Clue — $P g$ Fybdujuvef jo Tdjfodf, Jorge Luis Borges. + +# 18 + +# Complex + +“Make it as simple as possible, but no simpler.” + +Albert Einstein + +“'Can you do addition?' the White Queen asked. 'What's one and one and one and one and one and one and one and one and one and one?' 'I don't know' said Alice. 'I lost count.' 'She can't do addition,' the Red Queen interrupted.” + +Lewis Carroll, Through the Looking Glass and What Alice Found There. + +# Aims + +The aims of this chapter are: + +• To introduce the last predefined numeric data type in Fortran. • To illustrate with examples how to use this type. + +# Chapter 18 + +# 18 Complex + +This variable type reflects an extension of the real data type available in Fortran — the COMPLEX data type, where we can store and manipulate complex variables. Problems that require this data type are restricted to certain branches of mathemat- ics, physics and engineering. Complex numbers are defined as having a real and imaginary part, i.e., + +$$ +a=x+\mathrm{i}y +$$ + +where i is the square root of $^{-1}$ . + +They are not supported in many programming languages as a base type which makes Fortran the language of first choice for many people. + +To use this variable type we have to write the number as two parts, the real and imaginary elements of the number, for example, + +COMPLEX :: U U=(1.0,2.0) + +represents the complex number $1\,+\,\mathrm{i}2$ . Note that the complex number is enclosed in brackets. We can do arithmetic on variables like this, and most of the intrinsic functions such as LOG, SIN, COS, etc., accept a complex data type as argument. + +All the usual rules about mixing different variable types, like reals and integers, also apply to complex. Complex numbers are read in and written out in a similar way to real numbers, but with the provision that, for each single complex value, two format descriptors must be given. You may use either E or F formats (or in- deed, mix them), as long as there are enough of them. Although you use brackets around the pairs of numbers in a program, these must not appear in any input, nor will they appear on the output. + +Fortran has a number of functions which help to clarify the intent of mixed mode expressions. The functions REAL, CMPLX and INT can be used to 'force' any variable to real, complex or integer type. + +There are a number of intrinsic functions to enable complex calculations to be per- formed. The program segment below uses some of them: + +COMPLEX:: Z, Z1,Z2,Z3,ZBAR REAL :: X,Y,X1,Y1,X2,Y2,X3,Y3,ZMOD $\begin{array}{r l}{\mathrm{Z}\varOmega\varOmega}&{{}=}\end{array}$ CMPLX (1.0, 2.0) $\begin{array}{c c c c c}{{\bullet}}&{{\bullet}}&{{\bullet}}&{{\bullet}}&{{2}}\\ {{\bullet}}&{{\mathrm{X}2}}&{{+}}&{{\dot{\mathrm{~i~}}}}&{{\mathrm{Y}2}}\\ {{\bullet}}&{{\mathrm{X}3}}&{{+}}&{{\dot{\mathrm{~i~}}}}&{{\mathrm{Y}3}}\end{array}$ $\begin{array}{r l}{\mathrm{Z2}}&{{}=}\end{array}$ CMPLX(X2, Y2) Z3 = CMPLX (X3, Y3) Z = $\mathrm{z}1\!\times\!\mathrm{Z}2$ / Z3 + +# Complex + +X = REAL(Z) Y $=$ AIMAG (Z) ZMOD $=$ ABS(Z) ZBAR $=$ CONJG(Z) + +! real part of Z + + ! imaginary part of Z + + ! modulus of Z + + ! complex conjugate of Z + +# 18.1 Example + +The second order differential equation: + +$$ +{\frac{d^{2}\ y}{d\ t^{2}}}+2\,{\frac{d y}{d t}}+y=x(t) +$$ + +could describe the behaviour of an electrical system, where $x(t)$ is the input voltage and $y(t)$ is the output voltage and dy/dt is the current. The complex ratio + +$$ +\frac{y(w)}{x(w)}\!=\!1\!\quad\left(-w^{2}+2\ j\ w+1\right) +$$ + +is called the frequency response of the system because it describes the relationship between input and output for sinusoidal excitation at a frequency of $w$ and where $j$ is $\sqrt{\mathrm{~-~}1}$ . The following program segment reads in a value of $w$ and evaluates the frequency response for this value of $w$ together with its polar form (magnitude and phase): + +PROGRAM ch1801 IMPLICIT NONE + +! + + ! Program to calculate frequency response of a system + + ! for a given Omega + + ! and its polar form (magnitude and phase). + + ! REAL :: Omega ,Real_part , Imag_part , Magnitude, Phase COMPLEX:: Frequency response + + ! + + ! Input frequency Omega + + ! PRINT \*, 'Input frequency' READ \*,Omega + + ! Frequency response = 1.0 / & CMPLX( - Omega \* Omega $^+$ 1.0 , 2.0 \* Omega) Real_part $=$ REAL(Frequency response) Imag_part $=$ AIMAG(Frequency response) + + ! + +# Chapter 18 + +! Calculate polar coordinates (magnitude and phase) + + ! Magnitude $=$ ABS(Frequency response) Phase $=$ ATAN2 (Imag_part, Real_part) + + ! PRINT \*, ' At frequency ',Omega PRINT \*, 'Response $=$ ', Real_part,' + I ',Imag_part PRINT \*, 'in Polar form' PRINT \*, ' Magnitude $=$ ', Magnitude PRINT \*, ' Phase $=$ ', Phase END PROGRAM ch1801 + +# 18.2 Complex and kind type + +The standard requires that there be a minimum of two kind types for real numbers and this is also true of the complex data type. Chapter 8 must be consulted for a full coverage of real kind types. We would therefore use something like the fol- lowing to select a complex kind type other than the default: + +INTEGER , PARAMETER :: Long_Complex $\overleftarrow{\mathbf{\nabla}}$ SELECTED_REAL_KIND(15,307)COMPLEX (Long_Complex) :: Z + +Chapter 24 includes a good example of how to use modules to define and use pre- cision throughout a program and subprogram units. + +# 18.3 Summary + +COMPLEX is used to store and manipulate complex numbers: those with a real and an imaginary part. + +There are standard functions which allow conversion between the numerical data types — CMPLX, REAL and INT. + +# 18.4 Problems + +1. The program used in Chapter 15 which calculated the roots of a quadratic had to abandon the calculation if the roots were complex. You should now be able to remedy this, remembering that it is necessary to declare any complex variables. In- stead of raising the expression to the power 0.5 in order to take its square root, use the function SQRT. If you manage this to your satisfaction, try your skills on the roots of a cubic (see the problems in Chapter 15). + +Logical + +“A messenger yes/no semaphore her black/white keys in/out whirl of morse hoopooe signals salvation deviously.” + +Nathaniel Tarn, The Laurel Tree + +# Aims + +The aims of this chapter are: + +• To examine the last predefined type available in Fortran: logical. • To introduce the concepts necessary to use logical expressions effectively, namely: • Logical variables. • Logical operators. • The hierarchy of operations. • Truth tables. + +# Chapter 19 + +# 19 Logical + +Often we have situations where we need ON/OFF, TRUE/FALSE or YES/NO switches, and in such circumstances we can use LOGICAL type variables, e.g., + +LOGICAL :: FLAG Logicals may take only two possible values, as shown in the following: FLAG $=$ .TRUE. + +or + +FLAG $=$ .FALSE. + +Note the full stops, which are essential. With a little thought you can see why they are needed. You will already have met some of the ideas associated with logical variables from IF statements: + +IF( $\mathrm{~\AA~}\mathrm{~==~}\mathrm{~\AA~}$ ) THEN . ELSE . ENDIF + +The logical expression $\mathrm{(A==B)}$ ) returns a value true or false, which then deter- mines the route to be followed; if the quantity is true, then we execute the next statement, else we take the other route. + +Similarly, the following example is also legitimate: + +LOGICAL :: ANSWER ANSWER $=$ .TRUE. ... IF (ANSWER) THEN ... ELSE ... ENDIF + +Again the expression IF (ANSWER) is evaluated; here the variable ANSWER has been set to .TRUE., and therefore the statements following the THEN are exe- cuted. Clearly, conventional arithmetic is inappropriate with logicals. What does 2 times true mean? (very true?). There are a number of special operators for logicals: + +.NOT. which negates a logical value (i.e., changes true to false or vice versa). + +.AND. logical intersection. .OR. logical union. To illustrate the use of these operators, consider the following program extract: + +LOGICAL :: A,B,C $\mathtt{A}\!=$ .TRUE. $\scriptstyle\mathrm{\mathrm{~B}}=$ .NOT.A + +! (B now has the value 'false') $\mathrm{C}{=}\mathrm{\mathbb{A}}$ .OR.B + + ! (C has the value 'true') $\mathrm{C}{=}\mathrm{\mathbb{A}}$ .AND.B + + ! (C now has the value 'false') + +![To gauge the effect of these operators on logicals, we can consult a truth table: ](images/950a417dd2671b0a9c18448f38e3c15b6ac45c2ab468aa456cb8960e6a46855f.jpg) + +As with arithmetic operators, there is an order of precedence associated with the logical operators: + +.AND. is carried out before .OR. and .NOT. + +In dealing with logicals, the operations are carried out within a given level, from left to right. Any expressions in brackets would be dealt with first. The logical op- erators are a lower order of precedence than the arithmetic operators, i.e., they are carried out later. A more complete operator hierarchy is therefore: + +• Expressions within brackets. • Exponentiation. • Multiplication/division. + +# Chapter 19 + +• Addition/subtraction. • Relational logical $(==,>,<,>=,<=/=)$ . • .AND. • .OR. and .NOT. + +Although you can build up complicated expressions with mixtures of operators, these are often difficult to comprehend, and it is generally more straightforward to break ' big ' expressions down into smaller ones whose purpose is more readily ap- preciated. + +Historically, logicals have not been in evidence extensively in Fortran programs, although clearly there are occasions on which they are of considerable use. Their use often aids significantly in making programs more modular and comprehensible. They can be used to make a complex section of code involving several choices much more transparent by the use of one logical function, with an appropriate name. Logicals may be used to control output; e.g., + +LOGICAL :: DEBUG ... DEBUG $,=$ .TRUE. ... IF(DEBUG)THEN ... PRINT \*,'LOTS OF PRINTOUT' ... ENDIF + +ensures that, while debugging a program you have more output. Then, when the program is correct , run with DEBUG $=$ .FALSE. + +Note that Fortran does try to protect you while you use logical variables. You can- not do the following: + +LOGICAL :: UP, DOWN UP $=$ DOWN $^+$ .FALSE. + +LOGICAL :: A2 REAL DIMENSION(10):: OMEGA . A2=OMEGA(3) + +The compiler will note that this is an error, and will not permit you to run the pro- gram. This is an example of strong typing , since only a limited number of predetermined operations are permitted. The real, integer and complex variable types are much more weakly typed (which helps lead to the confusion inherent in mixing variable types in arithmetic assignments). + +# 19.1 I/O + +Since logicals may take only the values .TRUE. and .FALSE., the possibilities in reading and writing logical values are clearly limited. The L edit descriptor or for- mat allows logicals to be input and output. On input, if the first nonblank characters are either $\mathrm{T}$ or .T, the logical value .TRUE. is stored in the correspond- ing list item; if the first nonblank characters are F or .F, then .FALSE. is stored. (Note therefore that reading, say, TED and FAHR in an L4 format would be ac- ceptable.) If the first nonblank character is not F, T, .F or .T, then an error message will be generated. On output, the value $\mathrm{T}$ or F is written out, right justified, with blanks (if appropriate). Thus, + +LOGICAL :: FLAG FLAG $,=$ .TRUE. PRINT 100, FLAG, .NOT.FLAG 100 FORMAT(2L3) + +would produce + +T F + +at the terminal. + +Assigning a logical variable to anything other than a .TRUE. or .FALSE. value in your program will result in errors. The 'shorthand' forms of .T, .F, F and T are not acceptable in the program. + +# 19.2 Summary + +Another type of data — logical — is also recognised. A LOGICAL variable may take one of two values — true or false. + +• There are special operators for manipulating logicals: • .NOT. • .AND. • .OR. • Logical operators have a lower order of precedence than any others. + +# 262 + +# Chapter 19 + +# 19.3 Problems + +1. Why are the full stops needed in a statement like $\mathrm{A}=$ .TRUE.? + + 2. Generate a truth table like the one given in this chapter. + + 3. Write a program which will read in numerical data from the terminal, but will flag any data which is negative, and will also turn these negative values into posi- tive ones. + +# User Defined Types + +“Russell's theory of types leads to certain complexities in the foundations of math- ematics... Its interesting features for our purposes are that types are used to prevent certain erroneous expressions from being used in logical and mathematical formu- lae; and that a check against violation of type constraints can be made purely by scanning the text, without any knowledge of the value which a particular symbol might happen to have.” + +C.A.R. Hoare, Structured Programming + +# Aims + +The aim of this chapter is to introduce the concepts and ideas involved in using the facilities offered in Fortran 90 for the construction and use of user defined types: + +• The way in which we define our own types. • The way in which we declare variables to be of a user defined type. • The way in which we manipulate variables of our own types. • The way in which we can nest types within types. + +The examples are simple and are designed to highlight the syntax. More complex and realistic examples of the use of user defined data types are to be found in later chapters. + +# 20 User Defined Types + +In the coverage so far we have used the intrinsic types provided by Fortran. The only data structuring technique available has been to construct arrays of these in- trinsic types. Whilst this enables us to solve a reasonable variety of problems, it is inadequate for many purposes. In this chapter we look at the facilities offered by Fortran for the construction of our own types and how we manipulate data of these new, user defined types. + +With the ability to define our own types we can now construct aggregate data types that have components of a variety of base types. These are often given the name records in books on data structures. In mathematics the term cartesian prod- uct is often used, and this is the terminology adopted by Hoare. We will stick to the term records, as it is the one that is most commonly used in computing and texts on programming. + +There are two stages in the process of creating and using our own data types: we must first define the type, and then create variables of this type. + +# 20.1 Example 1: Dates + +PROGRAM ch2001 IMPLICIT NONE TYPE Date INTEGER :: Day $\mathbf{\omega=}\mathtt{1}$ INTEGER :: Month $\mathbf{\Psi}^{=}\mathbb{1}$ INTEGER :: Year $\scriptstyle=2\,0\,0\,0$ END TYPE Date TYPE (Date) :: D PRINT \*,D%Day, D%Month, D%Year PRINT \*,' Type in the date, day, month, year' READ \*,D%Day, D%Month, D%Year PRINT \*,D%Day, D%Month, D%Year END PROGRAM ch2001 + +This complete program illustrates both the definition and use of the type. It also shows how you can define initial values within the type definition. + +# 20.2 Type definition + +The type Date is defined to have three component parts, comprising a day , a month and a year , all of integer type. The syntax of a type construction comprises: + +TYPE Typename Data Type :: Component_name etc END TYPE Typename + +Reference can then be made to this new type by the use of a single word, Date , and we have a very powerful example of the use of abstraction. + +# 20.3 Variable definition + +This is done by + +TYPE (Typename) :: Variablename + +and we then define a variable $D$ to be of this new type. The next thing we do is have a READ \* statement that prompts the user to type in three integer values, and the data are then echoed straight back to the user. We use the notation Variablename%Component_Name to refer to each component of the new data type. + +# 20.4 Example 2: Address lists + +PROGRAM ch2002 + +IMPLICIT NONE TYPE Address CHARACTER ( $\mathrm{\Omega}.\mathrm{LEN}{=}40$ ) :: Name CHARACTER ( $\mathtt{L E N}\!\!=\!\!6\,0$ ) :: Street CHARACTER (LEN $\mathrm{\dot{=}}60$ ) :: District CHARACTER (LEN $\mathrm{\dot{=}}60$ ) :: City CHARACTER (LEN $\dot{=}8$ ) :: Post_Code END TYPE Address INTEGER , PARAMETER :: N_of_Address $=78$ TYPE (Address) , DIMENSION(N_of_Address):: Addr INTEGER :: I OPEN(UNIT $\vDash\vDash$ ,FILE $=$ "ADDRESS.DAT") DO $\mathbb{T}\!=\!1$ ,N_of_Address READ(UNIT $\vDash\vDash$ , $\mathrm{{FMT}=`}$ (A40)') Addr(I)%Name READ(UNIT $\vDash\vDash$ , $\mathrm{{FMT}=`}$ (A60)') Addr(I)%Street READ(UNIT $\vDash\vDash$ , $\mathrm{{FMT}=`}$ (A60)') Addr(I)%District READ(UNIT $\vDash\vDash$ ,FMT $"="$ (A60)') Addr(I)%City READ(UNIT $\vDash\vDash$ , $\mathrm{{FMT}=`}$ (A8)') Addr(I)%Post_Code END DO DO $\mathbb{T}\!=\!1$ ,N_of_Address PRINT \*,Addr(I)%Name + +# Chapter 20 + +PRINT \*,Addr(I)%Street PRINT \*,Addr(I)%District PRINT \*,Addr(I)%City PRINT \*,Addr(I)%Post_Code END DO END PROGRAM ch2002 + +In this example we define a type Address which has components that one would expect for a person's address. We then define an array Addr of this type. Thus we are now creating arrays of our own user defined types. We index into the array in the way we would expect from our experience with integer, real and character ar- rays. The complete example is rather trivial in a sense in that the program merely reads from one file and prints the file out to the screen. However, it highlights many of the important ideas of the definition and use of user defined types. + +# 20.5 Example 3: Nested user defined types + +The following example builds on the two data types already introduced. Here we construct nested user defined data types based on them and construct a new data type containing them both plus additional information: + +PROGRAM ch2003 IMPLICIT NONE TYPE Address CHARACTER ( $\mathtt{L E N}\!\!=\!\!6\,0$ ) :: Street CHARACTER ( $\mathtt{L E N}\!\!=\!\!6\,0$ ) :: District CHARACTER ( $\mathtt{L E N}\!\!=\!\!6\,0$ ) :: City CHARACTER (LEN $\dot{=}8$ ) :: Post_Code END TYPE Address TYPE Date_Of_Birth INTEGER :: Day INTEGER :: Month INTEGER :: Year END TYPE Date_Of_Birth TYPE Personal CHARACTER ( $\tt L E N\!\!=\!\!20$ ) :: First_Name CHARACTER (LEN $\scriptstyle{\dot{=}20}$ ) :: Other_Names CHARACTER (LEN $\mathrm{\dot{\rho}}\!=\!4\,0$ ) :: Surname TYPE (Date_Of_Birth) :: DOB CHARACTER (LEN $\scriptstyle{\stackrel{\cdot}{=}}1$ ) :: Sex TYPE (Address) :: Addr END TYPE Personal INTEGER , PARAMETER :: N_People=2 + +TYPE (Personal) , DIMENSION(N_People) :: P INTEGER :: I + +OPEN(UNIT $\vDash\vDash$ ,FILE $=$ 'PERSON.DAT') + +DO $\mathbb{T}\!=\!1$ ,N_People READ(1,FMT $\,=\,\!1\,0$ ) P(I)%First_Name,& P(I)%Other_Names,& P(I)%Surname,& P(I)%DOB%Day,& P(I)%DOB%Month,& P(I)%DOB%Year,& P(I)%Sex,& P(I)%Addr%Street,& P(I)%Addr%District,& P(I)%Addr%City,& + +P(I)%Addr%Post_Code + +10 FORMAT( A20,/,& A20,/,& A40,/,& I2,1X,I2,1X,I4,/,& A1,/,& A60,/,& A60,/,& A60,/,& A8) + +# END DO + +DO $\mathbb{T}\!=\!1$ ,N_People + +P(I)%Other_Names,& P(I)%Surname,& P(I)%DOB%Day,& P(I)%DOB%Month,& P(I)%DOB%Year,& P(I)%Sex,& P(I)%Addr%Street,& P(I)%Addr%District,& P(I)%Addr%City,& P(I)%Addr%Post_Code 20 FORMAT( A20,A20,A40,/,& I2,1X,I2,1X,I4,/,& A1,/,& A60,/,& A60,/,& + +# Chapter 20 + +# END DO + +END PROGRAM ch2003 + +Here we have a date of birth data type ( Date_Of_Birth ) based on the Date data type from the first example, plus a slightly modified address data type, incor- porated into a new data type comprising personal details. Note the way in which we reference the component parts of this new, aggregate data type. + +# 20.6 Problems + +1. Modify the last example to include a more elegant printed name. The current ex- ample will pad with blanks the first name, other names and surname and span 80 characters on one line, which looks rather ugly. + +Add a new variable name which will comprise all three subcomponents and write out this new variable, instead of the three subcomponents. + +# 20.7 Bibliography + +Dahl O.J., Dijkstra E.W., Hoare C.A.R., Structured Programming , Academic Press, 1972. + +• This is one of the earliest and best introductions to data structures and structured programming. The whole book hangs together very well, and the section on data structures is a must for serious programmers. + +Vowels R.A., Algorithms and Data Structures in $F$ and Fortran , Unicomp, 1989. + +• One of the few books looking at algorithms and data structures using For- tran. + +Wirth N., Algorithms $^+$ Data Structures $=$ Programs , Prentice-Hall, 1976. Wirth N., Algorithms $^+$ Data Structures , Prentice-Hall, 1986. • The first is in Pascal, and the second in Modula 2. Wood D., Paradigms and Programming in Pascal , Computer Science Press, 1984. + +• Contains a number of examples of the use of recursion in problem solv- ing. Also provides a number of useful case studies in problem solving. + +# An Introduction to Pointers + +“The question naturally arises whether the analogy can be extended to a data struc- ture corresponding to recursive procedures. A value of such a type would be permitted to contain more than one component that belongs to the same type as it- self; in the same way that a recursive procedure can call itself recursively from more than one place in its own body.” + +C.A.R. Hoare, Structured Programming + +# Aim + +The primary aim of the chapter is to introduce some of the key concepts of point- ers in Fortran. + +# 21 An Introduction to Pointers + +All of the data types introduced so far, with the exception of the allocatable array, have been static. Even with the allocatable array a size has to be set at some stage during program execution. The facilities provided in Fortran by the concept of a pointer combined with those offered by a user defined type enable us to address a completely new problem area, previously extremely difficult to solve in Fortran. There are many problems where one genuinely does not know what requirements there are on the size of a data structure. Linked lists allow sparse matrix problems to be soved with minimal storage requirements, two-dimensional spatial problems can be addressed with quad-trees and three-dimensional spatial problems can be addressed with oct-trees. Many problems also have an irregular nature, and pointer arrays address this problem. + +First we need to cover some of the technical aspects of pointers. A pointer is a variable that has the POINTER attribute. A pointer is associated with a target by allocation or pointer assignment. A pointer becomes associated as follows: + +• The pointer is allocated as the result of the successful execution of an AL- LOCATE statement referencing the pointer + +• The pointer is pointer-assigned to a target that is associated or is specified with the TARGET attribute and, if allocatable, is currently allocated. + +A pointer shall neither be referenced nor defined until it is associated. A pointer is disassociated following execution of a DEALLOCATE or NULLIFY statement, following pointer association with a disassociated pointer, or initially through pointer initialisation. + +A pointer may have a pointer association status of associated, disassociated, or un- defined. Its association status may change during execution of a program. Unless a pointer is initialised (explicitly or by default), it has an initial association status of undefined. A pointer may be initialised to have an association status of disassoci- ated. + +Let us look at some examples to clarify these points. + +# 21.1 Some basic pointer concepts + +With the introduction of pointers as a data type into Fortran we also have the intro- duction of a new assignment statement — the pointer assignment statement. Consider the following example: + +PROGRAM C2101 INTEGER , POINTER :: A,B + +INTEGER , TARGET :: C INTEGER :: D $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\AA~}=>\mathrm{~\AA~}\mathrm{~C~}$ $\textrm{C}=\textrm{2}$ $\mathrm{~\ensuremath~{~B~}~}=>\mathrm{~\ensuremath~{~C~}~}$ $\texttt{D}=\texttt{A}+\texttt{B}$ PRINT \*,A,B,C,D END PROGRAM C2101 + +The first declaration defines A and B to be variables, with the POINTER attribute. This means we can use A and B to refer or point to integer values. Note that in this case no space is set aside for the pointer variables A and B. A and B should not be referenced in this state. + +The second declaration defines C to be an integer, with the TARGET attribute, i.e., we can use pointers to refer or point to the value of the variable C. + +The last declaration defines D to be an ordinary integer variable. + +In the case of the last two declarations space is set aside to hold two integers. + +Let us now look at the various executable statements in the program, one at a time: + +$C=1$ This is an example of the normal assignment statement with which we are already familiar. We use the variable name C in our program and whenever we use that name we get the value of the variable C. $\mathrm{A}=>\mathrm{C}$ This is an example of a pointer assignment statement. This means that both A and C now refer to the same value, in this case 1. A becomes associated with the target C. A can now be referenced. C = 2 Conventional assignment statement, and C now has the value 2. B = > C Second example of pointer assignment. B now points to the value that C has, in this case 2. B becomes associated with the target C. B can now be referenced. $\mathrm{D}=\mathrm{A}+\mathrm{B}$ Simple arithmetic assignment statement. The value that A points to is added to the value that B points to and the result is assigned to D. + +The last statement prints out the values of A, B, C and D. + +The output is + +2 2 2 4 + +Chapter 21 + +# 21.2 The ASSOCIATED intrinsic function + +The ASSOCIATED intrinsic returns the association status of a pointer variable. Consider the following example: + +PROGRAM C2102 INTEGER , POINTER :: A,B INTEGER , TARGET :: C INTEGER :: D PRINT \*,ASSOCIATED(A) PRINT \*,ASSOCIATED(B) $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\AA~}=>\mathrm{~\AA~}\mathrm{~C~}$ $\textrm{C}=\textrm{2}$ $\mathrm{~\ensuremath~{~B~}~}=>\mathrm{~\ensuremath~{~C~}~}$ $\texttt{D}=\texttt{A}+\texttt{B}$ PRINT \*,A,B,C,D PRINT \*,ASSOCIATED(A) PRINT \*,ASSOCIATED(B)END PROGRAM C2102 + +The output from running this program with a number of compilers is shown below. + +# 21.2.1 CVF 6.6C + +F F T 2 2 2 + +# 21.2.2 Intel, Windows, 8.1 + +F F 2 2 2 4 T T + +# 21.2.3 Lahey, Windows 5.70f + +F F 2 2 2 4 T + +T + +# 21.2.4NAG, Windows, 4.2 + +T T 2 2 2 4 T T + +# 21.2.5Salford 4.6.0 + +We have some differences, and the actual answer as to why is rather subtle. The standard says that the ASSOCIATED function must not be called with a pointer whose status is undefined. So in this program we have declared the pointers A and B but their initial status is undefined. So in a sense all of the above could be re- garded as correct, as the program breaks the standard! + +The next example is a simple variant. + +# 21.3 Referencing A and B before assignment + +Consider the following example: + +PROGRAM C2103 INTEGER , POINTER :: A,B INTEGER , TARGET :: C INTEGER :: D PRINT \*,ASSOCIATED(A) PRINT \*,ASSOCIATED(B)PRINT $^{\star}\,,\,\mathtt{A}$ PRINT $^{\star}\,,\,\mathtt{B}$ $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\AA~}=>\mathrm{~\AA~}\mathrm{~C~}$ $\textrm{C}=\textrm{2}$ $\mathrm{~\ensuremath~{~B~}~}=>\mathrm{~\ensuremath~{~C~}~}$ $\texttt{D}=\texttt{A}+\texttt{B}$ PRINT \*,A,B,C,D PRINT \*,ASSOCIATED(A) + +# Chapter 21 + +PRINT \*,ASSOCIATED(B)END PROGRAM C2103 + +Here we are actually referencing the pointer, even though its status is undefined. Most compilers generate a run time error with this example. However, the error message tends to be a little cryptic. Some sample outputs with the default compila- tion options follow + +# 21.3.1 CVF + +F + +F forrtl: severe (157): Program Exception - access violation Image PC Routine Line Source ch2003cvf.exe 00401098 C2003 7 ch2003.f90 ch2003cvf.exe 004266A9 Unknown Unknown Unknown ch2003cvf.exe 0041D9E4 Unknown Unknown Unknown kernel32.dll 7C816D4F Unknown Unknown Unknown + +# 21.3.2 Intel, Windows 8.1 + +F forrtl: severe (157): Program Exception - access violation Image PC Routine Line Source ch2003intel.exe 0040106E Unknown Unknown Unknown ch2003intel.exe 0043DE2D Unknown Unknown Unknown ch2003intel.exe 00430D60 Unknown Unknown Unknown kernel32.dll 7C816D4F Unknown Unknown Unknown + +# 21.3.3 Lahey, Windows 5.70f + +F jwe0019i-u The program was terminated abnormally with Exception Code EXCEPTION ACCESS VIOLATION. Error occurs at or near line 6 of _MAIN__ error summary (Fortran) error number error level error count jwe0019i u 1 total error count $\begin{array}{r l}{\mathbf{\Sigma}=}&{{}\ \mathbb{1}}\end{array}$ + +# 21.3.4NAG, Windows 4.2 + +T T Segmentation fault (core dumped) + +# 21.3.5 Salford 4.6.0 + +Some of the compilers give a clue with a line number. The Salford output is interesting as the program actually ran to completion. Try and find compiler op- tions that will provide better diagnostic error messages with your compiler. + +# 21.4 The NULL intrinsic + +Fortran 95 introduced the NULL intrinsic. The three previous examples are to a degree examples of Fortran 90 style programming: + +PROGRAM C2104 INTEGER , POINTER :: $\mathbb{A}\!=\!>$ NULL(), $\mathrm{B}{=}\,\!>$ NULL() INTEGER , TARGET :: C INTEGER :: D PRINT \*,ASSOCIATED(A) PRINT \*,ASSOCIATED(B) $\begin{array}{l l l}{{\mathsf C}}&{{=}}&{{1}}\\ {{\mathsf{\bar{A}}}}&{{=>}}&{{\mathsf C}}\end{array}$ + +# Chapter 21 + +C = 2 B => C D = A + B PRINT \*,A,B,C,D PRINT \*,ASSOCIATED(A) PRINT \*,ASSOCIATED(B)END PROGRAM C2104 + +All compilers tested gave the same correct result. The recommendation is therefore to always use the NULL intrinsic to provide pointer variables with a known value of disassociated, rather than undefined. + +# 21.5 Assignment via $=$ + +Consider the following two examples: + +PROGRAM C2105 INTEGER , POINTER :: $\mathbb{A}\!=\!>$ NULL(), $\mathrm{B}{=}\,\!>$ NULL() INTEGER , TARGET :: C INTEGER :: D $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\textit~{~A~}~}=\mathrm{~\textit~{~21}~}$ $\textrm{C}=\textrm{2}$ $\mathrm{~\ensuremath~{~B~}~}=>\mathrm{~\ensuremath~{~C~}~}$ $\texttt{D}=\texttt{A}+\texttt{B}$ PRINT \*,A,B,C,D END PROGRAM C2105 + +# and + +PROGRAM C2106 INTEGER , POINTER :: $\mathbb{A}\!=\!>$ NULL(), $\mathrm{B}{=}\,\!>$ NULL() INTEGER , TARGET :: C INTEGER :: D $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\AA~}=>\mathrm{~\AA~}\mathrm{~C~}$ $\textrm{C}=\textrm{2}$ $\mathrm{~\textsc~{~B~}~}=\mathrm{~\AA~}$ $\texttt{D}=\texttt{A}+\texttt{B}$ PRINT \*,A,B,C,D END PROGRAM C2106 + +Both of these will compile but both will generate run time errors. In the first pro- gram the problems lies with the statement + +$$ +\mathrm{~\textit~{~A~}~}=\mathrm{~\textit~{~21}~} +$$ + +and in the second case the problem lies with the statement + +$$ +\mathrm{~\textsc~{~B~}~}=\mathrm{~\AA~} +$$ + +Below are the corrected versions of the programs + +PROGRAM C2107 INTEGER , POINTER :: $\mathbb{A}\!=\!>$ NULL(), $\mathrm{B}{=}\,\!>$ NULL() INTEGER , TARGET :: C INTEGER :: D ALLOCATE(A) $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\textit~{~A~}~}=\mathrm{~\textit~{~21}~}$ $\textrm{C}=\textrm{2}$ $\mathrm{~\ensuremath~{~B~}~}=>\mathrm{~\ensuremath~{~C~}~}$ $\texttt{D}=\texttt{A}+\texttt{B}$ PRINT \*,A,B,C,D END PROGRAM C2107 + +# and + +PROGRAM C2108 INTEGER , POINTER :: $\mathbb{A}\!=\!>$ NULL(), $\mathrm{B}{=}\,\!>$ NULL() INTEGER , TARGET :: C INTEGER :: D ALLOCATE(B) $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\AA~}=>\mathrm{~\AA~}\mathrm{~C~}$ $\textrm{C}=\textrm{2}$ $\mathrm{~\textsc~{~B~}~}=\mathrm{~\AA~}$ $\texttt{D}=\texttt{A}+\texttt{B}$ PRINT \*,A,B,C,D END PROGRAM C2108 + +Our recommendation when using pointers is to nullify them when declaring them and to explicitly allocate them before using them when assigning a value via nor- mal assignment. + +# Chapter 21 + +# 21.6 Singly linked list + +Conceptually a singly linked lists consists of a sequence of boxes with compart- ments. In the simplest case the first compartment holds a data item and the second contains directions to the next box. + +We can construct a data structure in Fortran to work with a singly linked list by combining the concept of a record from the previous chapter with the new concept of a pointer. A complete program to do this is given below: + +PROGRAM C2109 TYPE Link CHARACTER :: C TYPE (Link) , POINTER :: Next END TYPE Link TYPE (Link) , POINTER :: Root , Current INTEGER :: IO_Stat_Number $=\!0$ ALLOCATE(Root) READ (UNIT $\c=\quad\star$ , FMT = '(A)' , ADVANCE $=$ 'NO' , & IOSTAT $=$ IO_Stat_Number) Root%C IF (IO_Stat_Number $\begin{array}{r l}{\mathrm{{==}}}&{{}\end{array}$ ) THEN NULLIFY(Root%Next)ELSE ALLOCATE(Root%Next) ENDIF Current $=>$ Root DO WHILE (ASSOCIATED(Current%Next)) Current $=>$ Current%Next READ (UNIT $^{\ast}{=}^{\ast}$ , $\mathrm{{FMT}=`}$ (A)',ADVANCE $=$ 'NO', & IOSTAT $^{\ast}{=}$ IO_Stat_Number) Current%C IF (IO_Stat_Number $\begin{array}{r l}{\mathrm{{==}}}&{{}\end{array}$ ) THEN NULLIFY(Current%Next)ELSE ALLOCATE(Current%Next) ENDIF END DO Current $=>$ Root DO WHILE (ASSOCIATED(Current%Next)) PRINT \* , Current%C Current $=>$ Current%Next END DO END PROGRAM C2109 + +The behaviour of this program is system specific. You will have to look at your compiler documentation regarding the IO_Stat_Number. The first thing of interest is the type definition for the singly linked list. We have + +TYPE Link CHARACTER :: C TYPE (Link) , POINTER :: Next END TYPE Link + +and we call the new type Link . It comprises two component parts: the first holds a character C , and the second holds a pointer called Next to allow us to refer to another instance of type Link . Remember we are interested in joining together several boxes or Links . + +The next item of interest is the variable definition. Here we define two variables Root and Current to be pointers that point to items of type Link . In Fortran when we define a variable to be a pointer we also have to define what it is allowed to point to. This is a very useful restriction on pointers, and helps make using them more secure. + +The first executable statement + +ALLOCATE(Root) + +requests that the variable Root be allocated memory. At this time the contents of both the character component and the pointer component are undefined. + +The next statement reads a character from the keyboard. We are using a number of additional features of the READ statement, including + +ADVANCE $="$ NO' IOSTAT $^{\ast}{=}$ IO_Stat_Number + +and the two options combine to provide the ability to read an arbitrary amount of text from the user per line, and terminate only when end of file is encountered as the only input on a line, typically by typing CTRL Z. Note that the numbers re- turned by the IOSTAT option are implementation specific. A small program would have to be written to test the values returned for each platform. + +If an end of file is reached then the pointer Root%Next is nullified using the NULLIFY statement. This gives the pointer a status of disassociated, and this is a convenient way of saying that it doesn't point to anything valid. + +If the end of file is not detected then the next link in the chain is created. + +# Chapter 21 + +The statement + +Current $=>$ Root + +means that both Current and Root point to the same physical memory location, and this holds a character data item and a pointer. We must do this as we have to know where the start of the list is. This is now our responsibility, not the compil- ers. Without this statement we are not able to do anything with the list except fill it up — hardly very useful. + +The WHILE loop is then repeated until end of file is reached. If the user had typed an end of file immediately then Current%Next would not be ASSOCIATED, and the WHILE loop would be skipped. + +This loop allocates memory and moves down the chain of boxes one character at a time filling in the links between the boxes as we go. We then have + +Current $=>$ Root + +and this now means that we are back at the start of the list, and in a position to tra- verse the list and print out each character in the list. + +There is thus the concept with the pointer variable Current of it providing us with a window into memory where the complete linked list is held, and we look at one part of the list at a time. + +Both WHILE loops use the intrinsic function ASSOCIATED to check the associa- tion status of a pointer. + +It is recommended that this program be typed in, compiled and executed. It is sur- prisingly difficult to believe that it will actually read in a completely arbitrary number of characters from the user. Seeing is believing. + +# 21.7 Reading in an arbitrary quantity of numeric data + +In this example we will look at using a singly linked list to read in an arbitrary quantity of data and then allocating an array to copy it to for normal numeric cal- culations at run time: + +PROGRAM C2110 + +TYPE Link REAL :: N TYPE (Link) , POINTER :: Next END TYPE Link + +TYPE (Link) , POINTER :: Root, Current + +INTEGER :: $\mathbb{T}\!=\!0$ integer :: error $=\!0$ INTEGER :: IO_Stat_Number $=\!0$ integer :: blank_lines $=\!0$ + +real , allocatable , dimension(:) :: x + +ALLOCATE(Root) READ (UNIT $\c=\quad\star$ , $\begin{array}{l l l}{\mathrm{{FMT}}}&{=}&{{\star}}\end{array}$ , IOSTAT $=$ IO_Stat_Number) Root%N IF (IO_Stat_Number $>\quad0$ ) THEN error $=$ error $+1$ else if (io_stat_number $\begin{array}{r l}{\mathrm{{==}}}&{{}\end{array}$ ) then NULLIFY(Root%Next)else if (io_stat_number $\scriptstyle{\begin{array}{l l}{={}}&{-2}\end{array}}$ ) then blank_lines $=$ blank_lines $+1$ ELSE $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ ALLOCATE(Root%Next) ENDIF + +Current $=>$ Root + +DO WHILE (ASSOCIATED(Current%Next)) + +Current $=>$ Current%Next + +READ (UNIT $=\star$ , $\scriptstyle{\mathrm{FT}}={\star}$ , IOSTAT $^{\ast}{=}$ IO_Stat_Number) Current%N + +IF (IO_Stat_Number > 0) THEN error $=$ error $+1$ else if (io_stat_number $\begin{array}{r l}{\mathrm{{==}}}&{{}\end{array}$ ) then NULLIFY(current%Next) else if (io_stat_number $\scriptstyle{\begin{array}{l l}{={}}&{-2}\end{array}}$ ) then blank_lines $=$ blank_lines $+1$ ELSE $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ ALLOCATE(current%Next) ENDIF + +# Chapter 21 + +END DO + +print \*,i,' items read' print \*,blank_lines,' blank lines' print \*,error,' items in error' allocate(x(1:i)) $\dot{\bot}\!=\!\bot$ Current $=>$ Root DO WHILE (ASSOCIATED(Current%Next)) $\textsf{x}(\dot{\bot})=$ current%n $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ PRINT \* , Current%N Current $=>$ Current%Next END DO print \*,x + +END PROGRAM C2110 + +Below is a variant on this using the NAG compiler. Note the use of a module and meaningful names for the status of the read: + +PROGRAM C2111 + +use f90_iostat + +TYPE Link REAL :: N TYPE (Link) , POINTER :: Next END TYPE Link + +TYPE (Link) , POINTER :: Root, Current + +INTEGER :: $\mathbb{T}\!=\!0$ INTEGER :: IO_Stat_Number $=\!0$ + +ALLOCATE(Root) READ (UNIT $\c=\quad\star$ , FMT $\c=\quad\star$ , IOSTAT $=$ IO_Stat_Number) Root%N if (io_stat_number $==$ ioerr_eof) then NULLIFY(Root%Next) + +ELSE if(io_stat_number $==$ ioerr_ok) then $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ ALLOCATE(Root%Next) + +ENDIF Current $=>$ Root DO WHILE (ASSOCIATED(Current%Next)) Current $=>$ Current%Next READ (UNIT $=\star$ , $\scriptstyle{\mathrm{FT}}={\star}$ , IOSTAT $^{\ast}{=}$ IO_Stat_Number) Current%N if (io_stat_number $==$ ioerr_eof) then NULLIFY(current%Next) ELSE if(io_stat_number $==$ ioerr_ok) then $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ ALLOCATE(current%Next) ENDIF END DO print \*,i,' items read' Current $=>$ Root DO WHILE (ASSOCIATED(Current%Next)) PRINT \* , Current%N Current $=>$ Current%Next END DO + +# 21.8 Arrays of pointers + +Arrays in Fortran are rectangular, even when allocatable. So if you wish to set up a lower triangular matrix that uses minimal memory you have to use arrays of point- ers. The following examples show how to do this. + +PROGRAM C2112 IMPLICIT NONE TYPE Ragged + +REAL , DIMENSION(:) , POINTER :: Ragged_row END TYPE INTEGER :: i INTEGER , PARAMETER :: $\mathtt{n}\!\!=\!\!3$ TYPE (Ragged) , DIMENSION(1:n) :: Lower_Diag DO $\dot{\bot}\!=\!\bot$ ,n ALLOCATE(Lower_Diag(i)%Ragged_Row(1:i)) PRINT \*,' Type in the values for row ' , i READ \*,Lower_Diag(I)%Ragged_Row(1:i) END DO DO $\dot{\bot}\!=\!\bot$ ,n PRINT \*,Lower_Diag(i)%Ragged_Row(1:i) END DO END PROGRAM C2112 + +The type Ragged has a component that is a pointer to an array. Within the first DO loop we allocate a row at a time and each time we go around the loop the array al- located increases in size. + +# 21.9 Arrays of pointers and variable sized data sets — 1 + +In this example we use a parameter statement to set up the number of stations: + +PROGRAM C2113 IMPLICIT NONE TYPE Ragged REAL , DIMENSION(:) , POINTER :: rainfall END TYPE INTEGER :: i INTEGER , PARAMETER :: nr $=\!5$ INTEGER , DIMENSION (1:nr) :: nc TYPE (ragged) , DIMENSION(1:nr) :: station DO $\dot{\bot}\!=\!\bot$ ,nr PRINT \*,' enter the number of data values' & ' for station ',i READ \*,nc(i) ALLOCATE(station(i)%rainfall(1:nc(i)))PRINT \*,' Type in the values for station ' , i READ \*,station(i)%rainfall(1:nc(i)) END DO DO $\dot{\bot}\!=\!\bot$ ,nr PRINT \*,station(i)%rainfall(1:nc(i)) END DO + +END PROGRAM C2013 + +We read in the dimension or number of values for each station at run time, and al- locate the space at run time. + +# 21.10 Arrays of pointers and variable sized data sets — 2 + +In this example the number of stations is read in at run time: + +PROGRAM C2114 IMPLICIT NONE TYPE Ragged REAL , DIMENSION(:) , POINTER :: rainfall END TYPE INTEGER :: i INTEGER :: nr iNTEGER , ALLOCATABLE , DIMENSION (:) :: nc TYPE (ragged) , ALLOCATABLE , DIMENSION(:) :: station PRINT \*,' enter number of stations' READ \*,nr ALLOCATE(station(1:nr))ALLOCATE(nc(1:nr))DO $\mathbb{T}\!=\!1$ ,Nr PRINT \*,' enter the number of data values ' & ' for station ',i READ \*,nc(i) ALLOCATE(station(i)%rainfall(1:nc(i)))PRINT \*,' Type in the values for station ' , I READ \*,station(i)%rainfall(1:nc(i)) END DO DO $\dot{\bot}\!=\!\bot$ ,nr PRINT \*,station(i)%rainfall(1:nc(i)) END DO END PROGRAM C2114 + +In this example both the number of stations and the dimension for each station is read in at run time and allocated accordingly. + +# 21.11 Memory leak examples + +Dynamic memory brings greater versatility but requires greater responsibility. Con- sider the following example: + +# Chapter 21 + +PROGRAM C2115 IMPLICIT NONE INTEGER :: Allocate_status $=\!0$ REAL , DIMENSION(:) , POINTER :: X REAL , DIMENSION(1:10) , TARGET :: Y INTEGER , PARAMETER :: SIZE $=$ 10000000 INTEGER :: I ALLOCATE(X(1:SIZE),STAT $^{!}{=}$ Allocate_status) IF (allocate_status $>$ 0) THEN PRINT \*,' Allocate failed. Program ends.' STOP ENDIF + + ! initialise the memory that x points to DO $\mathbb{T}\!=\!1$ ,SIZE $\mathrm{X}\left(\,\mathbb{T}\,\right)=\mathbb{T}$ END DO + + ! print out the first 10 values DO $\mathbb{T}\!=\!1$ ,10 PRINT \*,X(I) END DO + + ! initialise the array y DO $\mathbb{T}\!=\!\mathbb{1}\,,\,\mathbb{1}\,0$ $\mathbb{Y}\left(\mathbb{T}\right)=\mathbb{T}^{\star}\mathbb{T}$ END DO + + ! print out y DO $\mathbb{T}\!=\!1$ ,10 PRINT \*,Y(I) END DO + + ! x now points to y $\tt X\!\!=\!\!>\!\tt Y + +$ ! print out what x now points to DO $\mathbb{T}\!=\!1$ ,10 PRINT \*,X(I) END DO + + ! what has happened to the memory that x + + ! used to point to? END PROGRAM C2115 + +The next is a simple variant on the above: + +PROGRAM C2116 IMPLICIT NONE INTEGER :: Allocate_status $=\!0$ + +REAL , DIMENSION(:) , POINTER :: X REAL , DIMENSION(1:10) , TARGET :: Y INTEGER , PARAMETER :: SIZE $=$ 10000000 INTEGER :: I DO ALLOCATE(X(1:SIZE),STAT $^{!}{=}$ Allocate_status) IF (allocate_status $>$ 0) THEN PRINT \*,' Allocate failed. Program ends.' STOP ENDIF + + ! initialise the memory that x points to DO $\mathbb{T}\!=\!1$ ,SIZE $\mathrm{X}\left(\,\mathbb{T}\,\right)=\mathbb{T}$ END DO + + ! print out the first 10 values DO $\mathbb{T}\!=\!1$ ,10 PRINT \*,X(I) END DO + + ! initialise the array y DO $\mathbb{T}\!=\!1$ ,10 $\mathbb{Y}\left(\mathbb{T}\right)=\mathbb{T}^{\star}\mathbb{T}$ END DO + + ! print out y DO $\mathbb{T}\!=\!1$ ,10 PRINT \*,Y(I) END DO + + ! x now points to y $\tt X\!\!=\!\!>\!\tt Y + +$ ! print out what x now points to DO $\mathbb{T}\!=\!1$ ,10 PRINT \*,X(I) END DO + + ! what has happened to the memory that x + + ! used to point to? end do END PROGRAM C2116 + +Before running this example we recommend starting up a memory monitoring pro- gram. + +Under Microsoft Windows XP Professional holding $[\mathrm{CTRE}]+[\mathrm{ALT}]+[\mathrm{DEL}]$ will bring up the Windows Task Manager. Choose the [Performance] tab to get a + +# Chapter 21 + +screen which will show CPU usage, PF Usage, CPU Usage History and Page File Usage History. You will also get details of Physical and Kernel memory usage. + +Under Linux type + +top + +in a terminal window. + +In these examples we also see the recommended form of the ALLOCATE state- ment when working with arrays. This enables us to test if the allocation has worked and take action accordingly. A positive value indicates an allocation error, zero indicates OK. + +# 21.12 Nonstandard pointer examples + +Some Fortran compilers provide a LOC intrinsic. The description from the CVF on line documentation follows: + +result $=\mathrm{LC}$ (x) x (Input) is a variable, an array or a record field reference, a procedure, or a con- stant; it can be of any data type. It must not be the name of an internal procedure or statement function. If it is a pointer, it must be defined and associated with a target. + +This returns the address of the variable passed. Below are four examples that show some of what is happening behind the scenes when using pointer variables. We have also included some sample output: + +PROGRAM C2117 INTEGER , POINTER :: A,B INTEGER , TARGET :: C INTEGER :: D PRINT \*,LOC(a) PRINT \*,LOC(b) PRINT \*,LOC(c) PRINT \*,LOC(d) $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\AA~}=>\mathrm{~\AA~}\mathrm{~C~}$ $\textrm{C}=\textrm{2}$ $\mathrm{~\ensuremath~{~B~}~}=>\mathrm{~\ensuremath~{~C~}~}$ $\texttt{D}=\texttt{A}+\texttt{B}$ PRINT \*,A,B,C,D PRINT \*,LOC(a) PRINT \*,LOC(b) PRINT \*,LOC(c) + +PRINT \*,LOC(d) END PROGRAM C2117 + +CVF Output: + +0 0 4424172 4424168 2 4 4424172 4424172 4424172 4424168 + +Lahey Output: + +0 4456968 4456972 2 2 2 4 4456968 4456968 4456968 4456972 + +The value zero is often used to signify a special memory value in computing. After the pointer assignments it is clear that all three variables point to the same value: + +PROGRAM C2018 INTEGER , POINTER :: $\mathbb{A}\!=\!>$ NULL(), $\mathrm{B}{=}\,\!>$ NULL() INTEGER , TARGET :: C INTEGER :: D PRINT \*,LOC(a) PRINT \*,LOC(b) PRINT \*,LOC(c) PRINT \*,LOC(d) $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\AA~}=>\mathrm{~\AA~}\mathrm{~C~}$ $\textrm{C}=\textrm{2}$ $\mathrm{~\ensuremath~{~B~}~}=>\mathrm{~\ensuremath~{~C~}~}$ $\texttt{D}=\texttt{A}+\texttt{B}$ + +# Chapter 21 + +PRINT \*,A,B,C,D PRINT \*,LOC(a) PRINT \*,LOC(b) PRINT \*,LOC(c) print \*,loc(d) END PROGRAM C2018 + +# CVF Output: + +0 0 4424168 4424164 2 + +4 4424168 4424168 4424168 4424164 + +# Lahey Output: + +0 0 4456968 4456972 2 2 2 4 4456968 4456968 4456968 4456972 + +We have again the use of zero as the special memory value: + +PROGRAM C2119 INTEGER , POINTER :: $\mathbb{A}\!=\!>$ NULL(), $\mathrm{B}{=}\,\!>$ NULL() INTEGER , TARGET :: C INTEGER :: D PRINT \*,LOC(a) PRINT \*,LOC(b) ALLOCATE(a) ALLOCATE(b) PRINT \*,LOC(a) PRINT \*,LOC(b) + +PRINT \*,LOC(c) PRINT \*,LOC(d) $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\AA~}=>\mathrm{~\AA~}\mathrm{~C~}$ $\textrm{C}=\textrm{2}$ $\mathrm{~\ensuremath~{~B~}~}=>\mathrm{~\ensuremath~{~C~}~}$ $\texttt{D}=\texttt{A}+\texttt{B}$ PRINT \*,A,B,C,D PRINT \*,LOC(a) PRINT \*,LOC(b) PRINT \*,LOC(c) PRINT \*,LOC(d) END PROGRAM C2119 + +CVF Output: + +3292304 3292328 4424152 4424148 + +2 + +4 4424152 4424152 4424152 4424148 + +Lahey Output: + +0 0 8915968 8916160 4457148 4457152 2 2 2 4 4457148 4457148 4457148 4457152 + +# Chapter 21 + +In this example we actually use the ALLOCATE statement to set aside space for the pointers. What is interesting in this example is that the original space set aside becomes lost after the pointer assignments. This indicates a small memory leak: + +PROGRAM C2020 INTEGER , POINTER :: $\mathbb{A}\!=\!>$ NULL(), $\mathrm{B}{=}\,\!>$ NULL() INTEGER , TARGET :: C INTEGER :: D PRINT \*,LOC(a) PRINT \*,LOC(b) ALLOCATE(a) ALLOCATE(b) PRINT \*,LOC(a) PRINT \*,LOC(b) PRINT \*,LOC(c) PRINT \*,LOC(d) $\mathrm{~\textit~{~C~}~}=\mathrm{~\1~}$ $\mathrm{~\textit~{~A~}~}=\mathrm{~\textit~{~21}~}$ $\textrm{C}=\textrm{2}$ B = A $\texttt{D}=\texttt{A}+\texttt{B}$ PRINT \*,A,B,C,D PRINT \*,LOC(a) PRINT \*,LOC(b) PRINT \*,LOC(c) PRINT \*,LOC(d) END PROGRAM C2020 + +# CVF Output: + +0 0 3292304 3292328 4424152 4424148 21 3292304 3292328 4424152 4424148 + +Lahey Output: + +0 8915968 8916160 4457152 4457156 21 21 2 42 8915968 8916160 4457152 4457156 + +In this case the addresses for A and B remain the same as for a normal assignment, not a pointer assignment. + +# 21.13 Problems + +1. Compile and run all of the example programs in this chapter with your compiler and examine the output. + +2. There are a number of ways of handling exceptions with the READ statement, and we have used the IOSTAT option in this chapter. Consider the following pro- gram: + +PROGRAM C2102p INTEGER :: IO_Stat_Number $=\!0$ INTEGER :: I DO READ (UNIT $^{\ast}{=}^{\ast}$ , $\scriptstyle{\mathrm{FMT}=10}$ , ADVANCE $=$ 'NO' & , IOSTAT $=$ IO_Stat_Number) I 10 FORMAT(I3) + + ! 0 $=$ no error + + ! no end of file (eof) + + ! no end of record (eor) + + ! - $=$ eor or eof + + ! + $=$ an error occurred PRINT \*,' iostat $=$ ',IO_Stat_Number PRINT \*,I END DO END PROGRAM C2102p + +This program is a simple test of the IOSTAT values of whatever system you work on. Try typing in a variety of values including minimally: + +• A valid three-digit number $^+$ [RETURN] key. • A three-digit number with an embedded blank, e.g., $^\textrm{\scriptsize12+}$ [RETURN] key. • [RETURN] key only. • $[\mathrm{CTRL}]+Z.$ . • Any other non-numeric character on the keyboard. • $100200300\;+$ [RETURN] key. • [CTRL] + C + +This will enable us to program exactly the kind of behaviour we want from I/O and can be used as a code segment for other programs. + +# Introduction to Subroutines + +“A man should keep his brain attic stacked with all the furniture he is likely to use, and the rest he can put away in the lumber room of his library, where he can get at it if he wants.” + +Sir Arthur Conan Doyle, Five Orange Pips + +# Aims + +The aims of this chapter are: + +• To consider some of the reasons for the inclusion of subroutines in a pro- gramming language. • To introduce with a concrete example some of the concepts and ideas in- volved with the definition and use of subroutines. • The INTERFACE statement and interface blocks. • Arguments or parameters. • The INTENT attribute for parameters. • The CALL statement. • Scope of variables. • Local variables and the SAVE attribute. • The use of parameters to report on the status of the action carried out in the subroutine. + +# 22 Introduction to Subroutines + +In the earlier chapter on functions we introduced two types of function + +• Intrinsic functions — which are part of the language. • User defined functions — by which we extend the language. + +We now introduce subroutines which collectively with functions are given the name procedures. Procedures provide a very powerful extension to the language by: + +• Providing us with the ability to break problems down into simpler more easily solvable subproblems. • Allowing us to concentrate on one aspect of a problem at a time. • Avoiding duplication of code. • Hiding away messy code so that a main program is a sequence of calls to procedures. • Providing us with the ability to put together collections of procedures that solve commonly occurring subproblems, often given the name libraries, and generally compiled. • Allowing us to call procedures from libraries written, tested and docu- mented by experts in a particular field. There is no point in reinventing the wheel! + +There are a number of concepts required for the successful use of subroutines and we met some of them in Chapter 15 when we looked at user defined functions. We will extend the ideas introduced there of parameters and introduce the additional concept of an interface block. The ideas are best explained with a concrete exam- ple. + +Note that we use the terms parameters and arguments interchangeably. + +# 22.1 Example 1 + +This example is one we met earlier that solves a quadratic equation, i.e., solves $\textit{a x}^{2}+\textit{b x}+\textit{c}\,{=}\,0$ + +The program to do this originally was just one program. In the example below we break that problem down into smaller parts and make each part a subroutine. The components are: + +• Main program or driving routine. • Interaction with user to get the coefficients of the equation. + +• Solution of the quadratic. + +Let us look now at how we do this with the use of subroutines: + +PROGRAM ch2201 IMPLICIT NONE + + ! Simple example of the use of a main program and two + + ! subroutines. One interacts with the user and the + + ! second solves a quadratic equation, + + ! based on the user input. + +REAL :: P, Q, R, Root1, Root2 INTEGER :: IFail $=\!0$ LOGICAL :: $\mathrm{CK}=$ .TRUE. CALL Interact(P,Q,R,OK) IF (OK) THEN CALL Solve(P,Q,R,Root1,Root2,IFail) IF (IFail $==\ \ \bot$ ) THEN PRINT \*,' Complex roots, PRINT \*,' calculation abandoned' ELSE PRINT \*,' Roots are ',Root1,' ',Root2 ENDIF ELSE PRINT\*,' Error in data input program ends' ENDIF END PROGRAM ch2201 + +SUBROUTINE Interact(A,B,C,OK) IMPLICIT NONE REAL , INTENT(OUT) :: A REAL , INTENT(OUT) :: B REAL , INTENT(OUT) :: C LOGICAL , INTENT(OUT) :: OK INTEGER :: IO_Status $=\!0$ PRINT\*,' Type in the coefficients A, B AND C' READ(UNIT $=\star$ , $\scriptstyle{\mathrm{FMT}=\star}$ ,IOSTAT $^{\ast}{=}$ IO_Status)A,B,C IF (IO_Status $==\quad0$ ) THEN $\mathrm{CK}=$ .TRUE. ELSE OK $\b=$ .FALSE. ENDIF END SUBROUTINE Interact + +SUBROUTINE Solve(E,F,G,Root1,Root2,IFail) IMPLICIT NONE REAL , INTENT(IN) :: E REAL , INTENT(IN) :: F REAL , INTENT(IN) :: G REAL , INTENT(OUT) :: Root1 REAL , INTENT(OUT) :: Root2 INTEGER , INTENT(INOUT) :: IFail + + ! Local variables REAL :: Term REAL :: A2 Term $=$ F\*F - 4.\*E\*G A2 = $\mathbf{E}^{\star}2\cdot0 + +$ ! if term $<\quad0$ , roots are complex IF(Term $<\quad0\cdot0$ )THEN IFail $\mathbf{\omega}=\mathbb{1}$ ELSE Term $=$ SQRT(Term) $\begin{array}{r l}{\mathsf{R o o t1}}&{=\quad\left(-\,\mathrm{F}\!+\!\mathrm{T}\!\mathrm{e}\,\mathtt{r}\!\mathrm{m}\right)\,/\!\mathrm{A}2}\end{array}$ Root2 $=$ (-F-Term)/A2 ENDIF + +END SUBROUTINE Solve + +# 22.1.1 Defining a subroutine + +A subroutine is defined as SUBROUTINE subroutine_name(optional list of dummy arguments) IMPLICIT NONE dummy argument type definitions with INTENT + +and from the earlier example we have the subroutine + +SUBROUTINE Interact(A,B,C,OK) IMPLICIT NONE REAL, INTENT(OUT)::A,B,C LOGICAL, INTENT(OUT)::OK + +END SUBROUTINE Interact + +# 22.1.2 Referencing a subroutine + +To reference a subroutine you use the CALL statement: + +CALL subroutine_name(optional list of actual arguments) + +and from the earlier example the call to subroutine Interact was of the form: + +CALL Interact(P,Q,R,OK) + +When a subroutine returns to the calling program unit control is passed to the statement following the CALL statement. + +# 22.1.3 Dummy arguments or parameters and actual arguments + +Procedures and their calling program units communicate through their arguments. We often use the terms parameter and arguments interchangeably throughout this text. The SUBROUTINE statement normally contains a list of dummy arguments, separated by commas and enclosed in brackets. The dummy arguments have a type associated with them; for example, in subroutine Solve X is of type REAL, but no space is put aside for this in memory. When the subroutine is referenced e.g., CALL Solve(P,Q,R,Root1,Root2,Ifail), then the dummy argument points to the ac- tual argument P, which is a variable in the calling program unit. The dummy argument and the actual argument must be of the same type — in this case REAL. + +# 22.1.4 Intent + +It is recommended that dummy arguments have an INTENT attribute. In the earlier example subroutine Solve has a dummy argument E with INTENT(IN), which means that when the subroutine is referenced or called it is expecting E to have a value, but its value cannot be changed inside the subroutine. This acts as an extra security measure besides making the program easier to understand. For each pa- rameter it may have one of three attributes: + +• INTENT(IN), where the parameter already has a value and cannot be al- tered in the called routine. • INTENT(OUT), where the parameter does not have a value, and is given one in the called routine. • INTENT(INOUT), where the parameter already has a value and this is changed in the called routine. + +# 22.1.5 Local variables + +We saw with functions that variables could be essentially local to the function and unavailable elsewhere. The concept of local variables also applies to subroutines. In the example above Term and A2 are both local variables to the subroutine Solve . + +# Chapter 22 + +# 22.1.6 Local variables and the SAVE attribute + +Local variables are usually created when a procedure is called and their value lost when execution returns to the calling program unit. To make sure that a local vari- able retains its values between calls to a subprogram the SAVE attribute can be used on a type statement; e.g., + +INTEGER , SAVE :: I + +means that when this statement appears in a subprogram the value of the local variable I is saved between calls. + +# 22.1.7 Scope of variables + +In most cases variables are only available within the program unit that defines them. The introduction of argument lists to functions and subroutines immediately opens up the possibility of data within one program unit becoming available in one or more other program units. + +In the main program we declare the variables P, Q, R, Root1, Root2, IFail and OK. + +Subroutine Interact has no variables locally declared. It works on the argu- ments A, B, C and OK; which map onto P, Q, R and OK from the main program, i.e., it works with those variables. + +Subroutine Solve has two locally defined variables, Term and A2. It works with the variables E, F, G, Root1, Root2 and IFail, which map onto P, Q, R, Root1, Root2 and IFail from the main program. + +# 22.1.8 Status of the action carried out in the subroutine + +It is also useful to use parameters that carry information regarding the status of the action carried out by the subroutine. With the subroutine Interact we use a log- ical variable OK to report on the status of the interaction with the user. In the subroutine Solve we use the status of the integer variable Ifail to report on the status of the solution of the equation. + +# 22.2 Example 2 + +Consider the following example: + +program ch2202 implicit none real :: a,b,c $\begin{array}{c c c}{\textmd{a}}&{=}&{\pounds\,0\,0\,0\,.\,0}\end{array}$ $\mathrm{~\textcent~{~b~}~}=\mathrm{~\textcent~{~2~0~.~0~}~}$ call divide(a,b,c) print \*,c + +end program ch2202 + +subroutine divide(a,b,c) implicit none integer , intent(in) :: a integer , intent(in) :: b integer , intent(out):: c $\tt C\!=\!a/b$ end subroutine divide + +There is a fundamental problem here. In the main program the variables A, B and C are declared to be of type real. In the subroutine DIVIDE they are integer. + +If the main program and subroutine are in one file when compiled then the com- piler has the oportunity of catching this mismatch. The Nag f95 compiler release 4.2 and the Salford FTN95 compiler release 4.6.0 both diagnose this error and will not compile the program. The following compilers compiled and executed the code generating the following answers: + +CVF 6.6C: 1.4012985E-45 Intel 9.0 1.4012985E-45 Lahey 5.7 1.40129846E-45 + +Fortran 90 introduced a number of language features to help in this area: + +• Interface blocks. • Contained procedures. • Modules. + +We will look at the first two in this chapter and at modules later on. + +# 22.3 Example 3 — Quadratic example with interface blocks + +This is the first example with the addition of interface blocks: + +PROGRAM ch2203 IMPLICIT NONE + + ! Simple example of the use of a main program and two + + ! subroutines. One interacts with the user and the + + ! second solves a quadratic equation, + + ! based on the user input. INTERFACE + +SUBROUTINE Interact(A,B,C,OK) + +# Chapter 22 + +IMPLICIT NONE REAL , INTENT(OUT) :: A REAL , INTENT(OUT) :: B REAL , INTENT(OUT) :: C LOGICAL , INTENT(OUT) :: OK END SUBROUTINE Interact + +SUBROUTINE Solve(E,F,G,Root1,Root2,IFail) IMPLICIT NONE REAL , INTENT(IN) :: E REAL , INTENT(IN) :: F REAL , INTENT(IN) :: G REAL , INTENT(OUT) :: Root1 REAL , INTENT(OUT) :: Root2 INTEGER , INTENT(INOUT) :: IFail END SUBROUTINE Solve + +END INTERFACE + +REAL :: P, Q, R, Root1, Root2 INTEGER :: IFail $.\!=\!0$ LOGICAL :: $\mathrm{CK}=$ .TRUE. CALL Interact(P,Q,R,OK) IF (OK) THEN CALL Solve(P,Q,R,Root1,Root2,IFail) IF (IFail $==\ \ \bot$ ) THEN PRINT \*,' Complex roots, calculation abandoned' ELSE PRINT \*,' Roots are ',Root1,' ',Root2 ENDIF ELSE PRINT\*,' Error in data input program ends' ENDIF + +END PROGRAM ch2203 + +SUBROUTINE Interact(A,B,C,OK) IMPLICIT NONE REAL , INTENT(OUT) :: A REAL , INTENT(OUT) :: B REAL , INTENT(OUT) :: C LOGICAL , INTENT(OUT) :: OK INTEGER :: IO_Status $=\!0$ + +PRINT\*,' Type in the coefficients A, B AND C' READ(UNIT $=\star$ , $\scriptstyle{\mathrm{FT}}={\star}$ ,IOSTAT $^{\ast}{=}$ IO_Status)A,B,C IF (IO_Status $==\quad0$ ) THEN OK $\b=$ .TRUE. ELSE $\mathrm{CK}=$ .FALSE. ENDIF + +SUBROUTINE Solve(E,F,G,Root1,Root2,IFail) IMPLICIT NONE REAL , INTENT(IN) :: E REAL , INTENT(IN) :: F REAL , INTENT(IN) :: G REAL , INTENT(OUT) :: Root1 REAL , INTENT(OUT) :: Root2 INTEGER , INTENT(INOUT) :: IFail + + ! Local variables REAL :: Term REAL :: A2 Term $=$ F\*F - 4.\*E\*G A2 = $\mathbf{E}^{\star}2\cdot0 + +$ ! if term $<\quad0$ , roots are complex IF(Term $<$ 0.0)THEN IFail $\underline{{\boldsymbol{\mathbf{\Pi}}}}_{\perp}$ ELSE Term $=$ SQRT(Term) Root1 $=$ (-F+Term)/A2 Root2 $=$ (-F-Term)/A2 ENDIF END SUBROUTINE Solve + +# The key code is given below: + +# INTERFACE + +SUBROUTINE Interact(A,B,C,OK) IMPLICIT NONE REAL , INTENT(OUT) :: A REAL , INTENT(OUT) :: B REAL , INTENT(OUT) :: C LOGICAL , INTENT(OUT) :: OK END SUBROUTINE Interact + +SUBROUTINE Solve(E,F,G,Root1,Root2,IFail) IMPLICIT NONE REAL , INTENT(IN) :: E REAL , INTENT(IN) :: F REAL , INTENT(IN) :: G REAL , INTENT(OUT) :: Root1 REAL , INTENT(OUT) :: Root2 INTEGER , INTENT(INOUT) :: IFail END SUBROUTINE Solve + +END INTERFACE + +Interface blocks in the above example provide us with the ability to do type check- ing between the calling routine and the called routine. One of the most common errors in programming is getting the sequence and type of the parameters wrong between subprograms. There is of course the editing overhead of duplicating the code in this example. We will look at software tools that can generate interface blocks for us in a later chapter. + +There are times when the use of interface blocks is mandatory in Fortran and we will cover this as and when required. However, it is good working practice to pro- vide interface blocks when dealing with legacy Fortran 77 style code. + +We will look at additional ways of providing explicit interfaces later on. + +As Fortran 95 libraries become more widely available interface blocks for library routines will be provided by the supplier on line, and this minimises much of the effort in using them. Nag, for example, already has interface blocks available for its library for many platforms. + +# 22.4 Example 4 — Quadratic example and the CONTAINS statement + +This example solves the problem of diagnosing mismatches between the calling and called routine by the CONTAINS statement. The two subroutines Interact and Solve are now part of the main program. This method has drawbacks with larger codes suites as we will end up recompiling all of the code within the main pro- gram: + +PROGRAM ch2204 IMPLICIT NONE + + ! Simple example of the use of a main program and two + + ! subroutines. One interacts with the user and the + + ! second solves a quadratic equation, + +! based on the use input. + +REAL :: P, Q, R, Root1, Root2 INTEGER :: IFail $=\!0$ LOGICAL :: $\mathrm{CK}=$ .TRUE. CALL Interact(P,Q,R,OK) IF (OK) THEN CALL Solve(P,Q,R,Root1,Root2,IFail) IF (IFail $==\ \ \bot$ ) THEN PRINT \*,' Complex roots, calculation abandoned' ELSE PRINT \*,' Roots are ',Root1,' ',Root2 ENDIF ELSE PRINT\*,' Error in data input program ends' ENDIF + +contains + +SUBROUTINE Interact(A,B,C,OK) IMPLICIT NONE REAL , INTENT(OUT) :: A REAL , INTENT(OUT) :: B REAL , INTENT(OUT) :: C LOGICAL , INTENT(OUT) :: OK INTEGER :: IO_Status $=\!0$ PRINT\*,' Type in the coefficients A, B AND C' READ(UNIT $=\star$ , $\scriptstyle{\mathrm{FT}}={\star}$ ,IOSTAT $^{\ast}{=}$ IO_Status)A,B,C IF (IO_Status $==\quad0$ ) THEN $\mathrm{CK}=$ .TRUE. ELSE OK $\b=$ .FALSE. ENDIF END SUBROUTINE Interact + +SUBROUTINE Solve(E,F,G,Root1,Root2,IFail) IMPLICIT NONE REAL , INTENT(IN) :: E REAL , INTENT(IN) :: F REAL , INTENT(IN) :: G REAL , INTENT(OUT) :: Root1 REAL , INTENT(OUT) :: Root2 + +INTEGER , INTENT(INOUT) :: IFail + + ! Local variables REAL :: Term REAL :: A2 Term $=$ F\*F - 4.\*E\*G A2 = $\mathbf{E}^{\star}2\cdot0 + +$ ! if term $<\quad0$ , roots are complex IF(Term < 0.0)THEN IFail $\mathbf{\tau}\!=\!1$ ELSE Term $=$ SQRT(Term) Root1 $=$ (-F+Term)/A2 Root2 $=$ (-F-Term)/A2 ENDIF END SUBROUTINE Solve + +END PROGRAM ch2204 + +Thus in this chapter we have seen three ways of using subroutines: + +• Classic Fortran 77 style as in the first example. The major disadvantage is the lack of checking of the parameters between the calling and called rou- tine. • Interface blocks — the major disadvantage is the code duplication • Contained subroutines — major disadvantage is that we have to recompile the program and all contained subroutines. + +We will look at using modules to address this problem in a later chapter. + +# 22.5 Why bother? + +Given the increase in the complexity of the overall program to solve a relatively straightforward problem, one must ask why bother. The answer lies in our ability to manage the solution of larger and larger problems. We need all the help we can get if we are to succeed in our task of developing large-scale reliable programs. + +We need to be able to break our problems down into manageable subcomponents and solve each in turn. We are now in a very good position to be able to do this. Given a problem that requires a main program, one or more functions and one or more subroutines we can work on each subcomponent in relative isolation, and know that by using features like interface blocks we will be able to glue all of the components together into a stable structure at the end. We can independently com- pile the main program and functions and subroutines and use the linker to generate the overall executable, and then test that. Providing we keep our interfaces the same we can alter the actual implementations of the functions and subroutines and just recompile the changed procedures. + +# 22.6 Summary + +We now have the following concepts for the use of subroutines: + +• INTERFACE blocks. • INTENT attribute for parameters. • Dummy parameters. • The use of the CALL statement to invoke a subroutine. • The concepts of variables that are local to the called routines and are un- available elsewhere in the overall program. • Communication between program units via the argument list. • The concept of parameters on the call that enable us to report back on the status of the called routine. + +# 22.7 Problems + +1. Type in the program example in this chapter as three files. Compile each indi- vidually. When you have successfully compiled each routine (there will be the inevitable typing mistakes) look at the file sizes of the object file. Now use the linker to produce one executable. Look at the file size of the executable. What do you notice? + +The development of large programs is eased considerably by the ability to compile small program units and eradicate the compilation errors from one unit at a time. + +The linker obviously also has an important role to play in the development process. + +2. Write a subroutine to calculate new coordinates $(x',y')$ from $(x,y)$ when the axes are rotated counterclockwise through an angle of a radians using: $\begin{array}{l c r}{{x^{\prime}=x\cos a+y\sin a}}\\ {{\qquad}}\\ {{y^{\prime}=-x\sin a+y\cos a}}\end{array}$ + +# Hint: + +The subroutine would look something like + +SUBROUTINE Change Coordinate(X,Y,A,XD,YD) + +Write a main program to read in values of $x,y,a$ , call the subroutine and print out the new coordinates. + +# Subroutines: 2 + +“It is one thing to show a man he is in error, and another to put him in possession of the truth.” + +John Locke + +# Aims + +The aims of this chapter are to extend the ideas in the earlier chapter on subrou- tines and look in more depth at parameter passing, in particular using a variety of ways of passing arrays. + +# Chapter 23 + +# 23 Subroutines: 2 + +# 23.1 More on parameter passing + +So far we have seen scalar parameters of type real, integer and logical. We will now look at numeric array parameters and character parameters. We need to intro- duce some technical terminology first. Don't panic if you don't fully understand the terminology the as examples should clarify things. + +# 23.1.1 Explicit-shape array + +An explicit-shape array is a named array that is declared with explicit values for the bounds in each dimension of the array. + +The following explicit-shape arrays can specify nonconstant bounds: + +• An automatic array (the array is a local variable). • An adjustable array (the array is a dummy argument to a subprogram). + +# 23.1.2 Assumed-shape array + +An assumed-shape array is a nonpointer dummy argument array that takes its shape from the associated actual argument array. + +# 23.1.3 Deferred-shape array + +A deferred-shape array is an allocatable array or an array pointer. An allocatable array is an array that has the ALLOCATABLE attribute and a specified rank, but its bounds, and hence shape, are determined by allocation or argument association. + +# 23.1.4 Automatic arrays + +An automatic array is an explicit-shape array that is a local variable. Automatic ar- rays are only allowed in function and subroutine subprograms, and are declared in the specification part of the subprogram. At least one bound of an automatic array must be a nonconstant specification expression. The bounds are determined when the subprogram is called . + +# 23.1.5 Assumed-size array — Fortran 77 style + +An assumed-size array is a dummy argument array whose size is assumed from that of an associated actual argument. The rank and extents may differ for the ac- tual and dummy arrays; only the size of the actual array is assumed by the dummy array. You would not use this type of parameter in modern Fortran code. We will come back to arrays of this type in the chapter on converting from Fortran 77 to modern Fortran. + +# 23.1.6 Adjustable arrays — Fortran 77 style + +An adjustable array is an explicit-shape array that is a dummy argument to a subprogram. At least one bound of an adjustable array must be a nonconstant spec- ification expression. The bounds are determined when the subprogram is called. You would not use this type of parameter in modern Fortran code. We will come back to arrays of this type in the chapter on converting from Fortran 77 to modern Fortran. + +# 23.2 Common code example + +We are going to use an example based on a main program and a subroutine that calculates the mean and standard deviation of an array of numbers. The subroutine has the following parameters: + +• x - the array containing the real numbers. • n - the number of elements in the array. • mean - the mean of the numbers. • std_dev - the standard deviaition of the numbers. + +We will look at some of the ways we can pass the array between the main program and the subroutine in both Fortran 77 and Fortran 90 styles. + +# 23.3 Explicit-shape example + +Consider the following program and subroutine. + +program ch2301 implicit none integer , parameter :: $\mathtt{n}\!=\!\mathtt{10}$ real , dimension(1:n) :: x real , dimension(-4:5) :: y real , dimension(10) :: z real , allocatable , dimension(:) :: t real :: m,sd integer :: i + +interface subroutine stats(x,n,mean,std_dev) implicit none integer , intent(in) :: n real , intent(in) , dimension(1:n) :: x real , intent(out) :: mean real , intent(out) :: std_dev + +# Chapter 23 + +end subroutine stats end interface + +do $\dot{\bot}\!=\!\bot$ ,n $\mathtt{x}\left(\,\mathtt{i}\,\right)=\mathtt{x}{\in}{\mathtt{e a l}}$ (i) end do call stats(x,n,m,sd) print \*,' x' print \*,' mean = ',m print \*,' Standard deviation $=$ ',sd $\mathtt{Y}^{=\mathtt{X}}$ call stats(y,n,m,sd) print \*,' y' print \*,' mean = ',m print \*,' Standard deviation $=$ ',sd $z\!=\!\overline{{\!\mathrm{x}}}$ call stats(z,10,m,sd) print \*,' z' print \*,' mean = ',m print \*,' Standard deviation $=$ ',sd allocate(t(10)) $\scriptstyle{\mathsf{t}}=\;\!\mathsf{x}$ call stats(t,10,m,sd) print \*,' t' print \*,' mean = ',m print \*,' Standard deviation $=$ ',sd end program ch2301 + +subroutine stats(x,n,mean,std_dev) implicit none integer , intent(in) :: n real , intent(in) , dimension(1:n) :: x real , intent(out) :: mean real , intent(out) :: std_dev real :: variance real:: sumxi,sumxi2 integer :: i + +variance $=\!0$ .0 sumxi $=\!0$ .0 sumxi $\gimel\b=0$ .0 do i $_\cdot\!=\!\!\perp$ ,n + +sumxi $=$ sumxi $^+$ x(i) sumxi2 $=$ sumxi2 + x(i)\*x(i) end do mean $\b=$ sumxi/n variance $=$ (sumxi2 - sumxi\*sumxi/n)/(n-1) std_dev ${}^{\prime}{=}$ sqrt(variance ) end subroutine stats + +The key line in the subroutine is + +real , intent(in) , dimension(1:n) :: x + +where the dummy array argument x is declared with explicit bounds and known as an explicit-shape dummy array. Even though it is not mandatory it is recommended that interface blocks be used so that the shape and size of actual and dummy argu- ments can be checked explicitly. + +Note also the use of a DO loop to calculate the sum of the elements and the sum of the squares of the elements. This is a Fortran 77 style solution to this problem. + +# 23.4 Assumed-shape example + +A fundamental rule in modern Fortran is that the shape of an actual array argument and its associated dummy arguments are the same, i.e., they both must have the same rank and the same extents in each dimension. The best way to apply this rule is to use assumed-shape dummy array arguments as shown in the example below. + +In the subroutine we have + +real , intent(in) , dimension(:) :: x + +where x is an assumed-shape dummy array argument, and it will assume the shape of the actual argument when the subroutine is called. + +In this example in the main program we have declared the actual array argument x to be allocatable to make the program more flexible. + +program ch2302 + +implicit none integer :: n real , allocatable , dimension(:) :: x real :: m,sd + +interface subroutine stats(x,n,mean,std_dev) + +implicit none integer , intent(in) :: n real , intent(in) , dimension(:) :: x real , intent(out) :: mean real , intent(out) :: std_dev end subroutine stats end interface + +print \*,' type in n' read \*,n allocate $\left(\,\mathbf{x}\;(\;\mathbb{1}:\mathbf{n}\,)\right.$ ) call random_number $(\mathbf{\nabla}\mathbf{x})$ $\mathrm{{x}{=}\mathrm{{x}}^{\star}\mathbb{1}_{0}}$ call stats(x,n,m,sd) print $^{\star}\,,\,\,^{\bullet}$ numbers were ' print $^{\star}\,,\,\mathtt{x}$ print \*,' Mean $=$ ',m print \*,' Standard deviation $=$ ',sd + +end program ch2302 + +subroutine stats(x,n,mean,std_dev) implicit none integer , intent(in) :: n real , intent(in) , dimension(:) :: x real , intent(out) :: mean real , intent(out) :: std_dev real :: variance real:: sumxi,sumxi2 integer :: i + +variance $=\!0$ .0 sumxi $=\!0$ .0 sumxi $\scriptstyle2\,=\,0$ .0 do $\dot{\bot}\!=\!\bot$ ,n sumxi $=$ sumxi $^+$ x(i) sumxi2 $=$ sumxi2 + x(i)\*x(i) end do mean $\b=$ sumxi/n variance $=$ (sumxi2 - sumxi\*sumxi/n)/(n-1) std_dev ${}^{\prime}{=}$ sqrt(variance) end subroutine stats + +# 23.4.1 Notes + +There are several restrictions when using assumed-shape arrays: + +• The rank is equal to the number of colons, in this case 1. • The lower bounds of the assumed-shape array are the specified lower bounds, if present, and 1 otherwise. In the example above it is 1 because we haven’t specified a lower bound. • The upper bounds will be determined on entry to the procedure and will be whatever values are needed to make sure that the extents along each dimension of the dummy argument are the same as the actual argument. In this case the upper bound will be n. • An assumed-shape array must not be defined with the POINTER or ALLOCATABLE attribute in Fortran 90 or Fortran 95. • When using an assumed-shape array an interface block is mandatory. + +Assumed-shape arraay parameter passing also works with Fortran 77 style stati- cally allocated arrays, i.e., + +real , dimension(1:10) :: x + +which is commonly seen in older code. + +# 23.5 Character arguments and assumed-length dummy arguments + +The types of parameters considered so far have been REAL, INTEGER and LOGI- CAL. CHARACTER variables are slightly different because they have a length associated with them. Consider the following program and subroutine which, given the name of a file, opens it and reads values into two REAL arrays, X and Y: + +PROGRAM ch2303 IMPLICIT NONE REAL,DIMENSION(1:100)::A,B INTEGER :: Nos,I CHARACTER(LEN $\scriptstyle{\dot{=}20}$ )::Filename INTERFACE SUBROUTINE Readin(Name,X,Y,N) IMPLICIT NONE INTEGER , INTENT(IN) :: N REAL,DIMENSION(1:N),INTENT(OUT)::X,YCHARACTER ( $\tt L E N\mathrm{=\star}$ ),INTENT(IN)::Name END SUBROUTINE Readin + +END INTERFACE PRINT \*,' Type in the name of the data file' READ '(A)' , Filename PRINT \*,' Input the number of items in the file' READ \* , Nos CALL Readin(Filename,A,B,Nos) PRINT \* , ' Data read in was' DO $\mathbb{T}\!=\!1$ ,Nos PRINT \*,' ',A(I),' ',B(I) ENDDO + +END PROGRAM ch2303 + +SUBROUTINE Readin(Name,X,Y,N) IMPLICIT NONE INTEGER , INTENT(IN) :: N REAL,DIMENSION(1:N),INTENT(OUT)::X,YCHARACTER (LEN $=\star$ ),INTENT(IN)::Name INTEGER::I OPEN(UNIT $\mathrm{\Sigma=}\beth0$ ,STATUS $=$ 'OLD',FILE $=$ Name) DO $\mathbb{T}\!=\!1$ ,N READ(10,\*)X(I),Y(I) END DO CLOSE(UNIT $\!=\!\underline{{1}}\,0$ )END SUBROUTINE Readin + +The main program reads the file name from the user and passes it to the subroutine that reads in the data. The dummy argument Name is of type assumed-length, and picks up the length from the actual argument Filename in the calling routine, which is in this case 20 characters. An interface block must be used with as- sumed-shape dummy arguments. + +# 23.6 Rank 2 and higher arrays as parameters + +# 23.6.1 Explicit-shape dummy arrays + +Consider the following example which uses a Fortran 77 style of two-dimensional array parameter passing. + +In the main program we have the following declaration of the rank 2 actual array arguments: + +REAL , DIMENSION (1:Max,1:Max)::One,Two,Three,One_T + +and in the subroutine Matrix_bits we declare the rank 2 dummy array arguments as follows: + +REAL, DIMENSION (1:Max,1:Max), INTENT(IN) :: A,B REAL, DIMENSION (1:Max,1:Max), INTENT(OUT) :: C,A_T + +We have split the declaration into two as the arrays have different intents. These dummy array arguments are explicit-shape, i.e., their bounds are declared in the subroutine. + +Note that in the main program $\mathbf{N}$ may be less than Max and because of the way Fortran stores arrays internally we must pass both variables as arguments to the subroutine Matrix_bits, N being used to control the DO loops and Max needed in the array declarations: + +PROGRAM ch2304 IMPLICIT NONE INTEGER, PARAMETER :: Max $\scriptstyle{\dot{=}}10$ REAL , DIMENSION (1:Max,1:Max)::One,Two,Three,One_T INTEGER :: I,N INTERFACE SUBROUTINE Matrix_bits(A,B,C,A_T,N,Max) IMPLICIT NONE INTEGER, INTENT(IN):: N, Max REAL, DIMENSION (1:Max,1:Max), INTENT(IN) :: A,B REAL, DIMENSION (1:Max,1:Max), INTENT(OUT) :: C,A_T END SUBROUTINE Matrix_bits END INTERFACE + +PRINT \*,'Input size of matrices' READ\*,N DO WHILE $(\mathtt{N}\;\;>\;\;\mathtt{M a x})$ ) PRINT\*,'size of matrices must be $<=$ ',Max PRINT \*,'Input size of matrices' READ\*,N END DO DO $\mathbb{T}\!=\!1$ ,N PRINT\*, 'Input row ', I,' of One' READ\*,One(I,1:N) END DO DO $\mathbb{T}\!=\!1$ ,N PRINT\*, 'Input row ', I,' of Two' READ\*,Two(I,1:N) END DO + +CALL Matrix_bits(One,Two,Three,One_T,N,Max) PRINT\*,' Matrix Three:' DO $\mathbb{T}\!=\!1$ ,N PRINT \*,Three(I,1:N) END DO PRINT \*,' Matrix One_T:' DO $\mathbb{T}\!=\!1$ ,N PRINT \*,One_T(I,1:N) END DO + +END PROGRAM ch2304 + +SUBROUTINE Matrix_bits(A,B,C,A_T,N,Max) IMPLICIT NONE INTEGER, INTENT(IN):: N, Max REAL, DIMENSION (1:Max,1:Max), INTENT(IN) :: A,B REAL, DIMENSION (1:Max,1:Max), INTENT(OUT) :: C,A_T + +INTEGER::I,J,K REAL:: Temp + +! + + ! matrix multiplication $\mathrm{C}{=}\mathrm{\mathbb{A}}$ B + + ! DO $\mathbb{T}\!=\!1$ ,N DO $\uptau\!\!=\!\!1$ ,N Temp $\iota\!=\!0\cdot($ DO $\mathtt{K}\!\!=\!\!1$ ,N Temp $=$ Temp $^+$ A(I,K) \* B (K,J) END DO $\begin{array}{r l r}{\mathsf{C}\left(\,\mathbb{T}\,,\,\mathbb{J}\,\right)}&{{}=}&{\mathsf{T e m p}}\end{array}$ END DO END DO + + ! + + ! set A_T to be transpose matrix A DO $\mathbb{T}\!=\!1$ ,N DO $\uptau\!\!=\!\!1$ ,N $\begin{array}{r l r}{\mathbb{A}\_{\mathbb{T}}\left(\,\mathbb{I}\,,\,\mathbb{J}\,\right)}&{{}=}&{\mathbb{A}\left(\,\mathbb{J}\,,\,\mathbb{I}\,\right)}\end{array}$ END DO END DO END SUBROUTINE Matrix_bits + +Note the use of DO loops to carry out the matrix multiplication and transpose. This is a Fortran 77 style solution to the problem. + +# 23.6.2Assumed-shape dummy array arguments + +With the introduction of assumed-shape dummy array arguments the necessity to pass through Max in the last program is removed. This is shown in the example below: + +PROGRAM ch2305 IMPLICIT NONE REAL , ALLOCATABLE , DIMENSION & (:,:)::One,Two,Three,One_T INTEGER :: I,N INTERFACE SUBROUTINE Matrix_bits(A,B,C,A_T,N) IMPLICIT NONE INTEGER, INTENT(IN):: N REAL, DIMENSION (:,:), INTENT(IN) :: A,B REAL, DIMENSION (:,:), INTENT(OUT) :: C,A_T END SUBROUTINE Matrix_bits END INTERFACE PRINT \*,'Input size of matrices' READ\*,N ALLOCATE(One(1:N,1:N))ALLOCATE(Two(1:N,1:N)) ALLOCATE(Three(1:N,1:N))ALLOCATE(One_T(1:N,1:N))DO $\mathbb{T}\!=\!1$ ,N PRINT\*, 'Input row ', I,' of One' READ\*,One(I,1:N) END DO DO $\mathbb{T}\!=\!1$ ,N PRINT\*, 'Input row ', I,' of Two' READ\*,Two(I,1:N) END DO CALL Matrix_bits(One,Two,Three,One_T,N) PRINT\*,' Matrix Three:' DO $\mathbb{T}\!=\!1$ ,N PRINT \*,Three(I,1:N) END DO PRINT \*,' Matrix One_T:' DO $\mathbb{T}\!=\!1$ ,N PRINT \*,One_T(I,1:N) END DO END PROGRAM ch2305 + +SUBROUTINE Matrix_bits(A,B,C,A_T,N) IMPLICIT NONE INTEGER, INTENT(IN):: N REAL, DIMENSION (:,:), INTENT(IN) :: A,B REAL, DIMENSION (:,:), INTENT(OUT) :: C,A_T INTEGER:: I,J, K REAL:: Temp + + ! + + ! matrix multiplication $\scriptstyle\mathrm{C}=\mathrm{AB} + +$ ! DO $\mathbb{T}\!=\!1$ ,N DO $\uptau\!\!=\!\!1$ ,N Temp $=\!0$ .0 DO $\mathtt{K}\!\!=\!\!1$ ,N Temp $=$ Temp $^+$ A(I,K) \* B (K,J) END DO $\begin{array}{r l r}{\mathsf{C}\left(\,\mathbb{T}\,,\,\mathbb{J}\,\right)}&{{}=}&{\mathsf{T e m p}}\end{array}$ END DO END DO + + ! + + ! Calculate A_T transpose of A + + ! + + ! + + ! set A_T to be transpose matrix A DO $\mathbb{T}\!=\!1$ ,N DO $\uptau\!\!=\!\!1$ ,N $\begin{array}{r l r}{\mathbb{A}\_{\mathbb{T}}\left(\,\mathbb{I}\,,\,\mathbb{J}\,\right)}&{{}=}&{\mathbb{A}\left(\,\mathbb{J}\,,\,\mathbb{I}\,\right)}\end{array}$ END DO END DO END SUBROUTINE Matrix_bits + +# 23.6.3 Notes + +The dummy array and actual array arguments look the same but there is a differ- ence: + +• The dummy array arguments A, B, C, A_T are all assumed-shape arrays and take the shape of the actual array arguments One, Two, Three and One_T, respectively. • The actual array arguments One, Two, Three and One_T in the main pro- gram are allocatable arrays or deferred-shape arrays. An allocatable array + +is an array that has an allocatable attribute. Its bounds and shape are de- clared when the array is allocated, hence deferred-shape. + +# 23.6.4 Using the intrinsic functions MATMUL and TRANSPOSE + +In the previous two examples the matrix multiplication and transpose were hand coded, and are what you would see in Fortran 77 style code. This example uses the built in intrinsics MATMUL and TRANSPOSE and is modern Fortran 90 style: + +PROGRAM ch2306 IMPLICIT NONE REAL , ALLOCATABLE , DIMENSION & (:,:)::One,Two,Three,One_T INTEGER :: I,N INTERFACE SUBROUTINE Matrix_bits(A,B,C,A_T) IMPLICIT NONE REAL, DIMENSION (:,:), INTENT(IN) :: A,B REAL, DIMENSION (:,:), INTENT(OUT) :: C,A_T END SUBROUTINE Matrix_bits END INTERFACE PRINT \*,'Input size of matrices' READ\*,N ALLOCATE(One(1:N,1:N))ALLOCATE(Two(1:N,1:N)) ALLOCATE(Three(1:N,1:N))ALLOCATE(One_T(1:N,1:N))DO $\mathbb{T}\!=\!1$ ,N PRINT\*, 'Input row ', I,' of One' READ\*,One(I,1:N) END DO DO $\mathbb{T}\!=\!1$ ,N PRINT\*, 'Input row ', I,' of Two' READ\*,Two(I,1:N) END DO CALL Matrix_bits(One,Two,Three,One_T) PRINT\*,' Matrix Three:' DO $\mathbb{T}\!=\!1$ ,N PRINT \*,Three(I,1:N) END DO PRINT \*,' Matrix One_T:' DO $\mathbb{T}\!=\!1$ ,N PRINT \*,One_T(I,1:N) END DO + +# Chapter 23 + +END PROGRAM ch2306 + +SUBROUTINE Matrix_bits(A,B,C,A_T) IMPLICIT NONE REAL, DIMENSION (:,:), INTENT(IN) :: A,B REAL, DIMENSION (:,:), INTENT(OUT) :: C,A_T $C{=}$ MATMUL(A,B) A_T $=$ TRANSPOSE(A)END SUBROUTINE Matrix_bits + +Fortran thus provides a variety of ways of passing array parameters. We have cov- ered both 77 and 90 styles, as you will see both in code that you work with. + +# 23.7 Automatic arrays and median calculation + +This example looks at the calculation of the median of a set of numbers and also illustrates the use of an automatic array. + +The median is the middle value of a list, i.e., the smallest number such that at least half the numbers in the list are no greater. If the list has an odd number of entries, the median is the middle entry in the list after sorting the list into ascending order. If the list has an even number of entries, the median is equal to the sum of the two middle (after sorting) numbers divided by two. One way to determine the median computationally is to sort the numbers and choose the item in the middle. + +Wirth classifies sorting into simple and advanced, and his three simple methods are as follows: + +• Insertion sorting — The items are considered one at a time and each new item is inserted into the appropriate position relative to the previously sorted item. If you have ever played bridge then you have probably used this method. • Selection sorting — First the smallest (or largest) item is chosen and is set aside from the rest. Then the process is repeated for the next smallest item and set aside in the next position. This process is repeated until all items are sorted. • Exchange sorting — If two items are found to be out of order they are in- terchanged. This process is repeated until no more exchanges take place. + +Knuth also identifies the above three sorting methods. For more information on sorting the Knuth and Wirth books are good starting places. Knuth is a little old (1973) compared to Wirth (1986), but it is still a very good coverage. Knuth uses mix assembler to code the examples whilst the Wirth book uses Modula 2, and is therefore easier to translate into modern Fortran. + +In the example below we use a selection sort: + +program ch2307 + +implicit none integer :: n real , allocatable , dimension(:) :: x real :: m,sd,median + +interface subroutine stats(x,n,mean,std_dev,median) implicit none integer , intent(in) :: n real , intent(in) , dimension(:) :: x real , intent(out) :: mean real , intent(out) :: std_dev real , intent(out) :: median end subroutine stats + +end interface + +print \*,' How many values ?' read \*,n allocate $\left(\,\mathbf{x}\;(\;\mathbb{1}:\mathbf{n}\,)\right.$ ) call random_number $(\mathbf{\nabla}\mathbf{x})$ $\tt{x=x^{\star}1000}$ call stats(x,n,m,sd,median) print \*,' mean = ',m print \*,' Standard deviation $=$ ',sd print \*,' median is $=$ ',median + +end program ch2307 + +subroutine stats(x,n,mean,std_dev,median) implicit none integer , intent(in) :: n real , intent(in) , dimension(:) :: x real , intent(out) :: mean real , intent(out) :: std_dev real , intent(out) :: median real , dimension(1:n) :: y real :: variance real :: sumxi, sumxi2 sumxi $=\!0$ .0 + +sumxi2 $=\!0$ .0 variance $=\!0$ .0 sumxi $=$ sum(x) sumxi2 $=$ sum(x\*x) mean $\b=$ sumxi/n variance $=$ (sumxi2-sumxi\*sumxi/n)/(n-1) std_dev $=$ sqrt(variance) $\mathtt{Y}^{=\mathtt{X}}$ call selection if (m $\begin{array}{r l r}{\mathsf{O d}\left(\mathfrak{n}\,,2\,\right)}&{{}==}&{0\,\}\end{array}$ ) then median $\mathsf{\tau}=\left(\mathsf{y}\left(\mathsf{n}/2\right)\mathsf{+y}\left(\mathsf{\}\mathsf{n}/2\right)\mathsf{+1}\right)\mathsf{\Omega}/2$ else median $\mathbf{\beta}.=\mathbb{Y}$ ((n/2)+1) endif + +contains + +subroutine selection implicit none integer :: i,j,k real :: minimum do $\dot{\bot}\!=\!\bot$ ,n-1 $\mathtt{k}\!=\!\dot{\mathtt{L}}$ minimum $\mathbf{\beta}^{=}\mathbb{Y}$ (i) do $\dot{\bar{\mathbf{\tau}}}\,{=}\,\dot{\mathbf{\varepsilon}}\,{+}\,\mathbf{\hat{\varepsilon}}$ ,n if (y(j) $<$ minimum) then $\upkappa{=}\dot{\uprho}$ minimum $\imath\mathrm{=}\mathrm{y}$ (k) end if end do $\mathtt{Y}\left(\mathtt{k}\right)=\mathtt{Y}$ (i) $\textsf{Y}(\dot{\bot})=$ minimum end do end subroutine selection + +end subroutine stats + +In the subroutine stats the array y is automatic. It will be allocated automatically when we call the subroutine. We use this array as a work array to hold the sorted data. We then use this sorted array to determine the median. + +Note the use of the SUM intrinsic in this example: + +sumxi $=$ sum(x) sumxi2 $=$ sum(x\*x) + +These statements replace the DO loop from the earlier example. A good optimising compiler would not make two passes over the data with these two statements. + +# 23.7.1 Internal subroutines and scope + +The stats subroutine contains the selection subroutine. The stats subroutine has ac- cess to the following variables + +• x,n,mean,std_dev, median — these are made available as they are passed in as parameters. • y, variance, sumxi, sumxi2 — are local to the subroutine stats. + +The subroutine selection has access to the above as it is contained within subrou- tine stats. It also has the following local variables that are only available within subroutine selection + +• i,j,k, minimum + +# 23.7.2 Timing the selection sort algorithm + +The selection sort is a simple algorithm and the following main program illustrates it limitations with increasing n. It uses the same stats subroutine as the previous example: + +program ch2308 + +implicit none integer :: n real , allocatable , dimension(:) :: x real :: m,sd,median integer , dimension(8) :: timing + +interface subroutine stats(x,n,mean,std_dev,median) implicit none integer , intent(in) :: n real , intent(in) , dimension(:) :: x real , intent(out) :: mean real , intent(out) :: std_dev real , intent(out) :: median end subroutine stats end interface + +do print $\mathrm{~\star~},\mathrm{~\iota~}\mathrm{~\mathfrak~{~n~}~}=\mathrm{~\iota~},\mathrm{~\mathfrak~{~n~}~}$ allocate $\left(\,\mathbf{x}\,(\,\mathbb{1}:\mathbf{n}\,)\right.$ ) call random_number(x) $\tt{x=x^{\star}1000}$ call date_and_time(values $=$ timing) print \*,' initial ' print \*, timing(6) , timing(7) , timing(8) call stats(x,n,m,sd,median) print \*,' mean $=$ ',m print \*,' Standard deviation $=$ ',sd print \*,' median is $=$ ',median call date_and_time(values $=$ timing) print \*,' after' print \*,timing(6),timing(7),timing(8) $\mathtt{n}\!=\!\mathtt{n}^{\star}\mathtt{10}$ deallocate(x) end do + +end program ch2308 + +# 23.7.2.1 Timing + +Dell Precision Workstation, 2 \* 933 MHz, 512 Mb ram: + +n = 1000 initial 9 13 906 mean $=$ 5.0895782E+02 Standard deviation $=$ 2.8708249E+02 median is $=$ 5.1872925E+02 after sort 9 13 950 $\mathrm{~\boldmath~n~}=\mathrm{~\boldmath~\nabla~}\,\mathbb{1}0\,0\,0\,0$ initial 9 13 951 mean $=$ 4.9967194E+02 Standard deviation $=$ 2.8635922E+02 median is $=$ 5.0259839E+02 after sort 9 14 689 $\begin{array}{r l r}{\mathrm{~n~}}&{{}=}&{10\,0\,0\,0\,0\,0}\end{array}$ initial 9 14 697 mean $=$ 5.0123392E+02 Standard deviation $=$ 2.8869482E+02 median is $=$ 4.9957404E+02 + +after sort 11 12 292 + +Dell Inspiron, 1 \* 3.4 Ghz, 1 Gb ram: + +$\mathrm{~\textit~{~n~}~}=\mathrm{~\textit~{~10~0~0~}~}$ initial 10 57 421 mean $=$ 5.0781586E+02 Standard deviation $=$ 2.9026807E+02 median is $=$ 5.1530060E+02 after sort 10 57 524 $\mathrm{~\boldmath~n~}=\mathrm{~\boldmath~\nabla~}\,\mathbb{1}0\,0\,0\,0$ initial 10 57 525 mean $=$ 4.9770724E+02 Standard deviation $=$ 2.8532513E+02 median is $=$ 4.9151331E+02 after sort 10 57 778 $\begin{array}{r l r}{\mathrm{~n~}}&{{}=}&{10\,0\,0\,0\,0\,0}\end{array}$ initial 10 57 781 mean $=$ 4.9930457E+02 Standard deviation $=$ 2.8866571E+02 median is $=$ 4.9931268E+02 after sort 11 23 374 + +This algorithm is approximately order $\imath^{\mathrm{~*~}}\log(\mathrm{n})$ . + +# 23.8 Alternative median calculation algorithm + +This program uses an algorithm developed by Hoare to determine the median. The number of computations required to find the median is approximately $2^{\mathrm{~*~n~}}$ . + +Timings are given at the end: + +program ch2309 + +implicit none integer :: n real , allocatable , dimension(:) :: x real :: m,sd,median integer , dimension(8) :: timing + +interface subroutine stats(x,n,mean,std_dev,median) implicit none integer , intent(in) + +real , intent(in) , dimension(:) :: x real , intent(out) :: mean real , intent(out) :: std_dev real , intent(out) :: median end subroutine stats end interface + +$\mathtt{n{=}1000}$ do print $\mathrm{~\star~},\mathrm{~\iota~}\mathrm{~\mathfrak~{~n~}~}=\mathrm{~\iota~},\mathrm{~\mathfrak~{~n~}~}$ allocate $\left(\,\mathbf{x}\,(\,\mathtt{L}:\mathtt{n}\,)\right.$ ) call random_number(x) $\tt{x=x^{\star}1000}$ call date_and_time(values $=$ timing) print \*,' initial ' print \*,timing(6),timing(7),timing(8) call stats(x,n,m,sd,median) print \*,' mean $=$ ',m print \*,' Standard deviation $=$ ',sd print \*,' median is $=$ ',median call date_and_time(values $=$ timing) print \*,' after sort' print \*, timing(6),timing(7),timing(8) $\mathtt{n}\!=\!\mathtt{n}^{\star}\mathtt{10}$ deallocate(x) end do + +end program ch2309 + +subroutine stats(x,n,mean,std_dev,median) implicit none integer , intent(in) :: n real , intent(in) , dimension(:) :: x real , intent(out) :: mean real , intent(out) :: std_dev real , intent(out) :: median real , dimension(1:n) :: y real :: variance real :: sumxi, sumxi2 integer:: k sumxi $=\!0$ .0 sumxi2 $=\!0$ .0 + +variance $=\!0$ .0 sumxi $=$ sum(x) sumxi2 $=$ sum $\left({{\bf{x}}^{\star}{\bf{x}}}\right)$ mean $\b=$ sumxi/n variance $=$ (sumxi2-sumxi\*sumxi/n)/(n-1) std_dev $=$ sqrt(variance) $\mathtt{Y}^{=\mathtt{X}}$ if $\begin{array}{r l r}{\mathsf{\Gamma}(\mathsf{m o d}\,(\mathsf{n}\,,2\,)}&{{}==}&{0\,)}\end{array}$ ) then median $=$ ( find(n/2) $^+$ find((n/2) $^{+1}$ ) )/2 else median $\cdot^{-}$ find( $(\mathsf{n}/2)+\beth]$ ) endif + +contains + +real function find(k) implicit none integer , intent(in) :: k integer :: l,r,i,j real :: t1,t2 $\bot\!=\!\bot$ $\scriptstyle\mathtt{T}=\mathtt{T}$ do while ( $\bot{<}\underline{{\tau}}$ ) $\tt t\Omega\tt=\!\tt y$ (k) $\dot{\bot}\!=\!\bot$ $\dot{\daleth}\,{=}\,\underline{{\boldsymbol{\Upsilon}}}$ do do while $\left(\,\mathtt{y}\left(\,\mathtt{i}\,\right)<\!\mathtt{t}\,\mathtt{i}\,\right)$ ) $\dot{\boldsymbol{\updownarrow}}\!=\!\dot{\boldsymbol{\updownarrow}}\!+\!\boldsymbol{\updownarrow}$ end do do while ( $\tt t\Omega\!<\!\mathtt{Y}$ (j)) $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ end do if ( $\dot{\Sigma}\!<\!=\!\dot{\Pi}$ ) then $\scriptstyle\tan\left(2\right)=\!$ (i) $\mathbb{y}\left(\stackrel{.}{\bot}\right)=\!\!\mathbb{y}$ (j) $\mathtt{Y}\left(\,\dot{\,}\,\right)=\!\mathsf{t}2$ $\dot{\boldsymbol{\updownarrow}}\!=\!\dot{\boldsymbol{\updownarrow}}\!+\!\boldsymbol{\updownarrow}$ $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ end if if ( $\dot{\perp}\!>\!\dot{\perp}$ ) exit end do + +# Chapter 23 + +if ( $\dot{\bar{\jmath}}\,{<}\mathrm{k}$ ) then $\perp\!=\!\dot{\perp}$ end if if ( $\qgtrless\dot{\qless1}$ ) then $\tt{T=\dot{I}}$ end if end do find $\mathbf{\beta}^{=}\mathbb{Y}$ (k) end function find + +end subroutine stats + +# 23.8.1 Timing + +Dell Precision Workstation, 2 \* 933 MHz, 512 Mb ram: + +$\begin{array}{r l r}{\mathrm{~n~}}&{{}=}&{10\,0\,0}\end{array}$ initial 19 37 421 mean $=$ 4.9430524E+02 Standard deviation $=$ 2.9318149E+02 median is $=$ 4.8815854E+02 after sort 19 37 426 $\mathrm{~\textit~{~n~}~}=\mathrm{~\ensuremath~{~\mathbb~1~0~0~0~0~}~}$ initial 19 37 427 mean $=$ $4\cdot9803854\mathrm{E}{+}02$ Standard deviation $=$ 2.9065613E+02 median is $=$ 4.9482861E+02 after sort 19 37 431 $\begin{array}{r l r}{\mathrm{~n~}}&{{}=}&{10\,0\,0\,0\,0\,0}\end{array}$ initial 19 37 439 mean $=$ 5.0035132E+02 Standard deviation $=$ 2.8867920E+02 median is $=$ 5.0099771E+02 after sort 19 37 468 $\begin{array}{r l r}{\mathrm{~n~}}&{{}=}&{10\,0\,0\,0\,0\,0\,0}\end{array}$ initial 19 37 542 mean $=$ 4.9944907E+02 Standard deviation $=$ 2.8857736E+02 median is $=$ 4.9952847E+02after sort 19 37 838 $\begin{array}{r l r}{\mathrm{~n~}=}&{{}}&{\mathbb{1}0\,0\,0\,0\,0\,0\,0\,0}\end{array}$ initial 19 38 626 mean $=$ 4.9974246E+02 Standard deviation $=$ 2.8268231E+02 + +median is $=$ 4.9996432E+02 after sort 19 41 722 + +Dell Inspiron, 1 \* 3.4 Ghz, 1 Gb ram: + +$\mathrm{~\textit~{~n~}~}=\mathrm{~\textit~{~10~0~0~}~}$ initial 21 38 734 mean $=$ 4.9351086E+02 Standard deviation $=$ 2.9076068E+02 median is $=$ 4.8678186E+02 after sort 21 38 734 $\mathrm{~\boldmath~n~}=\mathrm{~\boldmath~\nabla~}\,\mathbb{1}0\,0\,0\,0$ initial 21 38 734 mean $=$ 5.0000485E+02 Standard deviation $=$ 2.8909946E+02 median is $=$ 4.9666431E+02 after sort 21 38 765 $\begin{array}{r l r}{\mathrm{~n~}}&{{}=}&{10\,0\,0\,0\,0\,0}\end{array}$ initial 21 38 768 mean $=$ 5.0053433E+02 Standard deviation $=$ 2.8863885E+02 median is $=$ 4.9899475E+02 after sort 21 38 775 $\begin{array}{r l r}{\mathrm{~n~}}&{{}=}&{10\,0\,0\,0\,0\,0}\end{array}$ initial 21 38 797 mean $=$ 5.0039590E+02 Standard deviation $=$ 2.8852356E+02 median is $=$ 5.0053967E+02 after sort 21 38 855 $\begin{array}{r l r}{\mathrm{~n~}=}&{{}}&{\mathbb{1}0\,0\,0\,0\,0\,0\,0}\end{array}$ initial 21 39 67 mean $=$ 4.9973923E+02 Standard deviation $=$ 2.8260712E+02 median is $=$ 4.9987134E+02 after sort 21 39 806 $\begin{array}{r l r}{\mathrm{~n~}}&{{}=}&{10\,0\,0\,0\,0\,0\,0\,0\,0\,0}\end{array}$ initial 21 41 930 mean $=$ 1.7179869E+02 Standard deviation $=$ 3.8263177E+02 median is $=$ 5.0002957E+02 after sort 21 59 222 + +The differences between the two algorithms and systems are summarised below: + +Chapter 23 + +System N Selection Find Dual 100,000,000 NA NA 10,000,000 NA 3.096 1,000,000 NA 0.296 100,000 117.595 0.029 10,000 0.738 0.004 + +Single 100,000,000 NA 17.292 10,000,000 NA 0.739 1,000,000 NA 0.058 100,000 25.593 NA 10,000 0.253 NA + +Hoare's Find algorithm is obviously much faster, but far less easy to understand than the simple selection sort. + +The limiting factor with this algorithm on these systems is the amount of installed memory. The program crashes on both systems with a failure to allocate the auto- matic array. This is a drawback of automatic arrays in that there is no mechanism to handle this failure gracefully.You would then need to use allocatable local work arrays. The drawback here is that the programmer is then responsible for the deallocation of these arrays. Memory leaks are then possible. + +# 23.9 Recursive subroutines — Quicksort + +In Chapter 14 we saw an example of recursive functions. This example illustrates the use of recursive subroutines. It uses a simple implementation of Hoare's Quicksort. References are given in the bibliography. The overall problem is broken down into: + +• A main program that prompts the user for the name of the data file and n. The allocation of the array is carried out in the main program. • A subroutine to read the data. • A subroutine to sort the data. This subroutine contains the recursive sub- routine Quicksort. • A subroutine to write the sorted data to a file. + +Below is the complete program: + +PROGRAM ch2310 IMPLICIT NONE INTEGER :: How_Many CHARACTER (LEN $\scriptstyle{\dot{=}20}$ ) :: File_Name REAL , ALLOCATABLE , DIMENSION(:) :: Raw_Data integer , dimension(8) :: timing + +INTERFACE SUBROUTINE Read_Data(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER ( $\tt L E N\mathrm{=\star}$ ) , INTENT(IN) :: File_Name INTEGER , INTENT(IN) :: How_Many REAL , INTENT(OUT) , & DIMENSION(:) :: Raw_Data END SUBROUTINE Read_Data END INTERFACE + +INTERFACE SUBROUTINE Sort_Data(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many REAL , INTENT(INOUT) , & DIMENSION(:) :: Raw_Data END SUBROUTINE Sort_Data END INTERFACE + +INTERFACE SUBROUTINE Print_Data(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many REAL , INTENT(IN) , & DIMENSION(:) :: Raw_Data END SUBROUTINE Print_Data END INTERFACE + +PRINT \* , ' How many data items are there?' READ $\star$ , How_Many PRINT \* , ' What is the file name?' READ '(A)',File_Name call date_and_time(values $=$ timing) print \*,' initial' print \*,timing(6),timing(7),timing(8) + +ALLOCATE(Raw_Data(How_Many)) call date_and_time(values $=$ timing) print \*,' allocate' print \*,timing(6),timing(7),timing(8) CALL Read_Data(File_Name,Raw_Data,How_Many) call date_and_time(values $=$ timing) print \*,' read' print \*,timing(6),timing(7),timing(8) CALL Sort_Data(Raw_Data,How_Many) call date_and_time(values $=$ timing) print \*,' sort' print \*,timing(6),timing(7),timing(8) CALL Print_Data(Raw_Data,How_Many) call date_and_time(values $=$ timing) print \*,' print' print \*,timing(6),timing(7),timing(8) PRINT \* , ' ' PRINT \*, ' Data written to file SORTED.DAT' + +END PROGRAM ch2310 + +SUBROUTINE Read_Data(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER ( $\tt L E N\!\!=\!\star$ ) , INTENT(IN) :: File_Name INTEGER , INTENT(IN) :: How_Many REAL , INTENT(OUT) , DIMENSION(:) :: Raw_Data ! Local variables INTEGER :: I + +OPEN(FILE $_{\parallel}=$ File_Name,UNIT $\vDash\vDash$ ) DO $\mathbb{T}\!=\!1$ ,How_Many READ (UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) Raw_Data(I) ENDDO + +END SUBROUTINE Read_Data + +SUBROUTINE Sort_Data(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many REAL , INTENT(INOUT) , DIMENSION(:) :: Raw_Data + +CALL QuickSort(1,How_Many) + +CONTAINS + +RECURSIVE SUBROUTINE QuickSort(L,R) IMPLICIT NONE INTEGER , INTENT(IN) :: L,R ! Local variables INTEGER :: I,J + +REAL :: V,T $\dot{\bot}\!=\!\bot$ $\dot{\daleth}\,{=}\,\underline{{\boldsymbol{\Upsilon}}}$ $\mathbb{V}=$ raw_data( int((l+r)/2) ) do do while (raw_data(i) < v ) $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ enddo do while (v < raw_data(j) ) $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ enddo if ( $\dot{\Sigma}\!<\!=\!\dot{\Pi}$ ) then t $=$ raw_data(i) raw_data(i) $=$ raw_data(j) raw_data(j)=t $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ endif if ( $\dot{\perp}{>}\dot{\supset}$ ) exit enddo if ( $\perp<\!\dot{\!\!\perp}$ ) then call quicksort(l,j) endif if ( $\dot{\bot}\!<\!\tau$ ) then call quicksort(i,r) endif + +END SUBROUTINE QuickSort END SUBROUTINE Sort_Data + +SUBROUTINE Print_Data(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many REAL , INTENT(IN) , DIMENSION(:) :: Raw_Data ! Local variables INTEGER :: I OPEN(FILE $=$ 'SORTED.DAT',UNIT $^{1}{=}2$ ) DO $\mathbb{T}\!=\!1$ ,How_Many WRITE(UNIT $^{=2}$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) Raw_Data(I) ENDDO CLOSE(2)END SUBROUTINE Print_Data + +# 23.9.1 Note — Interface blocks + +We introduced interface blocks in Chapter 22, and in that chapter the parameters were scalars; in this example we have a mix of arrays and scalars. The above pro- gram is in Fortran 77 style with a main program and several distinct subroutines. The use of interface blocks is recommended when using this style of programming as it will minimise cross program unit (program, function, subroutine) errors. + +Interface blocks are mandatory under the following situations: + +• The procedure has optional arguments. • When a function returns an array. • When a function returns a pointer. • For character functions a result that is dynamic. • When the procedure has assumed-shape dummy arguments. • When the procedure has dummy arguments with the pointer attribute. • When the procedure has dummy arguments with the target attribute. • When the procedure has keyword arguments and/or optional arguments. • When the procedure is generic. You have already seen that some of the intrinsic procedures have this status, e.g., SINE will return a result when the argument is of a variety of types. This means that we can construct procedures that will accept arguments of a variety of types and all we need to do is provide a procedure that manipulates data of that type. We will look at the construction of a procedure that is generic in a later chap- ter. • When the procedure provides a user defined operator. + +• When the procedure provides user defined assignment. + +# 23.9.2 Note — Recursive subroutine + +The actual sorting is done in the recursive subroutine QuickSort. The actual al- gorithm is taken from the Wirth book. See the bibliography for a reference. + +Recursion provides us with a very clean and expressive way of solving many prob- lems. There will be instances where it is worthwhile removing the overhead of recursion, but the first priority is the production of a program that is correct. It is pointless having a very efficient but incorrect solution. + +We will look again at recursion and efficiency in a later chapter and see under what criteria we can replace recursion with iteration. + +# 23.9.3 Note — Flexible design + +The QuickSort recursive routine can be replaced with another sorting algorithm and we can maintain the interface to Sort_Data. We can thus decouple the imple- mentation of the actual sorting routine from the defined interface. We would only need to recompile the Sort_Data routine and we could relink using the already compiled main, read data and print data routines. + +# 23.9.4 Note — Timing information + +We call the date_and_time intrinsic subroutine to get timing information. A sum- mary table from runnning the above program on a 3.4 Ghz system with 1 Gb memory with four different compilers is given below: + +$$ +n{=}10\;,\,0\,0\;,\,0\,0\; +$$ + +![Compiler ](images/87eeee543358fa26392d8ca35081039d3120936ce8bf436ecc293e4f165a4fd0.jpg) + +As can be seen it is the I/O that dominates the overall running time of the program. In the 10 years since first running this program we have seen the data set size in- crease from tens of thousands to tens and hundreds of millions. + +# 23.10 Summary + +We now have a lot of the tools to start tackling problems in a structured and modu- lar way, breaking problems down into manageable chunks and designing subprograms for each of the tasks. + +# Chapter 23 + +# 23.11 Problems + +1. Below is the random number program that was used to generate the data sets for the Quicksort example: + +program ch2311 implicit none integer :: n integer :: i real , allocatable , dimension(:) :: x print \*,' how many values ?' read \*,n allocate $\left(\,\mathbf{x}\;(\;\mathbb{1}:\mathbf{n}\,)\right.$ ) call random_number(x) $\tt{x=x^{\star}1000}$ open(unit $=\!\perp0$ ,file $=$ 'random.txt') do $\dot{\bot}\!=\!\bot$ ,n write(10, 100)x(i) 100 format(f8.3) end do end program ch2311 + +Run the Quick_Sort program in this chapter with the data file as input. Obtain tim- ing details. + +What percentage of the time does the program spend in each subroutine? Is it worth trying to make the sort much more efficient given these timings? + +2. Find out if there is a subroutine library like the NAG library available. If there is replace the Quick_Sort recursive subroutine with a suitable routine from that li- brary. What times do you obtain? + +3. Try using the operating system SORT command to sort the file. What timing figures do you get now? + +Was it worth writing a program? + +4. Consider the following program: + +program ch2312 + +! + + ! Program to test array subscript checking + + ! when the array is passed as an argument. + + ! implicit none integer , parameter :: array_size=10 + +integer :: i integer , dimension(array_size) :: a do $\dot{\bot}\!=\!\bot$ ,array_size ${\sf a}\left(\,\dot{\bot}\,\right)=\dot{\bot}$ end do call sub01(a,array_size) end program ch2312 + +subroutine sub01(a,array_size) implicit none integer , intent(in) :: array_size integer , intent(in) , dimension(array_size) :: a integer :: i integer :: atotal $=\!0$ integer :: rtotal $=\!0$ do $\dot{\bot}\!=\!\bot$ ,array_size rtotal $=$ rtotal $+\,\textmd{a}$ (i) end do do $\dot{\bot}\!=\!\bot$ ,array_size $^{+1}$ atotal $=$ atotal+a(i) end do print \*,' Apparent total is ' , atotal print \*,' real total is ' , rtotal end subroutine sub01 + +The key thing to note is that we haven’t used interface blocks and we have an error in the subroutine where we go outside the array. Run this program. What answer do you get for the apparent total? + +Are there any compiler flags or switches which will enable you to trap this error? + +# Chapter 23 + +# 23.12 Bibliography + +Hoare C.A.R., Algorithm 63, Partition; Algorithm 64, Quicksort, p.321; Algo- rithm 65: FIND , Comm. of the ACM, 4 p.321–322, 1961. + +Hoare C.A.R., Proof of a Program: FIND , Comm A.C.M., 13, No 1 (1970) 39–45 Hoare C.A.R., Proof of a Recursive Program: Quicksort , Comp. J., 14, No 4 (1971) 391–95. + +Knuth D.E., The Art of Computer Programming , Volume 3 — Sorting and Searching , Addison-Wesley, 1973. + +Wirth N., Algorithms and Data Structures , Prentice-Hall, 1986. + +# 23.13 Commercial numerical and statistical subroutine libraries + +There are two major suppliers of commercial libraries: + +• NAG: Numerical Algorithms Group + +and + +• Visual Numerics + +They can be found at: + +• http://www.nag.co.uk/ + +and + +• http://www.vni.com/index.html + +respectively. Their libraries are written by numerical analysts, and are fully tested and well documented. They are under constant development and available for a wide range of hardware platforms and compilers. Parallel versions are also avail- able. + +# An Introduction to Modules + +“Common sense is the best distributed commodity in the world, for every man is convinced that he is well supplied with it.” + +Descartes + +# Aims + +The aims of this chapter are to look at the facilities found in Fortran provided by modules, in particular: + +• The use of a module to aid in the consistent definition of precision throughout a program and subprograms. • The use of modules for global data. • The use of modules for derived data types. • Two examples showing the use of modules with contained proecures and their use to package procedures. • A complete numerical example solving systems of linear equations using Gaussian elimination. + +# 24 An Introduction to Modules + +As summarised in the Chapter 23 we now have the tools to solve many problems using just a main program and one or more external and internal subprograms. Both external and internal subprograms communicate through their argument lists, whilst internal subprograms have access to data in their host program units. + +We now introduce another type of program unit, the module, which is probably one of the most important features of Fortran 90. The purpose of modules is quite different from that of subprograms. In their simplest form they exist so that any- thing required by more than one program unit may be packaged in a module and made available where needed. + +The form of a module is + +MODULE module_name + +END MODULE module_name + +and the information contained within it is made available in the program units that need to access it by + +USE module_name + +The USE statement must be the first statement after the PROGRAM or SUBROU- TINE or FUNCTION statement. + +In this chapter we will look at: + +• Modules for global data. • Modules for derived types. • Modules for explicit interfaces. • Modules containing procedures. + +Modules are another program unit and exist so that anything required by more than one program unit may be packaged in a module and made available where needed. + +# 24.1 Modules for global data + +So far the only way that a program unit can communicate with a procedure is through the argument list. Sometimes this is very cumbersome, especially if a number of procedures want access to the same data, and it means long argument lists. The problem can be solved using modules; e.g., by defining the precision to which you wish to work and any constants defined to that precision which may be needed by a number of procedures. + +# 24.2 Modules for precision specification and constant definition + +In the following example we use a module to define a parameter Long to specify the precision to which we wish to work, and another for a range of mathematical constants including a value for the parameter $\pi$ . Note that the parameter $\pi$ is de- fined to this working precision. We then import the module defining these parameters into the program units that need them: + +module precision definition implicit none integer , parameter :: long $=$ selected real kind(15,307) end module precision definition + +module maths_constants use precision definition implicit none real (long) , parameter :: c = 299792458.0_long ! units m s-1 real (long) , parameter :: & e = 2.71828182845904523 long real (long) , parameter :: $\texttt{g}=\texttt{9.812420\_l o n g}$ ! 9.780 356 m s-2 at sea level on the equator ! 9.812 420 m s-2 at sea level in London ! 9.832 079 m s-2 at sea level at the poles real (long) , parameter :: & pi = 3.14159265358979323 long end module maths_constants + +PROGRAM ch2401 USE Precision definition IMPLICIT NONE INTERFACE SUBROUTINE Sub1(Radius,Area,Circum) USE Precision definition IMPLICIT NONE REAL(Long),INTENT(IN)::Radius REAL(Long),INTENT(OUT)::Area,Circum END SUBROUTINE Sub1 END INTERFACE REAL(Long)::R,A,C INTEGER ::I DO $\mathbb{T}\!=\!1$ ,10 + +# Chapter 24 + +PRINT\*,'Radius?' READ\*,R CALL Sub1(R,A,C) PRINT \*,' For radius $\begin{array}{r l}{\mathbf{\chi}=}&{{}\left\Vert\mathbf{\chi},\,\right\Vert\mathbb{R}}\end{array}$ PRINT \*,' Area $=$ ',A PRINT \*,' Circumference $=$ ',C END DO + +END PROGRAM ch2401 + +SUBROUTINE Sub1(Radius,Area,Circum) USE Precision definition use maths_constants IMPLICIT NONE REAL(Long),INTENT(IN)::Radius REAL(Long),INTENT(OUT)::Area,Circum Area $=$ Pi\*Radius\*Radius Circum $_{:=2}$ .0_Long\*Pi\*Radius + +END SUBROUTINE Sub1 + +# 24.2.1 Note + +In this example we wish to work with the precision specified by the kind type pa- rameter Long in the module Precision definition. In order to do this we use the statement + +USE precision definition + +inside the program unit before any declarations. The kind type parameter Long is then used with all the REAL type declaration e.g., + +REAL (Long):: R ,A,C + +To make sure that all floating point calculations are performed to the working pre- cision specified by Long any constants such as 2.0 in subroutine Sub1 are specified as const_Long e.g., + +2.0_Long + +Note also that we define things once and use them on two occasions, i.e., we de- fine the precision once and use this definition in both the main program and the subroutine. + +# 24.3 Modules for sharing arrays of data + +The following example uses one module containing a number of constants and a second module containing an array definition: + +module data implicit none integer , parameter :: $\mathtt{n}\!\!=\!\!12$ real , dimension(1:n) :: rainfall real , dimension(1:n) :: sorted end module data + +program ch2402 use data implicit none + +call readdata call sortdata call printdata + +end program ch2402 + +subroutine readdata use data implicit none integer :: i character ( $\mathtt{l e n\!e\!40}$ ) :: filename print \*,' What is the filename ?' read \*,filename open(unit $=\!\perp00$ ,file $=$ filename) do $\dot{\bot}\!=\!\bot$ ,n read ( $^{\,1\,0\,0}\,,\,^{\star}\,)$ rainfall(i) end do end subroutine readdata + +subroutine sortdata use data sorted $=$ rainfall call selection + +contains subroutine selection + +# Chapter 24 + +implicit none integer :: i,j,k real :: minimum do $\dot{\bot}\!=\!\bot$ ,n-1 $\mathtt{k}\!=\!\dot{\mathtt{L}}$ minimum $\bumpeq$ sorted(i)do $\dot{\bar{\mathbf{\omega}}}=\dot{\mathbf{\omega}}_{+}\mathbf{\mathcal{1}}$ ,n if (sorted(j) $<$ minimum) then $\upkappa{=}\dot{\;}$ minimum $\bumpeq$ sorted(k) end if end do sorted(k) $=$ sorted(i) sorted(i) $=$ minimum end do end subroutine selection + +end subroutine sortdata + +subroutine printdata use data implicit none integer :: i print \*,' original data is ' do $\scriptstyle{\dot{\bot}}\,=\,\!1$ ,n print 100,rainfall(i)100 format(1x,f7.1)end do print \*,' Sorted data is ' do $\scriptstyle{\dot{\bot}}\,=\,\!1$ ,n print 100,sorted(i) end do end subroutine printdata + +Note that in this example the calls to the subroutines have no parameters. They work with the data contained in the module. + +# 24.4 Modules for derived data types + +When using derived data types and passing them as arguments to subroutines, both the actual arguments and dummy arguments must be of the same type, i.e., they must be declared with reference to the same type definition. The only way this can be achieved is by using modules. The user defined type is declared in a module and each program unit that requires that type uses the module. + +# 24.4.1 Person data type + +In this example we have a user defined type Person which we wish to use in the main program and pass arguments of this type to the subroutines Read_data and Stats. In order to have the type Person available to two subroutines and the main program we have defined Person in a module Personal details and then made the module available to each program unit with the statement + +USE Personal details + +We also have the use of an interface block to provide the ability to develop the overall solution in stages: + +MODULE Personal details IMPLICIT NONE TYPE Person REAL:: Weight INTEGER :: Age CHARACTER :: Sex END TYPE Person END MODULE Personal details + +PROGRAM ch2403 USE Personal details IMPLICIT NONE INTEGER ,PARAMETER:: Max_no $=\!\perp0\,0$ TYPE (Person), DIMENSION(1:Max_no) :: Patient INTEGER :: No_of_patients REAL :: Male_average, Female_average INTERFACE + +SUBROUTINE Read_data(Data,Max_no,No)USE Personal details IMPLICIT NONE TYPE (Person), DIMENSION (:), INTENT(OUT):: Data INTEGER, INTENT(OUT):: No INTEGER, INTENT(IN):: Max_no END SUBROUTINE Read_Data + +SUBROUTINE Stats(Data,No,M_a,F_a) USE Personal details + +IMPLICIT NONE TYPE(Person), DIMENSION (:) :: Data REAL:: M_a,F_a INTEGER :: No END SUBROUTINE Stats + +END INTERFACE ! + +CALL Read_data(Patient,Max_no,No_of_patients) CALL Stats( Patient , No_of_patients , & Male_average , Female_average) PRINT\*, 'Average male weight is ',Male_average PRINT\*, 'Average female weight is ',Female_average END PROGRAM ch2403 + +SUBROUTINE Read_Data(Data,Max_no,No)USE Personal details IMPLICIT NONE TYPE (PERSON), DIMENSION (:), INTENT(OUT)::Data INTEGER, INTENT(OUT):: No INTEGER, INTENT(IN):: Max_no INTEGER :: I DO PRINT \*,'Input number of patients' READ \*,No IF ( $\mathrm{~\sc~{~N~O~}~}>\mathrm{~\sc~{~0~}~}$ .AND. No $<=$ Max_no) EXIT END DO DO $\mathbb{T}\!=\!1$ ,No PRINT \*,'For person ',I PRINT \*,'Weight ?' READ\*,Data(I)%Weight PRINT\*,'Age ?' READ\*,Data(I)%Age PRINT\*,'Sex ?' READ\*,Data(I)%Sex END DO + +END SUBROUTINE Read_Data + +SUBROUTINE Stats(Data,No,M_a,F_a) USE Personal details IMPLICIT NONE TYPE(Person), DIMENSION(:)::Data + +REAL :: M_a,F_a INTEGER:: No INTEGER :: I,No_f,No_m $\mathtt{M\_a}{=}\mathtt{0}\cdot\mathtt{0}$ ; F_a $=\!0$ .0;No_f $\mathrm{\bar{=}}\,0$ ; No_m $=\!0$ DO $\mathbb{T}\!=\!1$ ,No IF ( Data(I)%Sex $==$ 'M' & .OR. Data(I)%Sex == 'm') THEN M_a $\tt=\tt M$ _a $^+$ Data(I)%Weight No_m $\mathbf{\beta}.\overline{{\overline{{\mathbf{\alpha}}}}}$ No_m+1 ELSEIF(Data(I)%Sex $==$ 'F' & .OR. Data(I)%Sex == 'f') THEN F_a $\mathrm{\mathbf{\mu=}}\mathrm{\mathbf{F}}.$ _a $^+$ Data(I)%Weight No_f $=$ No_f+1 ENDIF END DO IF (No_m > 0 ) THEN M_a $=$ M_a/No_m ENDIF IF (No_f > 0 ) THEN $\mathrm{~\tt~F~}\mathrm{~\tt~a~}=\mathrm{~\tt~F~}\mathrm{~\tt~a/NO~}\mathrm{~\tt~E~}$ ENDIF + +END SUBROUTINE Stats + +# 24.5 Modules containing procedures — Quicksort example + +In this example we rewrite the Quicksort example to use modules. Each subroutine is put into a module on its own. The program is given below: + +module read_data + +contains + +SUBROUTINE Read(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER ( $\tt L E N\mathrm{=\star}$ ) , INTENT(IN) :: File_Name INTEGER , INTENT(IN) :: How_Many REAL , INTENT(OUT) , DIMENSION(:) :: Raw_Data + +INTEGER :: I + +OPEN(FILE $_{\parallel}=$ File_Name,UNIT $\vDash\vDash$ ) DO $\mathbb{T}\!=\!1$ ,How_Many READ (UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) Raw_Data(I) + +# 350 + +# Chapter 24 + +ENDDO END SUBROUTINE Read + +end module read_data + +module sort_data + +contains + +SUBROUTINE Sort(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many REAL , INTENT(INOUT) , DIMENSION(:) :: Raw_data + +CALL QuickSort(1,How_Many) + +CONTAINS + +RECURSIVE SUBROUTINE QuickSort(L,R) IMPLICIT NONE INTEGER , INTENT(IN) :: L,R INTEGER :: I,J REAL :: V,T + +$\dot{\bot}\!=\!\bot$ $\dot{\daleth}\,{=}\,\underline{{\boldsymbol{\Upsilon}}}$ $\mathbb{V}=$ raw_data( int(( $\mathtt{l}_{+\mathtt{r}}$ )/2) ) do do while (raw_data(i) < v ) $\dot{\boldsymbol{\updownarrow}}\!=\!\dot{\boldsymbol{\updownarrow}}\!+\!\boldsymbol{\updownarrow}$ enddo do while (v < raw_data(j) ) $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ enddo if ( $\dot{\Sigma}\!<\!=\!\dot{\Pi}$ ) then t $=$ raw_data(i) raw_data(i) $=$ raw_data(j) raw_data $(\;\dot{\;})=\mathsf{t}$ $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ endif if ( $\dot{\perp}{>}\dot{\supset}$ ) exit + +enddo + +if ( $\perp<\!\dot{\!\!\perp}$ ) then call quicksort(l,j) endif if (i\!\dot{\perp}$ ) exit end do if ( $\dot{\bar{\jmath}}\,{<}\mathrm{k}$ ) then $\perp\!=\!\dot{\perp}$ end if if ( $\qgtrless\dot{\qless1}$ ) then $\tt{T=\dot{I}}$ end if end do find $\mathbf{\beta}^{=}\mathbb{Y}$ (k) end function find + +end function median + +end module statistics + +program ch2405 + +use statistics + +implicit none integer :: n real , allocatable , dimension(:) :: x real :: m,sd,med integer , dimension(8) :: v + +print \*,' How many values ?' read \*,n call date_and_time(values $=\!\bigtriangledown$ ) print \*,' initial ',v(6),v(7),v(8) allocate $\left(\,\mathbf{x}\,(\,\mathbb{1}:\mathbf{n}\,)\right.$ ) call date_and_time(values $=\!\bigtriangledown$ ) print \*,' allocate ',v(6),v(7),v(8) call random_number $(\mathbf{\nabla}\mathbf{x})$ call date_and_time(values $=\!\bigtriangledown$ ) print \*,' random ',v(6),v(7),v(8) $\tt{x=x^{\star}1000}$ call date_and_time(values $=\!\bigtriangledown$ ) print \*,' output ',v(6),v(7),v(8) $\mathbb{m}=$ mean(x,n) + +call date_and_time(values $=\!\bigtriangledown$ ) print \*,' mean print \*,' mean $\begin{array}{r l}{\mathbf{\Sigma}=}&{{}\ ^{\intercal}\mathbf{\Sigma},\mathbf{m}}\end{array}$ sd $=$ std_dev(x,n,m) call date_and_time(values $=\!\bigtriangledown$ ) print \*,' standard deviation print \*,' Standard deviation $=$ ',sd med $=$ median(x,n) call date_and_time(values $=\!\bigtriangledown$ ) print \*,' median print \*,' median is = ',med end program ch2405 + +',v(6),v(7),v(8) + + ',v(6),v(7),v(8) + + ',v(6),v(7),v(8) + +Note again that we do not need to have explicit interface blocks as the packaging of the procedures within a module provides the interface checking automatically. + +The program also has timing code added to allow profiling of the various parts of the program. + +# 24.7 The solution of linear equations using Gaussian elimination + +At this stage we have introduced many of the concepts needed to write numerical code, and have included a popular algorithm, Gaussian elimination, together with a main program which uses it and a module to bring together many of the features covered so far. + +Finding the solution of a system of linear equations is very common in scientific and engineering problems, either as a direct physical problem or indirectly, for ex- ample, as the result of using finite difference methods to solve a partial differential equation. We will restrict ourselves to the case where the number of equations and the number of unknowns are the same. The problem can be defined as: + +$$ +\begin{array}{l}{a_{11}\,x_{1}+a_{12}\,x_{2}+...+a_{1n}\,x_{n}=b_{1}}\\ {a_{12}\,x_{2}+a_{22}\,x_{2}+...+a_{2n}\,x_{n}=b_{2}}\\ {\cdots\qquad\cdots\qquad\cdots\quad\cdots\qquad=...}\\ {a_{n1}\,x_{1}+a_{n2}\,x_{2}+...+a_{n n}\,x_{n}=b_{n}}\end{array} +$$ + +or + +$$ +\left(\begin{array}{l l l l}{a_{11}}&{a_{12}}&{\ldots}&{a_{1n}}\\ {a_{21}}&{a_{22}}&{\ldots}&{a_{2n}}\\ {\ldots}&{\ldots}&{\ldots}&{\ldots}\\ {a_{n1}}&{a_{n2}}&{\ldots}&{a_{n n}}\end{array}\right)\left(\begin{array}{l}{x_{1}}\\ {x_{2}}\\ {\ldots}\\ {x_{n}}\end{array}\right)=\left(\begin{array}{l}{b_{1}}\\ {b_{2}}\\ {\ldots}\\ {b_{n}}\end{array}\right) +$$ + +which can be written as: + +$$ +\textit{A x}=\textit{b} +$$ + +where $A$ is the $n\ x\ n$ coefficient matrix, b is the right-hand-side vector and $x$ is the vector of unknowns. We will also restrict ourselves to the case where $A$ is a gen- eral real matrix. + +Note that there is a unique solution to (2) if the inverse, $A^{\cdot^{I}}$ , of the coefficient ma- trix $A$ , exists. However, the system should never be solved by finding $A^{\cdot^{I}}$ and then solving $\boldsymbol{A}^{-1}~\boldsymbol{b}=\boldsymbol{x}$ because of the problems of rounding error and the computational costs. + +A well-known method for solving (2) is Gaussian elimination, where multiples of equations are subtracted from others so that the coefficients below the diagonal be- come zero, producing a system of the form: + +$$ +\begin{array}{r}{\left(\begin{array}{c c c c}{a_{11}^{*}}&{a_{12}^{*}}&{\ldots}&{a_{1n}^{*}}\\ {0}&{a_{22}^{*}}&{\ldots}&{a_{2n}^{*}}\\ {\ldots}&{\ldots}&{\ldots}&{\ldots}\\ {0}&{0}&{0}&{a_{n n}^{*}}\end{array}\right)\left(\begin{array}{c}{x_{1}}\\ {x_{2}}\\ {\ldots}\\ {x_{n}}\end{array}\right)=\left(\begin{array}{c}{b_{1}^{*}}\\ {b_{2}^{*}}\\ {\ldots}\\ {b_{n}^{*}}\end{array}\right)}\end{array} +$$ + +where $A$ has been transformed into an upper triangular matrix. By a process of backward substitution the values of $x$ drop out. + +The subroutine Gaussian Elimination implements the Gaussian elimination algo- rithm with partial pivoting, which ensure that the multipliers are less than 1 in magnitude, by interchanging rows if necessary. This is to try and prevent the buildup of errors. + +This implementation is based on two LINPACK routines SGEFA and SGESL and a Fortran 77 subroutine written by Tim Hopkins and Chris Phillips and found in their book Numerical Methods in Practice. + +The matrix A and vector B are passed to the subroutine Gaussian Elimination and on exit both A and B are overwritten. Mathematically Gaussian elimination is de- scribed as working on rows, and using partial pivoting row interchanges may be necessary. Due to Fortran's row element ordering, to implement this algorithm effi- ciently it works on columns rather than rows by interchanging elements within a column if necessary. + +MODULE Precisions INTEGER,PARAMETER:: Long ${}^{\circ}\underline{{=}}$ SELECTED_REAL_KIND(15,307)END MODULE Precisions + +PROGRAM Solve + +USE Precisions IMPLICIT NONE INTEGER :: I,N REAL (Long), ALLOCATABLE:: A(:,:),B(:),X(:) LOGICAL:: Singular + +INTERFACE + +SUBROUTINE Gaussian Elimination(A,N,B,X,Singular) USE Precisions IMPLICIT NONE INTEGER, INTENT(IN)::N REAL (Long), INTENT (INOUT) :: A(:,:),B(:) REAL (Long), INTENT(OUT)::X(:) LOGICAL, INTENT(OUT) :: Singular END SUBROUTINE Gaussian Elimination + +END INTERFACE + +PRINT \*,'Number of equations?' READ \*,N ALLOCATE(A(1:N,1:N),B(1:N),X(1:N)) DO $\mathbb{T}\!=\!1$ ,N PRINT \*,'Input elements of row ',I,' of A' READ\*,A(I,1:N)PRINT\*,'Input element ',I,' of B' READ \*,B(I) END DO CALL Gaussian Elimination(A,N,B,X,Singular) IF(Singular) THEN PRINT\*, 'Matrix is singular' ELSE PRINT\*, 'Solution X:' PRINT\*,X(1:N)ENDIF + +END PROGRAM Solve + +SUBROUTINE Gaussian Elimination(A,N,B,X,Singular) + + ! Routine to solve a system $\scriptstyle{\mathrm{Ax}}={\mathrm{b}} + +$ ! using Gaussian Elimination + + ! with partial pivoting + +! The code is based on the Linpack routines + + ! SGEFA and SGESL + + ! and operates on columns rather than rows! USE Precisions IMPLICIT NONE + + ! Matrix A and vector B are over-written + + ! Arguments INTEGER, INTENT(IN):: N REAL (Long),INTENT(INOUT):: A(:,:),B(:) REAL (Long),INTENT(OUT)::X(:) LOGICAL,INTENT(OUT)::Singular + + ! Local variables INTEGER::I,J,K,Pivot_row REAL (Long):: Pivot,Multiplier,Sum,Element REAL (Long),PARAMETER::Eps $\mathbf{\omega}=\mathbb{1}$ .E-13_Long + + ! + + ! Work through the matrix column by column + + ! DO $\mathtt{K}\!=\!1$ ,N-1 + + ! + + ! Find largest element in column K for pivot + + ! Pivot_row $=$ MAXVAL( MAXLOC( ABS( A(K:N,K) ) ) ) & + K - 1 + + ! + + ! Test to see if A is singular + + ! if so return to main program + + ! IF(ABS(A(Pivot_row,K)) $<=$ Eps) THEN Singular ${}^{\circ}={}$ .TRUE. RETURN ELSE Singular $=$ .FALSE. ENDIF + + ! + + ! Exchange elements in column K if largest is + + ! not on the diagonal + + ! IF(Pivot_row $\begin{array}{r l}{\mathbf{\chi}/=}&{{}\,\mathbb{K}.}\end{array}$ ) THENElement $\mathrm{=A}$ (Pivot_row,K) A(Pivot_Row,K) $\mathrm{=A}$ (K,K)A(K,K) $=$ Element + +# Chapter 24 + +Element $\mathtt{=B}$ (Pivot_row) B(Pivot_row) $\mathtt{=B}$ (K)B(K) $=$ Element ENDIF + +! + + ! Compute multipliers + + ! elements of column K below diagonal + + ! are set to these multipliers for use + + ! in elimination later on + + ! $\begin{array}{r l r}{\mathbb{A}\left(\,\mathbb{K}\!+\!\mathbb{1}:\!\mathbb{N}\,,\,\mathbb{K}\right)}&{{}=}&{\mathbb{A}\left(\,\mathbb{K}\!+\!\mathbb{1}:\!\mathbb{N}\,,\,\mathbb{K}\right)\,/\,\mathbb{A}\left(\,\mathbb{K}\,,\,\mathbb{K}\right)}\end{array} + +$ ! + + ! Row elimination performed by columns for efficiency + + ! DO $\scriptstyle{\updownarrow}=\mathsf{K}+\sum$ ,N Pivot $=$ A(Pivot_row,J) IF(Pivot_row $/=~\texttt{K}$ ) THEN + +! Swap if pivot row is not K A(Pivot_row,J) $\mathrm{=A}$ (K,J)A(K,J) $=$ Pivot ENDIF A( $\mathrm{(K+1:N\,,\,\bar{U})=A}$ ( $\mathtt{K}\!+\!1$ :N,J)-Pivot\* A( $\mathtt{K}\!+\!1$ :N,K) END DO + + ! + + ! Apply same operations to B + + ! B $\left(\,\mathtt{K}\!+\!1:\!\mathtt{N}\,\right)=\!\mathtt{B}$ ( $\mathtt{K}\!+\!1$ :N)-A( $\mathtt{K}\!+\!1$ :N,K)\*B(K) END DO + + ! + + ! Backward substitution + + ! DO $\mathbb{T}\mathrm{=}\mathbb{N}$ ,1,-1 $\begin{array}{l l l}{\mathsf{S u m}}&{=}&{0\cdot0}\end{array}$ DO $\begin{array}{r l r}{)}&{{}\mathbb{J}=}&{\mathbb{I}\!+\!\mathbb{1}\,,\,\mathbb{N}}\end{array}$ $\mathrm{Sum}\!\!=\!\mathrm{Sum}\!\!+\!\!\mathrm{A}\left(\,\mathbb{I}\,,\,\mathbb{J}\,\right)\,{^{\star}\!\mathrm{X}\left(\,\mathbb{J}\,\right)}$ END DO $\mathrm{\tt~X}\left(\mathrm{\tt~I}\right)=\left(\mathrm{\tt~B}\left(\mathrm{\tt~I}\right)-\mathrm{Sum}\right)\big/\mathrm{\tt~A}\left(\mathrm{\tt~I}\,,\,\mathrm{\tt~I}\right)$ END DO END SUBROUTINE Gaussian Elimination + +# 24.7.1 Notes + +# 24.7.1.1 Module for kind type + +A module, Precisions, has been used to define a kind type parameter, Long, to specify the floating point precision to which we wish to work. This module is then used by the main program and the subroutine, and the kind type parameter Long is used with all the REAL type definitions and with any constants, e.g., + +REAL(Long), PARAMETER :: Eps $\mathbf{\omega}=\mathbb{1}$ .E-13_Long + +# 24.7.1.2 Deferred-shape arrays + +In the main program matrix A and vectors B and X are declared as deferred-shape arrays, by specifying their rank only and using the ALLOCATABLE attribute. Their shape is determined at run time when the variable $\mathbf{N}$ is read in and then the statement + +ALLOCATE(A(1:N,1:N), B(1:N), X(1:N)) + +is used. + +# 24.7.1.3 Intrinisic functions MAXVAL and MAXLOC + +In the context of subroutine Gaussian Elimination we have used: + +MAXVAL ( MAXLOC (ABS ( A ( K:N,K ) ) ) ) + K - 1 + +Breaking this down, + +MAXLOC ( ABS ( A (K:N,K) ) ) + +takes the rank 1 array + +$$ +(\:|\:A(K,K)\:|\,,|\:A(K+1,K)\:|\,,\ldots|\:A(N,K)\:|\:) +$$ + +where $\mid A(K,K)\mid\;=\;\mathrm{ABS}(A(K,K))$ )) and of length $\mathit{N-K+1}$ . It returns the position of the largest element as a rank 1 array of size one, e.g., $(L)$ + +Applying MAXVAL to this rank 1 array $(L)$ returns $L$ as a scalar, $L$ being the posi- tion of the largest element of array (1). + +What we actually want is the position of the largest element of (1) , but in the $K^{\mathrm{th}}$ column of matrix $A$ . We therefore have to add $K–I$ to $L$ to give the actual position in column $K$ of A. + +# 24.8 Notes on module usage and compilation + +If we only have one file comprising all of the program units (main program, mod- ules, functions and subroutines) then there is little to worry about. However, it is recommended that larger-scale programs be developed as a collection of files with related program units in each file, or even one program unit per file. This is more productive in the longer term, but it will lead to problems with modules unless we compile each module before we use it in other program units. + +Secondly, we must use one directory or subdirectory so that the compiler and linker can find each program unit. + +![Thirdly, we must be aware of the file naming conventions used by each compiler implementation we work with. Consider the following: ](images/ce4f45c970e9ab1c3920ab13127953da571c6643bbde81d0571db1064b63d8f7.jpg) + +Whilst in this case they are the same, this is not guaranteed. + +# 24.9 Summary + +We have now introduced the concept of a module, another type of program unit, probably one of of the most important features of Fortran 90. We have seen in this chapter how they can be used: + +• Define global data. • Define derived data types. • Contain explicit procedure interfaces. • Cackage together procedures. + +This is a very powerful addition to the language, especially when constructing large programs and procedure libraries. + +# 24.10 Problems + +1. Write two functions, one to calculate the volume of a cylinder $\pi\,r^{\,2}\,\,l$ where the radius is $r$ and the length is $l$ , and the other to calculate the area of the base of the cylinder $\pi\,r^{2}$ . Define $\pi$ as a parameter in a module which is used by the two func- tions. Now write a main program which prompts the user for the values of $r$ and $l$ , calls the two functions and prints out the results. + +2. Make all the real variables in the above problem have 15 significant digits and a range of $10^{-307}$ to ${10}^{+307}$ . Use a module. + +# 24.11 Bibliography + +Dongarra, J., Bunch, J.R., Moler, C.B., and Stewart, G.W. LINPACK User's Guide . SIAM Publications, 1979. + +• This Fortran 77 package is for the solution of simultaneous systems of lin- ear algebraic equations. Special subroutines are included for many com- mon types of coefficient matrices. The source is available through NETLIB. See Chapter 28 for more details. + +Hopkins T., Phillips C., Numerical Methods in Practice, using the NAG Library . Addison-Wesley. + +• This is a very good practical introduction to numerical analysis, with the aim of guiding users to the more commonly used routines in the NAG Fortran 77 library. It does this by introducing topics, giving some back- ground, advantages and disadvantages, and the Fortran 77 code for some of the more well-known algorithms. It then introduces the appropriate NAG routine with a brief discussion of its use, calling sequence and any error reporting facilities. We've found this invaluable for many of our stu- dents who are users of the NAG library but not well versed with numeri- cal analysis. Maybe we will see a Fortran 90 version of this book in the near future? + +NAG. Visit their web site for up to date details of their products: + +• http://www.nag.co.uk/ + +Visual Numerics. Visit their web site for details of their products: • http://www.vni.com/index.html + +# Converting from Fortran 77 + +“Twas brillig, and the slithy toves did gyre and gimble in the wabe; All mimsy were the borogoves, And the mome raths outgrabe.” + +Lewis Carroll + +This chapter looks at some of the options available when working with older For- tran code. + +# 25 Converting from Fortran 77 + +This chapter looks at converting Fortran 77 code to Fortran 90 and 95 styles. + +The aim here is to provide the Fortran 77 programmer (and in particular the person with legacy code) with some simple guidelines for conversion. + +The first thing that one must have is a thorough understanding of the newer, better language features of Fortran 95. It is essential that the material in the earlier chap- ters of this book be covered, and some of the problems attempted. This will provide a feel for Fortran 95. + +The second thing one must have is a thorough understanding of the language con- structs used in this legacy code. Use should be made of the compiler documentation for whatever Fortran 77 compiler you are using, as this will provide the detailed (often system specific) information required. The recommendations be- low are therefore brief. + +It is possible to move gradually from Fortran 77 to Fortran 95. In many cases ex- isting code can be quite simply recompiled by a suitable choice of compiler options. This enables us to mix and match old and new in one program. This pro- cess is likely to highlight nonstandard language features in your old code. There will inevitably be some problems here. + +The first thing to consider is what the standard says. The standard identifies two kinds of decremented features; deleted and obsolescent. It is extremely unwise to consider the long-term use of these features as they are candidates for removal from future standards. + +# 25.1 Deleted features + +The list of deleted features for Fortran 95 is empty, i.e., there are none. + +# 25.2 Obsolescent features + +The obsolescent features are those for which better methods are available. They are given below with alternatives. + +25.2.1 Arithmetic IF Use the IF statement. + +25.2.2 Real and double precision DO control variables + +Use integer. + +25.2.3 Shared DO termination and non-ENDDO termination + +Use an END DO. + +# 25.2.4 Alternate RETURN + +Use a CASE statement on return. An error code has to be returned. + +# 25.2.5 PAUSE statement + +System specific. Normally easily replaced with a suitable READ statement. + +25.2.6 ASSIGN and assigned GOTO statements Fortunately rarely used. + +# 25.2.7 Assigned FORMAT statements + +Use character arrays, arrays and constants. + +# 25.2.8 H editing + +Use character edit descriptor. + +# 25.3 Better alternatives + +Below we are looking at the new features of the Fortran 95 standard, and how we can replace our current coding practices with the better facilities that now exist. + +• DOUBLE PRECISION — use KIND, see Chapter 8, and examples throughout the book. • fixed format — use free format • implicit typing — use IMPLICIT NONE • BLOCK DATA — use modules • COMMON statement — use modules • EQUIVALENCE — Invariably the use of this feature requires consider- able system specific knowledge. There will be cases where there have been extremely good reasons why this feature has been used, normally ef- ficiency related. However with the rapid changes taking place in the power and speed of hardware these reasons are diminishing. •Assumed-size / explicit-shape dummy array arguments — If a dummy ar-gument is assumed-size or explicit-shape (the only ones available in For- tran 77) then the ranks of the actual argument and the associated argument don't have to be the same. With Fortran 95 arrays are now objects instead of a linear sequence of elements, as was the case with Fortran 77, and now for array arguments the fundamental rule is that actual and dummy arguments have the same rank and same extents in each dimension, i.e., the same shape, and this is done using assumed-shape dummy array argu- ments. An interface block is mandatory for assumed-shape arrays. + +# Chapter 25 + +• ENTRY statement — use module plus USE statement. • Statement functions — use internal function, see Chapter 14. • Computed GOTO — use CASE statement, see Chapter 15. • Alternate RETURN — use error flags on calling routine. • INCLUDE — use modules plus USE statement. • EXTERNAL statement for dummy procedure arguments. + +Use explicit interface blocks everywhere. This also provides argument checking and other benefits. + +# 25.4 Example 1 + +The first and simplest option is to do nothing. Any code that is valid standard For- tran 77 will compile as the various successor standards require backwards compatibility with the Fortran 77 standard. + +We will look at an example of leaving the Fortran 77 code alone using a sorting subroutine from netlib. To get hold of a copy of this code visit: + +•http://www.netlib.org/ + +and search using + +• sort + +as the keyword. + +One of the retrieved links should be + +•http://www.netlib.org/slatec/src/dsort.f + +Here is a complete listing of this subroutine as is. The code wraps in places at comment lines and this is intentional as we wanted to show you the code just as it is without any changes to fit the printed page + +\*DECK DSORT SUBROUTINE DSORT (DX, DY, N, KFLAG) C\*\*\*BEGIN PROLOGUE DSORT C\*\*\*PURPOSE Sort an array and optionally make the same interchanges in C an auxiliary array. The array may be sorted in increasing C or decreasing order. A slightly modified QUICKSORT C algorithm is used. + +C\*\*\*LIBRARY SLATEC + +C\*\*\*CATEGORY N6A2B C\*\*\*TYPE DOUBLE PRECISION (SSORT-S, DSORT-D, ISORT-I) C\*\*\*KEYWORDS SINGLETON QUICKSORT, SORT, SORTING C\*\*\*AUTHOR Jones, R. E., (SNLA) + +C Wisniewski, J. A., (SNLA) + +C C DSORT sorts array DX and optionally makes the same interchanges in C array DY. The array DX may be sorted in increasing order or C decreasing order. A slightly modified quicksort algorithm is used. C C Description of Parameters C DX - array of values to be sorted (usually abscissas) C DY - array to be (optionally) carried along C N - number of values in array DX to be sorted C KFLAG - control parameter C $\c=\qquad^{2}$ means sort DX in increasing order and carry DY along. C $\c=\quad\bot$ means sort DX in increasing order (ignoring DY) C $\begin{array}{r l}{\mathbf{\omega}=}&{{}-\mathbb{1}}\end{array}$ means sort DX in decreasing order (ignoring DY) C $\begin{array}{r l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}}\end{array}$ means sort DX in decreasing order and carry DY along. C C\*\*\*REFERENCES R. C. Singleton, Algorithm 347, An efficient algorithm C for sorting with minimal storage, Communications of C the ACM, 12, 3 (1969), pp. 185-187. C\*\*\*ROUTINES CALLED XERMSG C\*\*\*REVISION HISTORY (YYMMDD) C 761101 DATE WRITTEN C 761118 Modified to use the Singleton quicksort algorithm. (JAW) + +C 890531 Changed all specific intrinsics to generic. (WRB) C 890831 Modified array declarations. (WRB) C 891009 Removed unreferenced statement labels. (WRB) C 891024 Changed category. (WRB) C 891024 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) C 901012 Declared all variables; changed X,Y to DX,DY; changed C code to parallel SSORT. (M. McClain) C 920501 Reformatted the REFERENCES section. (DWL, WRB) C 920519 Clarified error messages. (DWL) C 920801 Declarations section rebuilt and code restructured to use CIF-THEN-ELSE-ENDIF.(RWC, WRB)C\*\*\*END PROLOGUE DSORT C .. Scalar Arguments .. INTEGER KFLAG, N C .. Array Arguments .. DOUBLE PRECISION DX(\*), DY(\*) C .. Local Scalars .. DOUBLE PRECISION R, T, TT, TTY, TY INTEGER I, IJ, J, K, KK, L, M, NN C .. Local Arrays .. INTEGER IL(21), IU(21)C .. External Subroutines .. EXTERNAL XERMSG C .. Intrinsic Functions .. INTRINSIC ABS, INT C\*\*\*FIRST EXECUTABLE STATEMENTDSORTNN = N IF (NN .LT. 1) THEN CALL XERMSG ('SLATEC', 'DSORT', + 'The number of values to be sorted is not positive.', 1, 1) RETURN ENDIF + +C KK $=$ ABS(KFLAG) IF (KK.NE.1 .AND. KK.NE.2) THEN CALL XERMSG ('SLATEC', 'DSORT', + 'The sort control parameter, K, is not 2, 1, -1, or -2.', 2, + 1) RETURN ENDIF C C Alter array DX to get decreasing order if needed C IF (KFLAG .LE. -1) THEN DO 10 $\mathbb{T}\!=\!\!1$ ,NN $\begin{array}{r l r}{\mathsf{D}\ensuremath{\mathbb{X}}\left(\,\ensuremath{\mathbb{T}}\,\right)}&{{}=}&{-\mathsf{D}\ensuremath{\mathbb{X}}\left(\,\ensuremath{\mathbb{T}}\,\right)}\end{array}$ 10 CONTINUE ENDIF C IF (KK .EQ. 2) GO TO 100 C C Sort DX only C $\begin{array}{r l}{{\mathbb M}}&{{}{}={\ \mathbb1}}\\ {{\mathbb I}}&{{}{}={\ \mathbb1}}\\ {{\mathbb J}}&{{}{}={\ \mathbb N}{\mathbb N}}\\ {{\mathbb R}}&{{}{}={\ 0}\cdot{37}}\end{array}$ C 20 IF (I .EQ. J) GO TO 60 IF (R .LE. 0.5898437D0) THEN R = R+3.90625D-2 ELSE R = R-0.21875D0 ENDIF C 30 K = I C C Select a central element of the array and save it in location T C $\begin{array}{r c l}{\mathbb{I}\mathbb{J}}&{=}&{\mathbb{I}\mathbb{\Gamma}\mathbf{\Gamma}+\mathbb{I}\mathbb{N}\mathbb{T}\left(\mathbf{\Gamma}\left(\mathbb{J}\!-\!\mathbb{I}\right)\mathbf{\Omega}^{\star}\mathbb{R}\right)}\\ {\mathbb{T}}&{=}&{\mathbb{D}\mathbb{X}\left(\mathbb{I}\mathbb{J}\right)}\end{array}$ C + +# Chapter 25 + +C If first element of array is greater than T, interchange with T C IF (DX(I) .GT. T) THEN DX(IJ) $=$ DX(I) $\mathrm{DX}\left(\,\mathbb{T}\,\right)\mathrm{~\ensuremath~{~=~}~}\mathrm{~T~}$ $\begin{array}{l l l}{\mathbb{T}}&{=}&{\tt D X\left(\tt I J\right)}\end{array}$ ENDIF $\begin{array}{r l r}{\mathbb{L}}&{{}=}&{\mathbb{J}}\end{array}$ C C If last element of array is less than than T, interchange with T C IF (DX(J) .LT. T) THEN DX(IJ) $=$ DX(J) $\mathrm{DX}\left(\mathbb{T}\right)\mathrm{~\ensuremath~{~=~}~}\mathrm{T}$ T = DX(IJ) C C If first element of array is greater than T, interchange with T C IF (DX(I) .GT. T) THEN DX(IJ) $=$ DX(I) DX(I) $=$ T $\begin{array}{r l}{\mathbb{T}}&{{}=}\end{array}$ DX(IJ) ENDIF ENDIF C C Find an element in the second half of the array which is smaller C than T C 40 $\begin{array}{l l l}{\mathbb{L}}&{=}&{\mathbb{L}\!-\!\mathbb{1}}\end{array}$ IF (DX(L) .GT. T) GO TO 40 C C Find an element in the first half of the array which is greater C than T C 50 $\begin{array}{l c l}{\mathbb{K}}&{=}&{\mathbb{K}\!+\!\bot}\end{array}$ IF (DX(K) .LT. T) GO TO 50 C + +C Interchange these elements C IF (K .LE. L) THEN TT $=$ DX(L) DX(L) $=$ DX(K) DX(K) $=$ TT GO TO 40 ENDIF C C Save upper and lower subscripts of the array yet to be sorted C IF (L-I .GT. J-K) THEN IL(M) $=\quad\mathbb{T}$ IU(M) $=~\texttt{L}$ $\mathrm{~\small~\mathbb~{~T~}~}=\mathrm{~\small~\mathbb~{~K~}~}$ $\mathrm{~\bf~M~}~=~\mathrm{~\bf~M+}\mathrm{~\bf~1~}$ ELSE $\begin{array}{r l r}{\mathbb{L}\mathbb{L}\left(\mathbb{M}\right)}&{{}=}&{\mathbb{K}}\\ {\mathbb{U}\left(\mathbb{M}\right)}&{{}=}&{\mathbb{J}}\end{array}$ $\begin{array}{l l l}{\top}&{=}&{\mathbb{L}}\end{array}$ $\mathrm{~\bf~M~}~=~\mathrm{~\bf~M+}\mathrm{~\bf~1~}$ ENDIF GO TO 70 C C Begin again on another portion of the unsorted array C 60 $\texttt{M}=\texttt{M}{-1}$ IF (M .EQ. 0) GO TO 190 $\begin{array}{l l l}{\mathbb{T}}&{=}&{\mathbb{T}\mathbb{L}}\end{array}$ (M) J = IU(M) C 70 IF (J-I .GE. 1) GO TO 30 IF (I .EQ. 1) GO TO 20 $\begin{array}{l l l}{\mathbb{T}}&{=}&{\mathbb{T}\!-\!\mathbb{1}}\end{array}$ C 80 $\begin{array}{l l l}{\mathbb{T}}&{=}&{\mathbb{T}\!+\!\mathbb{1}}\end{array}$ IF (I .EQ. J) GO TO 60 $\begin{array}{l l l}{\mathbb{T}}&{=}&{\tt D X\left(\,\mathbb{I}\!+\!\mathbb{1}\,\right)}\end{array}$ IF (DX(I) .LE. T) GO TO 80 $\textrm{\textsf{K}}=\textrm{\textsf{T}}$ + +C 90 DX(K+1) = DX(K) $\begin{array}{r c l}{\mathbb{K}}&{=}&{\mathbb{K}\!-\!\mathbb{1}}\end{array}$ IF (T .LT. DX(K)) GO TO 90 DX $\begin{array}{r l r}{(\,\mathrm{K}{+}\,\mathrm{\mathbb{1}}\,)}&{{}=}&{\mathrm{T}}\end{array}$ GO TO 80 C C Sort DX and carry DY along C 100 M = 1 $\begin{array}{r l}{\mathbb{M}}&{=\;\;1}\\ {\mathbb{I}}&{=\;\;1}\\ {\mathbb{J}}&{=\;\;\mathtt{N N}}\\ {\mathbb{R}}&{=\;\;0\cdot375\tt{D}0}\end{array}$ C 110 IF (I .EQ. J) GO TO 150 IF (R .LE. 0.5898437D0) THEN R = R+3.90625D-2 ELSE R = R-0.21875D0 ENDIF C 120 K = I C C Select a central element of the array and save it in location T C $\begin{array}{r l}{\mathbb{I}\mathbb{J}}&{=\mathrm{~\mathbb{I}~\mathbb{\Gamma}~}+\mathrm{~\mathbb{I}N T~}(\mathrm{~\mathbb{I}\Sigma~}(\mathbb{\Gamma}\mathbb{I}-\mathbb{I})\mathrm{~\star~\mathbb{R}~})}\\ {\mathbb{T}}&{=\mathrm{~\mathbb{D}X~}(\mathbb{I}\mathbb{J})}\\ {\mathbb{T}\mathbb{Y}}&{=\mathrm{~\mathbb{D}Y~}(\mathbb{I}\mathbb{J})}\end{array}$ C C If first element of array is greater than T, interchange with T C IF (DX(I) .GT. T) THEN DX(IJ) $=$ DX(I) $\mathrm{DX}\left(\,\mathbb{T}\,\right)\mathrm{~\ensuremath~{~=~}~}\mathrm{~T~}$ $\begin{array}{r l}{\mathbb{T}}&{{}=}\end{array}$ DX(IJ) DY(IJ) $=$ DY(I) DY(I) $=$ TY TY $=$ DY(IJ) ENDIF + +C C If last element of array is less than T, interchange with T C IF (DX(J) .LT. T) THEN DX(IJ) $=$ DX(J) $\mathrm{DX}\left(\mathbb{T}\right)\mathrm{~\ensuremath~{~=~}~}\mathrm{T}$ $\begin{array}{r l}{\mathbb{T}}&{{}=}\end{array}$ DX(IJ) $\begin{array}{r l r}{\mathrm{\sfDy}\left(\,\mathbb{I}\,\mathbb{J}\,\right)}&{{}=}&{\mathrm{\sfDy}\left(\,\mathbb{J}\,\right)}\end{array}$ $\begin{array}{r l r}{\mathrm{DY}\left(\,\daleth\,\right)}&{{}=}&{\mathrm{TY}}\end{array}$ TY $=$ DY(IJ) C C If first element of array is greater than T, interchange with T C IF (DX(I) .GT. T) THEN DX(IJ) $=$ DX(I) DX(I) $=~\ensuremath{\mathbb{T}}$ $\begin{array}{r l}{\mathbb{T}}&{{}=}\end{array}$ DX(IJ) DY(IJ) $=$ DY(I) DY(I) $=$ TY TY $=$ DY(IJ) ENDIF ENDIF C C Find an element in the second half of the array which is smaller C than T C $\begin{array}{r l r}{130\;\;\mathrm{~\mathbb~L~}\;=}&{{}\mathbb{L}\!-\!\mathbb{1}}\end{array}$ IF (DX(L) .GT. T) GO TO 130 C C Find an element in the first half of the array which is greater C than T C 140 $\begin{array}{l c l}{\mathbb{K}}&{=}&{\mathbb{K}\!+\!\bot}\end{array}$ IF (DX(K) .LT. T) GO TO 140 C C Interchange these elements C IF (K .LE. L) THEN + +$\mathrm{TT}\ \ =\ \ \mathrm{DX}$ (L) DX(L) $=$ DX(K) DX(K) $=$ TT TTY $=$ DY(L) DY(L) $=$ DY(K) DY(K) $=$ TTY GO TO 130 ENDIF + +C Save upper and lower subscripts of the array yet to be sorted C IF (L-I .GT. J-K) THEN IL(M) $=\quad\mathbb{T}$ IU(M) $=~\texttt{L}$ $\mathrm{~\small~\mathbb~{~T~}~}=\mathrm{~\small~\mathbb~{~K~}~}$ $\mathrm{~\bf~M~}~=~\mathrm{~\bf~M+}\mathrm{~\bf~1~}$ ELSE $\begin{array}{r l r}{\mathbb{L}\mathbb{L}\left(\mathbb{M}\right)}&{{}=}&{\mathbb{K}}\\ {\mathbb{U}\left(\mathbb{M}\right)}&{{}=}&{\mathbb{J}}\end{array}$ $\begin{array}{l l l}{\top}&{=}&{\mathbb{L}}\end{array}$ $\mathrm{~\bf~M~}~=~\mathrm{~\bf~M+}\mathrm{~\bf~1~}$ ENDIF GO TO 160 C C Begin again on another portion of the unsorted array + +C 160 IF (J-I .GE. 1) GO TO 120 IF (I .EQ. 1) GO TO 110 $\begin{array}{l l l}{\mathbb{T}}&{=}&{\mathbb{T}\!-\!\mathbb{1}}\end{array}$ + +$\begin{array}{r l r}{\mathbb{17}0\;\;\;\mathbb{I}}&{{}=}&{\mathbb{I}\!+\!\mathbb{1}}\end{array}$ IF (I .EQ. J) GO TO 150 $\begin{array}{l l l}{\mathbb{T}}&{=}&{\tt D X\left(\,\mathbb{I}\!+\!\mathbb{1}\,\right)}\end{array}$ $\begin{array}{l l l}{\mathbb{T Y}}&{=}&{\mathbb{D Y}\;(\;\mathbb{I}\!+\!\mathbb{1}\;)}\end{array}$ IF (DX(I) .LE. T) GO TO 170 + +K = I C 180 DX(K+1) = DX(K) $\begin{array}{r l r}{\mathrm{\ttDy}\left(\,\mathrm{\boldmath~K}{+}\,\mathrm{\boldmath~\downarrow~}\right)}&{{}=}&{\mathrm{\ttDy}\left(\,\mathrm{\boldmath~K}\right)}\end{array}$ $\begin{array}{r c l}{\mathbb{K}}&{=}&{\mathbb{K}\!-\!\mathbb{1}}\end{array}$ IF (T .LT. DX(K)) GO TO 180 DX(K+1) = T DY(K+1) = TY GO TO 170 C C Clean up C 190 IF (KFLAG .LE. $^{-1}$ ) THEN DO 200 $\scriptstyle{\mathbb{T}}=1$ ,NN $\begin{array}{r l r}{\mathsf{D}\ensuremath{\mathbb{X}}\left(\,\ensuremath{\mathbb{T}}\,\right)}&{{}=}&{-\mathsf{D}\ensuremath{\mathbb{X}}\left(\,\ensuremath{\mathbb{T}}\,\right)}\end{array}$ 200 CONTINUE ENDIF RETURN END + +Our aim is to replace a call to the Quicksort subroutine in an earlier example with a call to the dsort subroutine. The new program shall be called ch2501.f90 in what follows. + +If we follow the Fortran 77 route all we need to do is change all DOUBLE PRECI- SION variables to REAL and comment out the calls to the external error handling subroutine XERMSG. + +We then comment out the call to Quicksort: + +! CALL quicksort(1,how_many) + +and replace with + +CALL DSORT(Raw_data,Raw_Data,How_many,1) + +where the value of 1 for kflag means ignore the second array. + +Here is an example of using the Intel compiler to compile and run the complete program: + +• ifort -c dsort.f • ifort ch2501.f90 dsort.obj • ch2501 + +The first command compiles the dsort routine as a Fortran 77 fixed source form and generates an object file called dsort.obj. + +The second command compiles the main program and modules and links this with the dsort.obj file. + +The third line runs the program. + +This example shows how easy it is in practice to mix and match both Fortran 77 and Fortran 90 style code. + +Let us next look at converting the above dsort routine to Fortran 90/95. + +# 25.5 Example 2 + +Mike Metcalf provides a free program to convert from Fortran 77 to Fortran 90 syntax. A copy can be found at + +• http://www.kcl.ac.uk/fortran + +Below is the output from running this program. + +Type name of file, shift, max. indent level, T or F for blank treatment, T or F for interface blocks only. For simple use type only the name of the file followed by a slash (/) and RETUR + +dsort/ Loop bodies will be indented by 0 Maximum indenting level is 0 Processing complete in 0.000 seconds Maximum depth of DO-loop nesting 1 Maximum depth of IF-block nesting 2 No. of lines read 324 No. of program units read 1 Global syntax error flag F + +The program simply replaces the C in column 1 with the new comment symbol !. Here is what one needs to do to compile using the Intel compiler: + +• ifort ch2501.f90 dsort.f90 + +This couldn't really be much simpler. Both methods are completely straightfor- ward. + +Other free conversions tools include + +• http://www.owlnet.rice.edu/\~colby/f2f.html and here is a quote by the author about the software: + +“f2f is a Perl script, which does much of the tedious work of converting Fortran 77 source code into Fortran 90/95 form. There seems to be a lot of Fortran-hate in the world, and I think this comes from people who have been forced to use Fortran 77 at some time or another. Hopefully, this program will make you a less hateful per- son.” + +# 25.6 Commercial conversion tools + +There are a number of commercial conversion tools and some of them are given below. + +# 25.6.1 NAG + +Their home site is + +• http://www.nag.co.uk/ + +Here is a Fortran 77 program with several subroutines and common blocks. The comments in the original program have been removed due to space considerations. The complete programs can be found at: + +• http://www.kcl.ac.uk/fortran + +The tsunami plot file can be found at + +http://www.kcl.ac.uk/fortran + +The program was written by Ian whilst on an 18-month secondment to the United Nations Environment Programme. + +The code wraps in places and this is intentional as again we wanted to show you the code as it is without any changes to fit the printed page. + +PROGRAM Map01 LOGICAL TRIAL,SCREEN REAL LONG,LATSCREEN $=$ .FALSE. TRIAL $=$ .FALSE. CALL DATAIN(TRIAL) PRINT \*,' What resolution map do you want' PRINT $\star$ PRINT $\mathrm{~\star~,~}\,\mathrm{~\star~}\,\mathrm{~\star~}\,=\,\mathrm{~\star~}1.19\,,\,650~\mathrm{~\star~}$ PRINT \*,' 2 = 75,500' PRINT \*,' 3 = 43,100' PRINT \*,' 4 = 19,300' + +PRINT \*,' 5 = 4,420' PRINT \* 100 READ (UNIT $^{\ast}{=}^{\ast}$ , $\scriptstyle{\mathrm{FT}}={\star}$ ,END $\scriptstyle{1=2\,0\,0}$ ,ERR $_{\cdot=200}$ ) IRES 200 IF ((IRES.LT.1) .OR. (IRES.GT.5)) THEN PRINT \*,' Please input a number in the range 1 to 5' GO TO 100 END IF PRINT \*,' What projection would you like?' PRINT $\star$ PRINT \*,' 1 = Lambert - equal area + + - rectangle' PRINT \*,' 2 = Mercator - equal direction + + - rectangle' PRINT \*,' 3 = Hammer - equal area + + - oval' PRINT \*,' 4 = Bonne - + + - heart' PRINT \*,' 5 $=$ Orthographic - globe + + - round' 300 READ (UNIT $^{\ast}{=}^{\ast}$ , $\scriptstyle{\mathrm{FT}}={\star}$ ,END $\scriptstyle1=4\,0\,0$ ,ERR $\scriptstyle\cdot=4\,0\,0$ ) IPROJ 400 IF ((IPROJ.LT.1) .OR. (IPROJ.GT.5)) THEN PRINT \*,' Please input a number in the range 1 to 5' GO TO 300 END IF LAT $\begin{array}{r l}{\mathbf{\sigma}=}&{{}0\mathbf{\sigma}.\;0}\end{array}$ $\begin{array}{l l l}{\mathrm{{LONG}}}&{=}&{\mathbb{1}80\;.\;0}\end{array}$ PRINT \*,' Which region do you wish to plot?' PRINT \*,' 0 = all regions' PRINT $\mathrm{~\star~},\mathrm{~\iota~}_{\perp}$ = Hawaii' PRINT \*,' 2 = New Zealand and South Pacific Islands' PRINT \*,' 3 = Papua New Guinea and Solomon Islands' PRINT \*,' 4 = Indonesia' PRINT \*,' 5 = Philippines' PRINT \*,' 6 = Japan' PRINT \*,' 7 = Kuril Islands and Kamchatka' PRINT \*,' 8 = Alaska incluing Aleutian Islands' PRINT $\begin{array}{c c c c}{{\star\,\,,}}&{{}}&{{9}}&{{=}}\end{array}$ West Coast - North and Central America' + +PRINT \*,' $\begin{array}{r l}{\mathcal{10}}&{{}=}\end{array}$ West Coast - South America' READ (UNIT $=\star$ , $\scriptstyle{\mathrm{FT}}={\star}$ , $\mathrm{END}{=}8\,0\,0\,0$ ,ERR $\scriptstyle=8\,0\,0\,0$ ) NREG 8000 IF ((NREG.LT.0) .OR. (NREG.GT.10)) THENPRINT \*,' Please input a number between 0 and 10 inclusive' GO TO 8000 END IF PRINT \*,' Which colour table do you wish to use' PRINT \*,' HLS = 1' PRINT \*,' CMY = 2' PRINT \*,' RGB = 3' 500 READ (UNIT $^{\ast}{=}^{\ast}$ , $\scriptstyle{\mathrm{FT}}={\star}$ , $\mathrm{END}{=}6\,0\,0$ ,ERR $\scriptstyle\cdot=6\,0\,0$ ) ICOL 600 IF ((ICOL.LT.1) .OR. (ICOL.GT.3)) THEN PRINT \*,' Please input a number in the range 1 to $3~^{\prime}$ GO TO 500 END IF PRINT \*,' Select device, a list of valid devices maybe' PRINT \*,' obtained by typing' PRINT \*,' list \*' PRINT \*,' at the GROUTE prompt' CALL GROUTE(' ')CALL GOPEN CALL GSEGCR(1) CALL GSURFE IF (IRES.EQ.1) THEN CALL WEXTND ELSE IF (IRES.EQ.2) THEN CALL WRED1 ELSE IF (IRES.EQ.3) THEN CALL WRED2 ELSE IF (IRES.EQ.4) THEN CALL WRED3 ELSE IF (IRES.EQ.5) THEN CALL WRED4 END IF $\mathrm{~\ensuremath~{~\mathrm{~NR~}~}~}=\mathrm{~\ensuremath~{~7~}~}$ KOLOR $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\$ CALL WOPEN(0,NR) CALL WPROJ(IPROJ) + +CALL WCENTR(LONG,LAT) CALL WDEFC(KOLOR) CALL WPLOT(' ',0.0) CALL CONVRT(TRIAL) CALL PLOTEM(TRIAL,NREG)CALL GSEGCR(1) CALL GCLOSE END SUBROUTINE DATAIN(TRIAL) LOGICAL TRIAL CHARACTER $^{\star}8\,0$ FILNAM COMMON + +/TSUNAM/REG0LA(378),REG0LO(378),REG1LA(206),REG1LO(206), $^+$ REG2LA(41),REG2LO(41),REG3LA(54),REG3LO(54),REG4LA(60), $^+$ REG4LO(60),REG5LA(1540),REG5LO(1540),REG6LA(80),REG6LO(8 0), + +REG7LA(144),REG7LO(144),REG8LA(245),REG8LO(245), + REG9LA(285),REG9LO(285) IF (TRIAL.EQ..TRUE.) THEN PRINT \*,' Entering data input phase' END IF FILNAM $=$ 'tsunami.dat' OPEN (UNIT $\mathrm{\Sigma=}50$ ,FILE $=$ FILNAM,ERR $\scriptstyle\cdot=3\,0$ ,STATUS $=$ 'OLD') GO TO 40 30 PRINT \*,' Error opening data file' PRINT \*,' Program terminates' STOP 40 DO $\begin{array}{r c l}{\mathbb{1}00}&{\mathbb{I}}&{=}&{\mathbb{1}\,,\,378}\end{array}$ 100 READ (UNIT $\mathrm{\Sigma=}50$ , $\scriptstyle{\mathrm{FMT}=1000}$ ) REG0LA(I),REG0LO(I) 1000 FORMAT (1X,F7.2,2X,F7.2) DO 110 I = 1,206 110 READ (UNIT $\mathrm{\Sigma=}50$ , $\scriptstyle{\mathrm{FMT}=1000}$ ) REG1LA(I),REG1LO(I) DO 120 I = 1,41 120 READ (UNIT $\mathrm{\Sigma=}50$ , $\scriptstyle{\mathrm{FMT}=1000}$ ) REG2LA(I),REG2LO(I) DO 130 I = 1,54 130 READ (UNIT $\mathrm{\Sigma=}50$ , $\scriptstyle{\mathrm{FMT}=1000}$ ) REG3LA(I),REG3LO(I) DO 140 I = 1,60 140 READ (UNIT $\mathrm{\Sigma=}50$ , $\scriptstyle{\mathrm{FMT}=1000}$ ) REG4LA(I),REG4LO(I) + +DO 150 I = 1,1540 150 READ (UNIT $\mathrm{\Sigma=}50$ , $\scriptstyle{\mathrm{FMT}=1000}$ ) REG5LA(I),REG5LO(I) DO 160 I = 1,80 160 READ (UNIT $\mathrm{\Sigma=}50$ , $\scriptstyle{\mathrm{FMT}=1000}$ ) REG6LA(I),REG6LO(I) DO 170 I = 1,144 170 READ (UNIT $\mathrm{\Sigma=}50$ , $\scriptstyle{\mathrm{FMT}=1000}$ ) REG7LA(I),REG7LO(I) DO 180 I = 1,245 180 READ (UNIT $\mathrm{\Sigma=}50$ , $\scriptstyle{\mathrm{FMT}=1000}$ ) REG8LA(I),REG8LO(I) DO 190 I = 1,285 190 READ (UNIT $\mathrm{\Sigma=}50$ , $\scriptstyle{\mathrm{FMT}=1000}$ ) REG9LA(I),REG9LO(I) IF (TRIAL.EQ..TRUE.) THEN DO $200\;\mathrm{~\textperthousand~}$ 200 PRINT \*,REG0LA(I),' ',REG0LO(I) PRINT \*,' Exiting data input phase' READ \*,DUMMY END IF END SUBROUTINE CONVRT(TRIAL) LOGICAL TRIAL COMMON /TSUNAM/REG0LA(378),REG0LO(378),REG1LA(206),REG1LO(206), + +REG2LA(41),REG2LO(41),REG3LA(54),REG3LO(54),REG4LA(60), $^+$ REG4LO(60),REG5LA(1540),REG5LO(1540),REG6LA(80),REG6LO(8 0), + +REG7LA(144),REG7LO(144),REG8LA(245),REG8LO(245), + REG9LA(285),REG9LO(285) COMMON /MMTSUN/MM0LA(378),MM0LO(378),MM1LA(206),MM1LO(206), $^+$ MM2LA(41),MM2LO(41),MM3LA(54),MM3LO(54),MM4LA(60), $^+$ MM4LO(60),MM5LA(1540),MM5LO(1540),MM6LA(80),MM6LO(80), $^+$ MM7LA(144),MM7LO(144),MM8LA(245),MM8LO(245),MM9LA(285), + MM9LO(285) + +IF (TRIAL.EQ..TRUE.) THEN PRINT \*,' Entering convert' END IF + +# Chapter 25 + +DO 100 I = 1,378 100 CALL WGETMM(REG0LO(I),REG0LA(I),MM0LO(I),MM0LA(I)) DO 110 I = 1,206 110 CALL WGETMM(REG1LO(I),REG1LA(I),MM1LO(I),MM1LA(I)) DO $120\;\;\;\mathbb{I}\;\;\;=\;\;\;{\mathbb{1}}\,,\,4{\mathbb{1}}$ 120 CALL WGETMM(REG2LO(I),REG2LA(I),MM2LO(I),MM2LA(I)) DO $130\;\;\;\mathbb{I}\;\;\;=\;\;\;\mathbb{1}\,,\,54$ 130 CALL WGETMM(REG3LO(I),REG3LA(I),MM3LO(I),MM3LA(I)) DO 140 I = 1,60 140 CALL WGETMM(REG4LO(I),REG4LA(I),MM4LO(I),MM4LA(I)) DO 150 I $=$ 1,1540 150 CALL WGETMM(REG5LO(I),REG5LA(I),MM5LO(I),MM5LA(I)) DO $\begin{array}{r l r}{160}&{{}\mathbb{I}}&{=}&{1\,,\,80}\end{array}$ 160 CALL WGETMM(REG6LO(I),REG6LA(I),MM6LO(I),MM6LA(I)) DO 170 I = 1,144 170 CALL WGETMM(REG7LO(I),REG7LA(I),MM7LO(I),MM7LA(I)) DO 180 I = 1,245 180 CALL WGETMM(REG8LO(I),REG8LA(I),MM8LO(I),MM8LA(I)) DO 190 I = 1,285 190 CALL WGETMM(REG9LO(I),REG9LA(I),MM9LO(I),MM9LA(I)) IF (TRIAL.EQ..TRUE.) THEN PRINT \*,' Exiting convert' END IF END SUBROUTINE PLOTEM(TRIAL,NREG)LOGICAL TRIAL INTEGER NREG COMMON + +$^+$ + +MM2LA(41),MM2LO(41),MM3LA(54),MM3LO(54),MM4LA(60), MM4LO(60),MM5LA(1540),MM5LO(1540),MM6LA(80),MM6LO(80), $^+$ MM7LA(144),MM7LO(144),MM8LA(245),MM8LO(245),MM9LA(285), + MM9LO(285) DATA DWIDTH/1.0/ IF (TRIAL.EQ..TRUE.) THEN DWIDTH $=\mathrm{~\,~}5\cdot0$ PRINT \*,' Entering Plot points' END IF IF (NREG.EQ.0) THEN KOLOUR $=\textit{2}$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM0LO,MM0LA,378) KOLOUR $=\quad3\quad$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM1LO,MM1LA,206) KOLOUR $\c=4$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM2LO,MM2LA,41)KOLOUR $=~5$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM3LO,MM3LA,54) KOLOUR $=\quad6$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM4LO,MM4LA,60)KOLOUR $=~7$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM5LO,MM5LA,1540)KOLOUR $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM6LO,MM6LA,80)KOLOUR $=\quad24$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM7LO,MM7LA,144)KOLOUR $=\quad23$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM8LO,MM8LA,245)KOLOUR $=\quad22$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM9LO,MM9LA,285) + +ELSE IF (NREG.EQ.1) THEN KOLOUR $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM0LO,MM0LA,378) ELSE IF (NREG.EQ.2) THENKOLOUR $=\textit{2}$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM1LO,MM1LA,206) ELSE IF (NREG.EQ.3) THEN KOLOUR $\begin{array}{r l}{\mathbf{\varpi}=}&{{}\,\mathbb{1}\,2}\end{array}$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM2LO,MM2LA,41)ELSE IF (NREG.EQ.4) THEN KOLOUR $\c=4$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM3LO,MM3LA,54) ELSE IF (NREG.EQ.5) THEN KOLOUR $=~5$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM4LO,MM4LA,60)ELSE IF (NREG.EQ.6) THEN KOLOUR $=\quad6$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM5LO,MM5LA,1540)ELSE IF (NREG.EQ.7) THEN KOLOUR $=~7$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM6LO,MM6LA,80)ELSE IF (NREG.EQ.8) THEN KOLOUR $\begin{array}{r l}{\mathit{\Theta}}&{{}=\quad8}\end{array}$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM7LO,MM7LA,144)ELSE IF (NREG.EQ.9) THEN KOLOUR $\begin{array}{r l}{\mathbf{\chi}=}&{{}\,9\,}\end{array}$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM8LO,MM8LA,245)ELSE IF (NREG.EQ.10) THEN KOLOUR $=\quad\bot0$ CALL GWICOL(DWIDTH,KOLOUR) CALL GDOT(MM9LO,MM9LA,285)END IF IF (TRIAL.EQ..TRUE.) THEN + +PRINT \*,' Exiting Plot points' + +END IF + +END + +Below is the converted program after using the Nag tool suite. The code wraps in places and this is intentional as again we wanted to show you the code as it is without any changes to fit the printed page. + +MODULE mmtsun + +INTEGER :: mm0la(378), mm0lo(378), mm1la(206), mm1lo(206), mm2la(41), & mm2lo(41), mm3la(54), mm3lo(54), mm4la(60), mm4lo(60), mm5la(1540), & mm5lo(1540), mm6la(80), mm6lo(80), mm7la(144), mm7lo(144), mm8la(245), & mm8lo(245), mm9la(285), mm9lo(285) + +END MODULE mmtsun MODULE tsunam + +REAL :: reg0la(378), reg0lo(378), reg1la(206), reg1lo(206), reg2la(41), & reg2lo(41), reg3la(54), reg3lo(54), reg4la(60), reg4lo(60), reg5la(1540), & reg5lo(1540), reg6la(80), reg6lo(80), reg7la(144), reg7lo(144), & reg8la(245), reg8lo(245), reg9la(285), reg9lo(285) + +END MODULE tsunam + +PROGRAM map01 + + ! .. Local Scalars .. REAL :: lat, long INTEGER :: icol, iproj, ires, kolor, nr, nreg LOGICAL :: screen, trial + + ! .. + + ! .. External Subroutines .. EXTERNAL convrt, datain, gclose, gopen, groute, gsegcr, gsurfe, plotem, & wcentr, wdefc, wextnd, wopen, wplot, wproj, wred1, wred2, wred3, wred4 ! .. screen $=$ .FALSE. trial $=$ .FALSE. CALL datain(trial) PRINT \*, ' What resolution map do you want' PRINT $\star$ PRINT \*, ' 1 = 119,650' PRINT \*, ' 2 = 75,500' PRINT \*, ' 3 = 43,100' PRINT \*, ' 4 = 19,300' PRINT \*, ' 5 = 4,420' PRINT \* + +100 READ (unit $=\star$ ,fmt $=\star$ ,end $_{\cdot=2\,0\,0}$ ,err $=\!20\,0$ ) ires 200 IF ((ires ${<}\bot$ ) .OR. (ires ${>}5$ )) THEN PRINT \*, ' Please input a number in the range 1 to 5' GO TO 100 END IF PRINT \*, ' What projection would you like?' PRINT $\star$ PRINT \*, ' 1 = Lambert - equal area - rectangle' PRINT \*, ' $\begin{array}{r l}{_2}&{{}=}\end{array}$ Mercator - equal direction - rectangle' PRINT \*, ' 3 = Hammer - equal area + + - oval' PRINT \*, ' 4 = Bonne - + + - heart' PRINT \*, ' 5 $=$ Orthographic - globe - round' 300 READ (unit $=\star$ ,fmt $=\star$ ,end $\scriptstyle\operatorname{\rho}=\underline{{4}}\,0\,0$ ,err $\scriptstyle{\underline{{\mathbf{\tau}}}}\,\underline{{\mathbf{\tau}}}\,0\,0$ ) iproj 400 IF ((iproj $_{<1}$ ) .OR. (iproj ${>}5$ )) THEN PRINT \*, ' Please input a number in the range 1 to 5' GO TO 300 END IF lat $\begin{array}{r l}{\mathbf{\sigma}=}&{{}0\mathbf{\sigma}.\;0}\end{array}$ $\begin{array}{l l l}{\tt L o n g}&{=}&{\tt L80\,.\,0}\end{array}$ PRINT \*, ' Which region do you wish to plot?' PRINT \*, ' $\begin{array}{r l}{0}&{{}=}\end{array}$ all regions' PRINT \*, ' 1 = Hawaii' PRINT \*, ' $\begin{array}{r l}{_2}&{{}=}\end{array}$ New Zealand and South Pacific Islands' PRINT \*, ' 3 = Papua New Guinea and Solomon Islands' + +PRINT \*, ' $\begin{array}{r l}{\underbar{4}}&{{}=}\end{array}$ Indonesia' PRINT \*, ' 5 = Philippines' PRINT \*, ' 6 = Japan' PRINT \*, ' 7 = Kuril Islands and Kamchatka' PRINT \*, ' $\begin{array}{r l}{8}&{{}=}\end{array}$ Alaska incluing Aleutian Islands' PRINT \*, ' 9 = West Coast - North and Central America' PRINT \*, ' $\begin{array}{r l}{\mathcal{10}}&{{}=}\end{array}$ West Coast - South America' READ (unit $=\star$ ,fmt $=\star$ , $\mathsf{e n d}\!=\!8\,0\,0\,0$ ,err $\scriptstyle=8\,0\,0\,0$ ) nreg 8000 IF ((nreg $\cdot\!<\!0$ ) .OR. ( $.\mathtt{n r e g}{>}10$ )) THEN PRINT \*, ' Please input a number between 0 and 10 inclusive' GO TO 8000 END IF PRINT \*, ' Which colour table do you wish to use' PRINT \*, ' HLS = 1' PRINT \*, ' CMY = 2' PRINT \*, ' RGB $\begin{array}{r l}{\mathit{\Pi}}&{{}=\quad3{\mathit{\Pi}}^{\prime}}\end{array}$ 500 READ (unit $=\star$ ,fmt $=\star$ ,end $\scriptstyle{\underline{{\boldsymbol{\mathbf{\mathit{\tau}}}}}}\,6\,0\,0$ ,err $\mathrm{\Sigma=}600$ ) icol 600 IF (( $\scriptstyle{\dot{\perp}}_{C O{\perp}<1}$ ) .OR. ( $\scriptstyle{\dot{\perp}}_{C O{\perp}>3}$ )) THEN PRINT \*, ' Please input a number in the range 1 to 3' GO TO 500 END IF PRINT \*, ' Select device, a list of valid devices maybe' PRINT \*, ' obtained by typing' PRINT \*, ' list \*' PRINT \*, ' at the GROUTE prompt' CALL groute(' ') CALL gopen CALL gsegcr(1) CALL gsurfe IF (ires $==1$ ) THEN CALL wextnd ELSE IF ( $\mathtt{i}_{\mathtt{T C S}=2}$ ) THEN CALL wred1 ELSE IF (ires $==3$ ) THEN CALL wred2 ELSE IF (ires $==4$ ) THEN CALL wred3 ELSE IF (ires $==5$ ) THEN + +# Chapter 25 + +CALL wred4 END IF $\mathrm{~\tt~{~n~r~}~}=\mathrm{~\tt~{~7~}~}$ $\mathrm{\deltarccos~\dag~}=\mathrm{\deltar~\dag~}0$ CALL wopen(0,nr) CALL wproj(iproj) CALL wcentr(long,lat) CALL wdefc(kolor) CALL wplot(' ',0.0) CALL convrt(trial) CALL plotem(trial,nreg) CALL gsegcr(1) CALL gclose + +END PROGRAM map01 + +SUBROUTINE datain(trial) + +USE tsunam , ONLY : reg0la, reg0lo, reg1la, reg1lo, reg2la, reg2lo, reg3la, & reg3lo, reg4la, reg4lo, reg5la, reg5lo, reg6la, reg6lo, reg7la, reg7lo, & reg8la, reg8lo, reg9la, reg9lo + + ! .. Scalar Arguments .. LOGICAL :: trial + + ! .. + + ! .. Local Scalars .. REAL :: dummy INTEGER :: i CHARACTER (80) :: filnam + + ! .. + + ! .. Arrays in Common .. + + ! .. + + ! .. Common Blocks .. + + ! .. IF (trial) THEN PRINT \*, ' Entering data input phase' END IF filnam $=$ 'tsunami.dat' OPEN (unit $\mathtt{=50}$ ,file $=$ filnam,err $\mathrm{\Sigma=}\,3\,0$ ,status $\mathbf{\varepsilon}=\mathbf{\varepsilon}^{\mathsf{I}}$ 'OLD') GO TO 40 30 PRINT \*, ' Error opening data file' + +PRINT \*, ' Program terminates' + +STOP 40 DO 100 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 378 100 READ (unit $\mathtt{=50}$ ,fmt $\mathbf{\varepsilon}\!=\!\mathbb{1}000$ ) reg0la(i), reg0lo(i) 1000 FORMAT (1X,F7.2,2X,F7.2) DO 110 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 206 110 READ (unit $\mathtt{=50}$ ,fmt $\mathbf{\varepsilon}\!=\!\mathbb{1}000$ ) reg1la(i), reg1lo(i) DO 120 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 41 120 READ (unit $\mathtt{=50}$ ,fmt $\mathsf{\chi}\!=\!\mathsf{1000}$ ) reg2la(i), reg2lo(i) DO 130 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 54 130 READ (unit $\mathtt{=50}$ ,fmt=1000) reg3la(i), reg3lo(i) DO 140 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 60 140 READ (unit $\mathtt{=50}$ ,fm $\underline{{\underline{{\mathbf{\Pi}}}}}\!=\!\underline{{\mathbb{1}}}\,0\,0\,0$ ) reg4la(i), reg4lo(i) DO 150 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 1540 150 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg5la(i), reg5lo(i) DO 160 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 80 160 READ (unit $\mathtt{=50}$ ,fm $\scriptstyle{\mathsf{t}}=1\,0\,0\,0$ ) reg6la(i), reg6lo(i) DO 170 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 144 170 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg7la(i), reg7lo(i) DO 180 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 245 180 READ (unit $\mathtt{=50}$ ,fmt $\mathbf{\zeta}\mathbf{:=}\mathbf{1}\,0\,0\,0$ ) reg8la(i), reg8lo(i) DO 190 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 285 190 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg9la(i), reg9lo(i) IF (trial) THEN DO 200 i $\begin{array}{r l}{\mathbf{\Sigma}=}&{{}\ \mathbb{1}}\end{array}$ , 10 200 PRINT \*, reg0la(i), ' ', reg0lo(i) PRINT \*, ' Exiting data input phase' READ \*, dummy END IF + +END SUBROUTINE datain + +SUBROUTINE convrt(trial) + +USE tsunam, ONLY : reg0la, reg0lo, reg1la, reg1lo, reg2la, reg2lo, reg3la, & reg3lo, reg4la, reg4lo, reg5la, reg5lo, reg6la, reg6lo, reg7la, reg7lo, & reg8la, reg8lo, reg9la, reg9lo USE mmtsun, ONLY : mm0la, mm0lo, mm1la, mm1lo, mm2la, mm2lo, mm3la, mm3lo, & + +# Chapter 25 + +mm4la, mm4lo, mm5la, mm5lo, mm6la, mm6lo, mm7la, mm7lo, mm8la, mm8lo, & mm9la, mm9lo + + ! .. Scalar Arguments .. LOGICAL :: trial + + ! .. + + ! .. Local Scalars .. INTEGER :: i + + ! .. + + ! .. External Subroutines .. EXTERNAL wgetmm + + ! .. + + ! .. Arrays in Common .. + + ! .. + + ! .. Common Blocks .. + + ! .. IF (trial) THEN PRINT \*, ' Entering convert' END IF DO 100 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 378 100 CALL wgetmm(reg0lo(i),reg0la(i),mm0lo(i),mm0la(i)) DO 110 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 206 110 CALL wgetmm(reg1lo(i),reg1la(i),mm1lo(i),mm1la(i)) DO 120 i = 1, 41 120 CALL wgetmm(reg2lo(i),reg2la(i),mm2lo(i),mm2la(i)) DO 130 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 54 130 CALL wgetmm(reg3lo(i),reg3la(i),mm3lo(i),mm3la(i)) DO 140 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 60 140 CALL wgetmm(reg4lo(i),reg4la(i),mm4lo(i),mm4la(i)) DO 150 i = 1, 1540 150 CALL wgetmm(reg5lo(i),reg5la(i),mm5lo(i),mm5la(i)) DO 160 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 80 160 CALL wgetmm(reg6lo(i),reg6la(i),mm6lo(i),mm6la(i)) DO 170 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 144 170 CALL wgetmm(reg7lo(i),reg7la(i),mm7lo(i),mm7la(i)) DO 180 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 245 180 CALL wgetmm(reg8lo(i),reg8la(i),mm8lo(i),mm8la(i)) DO 190 $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 285 190 CALL wgetmm(reg9lo(i),reg9la(i),mm9lo(i),mm9la(i)) IF (trial) THEN PRINT \*, ' Exiting convert' END IF + +END SUBROUTINE convrt + +SUBROUTINE plotem(trial,nreg) + +USE mmtsun, ONLY : mm0la, mm0lo, mm1la, mm1lo, mm2la, mm2lo, mm3la, mm3lo, & mm4la, mm4lo, mm5la, mm5lo, mm6la, mm6lo, mm7la, mm7lo, mm8la, mm8lo, & mm9la, mm9lo + + ! .. Scalar Arguments .. INTEGER :: nreg LOGICAL :: trial + + ! .. + + ! .. Local Scalars .. REAL :: dwidth INTEGER :: kolour + + ! .. + + ! .. External Subroutines .. EXTERNAL gdot, gwicol + + ! .. + + ! .. Arrays in Common .. + + ! .. + + ! .. Common Blocks .. + + ! .. + + ! .. Data Statements .. DATA dwidth/1.0/ + + ! .. IF (trial) THEN dwidth $=\quad5\cdot0$ PRINT \*, ' Entering Plot points' END IF IF ( $\mathtt{n r e g}{=}0$ ) THEN kolour $=~^{2}$ CALL gwicol(dwidth,kolour) CALL gdot(mm0lo,mm0la,378) kolour $=\quad3\quad$ CALL gwicol(dwidth,kolour) CALL gdot(mm1lo,mm1la,206) kolour $\c=4$ CALL gwicol(dwidth,kolour) CALL gdot(mm2lo,mm2la,41) kolour = 5 CALL gwicol(dwidth,kolour) CALL gdot(mm3lo,mm3la,54) kolour $=\quad6$ CALL gwicol(dwidth,kolour) CALL gdot(mm4lo,mm4la,60) kolour $=~7$ CALL gwicol(dwidth,kolour) CALL gdot(mm5lo,mm5la,1540) kolour $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\$ CALL gwicol(dwidth,kolour) CALL gdot(mm6lo,mm6la,80) kolour $=\quad24$ CALL gwicol(dwidth,kolour) CALL gdot(mm7lo,mm7la,144) kolour $=\quad23$ CALL gwicol(dwidth,kolour) CALL gdot(mm8lo,mm8la,245) kolour $=\quad22$ CALL gwicol(dwidth,kolour) CALL gdot(mm9lo,mm9la,285) + +ELSE IF (nreg $==1$ ) THEN kolour $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\$ CALL gwicol(dwidth,kolour) CALL gdot( $\cdot{\sf m m}0\,\mathrm{L}\,\mathrm{O}$ ,mm0la,378) ELSE IF ( $\mathtt{n r e g}{=}{2}$ ) THEN kolour $=\textit{2}$ CALL gwicol(dwidth,kolour) CALL gdot(mm1lo,mm1la,206) ELSE IF ( $\mathtt{n r e g}{=}{3}$ ) THEN kolour $\begin{array}{r l}{\mathbf{\varpi}=}&{{}\,\mathbb{1}\,2}\end{array}$ CALL gwicol(dwidth,kolour) CALL gdot( $\cdot{\mathfrak{m}}2\,\mathbb{1}{\circ}$ ,mm2la,41) ELSE IF (nreg $==4$ ) THEN kolour $\c=4$ CALL gwicol(dwidth,kolour) CALL gdot(mm3lo,mm3la,54) ELSE IF ( $\mathtt{n r e g}{=}{5}$ ) THEN kolour $=~5$ CALL gwicol(dwidth,kolour) CALL gdot( $\cdot\mathsf{m}\mathsf{4}\,\mathsf{L}\,\mathsf{O}$ ,mm4la,60) ELSE IF ( $\mathtt{n r e g}{=}{6}$ ) THEN + +kolour = 6 CALL gwicol(dwidth,kolour) CALL gdot(mm5lo,mm5la,1540) ELSE IF ( $\mathtt{n r e g}{=}{7}$ ) THEN kolour $=~7$ CALL gwicol(dwidth,kolour) CALL gdot( $\mathsf{i}\mathsf{m}6\bot\mathsf{o}$ ,mm6la,80) ELSE IF ( $\mathtt{n r e g}{=}{8}$ ) THEN kolour $\begin{array}{r l}{\mathit{\Theta}}&{{}=\quad8}\end{array}$ CALL gwicol(dwidth,kolour) CALL gdot(mm7lo,mm7la,144) ELSE IF (nreg $==9$ ) THEN kolour $\begin{array}{r l}{\mathbf{\chi}=}&{{}\,9\,}\end{array}$ CALL gwicol(dwidth,kolour) CALL gdot(mm8lo,mm8la,245) ELSE IF (nreg $==\!\!\perp0$ ) THEN kolour $=\quad\bot0$ CALL gwicol(dwidth,kolour) CALL gdot(mm9lo,mm9la,285) END IF IF (trial) THEN PRINT \*, ' Exiting Plot points' END IF + +END SUBROUTINE plotem + +Some of the key points include: + +• Generation of modules from common blocks. • Generation of USE statements with the ONLY option to explicitly specify which variables are going to be made available in a particular subroutine. • Documenting of variable, intrinsic and external usage within a subroutine. • Code restructuring into a well laid out style. + +The Nag tool suite can obviously be used to help maintain code during develop- ment. + +# 25.6.2 Polyhedron + +Their home site is + +•http://www.polyhedron.com/ + +and the conversion of dsort.f is given below. + +# Chapter 25 + +The code wraps in places and this is intentional as again we wanted to show you the code as it is without any changes to fit the printed page. + +! $\mathrm{\star=}$ DSORT.f90 processed by SPAG 6.55Dc at 17:45 on 4 May 2005 + + !\*------------------ SPAG Configuration Options + + -------------------- + + !\*--0233,12 021101,-1 000000100000031111000002000020110201210,72 111 -- !\*--100000000012114110000000000,100,50,20,10 52,99000 12000000000031 -- !\*--99011000000000000,72,72 02,42,38,33 00011012110000100000000-- + +!\*------------------------------------------------------ + + ---------------- + + !DECK DSORT SUBROUTINE DSORT(Dx,Dy,N,Kflag) IMPLICIT NONE + + !\*--DSORT10 + + !\*\*\*BEGIN PROLOGUE DSORT + + !\*\*\*PURPOSE Sort an array and optionally make the same interchanges in ! an auxiliary array. The array may be sorted in increasing ! or decreasing order. A slightly modified QUICKSORT + + ! algorithm is used. + + !\*\*\*LIBRARY SLATEC + + !\*\*\*CATEGORY N6A2B + + !\*\*\*TYPE REAL (SSORT-S, DSORT-D, ISORT-I) + + !\*\*\*KEYWORDS SINGLETON QUICKSORT, SORT, SORTING + + !\*\*\*AUTHOR Jones, R. E., (SNLA) + + ! Wisniewski, J. A., (SNLA) + + !\*\*\*DESCRIPTION + + ! + + ! DSORT sorts array DX and optionally makes the same interchanges in ! array DY. The array DX may be sorted in increasing order or ! decreasing order. A slightly modified quicksort algorithm is used. + + ! + + ! Description of Parameters + +! DX - array of values to be sorted (usually abscissas) + + ! DY - array to be (optionally) carried along + + ! N - number of values in array DX to be sorted + + ! KFLAG - control parameter + + ! $\c=\qquad^{2}$ means sort DX in increasing order and carry DY along. + + ! $\c=\quad\bot$ means sort DX in increasing order + + (ignoring DY) + + ! $\begin{array}{r l}{\mathbf{\omega}=}&{{}-\mathbb{1}}\end{array}$ means sort DX in decreasing order + + (ignoring DY) + + ! $\begin{array}{r l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}{\l}}\end{array}$ means sort DX in decreasing order and carry DY along. + + ! + + !\*\*\*REFERENCES R. C. Singleton, Algorithm 347, An efficient algorithm ! for sorting with minimal storage, Communications of ! the ACM, 12, 3 (1969), pp. 185-187. + + !\*\*\*ROUTINES CALLED XERMSG + + !\*\*\*REVISION HISTORY (YYMMDD) + + ! 761101 DATE WRITTEN + + ! 761118 Modified to use the Singleton quicksort algorithm. (JAW) ! 890531 Changed all specific intrinsics to generic. (WRB) + + ! 890831 Modified array declarations. (WRB) + + ! 891009 Removed unreferenced statement labels. + + (WRB) + + ! 891024 Changed category. (WRB) + + ! 891024 REVISION DATE from Version 3.2 + + ! 891214 Prologue converted to Version 4.0 format. + + (BAB) + + ! 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) ! 901012 Declared all variables; changed X,Y to DX,DY; changed + + ! code to parallel SSORT. (M. McClain) + + ! 920501 Reformatted the REFERENCES section. (DWL, WRB) ! 920519 Clarified error messages. (DWL) + +# Chapter 25 + +! 920801 Declarations section rebuilt and code restructured to use + + !IF-THEN-ELSE-ENDIF.(RWC, WRB) + +!\*\*\*END PROLOGUE DSORT + + ! .. Scalar Arguments .. INTEGER Kflag , N ! .. Array Arguments .. REAL Dx(\*) , Dy(\*) ! .. Local Scalars .. REAL r , t , tt , tty , tyINTEGER i , ij , j , k , kk , l , m , nn ! .. Local Arrays .. INTEGER il(21) , iu(21) + +! .. External Subroutines .. + + ! EXTERNAL XERMSG + + ! .. Intrinsic Functions .. INTRINSIC ABS , INT CALL SB\$ENT('DSORT','D:\document\f2003\examples\ch25\polyhedr on\dsort.f'& & ) + + !\*\*\*FIRST EXECUTABLE STATEMENTDSORTCALL BL\$ENT(65) nn = N IF ( $\mathtt{n n}\!<\!\bot$ ) THEN + + ! CALL XERMSG ('SLATEC', 'DSORT', + + ! + 'The number of values to be sorted is not positive.', 1, 1) CALL BL\$ENT(69) CALL SB\$EXI RETURN ENDIF + + ! CALL BL\$ENT(72) kk $=$ ABS(Kflag) IF ( $\mathtt{k k}/\!=\!\mathtt{1}$ .AND. $\mathtt{k k}/\!=\!2$ ) THEN + + ! CALL XERMSG ('SLATEC', 'DSORT', + + ! $^+$ 'The sort control parameter, K, is not 2, 1, -1, or -2.', 2, ! $^+$ 1) CALL BL\$ENT(77) CALL SB\$EXI RETURN + +ENDIF + + ! + + ! Alter array DX to get decreasing order if needed + + ! CALL BL\$ENT(82) IF ( Kflag $\scriptstyle{\cdot<=-1}$ ) THEN CALL BL\$ENT(83) DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , nn CALL BL\$ENT(84) $\begin{array}{r l r}{\mathbb{D}{\mathbf x}\left(\dot{{\mathbf\Sigma}}\right)}&{{}=}&{-\mathbb{D}{\mathbf x}\left(\begin{array}{c c c}\end{array}\right)}\end{array}$ Dx(i) = -Dx(i) ENDDO ENDIF + + ! CALL BL\$ENT(88) IF ( $\mathrm{kker}\!=\!2$ ) GOTO 900 + + ! + + ! Sort DX only + + ! CALL BL\$ENT(92) $\begin{array}{l l l}{\mathrm{~\mathsf{m}~}=}&{\mathrm{1}}\\ {\dot{\mathrm{~\pmb{\mathscr{1}}~}}=}&{\mathrm{1}}\\ {\dot{\mathrm{~\pmb{\mathscr{1}}~}}=}&{\mathrm{nn}}\\ {\mathrm{~\mathsf{r}~}=}&{0\mathrm{\,.\mathsf{3}}}\end{array}$ $\begin{array}{l l l}{\underline{{\mathbf{\mathit{r}}}}}&{=}&{0\ .\ 37\,5\,\underline{{\mathbf{D}}}0}\end{array} + +$ ! 100 CALL BL\$ENT(97) IF ( $\dot{\Sigma}\!=\!=\!\dot{\mathrm{\Sigma}}$ ) GOTO 500 CALL BL\$ENT(98) IF ( $\mathtt{x}\!<\!=\!0$ .5898437D0 ) THEN CALL BL\$ENT(99) $\texttt{r}=\texttt{r}+\texttt{3.90625D-2}$ ELSE CALL BL\$ENT(101) r = r - 0.21875D0 ENDIF + + ! 200 CALL BL\$ENT(104) $\mathrm{~\texttt~{~k~}~}=\mathrm{~\texttt~{~i~}~} + +$ ! + + ! Select a central element of the array and save it in location T ! + +# Chapter 25 + +$\begin{array}{l}{{\dot{\mathrm{\boldmath~\xi~}}\,{\dot{\mathrm{\boldmath~\Xi~}}}}={\dot{\mathrm{\boldmath~\xi~}}}+{\dot{\mathrm{\boldmath~\Xi~}}}{\mathrm{INT}}\;({\mathrm{\boldmath~(\xi-\dot{\mathrm{\boldmath~\Sigma~}})~}}^{\star}{\mathrm{\boldmath~\mathfrak{r}~}})}\\ {{\mathrm{\boldmath~t~}}={\mathrm{\boldmath~\Sigma~}}({\dot{\mathrm{\boldmath~\xi~}}}{\dot{\mathrm{\boldmath~\mathfrak{j}~}}})}\end{array} + +$ ! + + ! If first element of array is greater than T, interchange with T + + ! IF ( $\mathrm{px}\,(\,\dot{\mathrm{~\!~\!~1~}})>\,\mathsf{t}$ ) THEN CALL BL\$ENT(114) Dx(ij) $=$ Dx(i) $\begin{array}{r l r}{\mathbb{D}\mathbf{x}\left(\,\dot{\mathbb{L}}\,\right)}&{{}=}&{\mathbf{t}}\end{array}$ $\begin{array}{r c l}{\mathsf{t}}&{=}&{\mathsf{D}\mathsf{x}\;(\,.}\end{array}$ ij) ENDIF CALL BL\$ENT(118) $\begin{array}{r l r}{\perp}&{{}=}&{\vec{\jmath}}\end{array} + +$ ! + + ! If last element of array is less than than T, interchange with T + + ! IF ( $\mathrm{Dx}\,(\,\dot{\,\textmd}\,)<\,\mathsf{t}$ ) THEN CALL BL\$ENT(123) Dx(ij) $=$ Dx(j) $\mathrm{Dx\(\xi)}\quad=\quad\mathsf{t}$ $\mathrm{~\textit~{~t~}~}=\mathrm{~\textit~{~D~x~}~}$ (ij) + + ! + + ! If first element of array is greater than T, interchange with T + + ! IF ( $\mathrm{px}\,(\,\dot{\mathrm{~\!~\!~1~}})>\,\mathsf{t}$ ) THEN CALL BL\$ENT(130) Dx(ij) $=$ Dx(i) $\begin{array}{r l r}{\mathbb{D}\mathbf{x}\left(\,\dot{\mathbb{L}}\,\right)}&{{}=}&{\mathbf{t}}\end{array}$ $\begin{array}{l l l}{\mathsf{t}}&{=}&{\mathsf{D x}\left(}\end{array}$ ij) ENDIF ENDIF + + ! + + ! Find an element in the second half of the array which is smaller + + ! than T + + ! 300 CALL BL\$ENT(139) $\texttt{l\_l}=\texttt{l\_l}-\texttt{\_l}$ IF ( $\mathrm{Dx}\left(\perp\right)>\mathtt{t}$ ) GOTO 300 + +! + + ! Find an element in the first half of the array which is greater + + ! than T + + ! 400 CALL BL\$ENT(145) $\texttt{k}=\texttt{k}+\texttt{1}$ IF ( $\mathtt{D x}\left(\mathtt{k}\right){<}\mathtt{t}$ ) GOTO 400 + + ! + + ! Interchange these elements + + ! CALL BL\$ENT(150) IF ( $\Bbbk<<\perp$ ) THEN CALL BL\$ENT(151) tt $=$ Dx(l) $\begin{array}{r l r}{\mathbb{D}{\mathbf{x}}\left(\,\mathbb{L}\,\right)}&{{}=}&{\mathbb{D}{\mathbf{x}}\left(\,\mathbb{k}\,\right)}\end{array}$ Dx(k) $=$ tt GOTO 300 ENDIF + + ! + + ! Save upper and lower subscripts of the array yet to be sorted + + ! CALL BL\$ENT(159) IF ( $\ensuremath{\mathbb{1}}-\dot{\ensuremath{\mathbb{1}}}\!>\!\dot{\ensuremath{\mathbb{1}}}-\ensuremath{\mathrm{k}}$ ) THEN CALL BL\$ENT(160) $\begin{array}{r l r}{\dot{\mathrm{~\textcent~}}\perp\mathrm{~(m~)~}}&{{}=}&{\dot{\mathrm{~\textcent~}}}\end{array}$ $\begin{array}{r l r}{\dot{\mathrm{\boldmath~\tau~}}\mathfrak{l}\mathfrak{u}\left(\mathfrak{m}\right)}&{{}=}&{\mathbb{1}}\end{array}$ $\mathrm{~\ensuremath~{~\dot{~}{~\varepsilon~}~}~}=\mathrm{~\ensuremath~{~\dot{~}{~\varepsilon~}~}~}$ m = m + 1 ELSE CALL BL\$ENT(165) $\begin{array}{r l r}{\dot{\mathrm{\boldmath~\Omega~}}_{\mathrm{\tiny~\left(m~\right)~}}}&{{}=}&{\mathrm{\boldmath~k~}}\end{array}$ $\begin{array}{r l r}{\dot{\mathrm{\boldmath~\tau~}}\mathrm{\boldmath~\dot{\Omega}~}(\mathrm{\boldmath~m~})}&{{}=}&{\dot{\mathrm{\boldmath~\omega~}}}\end{array}$ $\begin{array}{r l r}{\dot{\quad}\dot{\quad}}&{{}=}&{\mathbb{1}}\end{array}$ $\texttt{m}=\texttt{m}+\texttt{1}$ ENDIF CALL BL\$ENT(170) GOTO 600 + + ! + + ! Begin again on another portion of the unsorted array + +# Chapter 25 + +! 500 CALL BL\$ENT(174) $\textrm{\textbf{m}}=\textrm{\textbf{m}}$ - 1 IF ( $\scriptstyle{\mathfrak{m}}==0$ ) GOTO 1800 CALL BL\$ENT(176) $\begin{array}{r l}{\dot{\bot}}&{{}=}\end{array}$ il(m) j $=$ iu(m) + + ! 600 CALL BL\$ENT(179) IF ( $\dot{\daleth}-\dot{\Sigma}\!>=\!\beth$ ) GOTO 200 CALL BL\$ENT(180) IF ( $\scriptstyle{\dot{\bot}}\,=\,=\,\bot$ ) GOTO 100 CALL BL\$ENT(181) i = i - 1 + + ! 700 CALL BL\$ENT(183) $\mathrm{~\normalfont~{\dot{~}}~}=\mathrm{~\normalfont~{~\dot{~}}~}+\mathrm{~\normalfont~{~\dot{~}}~}1$ IF ( $\dot{\Sigma}\,{=}\,{\dot{\jmath}}$ ) GOTO 500 CALL BL\$ENT(185) t = $\mathbb{D}\mathbf{x}\left(\,\dot{\bot}\!+\!\bot\,\right)$ IF ( $\mathtt{D x}\left(\,\mathtt{i}\,\right)<=\mathtt{t}$ ) GOTO 700 CALL BL\$ENT(187) $\mathrm{~\texttt~{~k~}~}=\mathrm{~\texttt~{~i~}~} + +$ ! 800 CALL BL\$ENT(189) $\begin{array}{r l r}{\mathrm{D}{\bf x}\left(\,\mathrm{\boldmath~k+1~}\right)}&{{}=}&{\mathrm{D}{\bf x}\left(\,\mathrm{\boldmath~k~}\right)}\end{array}$ $\texttt{k}=\texttt{k}-\texttt{1}$ IF ( $\sf t\!<\!D x~(~k~)$ ) GOTO 800 CALL BL\$ENT(192) $\begin{array}{r l r}{\mathrm{D}\mathbf{x}\left(\,\mathbf{k}{+}\,\mathbb{1}\,\right)}&{{}=}&{\mathbf{t}}\end{array}$ GOTO 700 + + ! + + ! Sort DX and carry DY along + + ! 900 CALL BL\$ENT(197) $\begin{array}{r l r}{\mathrm{~\boldmath~m~}=}&{{}\mathbb{1}}\\ {\dot{\mathrm{\boldmath~\cal~\Sigma~}}=}&{{}\mathbb{1}}\\ {\dot{\mathrm{\boldmath~\cal~\Sigma~}}=}&{{}\mathrm{nn}}\\ {\mathrm{\boldmath~\cal~r~}=}&{{}0\cdot375\mathbb{D}0}\end{array} + +$ ! 1000 CALL BL\$ENT(202) IF ( $\dot{\Sigma}\,{=}\,{\dot{\jmath}}$ ) GOTO 1400 + +CALL BL\$ENT(203) IF ( $\mathtt{x}\!<\!=\!0$ .5898437D0 ) THEN CALL BL\$ENT(204) $\texttt{r}=\texttt{r}+\texttt{3.90625D-2}$ ELSE CALL BL\$ENT(206) r = r - 0.21875D0 ENDIF + + ! 1100 CALL BL\$ENT(209) $\mathrm{~\texttt~{~k~}~}=\mathrm{~\texttt~{~i~}~} + +$ ! + + ! Select a central element of the array and save it in location T + + ! ij = i + INT((j-i)\*r) $\begin{array}{r l r}{{\sf t}}&{{}=}&{\mathrm{Dx}}\end{array}$ (ij) ty $=$ Dy(ij) + + ! + + ! If first element of array is greater than T, interchange with T + + ! IF ( $\mathrm{px}\,(\,\dot{\mathrm{~\!~\!~1~}})>\,\mathsf{t}$ ) THEN CALL BL\$ENT(220)Dx(ij) $=$ Dx(i) $\begin{array}{r l r}{\mathbb{D}\mathbf{x}\left(\,\dot{\mathbb{L}}\,\right)}&{{}=}&{\mathbf{t}}\end{array}$ $\begin{array}{r l r}{{\sf t}}&{{}=}&{\mathrm{Dx}}\end{array}$ (ij) $\begin{array}{r l r}{\mathbb{D}\mathbb{y}\left(\,\dot{\mathbb{1}}\,\dot{\mathbb{\sigma}}\,\right)}&{{}=}&{\mathbb{D}\mathbb{y}\left(\,\dot{\mathbb{1}}\,\right)}\end{array}$ $\begin{array}{r l r}{\mathbb{D}\mathbb{y}\left(\dot{\perp}\right)}&{{}=}&{\mathbb{t}\mathbb{y}}\end{array}$ ty $=$ Dy(ij) ENDIF CALL BL\$ENT(227) $\begin{array}{r l r}{\perp}&{{}=}&{\vec{\jmath}}\end{array} + +$ ! + + ! If last element of array is less than T, interchange with T ! IF ( $\mathrm{Dx}\,(\,\dot{\,\textmd}\,)<\,\mathsf{t}$ ) THEN CALL BL\$ENT(232)Dx(ij) $=$ Dx(j) Dx(j) $=$ t $\begin{array}{r l r}{{\sf t}}&{{}=}&{\mathrm{Dx}}\end{array}$ (ij) + +# Chapter 25 + +$\begin{array}{r l r}{\mathbb{D}\mathbb{y}\left(\stackrel{.}{\supset}\right)}&{{}=}&{\mathbb{t}\mathbb{y}}\end{array}$ $\begin{array}{l l l}{{\tt t y}}&{{=}}&{{\tt D y}\,(\,\dot{\tt L}\,\dot{\tt J}\,)}\end{array} + +$ ! + + ! If first element of array is greater than T, interchange with T + + ! IF ( $\mathrm{px}\,(\,\dot{\mathrm{~\!~\!~1~}})>\,\mathsf{t}$ ) THEN CALL BL\$ENT(242)Dx(ij) $=$ Dx(i) $\begin{array}{r l r}{\mathbb{D}\mathbf{x}\left(\,\dot{\mathbb{L}}\,\right)}&{{}=}&{\mathbf{t}}\end{array}$ $\begin{array}{r l r}{{\sf t}}&{{}=}&{\mathrm{Dx}}\end{array}$ (ij) $\begin{array}{r l r}{\mathbb{D}\mathbb{y}\left(\,\dot{\mathbb{1}}\,\dot{\mathbb{\sigma}}\,\right)}&{{}=}&{\mathbb{D}\mathbb{y}\left(\,\dot{\mathbb{1}}\,\right)}\end{array}$ $\begin{array}{r l r}{\mathbb{D}\mathbb{y}\left(\dot{\perp}\right)}&{{}=}&{\mathbb{t}\mathbb{y}}\end{array}$ ty $=$ Dy(ij) ENDIF ENDIF + + ! + + ! Find an element in the second half of the array which is smaller + + ! than T + + ! 1200 CALL BL\$ENT(254) $\texttt{l\_l}=\texttt{l\_l}-\texttt{\_l}$ IF ( $\mathrm{Dx}\left(\perp\right)>\mathtt{t}$ ) GOTO 1200 + + ! + + ! Find an element in the first half of the array which is greater + + ! than T + + ! 1300 CALL BL\$ENT(260) $\texttt{k}=\texttt{k}+\texttt{1}$ IF ( $\mathtt{D x}\left(\mathtt{k}\right){<}\mathtt{t}$ ) GOTO 1300 + + ! + + ! Interchange these elements + + ! CALL BL\$ENT(265) IF ( $\Bbbk<<\perp$ ) THEN CALL BL\$ENT(266) tt $=$ Dx(l) Dx(l) = Dx(k) $\begin{array}{r l r}{\mathrm{D}\mathbf{x}\left(\mathbf{\vec{k}}\right)}&{{}=}&{\mathbf{t}\,\mathbf{t}}\end{array}$ + +$$ +\begin{array}{r l r}{\mathrm{\sfDy}\left(\,\right]}&{{}=}&{\mathrm{\sfDy}\left(\,\mathrm{\boldmath~k~}\right)}\end{array} +$$ + +$$ +\begin{array}{r l r}{\mathrm{Dy}\left(\mathrm{k}\right)}&{{}=}&{\mathsf{t}\,\mathsf{t}\,\mathsf{y}}\end{array} +$$ + +GOTO 1200 ENDIF + + ! + + ! Save upper and lower subscripts of the array yet to be sorted + + ! CALL BL\$ENT(277) IF ( $\ensuremath{\mathbb{1}}-\dot{\ensuremath{\mathbb{1}}}\!>\!\dot{\ensuremath{\mathbb{1}}}-\ensuremath{\mathrm{k}}$ ) THEN CALL BL\$ENT(278) $\begin{array}{r l r}{\dot{\mathrm{~\textcent~}}\perp\mathrm{~(m~)~}}&{{}=}&{\dot{\mathrm{~\textcent~}}}\end{array}$ $\begin{array}{r l r}{\mathfrak{l}\mathfrak{u}\left(\mathfrak{m}\right)}&{{}=}&{\mathbb{1}}\end{array}$ $\mathrm{~\ensuremath~{~\dot{~}{~\varepsilon~}~}~}=\mathrm{~\ensuremath~{~\dot{~}{~\varepsilon~}~}~}$ $\texttt{m}=\texttt{m}+\texttt{1}$ ELSE CALL BL\$ENT(283) $\begin{array}{r l r}{\dot{\mathrm{\boldmath~\Omega~}}_{\mathrm{\tiny~\left(m~\right)~}}}&{{}=}&{\mathrm{\boldmath~k~}}\end{array}$ $\begin{array}{r l r}{\dot{\mathrm{\boldmath~\tau~}}\mathrm{\boldmath~\dot{\Omega}~}(\mathrm{\boldmath~m~})}&{{}=}&{\dot{\mathrm{\boldmath~\omega~}}}\end{array}$ $\begin{array}{r l r}{\dot{\quad}\dot{\quad}}&{{}=}&{\mathbb{1}}\end{array}$ $\texttt{m}=\texttt{m}+\texttt{1}$ ENDIF CALL BL\$ENT(288) GOTO 1500 + + ! + + ! Begin again on another portion of the unsorted array + + ! 1400 CALL BL\$ENT(292) $\textrm{\textbf{m}}=\textrm{\textbf{m}}$ - 1 IF ( $\scriptstyle{\mathfrak{m}}==0$ ) GOTO 1800 CALL BL\$ENT(294) $\begin{array}{r l}{\dot{\bot}}&{{}=}\end{array}$ il(m) j $=$ iu(m) + + ! 1500 CALL BL\$ENT(297) IF ( $\dot{\daleth}-\dot{\Sigma}\!>=\!\beth$ ) GOTO 1100 CALL BL\$ENT(298) IF ( $\scriptstyle{\dot{\bot}}\,=\,=\,\bot$ ) GOTO 1000 CALL BL\$ENT(299) i = i - 1 + +# Chapter 25 + +! 1600 CALL BL\$ENT(301) $\mathrm{~\normalfont~{\dot{~}}~}=\mathrm{~\normalfont~{~\dot{~}}~}+\mathrm{~\normalfont~{~\dot{~}}~}1$ IF ( $\dot{\Sigma}\,{=}\,{\dot{\jmath}}$ ) GOTO 1400 CALL BL\$ENT(303) t = $\mathbb{D}\mathbf{x}\left(\,\dot{\bot}\!+\!\bot\,\right)$ ty = $\mathrm{Dy}\,(\,\dot{\bot}\,+\,\bot\,)$ IF ( $\mathtt{D x}\left(\,\mathtt{i}\,\right)<=\mathtt{t}$ ) GOTO 1600 CALL BL\$ENT(306) $\mathrm{~\texttt~{~k~}~}=\mathrm{~\texttt~{~i~}~} + +$ ! 1700 CALL BL\$ENT(308) $\begin{array}{r l r}{\mathrm{D}{\bf x}\left(\,\mathrm{\boldmath~k+1~}\right)}&{{}=}&{\mathrm{D}{\bf x}\left(\,\mathrm{\boldmath~k~}\right)}\end{array}$ $\begin{array}{r l r}{\mathbb{D}\mathbb{y}\left(\mathbb{k}\!+\!\mathbb{1}\right)}&{{}=}&{\mathbb{D}\mathbb{y}\left(\mathbb{k}\right)}\end{array}$ $\texttt{k}=\texttt{k}-\texttt{1}$ IF ( $\sf t\!<\!D x~(~k~)$ ) GOTO 1700 CALL BL\$ENT(312) Dx(k+1) = t $\begin{array}{r l r}{\mathrm{Dy}\left(\,\mathrm{k}{+}\,\mathrm{\mathbb{1}}\,\right)}&{{}=}&{\mathrm{tby}}\end{array}$ GOTO 1600 + + ! + + ! Clean up + + ! 1800 CALL BL\$ENT(318) IF ( Kflag $<=-1$ ) THEN CALL BL\$ENT(319) DO i $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ , nn CALL BL\$ENT(320) $\begin{array}{r l r}{\mathbb{D}{\mathbf x}\left(\,\dot{\mathbf{\rho}}_{}\,\right)}&{{}=}&{-\mathbb{D}{\mathbf x}\left(\,\dot{\mathbf{\rho}}_{}\,\right)}\end{array}$ ENDDO ENDIF CALL BL\$ENT(323) CONTINUE CALL SB\$EXI END SUBROUTINE DSORT + +Changing the subroutine using an editor would obviously be much more tedious and error prone. + +Below is an example from their site that looks at the same subroutine in Fortran 66, 77 and 90 styles. + +# 25.6.3 Original Fortran 66 + +This subroutine picks off digits from an integer and branches depending on their value. + +SUBROUTINE OBACT(TODO) INTEGER TODO,DONE,IP,BASE COMMON /EG1/N,L,DONE PARAMETER ( $\mathtt{B A S E}{=}10$ ) 13 IF(TODO.EQ.0) GO TO 12 $\mathbb{T}=$ MOD(TODO,BASE) TODO $=$ TODO/BASE GO TO(62,42,43,62,404,45,62,62,62),I GO TO 13 42 CALL COPY GO TO 127 43 CALL MOVE GO TO 144 404 $\scriptstyle{\mathrm{N}}=-{\mathrm{N}}$ 44 CALL DELETE GO TO 127 45 CALL PRINT GO TO 144 62 CALL BADACT(I) GO TO 12 127 $\tt L{=}{\tt L{+}N}$ 144 DONE $=$ DONE+1 CALL RESYNC GO TO 13 12 RETURN END + +# 25.6.4 Fortran 77 Version + +In addition to restructuring, SPAG has renamed some variables, removed the un- used variable IP, inserted declarations, and used upper and lower case to distinguish different types of variable: + +SUBROUTINE OBACT(Todo) IMPLICIT NONE C\*\*\* Start of declarations inserted by SPAG INTEGER act , LENgth , NCHar C\*\*\* End of declarations inserted by SPAG INTEGER Todo , DONe , BASE COMMON /EG1 / NCHar , LENgth , DONe + +PARAMETER (BASE $=\!\perp0$ ) 100 IF ( Todo.NE.0 ) THEN act $=$ MOD(Todo,BASE) Todo $=$ Todo/BASE IF ( act.EQ.1 .OR. act.EQ.4 .OR. & act.EQ.7 .OR. act.EQ.8 .OR. & act.EQ.9 ) THEN CALL BADACT(act) GOTO 200 ELSEIF ( act.EQ.2 ) THEN CALL COPY LENgth $=$ LENgth $^+$ NCHar ELSEIF ( act.EQ.3 ) THEN CALL MOVE ELSEIF ( act.EQ.5 ) THEN NCHar $=$ -NCHar CALL DELETE LENgth $=$ LENgth $^+$ NCHar ELSEIF ( act.EQ.6 ) THEN CALL PRINT ELSE GOTO 100 ENDIF DONe $=$ DONe + 1 CALL RESYNC GOTO 100 ENDIF 200RETURNEND + +# 25.6.5 Fortran 90 Version + +SPAG has used DO WHILE, SELECT CASE, EXIT and CYCLE. No GOTOs or labels remain. + +SUBROUTINE OBACT(Todo) IMPLICIT NONE C\*\*\* Start of declarations inserted by SPAG INTEGER act , LENgth , NCHar C\*\*\* End of declarations inserted by SPAG INTEGER Todo , DONe , BASE COMMON /EG1 / NCHar , LENgth , DONe PARAMETER ( $\mathtt{B A S E}{=}10$ ) DO WHILE ( Todo.NE.0 ) + +act $=$ MOD(Todo,BASE) Todo $=$ Todo/BASE SELECT CASE (act) CASE (1,4,7,8,9) CALL BADACT(act) EXIT CASE (2) CALL COPY LENgth $=$ LENgth $^+$ NCHar CASE (3) CALL MOVE CASE (5) NCHar $=$ -NCHar CALL DELETE LENgth $=$ LENgth $^+$ NCHar CASE (6) CALL PRINT CASE DEFAULT CYCLE END SELECT DONe $=$ DONe + 1 CALL RESYNC ENDDO RETURN END + +This tool suite can also be used in the maintenance of code during development. + +# 25.7 Summary + +This chapter has shown some of the options open to you when working with leg- acy code. The emphasis has been on relatively straightforward code restructuring. The use of software tools to aid in this is highly recommended as converting man- ually using an editor is obviously going to involve much more work. + +In Chapter 26 we will look at an example that involves a major rewrite using user defined data types. + +# 25.8 Problems + +1. Try out example 1 with your compiler. What compiler and linker options did you need? + +2. Get hold of the Metcalf conversion program and try example 2. What compiler options did you need? + +# Case Studies + +“The good teacher is a guide who helps others to dispense with his services.” + +R. S. Peters, Ethics and Education + +# Aims + +The aims of this chapter are to look at several complete examples highlighting a variety of aspects of the use of Fortran 95: + +• Using linked lists for sparse matrix problems. • The solution of a set of ordinary differential equations using the Runga–Kutta–Merson method, with the use of a procedure as a parameter, and the use of work arrays. • Generic procedures. • A function that returns a variable length array. • Operator and assignment overloading. • Diagonal extraction of a matrix. • Modules and packaging. • Pure and elemental functions. • Elemental subroutines. + +# Chapter 26 + +# 26 Case Studies + +This chapter looks at more realistic case studies of the use of Fortran 95 and its new features. There are examples of: + +• Using linked lists for sparse matrix problems. • The solution of a set of ordinary differential equations using the Runga–Kutta–Merson method, with the use of a procedure as a parameter, and the use of work arrays. • The construction of generic procedures in Fortran 95. Many of the internal functions will take arguments of a variety of data types and return a result of the same type e.g., SINE will take an integer argument, real argument of whatever precision, complex argument and return the appropriate result. • Operator and assignment overloading and the use of a MODULE PROCE- DURE. • The extraction of the diagonal elements of a matrix. • Pure and elemental functions. • Elemental subroutines. + +The examples have been chosen to highlight the better features of Fortran 95 and what is possible with a modern language. + +# 26.1 Using linked lists for sparse matrix problems + +A matrix is said to be sparse if many of its elements are zero. Mathematical mod- els in areas such as management science, power systems analysis, circuit theory and structural analysis consist of very large sparse systems of linear equations. It is not possible to solve these systems with classical methods because the sparsity would be lost and the eventual system would become too large to solve. Many of these systems consist of tens of thousands, hundreds of thousands and millions of equations. As computer systems become ever more powerful with massive amounts of memory the solution of even larger problems becomes feasible. + +Direct Methods for Sparse Matrices , by Duff I.S., Erismon A.M. and Reid J.K., looks at direct methods for solving sparse systems of linear equations. + +Sparse matrix techniques lend themselves to the use of dynamic data structures in Fortran 95. Only the nonzero elements of a sparse matrix need be stored, together with their positions in the matrix. Other information also needs to be stored so that row or column manipulation can be performed without repeated scanning of a po- tentially very large data structure. Sparse methods may involve introducing some new nonzero elements, and a way is needed of inserting them into the data struc- ture. This is where the Fortran 95 pointer construct can be used. The sparse matrix can be implemented using a linked list to which entries can be easily added and from which they can be easily deleted. + +As a simple introduction, consider the storage of sparse vectors. What we learn here can easily be applied to sparse matrices, which can be thought of as sets of sparse vectors. + +# 26.1.1 Inner product of two sparse vectors + +Assume that we have two sparse vectors $x$ and $y$ , for example: + +$$ +\underline{{x}}=\left[\begin{array}{l}{3}\\ {0}\\ {5}\\ {0}\\ {0}\\ {4}\end{array}\right]\qquad\quad\underline{{y}}=\left[\begin{array}{l}{0}\\ {1}\\ {3}\\ {0}\\ {2}\\ {1}\end{array}\right] +$$ + +and we wish to calculate the inner product $x^{\mathit{T}}\;{\mathit{y}}\equiv\sum_{i\;=\;1}^{n}\;x_{i}\;{\mathit{y}}_{i}$ . There are a number of approaches to doing this and the one we use in the program below stores them as two linked lists. Only the nonzero elements are stored (together with their indi- ces): + +x data file y data file 3 1 1 2 5 3 3 3 4 6 2 5 1 6 + +PROGRAM ch2601 + +! + + ! This program reads the non-zero elements of + + ! two sparse vectors x and y together with their + + ! indices, and stores them in two linked lists. + + ! Using these linked lists it then calculates + + ! and prints out the inner product. + + ! It also prints the values. + + ! + + ! updated 21/3/00 to initialise pointers to + + ! be disassociated using intrinsic function NULL + + ! plus minor updates + + ! + +IMPLICIT NONE CHARACTER (LEN $\dot{=}30$ ):: Filename TYPE sparse_vector INTEGER :: index REAL:: value TYPE (sparse_vector), POINTER ::next $=>$ NULL ( ) END TYPE sparse_vector TYPE(sparse_vector), POINTER :: Root_x,Current_x, & Root_y,Current_y REAL :: Inner_prod $\scriptstyle{.=0}$ .0 INTEGER::IO_status + +! + + ! Read non-zero elements of vector x together + + ! with indices into a linked list + + ! PRINT \*,'input file name for vector x' READ '(A)',Filename OPEN(UNIT $\vDash\vDash$ , FILE $=$ Filename , STATUS $=$ 'OLD' & , IOSTAT $=$ IO_status) IF(IO_status $/=~\,\,0$ )THEN PRINT\*,'Error opening file ',Filename STOP ENDIF ALLOCATE(Root_x) READ (UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FT}}={\star}$ , IOSTAT $^{\ast}{=}$ IO_status) & Root_x%value,Root_x%index IF(IO_status $/=~\,\,0$ ) THEN PRINT\*,' Error when reading from file ' & , Filename, ' or file empty' STOP ENDIF + + ! + + ! Read data for vector x from file until eof + +CYCLE ELSEIF(IO_status $>$ 0 )THEN + +! + + ! Error on reading + + ! PRINT \* ,'Error occurred when reading from ' & , Filename STOP ELSE + + ! + + ! End of file + + ! NULLIFY (Current_x%next) END IF END DO + +! + + ! Read non-zero elements of vector y together + + ! with indices into a linked list + + ! PRINT \*,'input file name for vector y' READ '(A)',Filename OPEN(UNIT $\vDash\vDash$ , FILE $=$ Filename , STATUS $=$ 'OLD' & , IOSTAT $^1\!=$ IO_status) IF(IO_status $/=~\,0$ )THEN PRINT\*,'Error opening file ',Filename STOP ENDIF ALLOCATE(Root_y) READ (UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FMT}=\star}$ , IOSTAT $^{\ast}{=}$ IO_status) & Root_y%value,Root_y%index IF(IO_status $/=~\,0$ ) THEN PRINT\*,' Error when reading from ' & , Filename, ' or file empty' STOP ENDIF + + ! + + ! Read data for vector y from file until eof + + ! Current_y $=>$ Root_y ALLOCATE(Current_y%next) + +DO WHILE(ASSOCIATED(Current_y%next)) Current_y $=>$ Current_y%next READ (UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FT}}={\star}$ , IOSTAT $^{\ast}{=}$ IO_status) & Current_y%value, Current_y%index IF(IO_status $==\quad0$ )THEN ALLOCATE(Current_y%next) CYCLE ELSEIF(IO_status $>$ 0 )THEN + + ! + + ! Error on reading + + ! PRINT \* , 'Error occurred when reading from' & , Filename STOP ELSE + + ! + + ! End of file + + ! NULLIFY (Current_y%next) END IF END DO + + ! + + ! Data has now been read and stored in 2 linked lists + + ! start at the beginning of x linked list and + + ! y linked list and compare indices + + ! in order to perform inner product + + ! Current_x $=>$ Root_x Current_y $=>$ Root_y DO WHILE (ASSOCIATED(Current_x%next)) DO WHILE (Associated(Current_y%next) & .AND. Current_y%index $<$ Current_x%index) + + ! + + ! move through 2nd list + + ! Current_y $=>$ Current_y%next END DO + + ! + + ! At this point Current_y%index $>=$ Current_x%index + + ! or 2nd list is exhausted + + ! IF (Current_y%index $=$ Current_x%index) THEN + +Inner_prod $=$ Inner_prod & $^+$ Current_x%value \* Current_y%value END IF Current_x $=>$ Current_x%next END DO + + ! + + ! Print out inner product + + ! PRINT \*,'Inner product of two sparse vectors is :' & , Inner_prod + + ! + + ! Print non-zero values of vector x and indices + + ! PRINT\*, 'non-zero values of vector x and indices:' Current_x $=>$ Root_x DO WHILE (ASSOCIATED(Current_x%next)) PRINT\*,Current_x%value,Current_x%index Current_x $=>$ Current_x%next END DO + + ! + + ! Print non-zero values of vector y and indices + + ! PRINT\*, 'non-zero values of vector y and indices:' Current_y $=>$ Root_y DO WHILE (ASSOCIATED(Current_y%next)) PRINT\*,Current_y%value,Current_y%index Current_y $=>$ Current_y%next END DO ! + +# 26.2 Solving a system of first-order ordinary differential equations using Runga–Kutta–Merson + +Simulation and mathematical modelling of a wide range of physical processes of- ten leads to a system of ordinary differential equations to be solved. Such equations also occur when approximate techniques are applied to more complex problems. We will restrict ourselves to a class of ordinary differential equations called initial value problems. These are systems for which all conditions are given at the same value of the independent variable. We will further restrict ourselves to first-order initial value problems of the form: + +$$ +\begin{array}{l}{\displaystyle\frac{d y_{1}}{d t}=f_{1}(\underline{{y}},t)}\\ {\displaystyle\frac{d y_{2}}{d t}=f_{2}(\underline{{y}},t)}\\ {\displaystyle\cdots}\\ {\displaystyle\frac{d y_{n}}{d t}=f_{n}(\underline{{y}},t)}\end{array} +$$ + +or + +$$ +\underline{{\dot{y}}}=\underline{{f}}\mathrm{~\left(~}\underline{{y}}\mathrm{~,~}t\mathrm{~\right)~} +$$ + +with initial conditions + +$$ +\underline{{y}}(t0)=\underline{{y}}0 +$$ + +where + +$$ +\underline{{\boldsymbol{y}}}=\left(\begin{array}{c}{y_{1}}\\ {\vdots}\\ {y_{n}}\end{array}\right)\qquad\underline{{\boldsymbol{f}}}=\left(\begin{array}{c}{f_{1}}\\ {\vdots}\\ {f_{n}}\end{array}\right)\qquad\underline{{\boldsymbol{y}}}_{0}=\left(\begin{array}{c}{y_{1}(t_{0})}\\ {\vdots}\\ {y_{n}(t_{0})}\end{array}\right) +$$ + +If we have a system of ordinary differential equations of higher order then they can be reformulated to a system of order one. See the NAG library documentation for solving ordinary differential equations. + +One well-known class of methods for solving initial value ordinary differential equations is Runge–Kutta. In this example we have coded the Runga–Kutta–Merson algorithm, which is a fourth-order method and solves (1) from a point $t=A$ to a point $t=B$ . + +It starts with a step length $h=\left(B-A\right)\;\;\;\;100$ and includes a local error control strategy such that the solution at $_{t+h}$ is accepted if: + + error estimate   user defined tolerance . + +If this isn't satisfied the step length $h$ is halved and the solution attempt is repeated until the above is satisfied or the step length is too small and the problem is left unsolved. If the error criterion is satisfied the algorithm progresses with a suitable step length solving the equations at intermediate points until the end point $B$ is reached. For a full discussion of the algorithm and the error control mechanism used see Numerical Methods in Practice by Tim Hopkins and Chris Phillips: + +MODULE Precisions INTEGER,PARAMETER:: Long $^{\prime}{=}$ SELECTED_REAL_KIND(15,307)END MODULE Precisions + +SUBROUTINE Run ge Kut ta Mers on(Y,FUN,IFAIL,N,A,B,Tol) + + ! + + ! Runge-Kutta-Merson method for the solution + + ! of a system of N + + ! 1st order initial value ordinary + + ! differential equations. + + ! The routine tries to integrate from $\scriptstyle{\mathrm{T}}={\mathrm{\bar{A}}}$ to $\scriptstyle{\mathrm{T}}={\mathrm{B}}$ with + + ! initial conditions in Y, subject to the + + ! condition that the + + ! Absolute Error Estimate $<=\ \ \mathrm{T}\circ\mathbb{L}$ . The step length is + + ! adjusted automatically to meet this condition. + + ! If the routine is successful it returns with + + ! IFAIL $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\$ , $\scriptstyle{\mathrm{T}}={\mathrm{B}}$ and + + ! the solution in Y. + + ! USE Precisions ! IMPLICIT NONE + + ! Define arguments + + ! REAL (Long),INTENT(INOUT):: Y(:) REAL(Long), INTENT(IN)::A,B,Tol INTEGER,INTENT(IN)::N INTEGER,INTENT(OUT)::IFAIL ! INTERFACE SUBROUTINE FUN(T,Y,F,N) USE Precisions IMPLICIT NONE REAL(Long),INTENT(IN),DIMENSION(:)::Y REAL(Long),INTENT(OUT),DIMENSION(:)::F REAL(Long),INTENT(IN)::T INTEGER,INTENT(IN)::N END SUBROUTINE FUN END INTERFACE + + ! + + ! Local variables + + ! REAL(Long), DIMENSION(1:SIZE(Y)):: & S1,S2,S3,S4,S5,New_Y_1,New_Y_2,Error REAL(Long)::T,H,H2,H3,H6,H8,Factor $\mathbf{\omega}=\mathbb{1}$ .E-2_Long + +# Chapter 26 + +REAL(Long)::Smallest_step $\r=\!1$ .E-6_Long,Max_Error INTEGER::No_of_steps $=\!0 + +$ + +! IFAIL $=\!0 + +$ ! + + ! Check input parameters + + ! IF(N $<=~\mathrm{~\textmu~}0$ .OR. $\mathrm{~\textit~{~A~}~}==\mathrm{~\textit~{~B~}~}$ .OR. Tol $<=\quad0\;.\;0$ ) THEN IFAIL $\begin{array}{r l}{\mathbf{\varepsilon}=}&{{}\,\mathbb{1}}\end{array}$ RETURN ENDIF + + ! + + ! Initialize T to be start of interval and + + ! H to be 1/100 of interval $\scriptstyle{\mathrm{T}}={\mathrm{\bar{A}}}$ $\mathrm{H}{=}$ (B-A)/100.0_Long DO ! Beginning of Repeat loop H2=H/2.0_Long H3=H/3.0_Long H6=H/6.0_Long H8=H/8.0_Long + + ! + + ! Calculate S1,S2,S3,S4,S5 + + ! + + ! $S1\!=\!\mathbb{F}$ (T,Y) CALL FUN(T,Y,S1,N) New_Y_1=Y+H3\*S1 + + ! S2 $=$ F(T+H/3,Y+H/3\*S1) CALL FUN( $\mathrm{T}{+}\mathrm{H}3$ ,New_Y_1,S2,N) $\mathtt{N e w\_Y\_1}\mathtt{=Y+H6}\,^{\star}\,\mathtt{S1}\mathtt{+H6}\,^{\star}\,\mathtt{S2} + +$ ! $S3\!=\!\mathrm{{F}}$ (T+H/3,Y+H/6\*S1+H/6\*S2) CALL FUN( $\mathrm{T}{+}\mathrm{H}3$ ,New_Y_1,S3,N)New_Y_1=Y+H8\*(S2+3.0_Long\*S3) + + ! $S4\!=\!\mathbb{F}$ (T+H/2,Y+H/8\*(S2+3\*S3)) CALL FUN( $\mathrm{T}{+}\mathrm{H}{\cal2}$ ,New_Y_1,S4,N)New_Y_1=Y+H2\*(S1-3.0_Long\*S3+4.0_Long\*S4) + +! $S5\!=\!\mathrm{{F}}$ ( $\mathrm{T+H}$ , $\tt Y{+}\mathrm{H}$ /2\*(S1-3\*S3+4\*S4))CALL FUN( $\mathrm{T+H}$ ,New_Y_1,S5,N) + +! + + ! Calculate values at $\mathrm{T+H} + +$ ! $\mathsf{N e w\_Y\_1=Y+H6^{\star}\ (S1+4\cdot0\_L o n g^{\star}S4+S5)}$ ) New_ $\scriptstyle\mathrm{Y}\_2=\mathrm{Y}+\mathrm{H}2\,^{\star}$ (S1-3.0_Long\*S3+4.0\*S4) + + ! + + ! Calculate error estimate + + ! Error $=$ ABS(0.2_Long\*(New_Y_1-New_Y_2)) Max_Error ${}^{\ast}=$ MAXVAL(Error) IF(Max_Error $>$ Tol) THEN + +! + + ! Halve step length and try again + + ! IF(ABS(H2) $<$ Smallest_step) THEN IFAIL $=$ 2 RETURN ENDIF $\scriptstyle{\mathrm{H}}={\mathrm{H}}2$ ELSE + + ! + + ! Accepted approximation so overwrite Y with Y_new_1, + + ! and T with $\mathrm{T+H} + +$ ! Y=New_Y_1 $\scriptstyle\mathrm{T}=\mathrm{T}+\mathrm{H} + +$ ! + + ! Can next step be doubled? + + ! IF(Max_Error\*Factor $<$ Tol)THEN $\scriptstyle{\mathrm{H}=\mathrm{H}^{\star}2}$ .0_Long ENDIF + + ! + + ! Does next step go beyond interval end B, + + ! if so set $\begin{array}{l l l}{\mathrm{H}}&{=}&{\mathrm{B-T}}\end{array}$ + +! IF( $\mathrm{~T+H~}\,>\,\mathrm{~B~}$ ) THEN $\scriptstyle\mathrm{H=B-T}$ ENDIF No_of_steps $=$ No_of_steps+1 ENDIF IF( $\mathrm{~\mathsf~{~T~}~}>=\mathrm{~\mathsf~{~B~}~}$ ) EXIT ! End of repeat loop END DO END SUBROUTINE Runge_Kutta_Merson + +A main program to use this subroutine is of the form: + +PROGRAM ch2602 USE Precisions IMPLICIT NONE REAL(Long),Dimension(:),Allocatable::Y REAL(Long)::A,B,Tol INTEGER::N,IFAIL,All_stat INTERFACE SUBROUTINE Run ge Kut ta Mers on(Y,FUN,IFAIL,N,A,B,Tol) USE Precisions IMPLICIT NONE REAL(Long),INTENT(INOUT) :: Y(:) REAL(Long),INTENT(IN)::A,B,Tol INTEGER,INTENT(IN)::N INTEGER,INTENT(OUT)::IFAIL INTERFACE SUBROUTINE FUN(T,Y,F,N) USE Precisions IMPLICIT NONE REAL(Long), INTENT(IN),DIMENSION(:)::Y REAL(Long), INTENT(OUT),DIMENSION(:)::F REAL(Long), INTENT(IN)::T INTEGER,INTENT(IN)::N END SUBROUTINE FUN END INTERFACE END SUBROUTINE Runge_Kutta_Merson END INTERFACE + +INTERFACE SUBROUTINE Fun1(T,Y,F,N) USE Precisions IMPLICIT NONE + +REAL(Long), INTENT(IN),DIMENSION(:)::Y REAL(Long), INTENT(OUT),DIMENSION(:)::F REAL(Long),INTENT(IN):: T INTEGER, INTENT(IN):: N END SUBROUTINE Fun1 END INTERFACE + + ! PRINT \*,'Input no of equations' READ\*,N + + ! + + ! Allocate space for Y - checking to see that it + + ! allocates properly + + ! ALLOCATE(Y(1:N),STAT $^{!}{=}$ All_stat) IF(All_stat $/=~\,0$ ) THEN PRINT \* , ' Not enough memory' PRINT \* , ' array Y is not allocated' STOP ENDIF PRINT \*,' Input start and end of interval over' PRINT \*,' which equations to be solved' READ \*,A,B PRINT \*,"Input ic's" READ \*,Y(1:N) PRINT \*,'Input Tolerance' READ \*,Tol PRINT \*,'At T= ',A PRINT \*,'Initial conditions are :',Y(1:N) CALL Run ge Kut ta Mers on(Y,Fun1,IFAIL,N,A,B,Tol) IF(IFAIL $/=~\,0$ ) THEN PRINT \*,'Integration stopped with IFAIL $=$ ',IFAIL ELSE PRINT \*,'at $\mathrm{T}{=}$ ',B PRINT\*,'Solution is: ',Y(1:N) ENDIF END PROGRAM ch2602 Consider trying to solve the following system of first-order ordinary differential equations: $\dot{y}_{1}=\tan y_{3}$ + +$$ +\begin{array}{l}{{\dot{y}}_{2}=\displaystyle\frac{-\,0.032\,\tan y_{3}}{y_{2}}-\frac{0.02\,\,y_{2}}{\cos y_{3}}}\\ {{\dot{y}}_{3}=\displaystyle-\frac{0.032}{y_{2}^{2}}}\end{array} +$$ + +over an interval $t=0.0$ to $t=8.0$ with initial conditions $y1=0\;\;\;\;\;\;y2=0.5\;\;\;\;\;\;y3=\frac{\pi}{5}$ + +The user supplied subroutine is: + +Subroutine Fun1(T,Y,F,N) USE Precisions IMPLICIT NONE REAL(Long),INTENT(IN),DIMENSION(:)::Y REAL(Long),INTENT(OUT),DIMENSION(:)::F REAL(Long),INTENT(IN)::T INTEGER,INTENT(IN)::N ! F(1)=TAN(Y(3)) F(2)=-0.032_Long\*F(1)/Y(2)-0.02_Long\*Y(2)/COS(Y(3))F(3)=-0.032_Long/(Y(2)\*Y(2)) + +END SUBROUTINE Fun1 + +# 26.2.1 Note: Alternative form of the ALLOCATE statement + +In the main program Odes we have defined Y to be a deferred-shape array, allocat- ing it space after the variable $_\mathrm{N}$ is read in. In order to make sure that enough memory is available to allocate space to array $\mathrm{Y}$ the ALLOCATE statement is used as follows: + +ALLOCATE(Y(1:N),STAT $\leftleftarrows$ All_stat) + +If the allocation is successful variable All_stat returns zero; otherwise it is given a processor dependent positive value. We have included code to check for this and the program stops if All_stat is not zero. + +# 26.2.2 Note: Automatic arrays + +The subroutine Run ge Kut ta Mers on needs a number of local rank 1 arrays S1, S2, S3, S4 and S5 for workspace, their shape and size being the same as the dummy argument Y. Fortran 95 supplies automatic arrays for this purpose and can be declared as + +REAL(Long), DIMENSION (1:SIZE(Y)) :: S1, S2, S3, S4, S5 The size of automatic arrays can depend on the size of actual arrays: in our exam- ple they are the same shape and size as the dummy array Y, or some other dummy arguments. Automatic arrays are created when the procedure is called and de- stroyed when control passes back to the calling program unit. They may have different shapes and sizes with different calls to the procedure, and because of this automatic arrays cannot be saved or initialised. + +A word of warning should be given at this point. If there isn't enough memory available when an automatic array needs to be created problems will occur. Unlike allocatable arrays there is no way of testing to see if an automatic array has been created successfully. The general feeling is that even though they are nice, auto- matic arrays should be used with care and perhaps shouldn't be used in production code! + +# 26.2.3 Note: Dummy procedure arguments + +In order to make the use of the subroutine Run ge Kut ta Mers on as general as pos- sible, one of its dummy arguments (FUN) is the name of a subroutine which the user supplies with the actual system of ordinary differential equations he or she wishes to solve. This means that the main program which calls Run ge Kut ta Mers on passes as an actual argument the name of the subroutine containing the definition of the equations to be solved. In this example the subrou- tine is called Fun1. In order to do this in the main program Odes we have an interface block for the Run ge Kut ta Mers on subroutine and within this interface block we have another interface block for the dummy routine FUN. We also have an interface block for the actual subroutine Fun1 in the main program. + +# 26.2.4 Keyword and optional arguments + +The examples of procedures so far have assumed that the dummy arguments and the corresponding arguments are in the same position, i.e., we are using positional arguments. Fortran 95 also provides the ability to supply the actual arguments to a procedure by keyword, and hence in any order. To do this the name of the dummy argument is referred to as the keyword and is specified in the actual argument list in the form dummy-argument $=$ actual-argument To illustrate this, let us consider a subroutine to solve ordinary differential equa- tions. The full subroutine and explanation are given in Chapter 27: + +SUBROUTINE Run ge Kut ta Mers on(Y,FUN,IFAIL,N,A,B,TOL) + +where A is the initial point, B is the end point at which the solution is required, TOL is the accuracy to which the solution is required and N is the number of equa- tions. The rest of the dummy arguments are explained in Chapter 27. + +The subroutine can be called as follows: + +CALL Runge_Kutta_Merson( Y , Fun1 , IFAIL , $\mathbb{A}\!=\!0\;.\;0$ ,& $\scriptstyle{\mathrm{B}}=8\,.\ 0$ , Tol $\mathbf{\omega}=\mathbb{1}$ .0E-6 , $\scriptstyle{\mathrm{N}}=3$ ) + +where the dummy arguments A, B, Tol and $_\mathrm{N}$ are now being used as keywords. The use of keyword arguments makes the code easier to read and decreases the need to remember their precise position in the argument list. + +Also with Fortran 95 comes the ability to specify that an argument is optional. This is very useful when designing procedures for use by a range of programmers. In- side a procedure defaults can be set for the optional arguments providing an easy-to-use interface, while at the same time allowing sophisticated users a more comprehensive one. + +To declare a dummy argument to be optional the OPTIONAL attribute can be used. For example, the last dummy argument Tol for the subroutine Run ge Kut ta Mers on could be declared to be optional (although internally in the subroutine the code would have to be changed to allow for this), e.g., + +SUBROUTINE Run ge Kut ta Mers on(Y,FUN,IFAIL,N,A,B,Tol) USE Precisions REAL(Long), INTENT(INOUT), OPTIONAL :: Tol + +and because it is at the end of the dummy argument list, calling the subroutine with a positional argument list, Tol can be omitted, e.g., + +CALL Run ge Kut ta Mers on(Y,Fun1,IFAIL,N,A,B) + +The code of the subroutine will need to be changed to check to see if the argument Tol is supplied, the intrinsic function PRESENT being available for this purpose. Sample code is given below: + +SUBROUTINE Runge_Kutta_Merson(Y,FUN, IFAIL, N,A,B,Tol) USE Precisions ! code left out REAL(Long),INTENT(IN),OPTIONAL::Tol REAL(Long)::Internal_tol $=$ 1.0D-3 IF(PRESENT(Tol)) THEN Internal_tol $=$ Tol PRINT\*,'Tol $=$ ', Internal_tol,' is supplied' ELSE + +PRINT\*,"Tol isn't supplied, default tolerance = " PRINT \*,Internal_tol,' is used' ENDIF + + ! code left out but all references to tol + + ! would have to be changed to internal_tol END SUBROUTINE Runge_Kutta_Merson + +A number of points need to be noted when using keyword and optional arguments: + +• If all the actual arguments use keywords, they may appear in any order. • When only some of the actual arguments use keywords, the first part of the list must be positional followed by keyword arguments in any order. • When using a mixture of positional and keyword arguments, once a key- word argument is used all subsequent arguments must be specified by keyword. • If an actual argument is omitted the corresponding optional dummy argu- ment must not be redefined or referenced, except as an argument to the PRESENT intrinsic function. • If an optional dummy argument is at the end of the argument list then it can just be omitted from the actual argument list. • Keyword arguments are needed when an optional argument not at the end of an argument list is omitted, unless all the remaining arguments are omitted as well. • Keyword and optional arguments require explicit procedure interfaces, i.e., the procedure must be internal, a module procedure or have an inter- face block available in the calling program unit. + +A number of the intrinsic procedures we have used have optional arguments. Con- sult Appendix D for details. + +# 26.3 Generic procedures + +There has always been a degree of support in Fortran for the concept of a generic procedure. Simplistically, a procedure is generic if it can handle arguments of more than one data type. This concept is one that is probably taken for granted with the intrinsic procedures. + +With Fortran 95 we can now define our own generic procedures. The example we will use is based on the earlier one of sorting. In the original example the program worked with real data. In the example below we have extended the program to handle both integer and real data. + +What is not obvious from our use of the internal procedures is that there will be specific procedures to handle each data type, i.e., if a function can take integer, real and complex arguments then there will be one implementation of that function for each data type, i.e., three separate functions. + +In the example below we add the ability to handle integer data. This means that where we had: + +• read data • sort data • print data + +and one subroutine to implement the above we now have two subroutines to do each of the above, one to handle integers and one to handle reals: + +PROGRAM ch2603 + +use read data module use sort data module use print data module + +IMPLICIT NONE INTEGER :: How_Many CHARACTER ( $\tt L E N\!\!=\!\!20$ ) :: File_Name INTEGER , ALLOCATABLE , DIMENSION(:) :: integer_data REAL , ALLOCATABLE , DIMENSION(:) :: real_Data + +PRINT \* , ' How many data items are there?' READ $\star$ , How_Many PRINT \* , ' What is the file name?' READ '(A)',File_Name ALLOCATE(integer_data(How_Many)) CALL Read_Data(File_Name,integer_data,How_Many) CALL Sort_Data(integer_data,How_Many) CALL Print_Data(integer_data,How_Many) PRINT \*, ' Phase 1 ends.' PRINT \*, ' data written to file name ISORTED.DAT' DEALLOCATE(integer_data) + +PRINT \* , ' How many data items are there?' READ \* , How_Many PRINT \* , ' What is the file name?' READ '(A)',File_Name + +ALLOCATE(real_data(How_Many)) CALL Read_Data(File_Name,real_data,How_Many) CALL Sort_Data(real_data,How_Many) CALL Print_Data(real_data,How_Many) PRINT \*, ' Program ends.' PRINT \*, ' data written to file name RSORTED.DAT' END PROGRAM ch2603 + +module read_data_module + +interface read_data + +module procedure read_integer module procedure read_real + +end interface read_data + +contains + +SUBROUTINE read_real(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER (LEN $=\star$ ) , INTENT(IN) :: File_Name INTEGER , INTENT(IN) :: How_Many REAL , INTENT(OUT) , & DIMENSION(:) :: Raw_Data INTEGER :: I OPEN(FILE $_{\parallel}=$ File_Name,UNIT $\vDash\vDash$ ) DO $\mathbb{T}\!=\!1$ ,How_Many READ (UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) Raw_Data(I) ENDDO + +END SUBROUTINE read_real + +SUBROUTINE read_integer(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER ( $\tt L E N\mathrm{=\star}$ ) , INTENT(IN):: File_Name INTEGER , INTENT(IN) :: How_Many INTEGER , INTENT(OUT) , & DIMENSION(:) :: Raw_Data INTEGER :: I OPEN(FILE $_{\parallel}=$ File_Name,UNIT $\vDash\vDash$ ) DO $\mathbb{T}\!=\!1$ ,How_Many READ (UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) Raw_Data(I) + +# 430 + +# Chapter 26 + +ENDDO + +END SUBROUTINE read_integer + +end module read_data_module + +module sort_data_module + +interface sort_data + +module procedure sort_integer module procedure sort_real + +end interface sort_data + +contains + +SUBROUTINE sort_real(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many REAL , INTENT(INOUT) , & DIMENSION(:) :: Raw_Data CALL QuickSort(1,How_Many) + +CONTAINS + +RECURSIVE SUBROUTINE QuickSort(L,R) IMPLICIT NONE INTEGER , INTENT(IN) :: L,R INTEGER :: I,J REAL :: V,T + +$\dot{\bot}\!=\!\bot$ $\dot{\daleth}\,{=}\,\underline{{\boldsymbol{\Upsilon}}}$ v=raw_data( int(( $\dot{\mathbf{\nabla}}\perp+\mathtt{r}\dot{\mathbf{\nabla}}.$ )/2) ) do do while (raw_data(i) < v ) $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ enddo do while (v < raw_data(j) ) $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ enddo if ( $\dot{\Sigma}\!<\!=\!\dot{\Pi}$ ) then + +t $=$ raw_data(i) raw_data(i) $=$ raw_data(j) raw_data $(\,\dot{\,}\,)=\,\!\mathsf{C}$ $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ endif if ( $\dot{\perp}{>}\dot{\supset}$ ) exit enddo if ( $\perp<\!\dot{\!\!\perp}$ ) then call quicksort(l,j) endif if ( $\dot{\bot}\!<\!\underline{{\boldsymbol{\tau}}}.$ ) then call quicksort(i,r) endif + +END SUBROUTINE QuickSort + +END SUBROUTINE sort_real + +SUBROUTINE sort_integer(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many INTEGER , INTENT(INOUT) , & DIMENSION(:) :: Raw_Data CALL QuickSort(1,How_Many) + +CONTAINS + +RECURSIVE SUBROUTINE QuickSort(L,R) IMPLICIT NONE INTEGER , INTENT(IN) :: L,R INTEGER :: I,J INTEGER :: V,T + +$\dot{\bot}\!=\!\bot$ $\dot{\daleth}\,{=}\,\underline{{\boldsymbol{\Upsilon}}}$ $\mathbb{V}=$ raw_data( int((l+r)/2) ) do do while (raw_data(i) < v ) $\dot{\ensuremath{\boldsymbol{\Sigma}}}\!=\!\dot{\ensuremath{\boldsymbol{\Sigma}}}\!+\!\boldsymbol{\mathbb{1}}$ + +enddo do while (v < raw_data(j) ) $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ enddo if ( $\dot{\Sigma}\!<\!=\!\dot{\Pi}$ ) then t $=$ raw_data(i) raw_data(i) $=$ raw_data(j) raw_data $(\,\dot{\,}\,)=\,\!\mathsf{C}$ $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ endif if ( $\dot{\perp}{>}\dot{\supset}$ ) exit enddo + +if ( $\perp<\!\dot{\!\!\perp}$ ) then call quicksort(l,j) endif + +END SUBROUTINE QuickSort + +END SUBROUTINE sort_integer + +end module sort_data_module + +module print data module + +interface print_data + +module procedure print_integer module procedure print_real + +end interface print_data + +contains + +SUBROUTINE print_real(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many + +REAL , INTENT(IN) , & DIMENSION(:) :: Raw_Data INTEGER :: I OPEN(FILE $=$ 'RSORTED.DAT',UNIT $^{=2}$ ) DO $\mathbb{T}\!=\!1$ ,How_Many WRITE(UNIT $^{=2}$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) Raw_Data(I) END DO CLOSE(2)END SUBROUTINE print_real SUBROUTINE print_integer(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many INTEGER , INTENT(IN) , & DIMENSION(:) :: Raw_Data INTEGER :: I OPEN(FILE $="$ 'ISORTED.DAT',UNIT $^{1}{=}2$ ) DO $\mathbb{T}\!=\!1$ ,How_Many WRITE(UNIT $^{=2}$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) Raw_Data(I) END DO CLOSE(2)END SUBROUTINE print_integer end module print data module The key code is given below for each module: interface read_data module procedure read_integer module procedure read_real end interface read_data interface sort_data module procedure sort_integer module procedure sort_real end interface sort_data interface print_data + +# Chapter 26 + +module procedure print_integer module procedure print_real + +end interface print_data + +The interface block name is used in the calling routine and the appropriate module procedure will be called, based on a signature match of the actual and dummy pa- rameters. + +# 26.4 A function that returns a variable length array + +The following program illustrates the use of a function that returns a variable length array. This is in Fortran 90 style. The program also shows a second way of generic programming using explicit interfaces. In this case you again use the inter- face name in the calling routine, and the appropriate routine is called on the basis of a parameter signature match: + +PROGRAM ch2604 IMPLICIT NONE + +INTERFACE FUNCTION Running_Average(R,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many REAL , DIMENSION (:) , INTENT(IN) :: R REAL , DIMENSION(how_many) :: Running_Average END FUNCTION Running_Average END INTERFACE + +INTERFACE Read_Data + +SUBROUTINE RR(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER ( $\tt L E N\mathrm{=\star}$ ) , INTENT(IN) :: File_Name INTEGER , INTENT(IN) :: How_Many REAL , INTENT(OUT) , $\&$ DIMENSION(:) :: Raw_Data END SUBROUTINE RR + +SUBROUTINE RI(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER ( $\tt L E N\mathrm{=\star}$ ) , INTENT(IN) :: File_Name INTEGER , INTENT(IN) :: How_Many + +INTEGER , INTENT(OUT) , & DIMENSION(:) :: Raw_Data END SUBROUTINE RI + +END INTERFACE + +INTEGER :: How_Many CHARACTER (LEN $\scriptstyle{\dot{=}20}$ ) :: File_Name REAL , ALLOCATABLE , DIMENSION(:) :: Raw_Data REAL , ALLOCATABLE , DIMENSION(:) :: RA INTEGER :: I PRINT $\star$ , ' How many data items are there?' READ $\star$ , How_Many PRINT \* , ' What is the file name?' READ '(A)',File_Name ALLOCATE(Raw_Data(how_many)) ALLOCATE(RA(how_many)) CALL Read_Data(File_Name,Raw_Data,How_Many) RA $\b=$ Running_Average(Raw_Data,How_Many) DO $\scriptstyle{\mathbb{T}}=1$ ,How_Many PRINT \*,Raw_Data(I),' ' ,RA(I) END DO + +END PROGRAM ch2604 + +FUNCTION Running_Average(R,How_Many) INTEGER , INTENT(IN) :: How_Many REAL , INTENT(IN) , DIMENSION(:) :: R REAL , DIMENSION(how_many) :: Running_Average INTEGER :: I REAL :: Sum $\scriptstyle{\underline{{\boldsymbol{\mathbf{\Pi}}}}}\,0$ .0 DO $\scriptstyle{\mathbb{T}}=1$ ,How_Many Sum $\begin{array}{r l r}{\updownarrow}&{{}=}&{\mathrm{Sum}}\end{array}$ $^+$ R(I) Running_Average(I) $=$ Sum/I END DO + +SUBROUTINE RR(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER ( $\tt L E N\mathrm{=\star}$ ) , INTENT(IN) :: File_Name INTEGER , INTENT(IN) :: How_Many REAL , INTENT(OUT) , & DIMENSION(:) :: Raw_Data + +# Chapter 26 + +INTEGER :: I OPEN(FILE $_{\parallel}=$ File_Name,UNIT $\vDash\vDash$ ) DO $\mathbb{T}\!=\!1$ ,How_Many READ (UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) Raw_Data(I) ENDDO END SUBROUTINE RR + +SUBROUTINE RI(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER ( $\tt L E N\!\!=\!\star$ ) , INTENT(IN):: File_Name INTEGER , INTENT(IN) :: How_Many INTEGER , INTENT(OUT) , & DIMENSION(:) :: Raw_Data INTEGER :: I OPEN(FILE $_{\parallel}=$ File_Name,UNIT $\vDash\vDash$ ) DO $\mathbb{T}\!=\!1$ ,How_Many READ (UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) Raw_Data(I) ENDDO END SUBROUTINE RI + +The preferred way of doing this is shown in Chapter 28, but it only works with compilers that support ISO TR 15581. + +# 26.5 Operator and assignment overloading + +It is sometimes convenient to extend the meaning of an operator and the assign- ment symbol beyond that provided by the language. This can be done in Fortran 95 using module procedures. The example below is based on moving around a three-dimensional space: + +MODULE T_Position IMPLICIT NONE TYPE Position INTEGER :: X INTEGER :: Y INTEGER :: Z END TYPE Position + +INTERFACE OPERATOR $(+)$ MODULE PROCEDURE New_Position END INTERFACE + +CONTAINS + +TYPE (Position) ,INTENT(IN) :: A,B TYPE (Position) :: New_Position New_Position % X = A % X + B % X ${\begin{array}{l l l l l l l l l}{{\mathfrak{s}}}&{\mathrm{X}}&{=}&{\mathbb{A}}&{{\circ\!\!\!}}&{\mathrm{X}}&{+}&{\mathbb{B}}&{{\circ\!\!\!}}&{\mathrm{X}}\\ {{\mathfrak{s}}}&{\mathrm{Y}}&{=}&{\mathbb{A}}&{{\circ\!\!\!}}&{\mathrm{Y}}&{+}&{\mathbb{B}}&{{\circ\!\!\!}}&{\mathrm{Y}}\\ {{\mathfrak{s}}}&{\mathrm{Z}}&{=}&{\mathbb{A}}&{{\circ\!\!\!}}&{\mathrm{Z}}&{+}&{\mathbb{B}}&{{\circ\!\!\!}}&{\mathrm{Z}}\end{array}}$ New_Position % Y = A % Y + B % Y New_Position % Z = A % Z + B % Z END FUNCTION New_Position + +END MODULE T_Position + +PROGRAM ch2605 USE T_Position IMPLICIT NONE TYPE (Position) :: A,B,C A%X=10 A%Y=10 A%Z=10 B%X=20 B%Y=20 B%Z=20 C=A+B PRINT \*,A PRINT \*,B PRINT \*,C END PROGRAM ch2605 + +We have extended the meaning of the addition operator so that we can write sim- ple expressions in Fortran based on it and have our new position calculated using a user supplied function that actually implements the calculation of the new position. + +# 26.6 A subroutine to extract the diagonal elements of a matrix + +A common task mathematically is to extract the diagonal elements of a matrix. For example if + +$$ +A=\!\!\left(\!\!{\begin{array}{l l l}{21}&{6}&{7}\\ {9}&{3}&{2}\\ {4}&{1}&{8}\end{array}}\!\!\right) +$$ + +the diagonal elements are ( 21, 3, 8 ). + +This can be thought of as extracting an array section, but the intrinsic function PACK is needed. In its simplest form PACK (Array,Vector) packs an array, Array, into a rank 1 array, Vector, according to Array's array element order. + +Below is a complete program to demonstrate this: + +PROGRAM ch2606 IMPLICIT NONE INTEGER::I, N REAL, ALLOCATABLE, DIMENSION(:,:) :: A REAL, ALLOCATABLE, DIMENSION(:):: Adiag LOGICAL:: OK CHARACTER(LEN $\scriptstyle{\dot{=}20}$ )::Filename + +INTERFACE SUBROUTINE Matrix_Diagonal (A, Diag, N, OK) IMPLICIT NONE REAL, INTENT(IN), DIMENSION(:,:) ::A REAL, INTENT(OUT), DIMENSION(:) :: Diag INTEGER, INTENT(IN) ::N LOGICAL, INTENT(OUT):: OK END SUBROUTINE Matrix_Diagonal + +END INTERFACE + +PRINT\*,'input name of data file' READ '(A)',Filename OPEN(UNIT $\vDash\vDash$ ,FILE $=$ Filename) READ(1,\*) N ALLOCATE(A(1:N,1:N), Adiag(1:N)) DO $\mathbb{T}\!=\!1$ ,N READ $(\,\mathtt{1}\,,\,^{\star}\,)\,\mathtt{A}\,(\,\mathtt{I}\,,\,\mathtt{1}:\mathtt{N}\,)$ END DO CALL Matrix_Diagonal(A,Adiag,N, OK) IF(OK) THEN PRINT\*,' Diagonal elements of A are:' PRINT \*,Adiag ELSE PRINT\*,'Matrix A is not square' END IF END PROGRAM ch2606 + +SUBROUTINE Matrix_Diagonal (A, Diag, N, OK) IMPLICIT NONE + +REAL, INTENT(IN), DIMENSION(:,:) ::A REAL, INTENT(OUT), DIMENSION(:) :: Diag INTEGER, INTENT(IN) ::N LOGICAL, INTENT(OUT):: OK REAL, DIMENSION (1:SIZE(A,1)\*SIZE(A,1)) :: Temp + + ! + + ! Subroutine to extract the diagonal elements of + + ! an N \* N matrix A + + ! IF(SIZE $\begin{array}{r l}{(\mathbb{A},\,\mathbb{1})}&{{}==\mathrm{~\mathbb~N~}}\end{array}$ .AND. SIZE(A,2) == N) THEN + + ! Matrix is square $\mathrm{CK}=$ .TRUE. Temp $=$ PACK(A,.TRUE.) Diag $=$ Temp( $\mathbb{1}:\!\mathbb{N}^{\star}\mathbb{N}:\!\mathbb{N}\!+\!\mathbb{1}$ ) ELSE + + ! Matrix isn't square $\mathrm{CK}=$ .FALSE. END IF END SUBROUTINE Matrix_Diagonal + +# 26.7 Perfectly balanced tree + +Let us now look at a more complex example that builds a perfectly balanced tree and prints it out. A loose definition of a perfectly balanced tree is one that has minimum depth for $n$ nodes. More accurately a tree is perfectly balanced if for each node the number of nodes in its left and right subtrees differ by at most 1: + +MODULE Node_Type_Def IMPLICIT NONE TYPE Tree_Node INTEGER :: Number TYPE (Tree_Node) , POINTER :: Left,Right END TYPE Tree_Node END MODULE Node_Type_Def + +PROGRAM ch2607 ! Construction of a perfectly balanced tree USE Node_Type_Def IMPLICIT NONE TYPE (Tree_Node) , POINTER :: Root INTEGER :: N_of_Items + +INTERFACE + +RECURSIVE FUNCTION Tree(N) RESULT(Answer) USE Node_Type_Def IMPLICIT NONE INTEGER , INTENT(IN) :: N TYPE (Tree_Node) , POINTER :: Answer END FUNCTION Tree + +SUBROUTINE Print_Tree(Trees,H) USE Node_Type_Def IMPLICIT NONE TYPE (Tree_Node) , POINTER :: Trees INTEGER :: H END SUBROUTINE Print_Tree + +END INTERFACE + +PRINT \*,' Enter number of items' READ \*,N_Of_Items Root $=>$ Tree(N_Of_Items) CALL Print_Tree(Root,0) + +END PROGRAM ch2607 + +RECURSIVE FUNCTION Tree(N) RESULT (Answer) USE Node_Type_Def IMPLICIT NONE INTEGER , INTENT(IN) :: N TYPE (Tree_Node) , POINTER :: Answer TYPE (Tree_Node) , POINTER :: New_Node + +INTEGER :: L,R,X IF ( $\mathrm{~{~\bf~\cdot~}~N~}==\mathrm{~{~\bf~0~}~}$ ) THEN print \*,' terminate tree' NULLIFY(Answer) ELSE $\scriptstyle{\mathbb{L}}=\mathrm{N}/2$ $\scriptstyle\mathrm{R=N-L-}\,1$ PRINT \*,L,R,N PRINT \*,' Next item' READ \*,X ALLOCATE(New_Node) + +New_Node%Number $\tt:=\tt X$ print \*, ' left branch' New_Node%Left $=>$ Tree(L) print \*, ' right branch' New_Node%Right $=>$ Tree(R) Answer $=>$ New_Node ENDIF PRINT \*, ' Function tree ends' END FUNCTION Tree + +RECURSIVE SUBROUTINE Print_Tree(T,H) USE Node_Type_Def IMPLICIT NONE TYPE (Tree_Node) , POINTER :: T INTEGER :: I INTEGER :: H IF (ASSOCIATED(T)) THEN CALL Print_Tree(T%Left, $\mathrm{H}{+}1$ )DO $\mathbb{T}\!=\!1$ ,H WRITE(UNIT $=\star$ , $\mathrm{FMT}{=}10$ ,ADVANCE $=$ 'NO') 10 FORMAT('')ENDDO PRINT \*,T%Number CALL Print_Tree(T%Right, $\mathrm{H}{+}1$ ) ENDIF END SUBROUTINE Print_Tree + +There are a number of very important concepts contained in this example and they include: + +• The use of a module to define a type. For user defined data types we must create a module to define the data type if we want it to be available in more than one program unit. • The use of a function that returns a pointer as a result. • As the function returns a pointer we must determine the allocation status before the function terminates. This means that in the above case we use the NULLIFY(Result) statement. The other option is to TARGET the pointer. • The use of ASSOCIATED to determine if the node of the tree is termi- nated or points to another node. + +Type the program in and compile, link and run it. Note that the tree only has the minimal depth necessary to store all of the items. Experiment with the number of items and watch the tree change its depth to match the number of items. + +# 26.8 Pure function example + +We recommended in Chapter 14 that you should ensure that your functions do not have side effects, for safety reasons. With the ability to run your code on parallel systems we have the additional problem in that the code may not actually work! We would also like to be able to take advantage of automatic parallelisation if pos- sible. In the following example we show how to do this using the PURE function prefix: + +PROGRAM ch2608 IMPLICIT NONE INTEGER :: I,J,Result INTEGER :: GCD PRINT \*,' Type in two integers' READ \*,I,J Result $=$ GCD(I,J) PRINT \*,' GCD is ',Result END PROGRAM ch2608 + +PURE INTEGER FUNCTION GCD(A,B) IMPLICIT NONE INTEGER , INTENT(IN) :: A,B INTEGER :: Temp IF ( $\mathrm{~\AA~}<\mathrm{~\textmu~}$ ) THEN Temp $\mathrm{=A}$ ELSE Temp $\mathtt{=B}$ ENDIF DO WHILE ((MOD(A,Temp) /= 0) .OR. (MOD(B,Temp) / $=\!0$ )) Temp $=$ Temp-1 END DO GCD ${}^{1=}$ Temp END FUNCTION GCD + +Procedures can also be made pure. + +# 26.8.1 Pure constraints + +The following are some of the constraints on pure procedures: • A dummy argument must be INTENT(IN) for a pure function. + +• A dummy argument must have an INTENT attribute in a pure subroutine. • Local variables may not have the save attribute. • No I/O must be done in the procedure. • Any functions and procedures referenced must be pure. • You cannot have a stop statement with a pure procedure. + +The above information should be enough to enable you to write simple pure func- tions and procedures. + +# 26.9 Elemental function example + +The intrinsic trigonometric functions are elemental in that they can take an argu- ment that is scalar or array valued, and of any of the supported numeric kind types. With Fortran 95 we can make our own user defined functions elemental. Consider the following example, which is an extension of the earlier example that calculated e\*\*x: + +PROGRAM ch2609 IMPLICIT NONE + + ! + + ! Elemental function example + + ! INTEGER :: I REAL :: X REAL , DIMENSION(10) :: Y INTERFACE ELEMENTAL REAL FUNCTION ETOX(X) IMPLICIT NONE REAL , INTENT(IN) :: X END FUNCTION ETOX END INTERFACE $\mathrm{Xe}{=}1$ .0 DO $\mathbb{T}\!=\!1$ ,10 $\mathtt{Y}\left(\,\mathtt{T}\,\right)=\mathtt{T}$ END DO PRINT \*,Y $\mathrm{X}{=}$ ETOX(X) PRINT \*,X Y $\bf{\dot{\rho}}=\bf{\rho}$ ETOX(Y) PRINT \*,Y + +# Chapter 26 + +END PROGRAM ch2609 + +ELEMENTAL REAL FUNCTION ETOX(X) IMPLICIT NONE REAL , INTENT(IN) :: X REAL :: TERM INTEGER :: NTERM REAL , PARAMETER :: TOL $=$ 1.0E-6 ETOX $\v{=}\,\v{1}$ .0 TERM $\mathbf{\omega}=\mathbb{1}$ .0 NTERM $\dot{=}0$ DO NTERM $\fallingdotseq$ NTERM $+1$ TERM $=$ (X/NTERM)\*TERM ETOX $\b=$ ETOX $^+$ TERM IF (TERM $<=$ TOL) EXIT END DO END FUNCTION ETOX + +Note the following: + +• We have added an interface block for the etox function. • We have added the ELEMENTAL prefix to the function header. • The dummy argument is scalar and INTENT(IN). • The function result is scalar. + +In this example we call the function with a scalar argument and a rank 1 array. Run the program to see what happens. + +You can have the PURE prefix but it is redundant as ELEMENTAL implies PURE. + +# 26.9.1 Elemental constraints + +Some of the restrictions include: + +• An elemental procedure must not be recursive. • A dummy argument must not be a pointer. • The result of an elemental function must not be a pointer. • Elemental procedures must have explicit interfaces in all program units that reference them. + +In a parallel environment this means that the array calculations could be carried out simultaneously on two or more processors. + +# 26.10 Elemental subroutine example + +It is also possible to make subroutines elemental. Consider the following example: + +PROGRAM ch2610 IMPLICIT NONE + +INTERFACE ELEMENTAL SUBROUTINE SWAP(X,Y) INTEGER , INTENT(INOUT) :: X,Y END SUBROUTINE SWAP END INTERFACE + +INTEGER , DIMENSION(10) :: A,B INTEGER :: I DO $\mathbb{T}\!=\!1$ ,10 $\mathtt{A}\left(\,\mathtt{T}\,\right)=\mathtt{T}$ $\mathtt{B}\left(\,\mathbb{T}\,\right){=}\mathbb{T}^{\star}\mathbb{T}$ END DO PRINT \*,A PRINT \*,B CALL SWAP(A,B) PRINT \*,A PRINT \*,B END PROGRAM ch2610 + +ELEMENTAL SUBROUTINE SWAP(X,Y) INTEGER , INTENT(INOUT) :: X,Y INTEGER :: TEMP TEMP $\tt=\tt X$ $\tt X\!=\!\tt Y$ Y $\bf{\dot{\rho}}=\bf{\rho}$ TEMP END SUBROUTINE SWAP + +Note the following: + +• We have an interface block for the elemental subroutine. • The subroutine dummy arguments are scalar with an INTENT attribute, in this case INTENT(INOUT) as we are swapping them over. + +# Chapter 26 + +In a parallel environment the swapping of array elements may be done simulta- neously on two or more processors. + +# 26.11 Date class + +The following is a complete manual rewrite of Skip Noble and Alan Millers date module. The original worked with the built-in Fortran intrinsic data types. It has been rewritten to work with a user defined or derived date data type. + +The first key code segment is + +TYPE, PUBLIC :: date PRIVATE INTEGER :: day INTEGER :: month INTEGER :: year END TYPE date + +where the date data type is public but its components are private. This means that access to the components must be done via subroutines and functions within the date_module module. + +The next key code segment is + +PUBLIC :: calendar to julian,& date_, & date_stamp, & date to day in year, & date_to_weekday_number, & get_day, & get_month, & get_year, & julian_to_date, & julian to date and week and day, & ndays, & year and day to date + +where we explicitly make the listed subroutines and functions public, as the code segment from the top of the module, + +! .. + + ! .. Default Accessibility .. PRIVATE + +defines everything to be private. + +We have to provide a user defined constructor when the components of the derived type are private. This is given below: + +FUNCTION date_(dd,mm,yyyy) RESULT (x) + + ! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Function Return Value .. TYPE (date) :: x + + ! .. + + ! .. Scalar Arguments .. INTEGER, INTENT (IN) :: dd, mm, yyyy + + ! .. $\mathrm{~\v~{~x~}~}=$ date(dd,mm,yyyy) END FUNCTION date_ + +This in turn calls the built-in constructor date. + +We also provide three additional procedures to access the components of the date class: + +get_day get_month get_year + +This is common programming practice in object oriented and object based pro- gramming. + +The program has also been through the Nag tool suite and this has helped to sys- tematically lay out the code. + +MODULE date_module + + ! Collected and put together january 1972, + + ! h. d. knoble. + + ! Original references are cited in each routine. + + ! Code converted using to_f90 by alan miller + + ! Date: 1999-12-22 time: 10:23:47 + + ! Compatible with imagine1 f compiler: 2002-07-19 + + ! At this time the functions and + + ! subroutines were as described below + + ! FUNCTION iday(yyyy, mm, dd) RESULT(ival) + + ! FUNCTION izlr(yyyy, mm, dd) RESULT(ival) + + ! SUBROUTINE calend(yyyy, ddd, mm, dd) + + !SUBROUTINE cdate(jd, yyyy, mm, dd) + +# Chapter 26 + +!SUBROUTINE daysub(jd, yyyy, mm, dd, wd, ddd) + +!FUNCTION jd(yyyy, mm, dd) RESULT(ival) + +! FUNCTION ndays(mm1, dd1, yyyy1, + + ! mm2, dd2, yyyy2) RESULT(ival) + + ! SUBROUTINE date_stamp( string, want_day, short ) + + ! Code converted by ian chivers and jane sleightholme + + ! November 2004 - May 2005 + + ! The changes are to go from + + ! working with integer variables + + ! for year, day and month to + + ! user defined date variables. + + ! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Default Accessibility .. PRIVATE + + ! .. + + ! .. Derived Type Declarations .. TYPE, PUBLIC :: date PRIVATE INTEGER :: day INTEGER :: month INTEGER :: year END TYPE date + + ! .. + + ! .. Public Statements .. PUBLIC :: calendar to julian,& date_, & date_stamp, & date to day in year, & date_to_weekday_number, & get_day, & get_month, & get_year, & julian_to_date, & julian to date and week and day, & ndays, & year and day to date + + ! .. + + ! The above are the contained + + ! functions and subroutines + + ! in this module. + +! Here is a short description of each one + + ! date to day in year - function + + ! returns the day in the year + + ! original arguments of day,month,year + + ! now date + + ! dayinyear + + ! date_to_weekday_number - function + + ! returns the week day number + + ! original argument d,m,y + + ! now date + + ! weekdaynum + + ! year and day to date - subroutine + + ! returns the day and month from + + ! year and day in year + + ! julian_to_date - subroutine + + ! returns a year and day to date ar date from + + ! a julian date + + !ndays- function + +! returns the number of days between + + ! two dates + + ! julian to date and week and day - subroutine + + ! given a julian day this routine + + ! calculates year, month day and + + ! week day number and day number + + ! calendar to julian - function + + ! returns julian date from + + ! year and day to date ar date + +CONTAINS + +! arithmetic functions "izlr" and "iday" + + ! are taken from remark on + + ! algorithm 398, by j. douglas robertson, + + ! cacm 15(10):918. FUNCTION date to day in year(x) + + ! Convert from date to day in year + + ! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Function Return Value .. INTEGER :: date to day in year + + ! .. + +! .. Structure Arguments .. TYPE (date), INTENT (IN) :: x + + ! .. + + ! .. Intrinsic Functions .. INTRINSIC modulo + + ! .. date to day in year $=$ $3055\,\star$ (x%month $^{.+2}$ )/100 & - (x%month+10)/13\*2 - 91 + & $\begin{array}{l l l l l}{(\mathrm{x\mathrm{\textquigarrow}}+10\,)\ /\,13\,\mathrm{\textstar}\,2}&{-}&{9\,\mathrm{\textbar{~}}+}&{\&}\end{array}$ (1-(modulo(x%year,4) $^{+3}$ )/4 & + (modulo(x%year,100) ${}+99$ )/100 & - (modulo(x%year, & 400) $+399$ )/400)\*(x%month+10)/13 $^+$ x%day END FUNCTION date to day in year FUNCTION date_to_weekday_number(x) + + ! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Function Return Value .. INTEGER :: date_to_weekday_number + + ! .. + + ! .. Structure Arguments .. TYPE (date), INTENT (IN) :: x + + ! .. + + ! .. Intrinsic Functions .. INTRINSIC modulo + + ! .. date_to_weekday_number $=\quad\&$ modulo(( $\beth3\,\star$ (x%month+10 $\&$ -(x%month $_{\cdot+10}$ ) $/\,\bot\,3^{\star}\bot2$ )-1) $\,/\,5\,{+}\,\mathrm{x}$ & %day $\cdot+77\,{+}\,5\,^{\star}$ (x%year $^+$ (x%month-14)/12 & -(x%year $^+$ (x%month-14)/12)/ $^{\prime}\,\mathbb{1}\,0\,0\,{^\star}\,\mathbb{1}\,0\,0$ )/4 & +(x%year $^+$ (x%month-14)/12)/400 & -(x%year $^+$ (x%month-14)/12)/100\*2,7) END FUNCTION date_to_weekday_number FUNCTION year and day to date(year,day) RESULT (x) + + ! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + +! .. Function Return Value .. TYPE (date) :: x + + ! .. + + ! .. Scalar Arguments .. INTEGER, INTENT (IN) :: day, year + + ! .. + + ! .. Local Scalars .. INTEGER :: t + + ! .. + + ! .. Intrinsic Functions .. INTRINSIC modulo + + ! .. x%year $=$ year $\mathrm{~\ensuremath~{~\mathsf~{~t~}~}~}=\mathrm{~\ensuremath~{~0~}~}$ IF (modulo(year,4) $==0$ ) THEN $\mathrm{~\textit~{~\textcent~}~}=\mathrm{~\it~1~}$ END IF + + !------the following statement is + + ! necessary IF year is $<$ 1900 or $>$ 2100. IF (modulo(year,400)/ $=\!0$ & .AND. modulo(year, $\perp0\,0\,)==0$ ) THEN $\mathrm{~\ensuremath~{~\mathsf~{~t~}~}~}=\mathrm{~\ensuremath~{~0~}~}$ END IF x%day $=$ day IF ( $\mathtt{d a y}{>}59\mathtt{+t}$ ) THEN x%day $=$ x%day $^+$ 2 - t END IF x%month $=$ ((x%day+91)\*100)/3055 x%day $=$ (x%day+91) - (x%month\*3055)/100 x%month $=$ x%month - 2 IF (x%month> $\mathbf{\omega}=\mathbb{1}$ .AND. x%month $_<=\!\perp\!2$ ) THEN RETURN END IF + + ! x%month will be correct WRITE (unit $=\star$ ,fmt $="$ (a,i11,a)') & '\$\$year and day to date: day of the year input $="$ , + +& day, ' is out of range.' + +END FUNCTION year and day to date FUNCTION julian_to_date(julian) RESULT (x) + +! Given a julian day number the date is returned. + + ! julian is the julian date from an epocch + + ! in the very distant past. see cacm 1968 11(10):657, + + ! letter to the editor by fliegel and van flandern. + + ! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Scalar Arguments .. INTEGER, INTENT (IN) :: julian + + ! .. + + ! .. Local Scalars .. INTEGER :: l, n + + ! .. + + ! .. Function Return Value .. TYPE (date) :: x + + ! .. $\begin{array}{r l}{\bot}&{{}=}\end{array}$ julian + 68569 n = $\underline{{4}}\,{}^{\star}\,\underline{{1}}\,{}/\,\underline{{1}}\,4\,6\,0\,9\,7$ $\begin{array}{r l r}{\perp}&{{}=}&{\perp}\end{array}$ - (146097\*n+3)/4 $\begin{array}{l l l}{\tt x\mathrm{\textdegree}\tt a r}&{=}&{4\,0\,0\,0\,^{\star}\,\left(\,\tt l\!+\!\!1\,\right)\,/\,1\,4\,6\,\tt l\,0\,0\,\tt l}\end{array}$ $\texttt{l}=\texttt{l}-\texttt{l}461^{\star}\mathbf{x}_{\circ}^{\circ}\mathbf{y e a r}/4\,\,\,+\,\,\,31$ x%month = $80^{\star}\mathbb{1}/2447$ $\mathbf{x\,\%\,day}\ =\ \mathrm{\texttt{l}}\ -\ 2447\,\star\mathbf{x\,\%m o n t h}/80$ $\begin{array}{r l}{\bot}&{{}=}\end{array}$ x%month/11 x%month $=$ x%month $^+$ 2 - 12\*l x%year $=$ 100\*(n-49) $^+$ x%year + l + +END FUNCTION julian_to_date + +SUBROUTINE & julian to date and week and day(jd,x,wd,ddd) + +! given jd, a julian day # (see asf jd), + + ! this routine calculates dd, + + ! the day number of the month; + + ! mm, the month number; yyyy the year; + + ! wd the weekday number, and + + ! ddd the day number of the year. + + ! example: + + ! CALL julian to date and week and day + + ! (2440588, yyyy, mm, dd, wd, ddd) + + ! yields 1970 1 1 4 1. + + ! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Scalar Arguments .. INTEGER, INTENT (OUT) :: ddd, wd INTEGER, INTENT (IN) :: jd + + ! .. + + ! .. Structure Arguments .. TYPE (date), INTENT (OUT) :: x + + ! .. $\mathrm{~\v~{~x~}~}=$ julian_to_date(jd) wd $=$ date_to_weekday_number(x) ddd $=$ date to day in year(x) + +END SUBROUTINE julian to date and week and day FUNCTION calendar to julian(x) RESULT (ival) + +! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Function Return Value .. INTEGER :: ival + + ! .. + + ! .. Structure Arguments .. TYPE (date), INTENT (IN) :: x + + ! .. + + ! date routine calendar to julian converts date to + + ! julian date. see cacm 1968 11(10):657, + + ! letter to the + + ! editor by henry f. fliegel and + +! example calendar to julian(1970, 1, 1) = 2440588 ival $=$ x%day - 32075 & $^+$ 1461\*(x%year+4800+(x%month-14)/12) $/\,4\,\,\,\,+\,\,\,\,\,\&$ 367\*(x%month-2-((x%month-14)/12) $^{\star}\bar{\bot}2$ )/12 & - 3\*((x%year+4900+(x%month-14)/ & 12)/100)/4 + +END FUNCTION calendar to julian FUNCTION ndays(date1,date2) + +! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Function Return Value .. INTEGER :: ndays + + ! .. + + ! .. Structure Arguments .. TYPE (date), INTENT (IN) :: date1, date2 + + ! .. + + ! dates; that is mm1/dd1/yyyy1 minus + + ! mm2/dd2/yyyy2, + + ! where datei and datej have elements mm, dd, yyyy. + + ! ndays will be positive iff + + ! date1 is more recent than date2. ndays $=$ calendar to julian(date1) & - calendar to julian(date2) + +END FUNCTION ndays + +SUBROUTINE date_stamp(string,want_day,short) + +! Returns the current date as a character string + + ! e.g. + +! want_day short string + + ! .TRUE. .TRUE. Thursday, 23 Dec 1999 + + ! .TRUE. .FALSE. Thursday, 23 December 1999 + +! <- defaul/ + +! .FALSE. .TRUE. 23 Dec 1999 + + ! .FALSE. .FALSE. 23 December 1999 + + ! .. Implicit None Statement .. + +# IMPLICIT NONE + +! .. + + ! .. Scalar Arguments .. LOGICAL, OPTIONAL, INTENT (IN) :: short, want_day CHARACTER (\*), INTENT (OUT) :: string + + ! .. + + ! .. Local Scalars .. INTEGER :: pos LOGICAL :: sh, want_d + + ! .. + + ! .. Local Arrays .. INTEGER :: val(8) CHARACTER (9) :: day(0:6) = (/ 'Sunday ' & , 'Monday ' & , 'Tuesday ' & , 'Wednesday' & , 'Thursday ' & , 'Friday ' & , 'Saturday '/) CHARACTER (9) :: month(1:12) = & (/ 'January ' & , 'February ' & , 'March ' & , 'April ' & , 'May ' & , 'June ' & , 'July ' & , 'August ' & , 'September' & , 'October ' & , 'November ' & , 'December '/) + + ! .. + + ! .. Intrinsic Functions .. INTRINSIC date_and_time, len_trim, present, trim + + ! .. + + ! .. Local Structures .. TYPE (date) :: x + + ! .. want_d $=$ .TRUE. IF (present(want_day)) want_d $=$ want_day sh $=$ .FALSE. IF (present(short)) sh $=$ short + +CALL date_and_time(values $=$ val) x = date_(val(3),val(2),val(1)) IF (want_d) THEN pos $=$ date_to_weekday_number(x) string $=$ trim(day(pos)) // ',' pos $=$ len_trim(string) $^+$ 2 ELSE $\begin{array}{r l r}{\mathsf{p o s}}&{{}=}&{\mathbb{1}}\end{array}$ string $=$ ' ' END IF WRITE (string(pos:pos $+1$ ),'(i2)') val(3) IF (sh) THEN string( $\mathtt{(p o s+3:p o s+5)}$ ) $=$ month(val(2)) (1:3) $\mathsf{p o s}\ \ =\ \ \mathsf{p o s}\ \ +\ \ 7$ ELSE string $\begin{array}{r l}{(\underbrace{\mathrm{pOS}}+3:)}&{{}=}\end{array}$ month(val(2))pos $=$ len_trim(string) $^+$ 2 END IF WRITE (string(pos:pos $^{+3}$ ),'(i4)') val(1) RETURN END SUBROUTINE date_stamp FUNCTION date_(dd,mm,yyyy) RESULT (x) + + ! .. Implicit None Statement .. IMPLICIT NONE + +! .. Function Return Value .. TYPE (date) :: x + + ! .. + + ! .. Scalar Arguments .. INTEGER, INTENT (IN) :: dd, mm, yyyy + + ! .. $\mathrm{~\v~{~x~}~}=$ date(dd,mm,yyyy) END FUNCTION date_ + +# Case Studies + +FUNCTION get_year(x) + +! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Function Return Value .. INTEGER :: get_year + + ! .. + + ! .. Structure Arguments .. TYPE (date), INTENT (IN) :: x + + ! .. get_year $=$ x%year END FUNCTION get_year FUNCTION get_month(x) + + ! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Function Return Value .. INTEGER :: get_month + + ! .. + + ! .. Structure Arguments .. TYPE (date), INTENT (IN) :: x + + ! .. get_month $=$ x%month END FUNCTION get_month FUNCTION get_day(x) + + ! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Function Return Value .. INTEGER :: get_day + + ! .. + + ! .. Structure Arguments .. TYPE (date), INTENT (IN) :: x + + ! .. get_day $=$ x%day END FUNCTION get_day + +END MODULE date_module + +PROGRAM ch2611 + +! .. Use Statements .. USE date_module, ONLY : calendar_to_julian, & date, date_, & date_stamp, & date to day in year, & date_to_weekday_number, & get_day, & get_month, & get_year, & julian to date and week and day, & ndays, & year and day to date + + ! .. + + ! .. Implicit None Statement .. IMPLICIT NONE + + ! .. + + ! .. Local Scalars .. INTEGER :: dd, ddd, i, mm, ndiff, wd, yyyy CHARACTER (50) :: message + + ! .. + + ! .. Local Arrays .. INTEGER :: val(8) + + ! .. + + ! .. Intrinsic Functions .. + + ! compute date this year for changing clocks + + ! back to est. + + ! i.e.compute date for the last + + ! sunday in october for this year. INTRINSIC date_and_time + + ! .. + + ! .. Local Structures .. TYPE (date) :: date1, date2, x + + ! .. + + ! Test date_stamp message $=$ ' date_stamp $=$ ' CALL date_stamp(message(15:)) WRITE (\*,'(a)') message message $=$ ' date_stamp $=$ ' CALL date_stamp(message(15:),want_day $=$ .FALSE.) WRITE $(\mathrm{~\d~}^{\star}\mathrm{~,~}^{\bullet}\mathrm{~(~a~)~}^{\bullet}\mathrm{~.~}$ ) message + +message $=$ ' date_stamp $=$ ' CALL date_stamp(message(15:),short $=$ .TRUE.) WRITE (\*,'(a)') message message $=$ ' date_stamp $=$ ' CALL date_stamp & (message(15:),want_day ${}^{\ast}{=}$ .FALSE.,short $=$ .TRUE.) WRITE $(\mathrm{\boldmath~\star~},\mathrm{\boldmath~\iota~}({\mathsf{a}})\mathrm{\boldmath~\iota~})$ ) message + +CALL date_and_time(values $=$ val) + +yyyy $=$ val(1) $\mathrm{{mm}~{~=~}~}10$ + +DO $\begin{array}{r l r}{\dot{\bot}}&{{}=}&{3\bot}\end{array}$ , 26, -1 x $=$ date_(i,mm,yyyy) IF (date to weekday number $\left({{\bf x}}\right)\mathrm{{==}}0$ ) THEN PRINT \*, 'turn clocks back to est on: ' print \*, i, ' october ', get_year(x) EXIT END IF END DO + +! compute date this year for + + ! turning clocks ahead to dst + + ! i.e., compute date for the first + + ! sunday in april for this year. + +CALL date_and_time(values $=$ val) + +yyyy $=$ val(1) $\mathrm{{mm}~{~=~}~{4}}$ + +DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 8 $\mathrm{~\v~{~x~}~}=$ date_(i,mm,yyyy) IF (date to weekday number $\left({{\bf x}}\right)\mathrm{{==}}0$ ) THEN PRINT \*, 'turn clocks ahead to dst on: ' print \*, i, ' april ', get_year(x) EXIT END IF END DO + +CALL date_and_time(values $=$ val) yyyy $=$ val(1) $\mathrm{{mm}}~=~12$ $\l{\mathrm{~\tt~{~d~d~}~}}=\mathrm{~\tt~{~3~}~}\!\mathrm{~\tt~{~1~}~}$ $\mathrm{~\v~{~x~}~}=$ date_(dd,mm,yyyy) + +! is this a leap year? i.e., is + + ! 12/31/yyyy the 366th day of the year? + +IF (date to day in year $(\mathtt{x})\==366$ ) THEN PRINT \*, get_year(x), ' is a leap year' ELSE PRINT \*, get_year(x), ' is not a leap year' END IF + +$\mathrm{~\v~{~x~}~}=$ date_(1,1,1970) + +CALL julian to date and week and day & (calendar to julian(x),x,wd,ddd) + +IF (get_year(x) $/\!=\!\!1970$ .OR. & get_month $\mathrm{(\,x\,)~/\,=\,1\,}$ .OR. & get_day(x) $'\!=\!\perp$ .OR. &wd/ $\l=\!4$ .OR. $\mathsf{d}\mathsf{d}\mathsf{d}/\mathsf{=}\mathsf{1}\,,$ ) THEN PRINT \*, 'julian to date and week and day failed' print \*,' date, wd, ddd $=$ ', & get_year(x), get_month(x), get_day(x), wd, ddd STOP END IF + +! difference between to same + + ! months and days over 1 leap year is 366. + +date1 $=$ date_(22,5,1984) date2 $=$ date_(22,5,1983) ndiff $=$ ndays(date1,date2) yyyy $=\phantom{-}1970$ $\mathrm{~\v~{~x~}~}=$ year and day to date(yyyy,ddd) + +IF (ndiff/ $=\!3\,6\,6$ ) THEN PRINT \*, 'ndays failed; ndiff $=$ ', ndiff ELSE + +! recover month and day + + ! from year and day number. IF (get_month $\mathrm{(\,x\,)~/\,=\,1\,}$ .AND. get_day(x)/ $'\!=\!\perp$ ) THEN PRINT \*, 'year and day to date failed' print \*,' mma, dda $=$ ', get_month(x), & get_day(x) ELSE PRINT \*, '\*\* date manipulation subroutines' print \*, '\*\* simple test ok.' END IF END IF + +END PROGRAM ch2611 + +We also have an alternate form of array declaration in this program, which is given below. It is common in Fortran 77 style code: + +INTEGER :: val(8) + +The next major addition to this code would be a date checking routine to test the validity of dates. This would be called from within our constructor date_. This would mean that we could never have an invalid date when using the date_module. This is left as a programming exercise. + +# 26.12 Graphics example — dislin + +The following is a rewrite of the earlier tsunami plotting program. It now uses a publically available graphics library called dislin. This is available from + +• http://www.mps.mpg.de/dislin/ It is free for Linux operating systems. The tsunami plot can be found at + +• http://www.kcl.ac.uk/fortran + +There are some minor wrap problems with the code: + +PROGRAM ch2612 + +USE DISLIN + +LOGICAL :: trial, screen REAL :: long, lat screen $=$ .FALSE. trial $=$ .FALSE. + +CALL datain(trial) + +PRINT \*, ' Which region do you wish to plot?' PRINT \*, ' $\begin{array}{r l}{0}&{{}=}\end{array}$ all regions' PRINT \*, ' 1 = Hawaii' PRINT \*, ' 2 = New Zealand and South Pacific Islands' PRINT \*, ' $\begin{array}{r l}{3}&{{}=}\end{array}$ Papua New Guinea and Solomon Islands' PRINT \*, ' 4 = Indonesia' PRINT \*, ' 5 = Philippines' PRINT \*, ' 6 = Japan' PRINT \*, ' $7\ \mathrm{~=~}$ Kuril Islands and Kamchatka' PRINT \*, ' $\begin{array}{r l}{8}&{{}=}\end{array}$ Alaska incluing Aleutian Islands' PRINT \*, ' 9 = West Coast - North and Central America' PRINT \*, ' $\begin{array}{r l}{\mathcal{10}}&{{}=}\end{array}$ West Coast - South America' 120 READ (unit $=\star$ ,fmt $=\star$ ,end $\scriptstyle\varepsilon=\perp3\,0$ ,err $=\!\perp\!3\,0$ ) nreg 130 IF ((nreg $<\!0$ ) .OR. (nreg $\cdot{>}10$ )) THEN PRINT \*, ' Please input a number between 0 and 10 inclusive' GO TO 120 END IF + +! dislin initialisation routines + +! and setting of some basic components + +! of the plot + +! These are based on two program examples. + +! Choose a file format + +CALL METAFL('PDF') + +! da4l $=$ din a4 landscape 2970\*2100 points + +CALL SETPAG('DA4L') + +! Initialise dislin + +CALL DISINI + +! Plot a border round the page + +CALL PAGERA + + ! Choose font CALL PSFONT('HELVETICA') + + ! argument is the thickness of the frame in plot coordinates. CALL FRAME(3) + + ! determines the position of an axis system. + + ! the lower left corner of the axis system CALL AXSPOS(400,1850) + + ! The size of the axis system + + ! are the length and height of an axis system in plot coordinates. The default + + ! values are set to 2/3 of the page length and height. CALL AXSLEN(2400,1400) + + ! Define axis title CALL NAME('Longitude','X') + + ! Define axis title CALL NAME('Latitude','Y') + + ! This routine plots a title over an axis system. CALL TITLIN('Plot of 3034 Tsunami events ',3) + + ! determines which label types will be plotted on an axis. ! MAP defines geographical labels which are plotted as non negative floating-point + +# Chapter 26 + +! numbers with the following characters ’W’, ’E’, ’N’ and ’S’. + +CALL LABELS('MAP','XY') + +! plots a geographical axis system. + +CALL GRAFMP(-180.,180.,-180.,90.,-90.,90.,-90.,30.) + +! The statement CALL GRIDMP (I, J) overlays an axis system with a longitude ! and latitude grid where I and J are the number of grid lines between labels in + + ! the X- and Y-direction. + +CALL GRIDMP(1,1) + +! The routine WORLD plots coastlines and lakes. + +CALL WORLD + +! The angle and height of the characters can be changed with the routines AN-GLE + +! and HEIGHT. + +CALL HEIGHT(50) + +! This routine plots a title over an axis system. The title may contain up to four lines of text + +designated + +! with TITLIN. + +CALL TITLE + +! This is a call to the convert routine. + + ! This was required by UNIRAS + + ! CALL convrt(trial) + +! This is a call to the routine that actually plots each event. + +CALL plotem(trial,nreg) + +! DISFIN terminates DISLIN and prints a message on the screen. The level is set back to 0. + +CALL DISFIN + +END PROGRAM ch2612 + +SUBROUTINE datain(trial) + +COMMON /TSUNAM/ & reg0la(378) , & reg0lo(378) , & reg1la(206) , & reg1lo(206) , & reg2la(41) , & reg2lo(41) , & reg3la(54) , & reg3lo(54) , & reg4la(60) , & reg4lo(60) , & reg5la(1540) , & reg5lo(1540) , & reg6la(80) , ®6lo(80) , & reg7la(144) , & reg7lo(144) , & reg8la(245) , & reg8lo(245) , & reg9la(285) , & reg9lo(285) + +LOGICAL :: trial CHARACTER (80) :: filnam + +IF (trial) THEN PRINT \*, ' Entering data input phase' END IF filnam $=$ 'tsunami.dat' OPEN (unit $=\!5\,0$ ,file $=$ filnam,err $\scriptstyle{\stackrel{\cdot}{=}\,}1\,0\,0$ ,status $=$ 'OLD') + +# Chapter 26 + +GO TO 110 100 PRINT \*, ' Error opening data file' PRINT \*, ' Program terminates' STOP 110 DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 378 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg0la(i), reg0lo(i) END DO 1000 FORMAT (1X,F7.2,2X,F7.2) DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 206 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg1la(i), reg1lo(i) END DO DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 41 READ (unit $\mathtt{=50}$ ,fmt $\mathbf{\varepsilon}\!=\!\mathbb{1}000$ ) reg2la(i), reg2lo(i) END DO DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 54 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg3la(i), reg3lo(i) END DO DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 60 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg4la(i), reg4lo(i) END DO DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 1540 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg5la(i), reg5lo(i) END DO DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 80 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg6la(i), reg6lo(i) END DO DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 144 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg7la(i), reg7lo(i) END DO DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 245 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg8la(i), reg8lo(i) END DO DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 285 READ (unit $\mathtt{=50}$ ,fmt $\b=\beth\,0\,0\,0$ ) reg9la(i), reg9lo(i) END DO IF (trial) THEN DO $\begin{array}{r l r}{\dot{\mathrm{~\boldmath~\lambda~}}_{\perp}}&{{}=}&{\mathbb{1}}\end{array}$ , 10 PRINT \*, reg0la(i), ' ', reg0lo(i) END DO PRINT \*, ' Exiting data input phase' READ \*, dummy END IF + +END SUBROUTINE datain + +SUBROUTINE plotem(trial,nreg) + +# USE DISLIN + +COMMON /TSUNAM/ & reg0la(378) , & reg0lo(378) , & reg1la(206) , & reg1lo(206) , & reg2la(41) , & reg2lo(41) , & reg3la(54) , & reg3lo(54) , & reg4la(60) , & reg4lo(60) , & reg5la(1540) , & reg5lo(1540) , & reg6la(80) , ®6lo(80) , & reg7la(144) , & reg7lo(144) , & reg8la(245) , & reg8lo(245) , & reg9la(285) , & reg9lo(285) LOGICAL :: trial INTEGER :: nreg INTEGER :: kolour=10 DATA dwidth/1.0/ IF (trial) THEN dwidth $=\quad5\cdot0$ PRINT \*, ' Entering Plot points' END IF + +CALL INCMRK(-1) + +IF ( $\mathtt{n r e g}{=}0$ ) THEN + +CALL SETCLR(kolour) CALL CURVMP(reg0lo,reg0la,378) kolour $=$ kolour $+30$ CALL SETCLR(kolour) CALL CURVMP(reg1lo,reg1la,206) kolour $=$ kolour $+30$ CALL SETCLR(kolour) CALL CURVMP(reg2lo,reg2la,41) kolour $=$ kolour $+30$ CALL SETCLR(kolour) CALL CURVMP(reg3lo,reg3la,54) kolour $=$ kolour $+30$ CALL SETCLR(kolour) CALL CURVMP(reg4lo,reg4la,60) kolour $=$ kolour $+30$ CALL SETCLR(kolour) CALL CURVMP(reg5lo,reg5la,1540) kolour $=$ kolour $+30$ CALL SETCLR(kolour) CALL CURVMP(reg6lo,reg6la,80) kolour $=$ kolour $+30$ CALL SETCLR(kolour) CALL CURVMP(reg7lo,reg7la,144) kolour $=$ kolour $+30$ CALL SETCLR(kolour) CALL CURVMP(reg8lo,reg8la,245) kolour $=$ kolour $+30$ CALL SETCLR(kolour) CALL CURVMP(reg9lo,reg9la,285) ELSE IF ( $\mathfrak{n r e g}{=}\mathbb{1}$ ) THEN kolour $=\quad\bot0$ CALL SETCLR(kolour) CALL CURVMP(reg0lo,reg0la,378) ELSE IF (nreg $==2$ ) THEN kolour $=\quad2\,0$ CALL SETCLR(kolour) CALL CURVMP(reg1lo,reg1la,206) ELSE IF ( $\mathtt{n r e g}{=}{3}$ ) THEN kolour $=\quad3\,0$ CALL SETCLR(kolour) CALL CURVMP(reg2lo,reg2la,41) ELSE IF ( $\mathtt{n r e g}{=}{4}$ ) THEN + +kolour = 40 CALL SETCLR(kolour) CALL CURVMP(reg3lo,reg3la,54) ELSE IF ( $\mathtt{n r e g}{=}{5}$ ) THEN kolour $=\quad5\,0$ CALL SETCLR(kolour) CALL CURVMP(reg4lo,reg4la,60) ELSE IF ( $\mathtt{n r e g}{=}{6}$ ) THEN kolour $=\quad6\,0$ CALL SETCLR(kolour) CALL CURVMP(reg5lo,reg5la,1540) ELSE IF (nreg $==7$ ) THEN kolour $=\quad7\,0$ CALL SETCLR(kolour) CALL CURVMP(reg6lo,reg6la,80) ELSE IF ( $\mathtt{n r e g}{=}{8}$ ) THEN kolour $=\quad8\,0$ CALL SETCLR(kolour) CALL CURVMP(reg7lo,reg7la,144) ELSE IF ( $\mathtt{n r e g}{=}{=}9$ ) THEN kolour $=\quad9\,0$ CALL SETCLR(kolour) CALL CURVMP(reg8lo,reg8la,245) ELSE IF (nreg $==\!\!\perp0$ ) THEN kolour $=\quad\mathbb{1}_{}00$ CALL SETCLR(kolour) CALL CURVMP(reg9lo,reg9la,285) END IF IF (trial) THEN PRINT \*, ' Exiting Plot points' END IF + +END SUBROUTINE plotem + +# 26.13 Problems + +1. Compile and run ch2606. Try running it with matrices of your own choice. + +2. Write a generic subroutine Swap which two takes arguments a and b (real or in- teger) and swaps them. Write a main program that reads two real values and calls Swap and then reads two integer values and again calls Swap. + +3. Using ch2605 as a starting point extend the program to overload the subtraction operator $(-)$ , for the user defined type position. + +4. Modify the elemental function example to include usage of the function etox with a rank 2 or higher array. Also add a call to the intrinsic function exp and compare the results. + +5. Using the balanced tree example as a basis and modify it to work with a charac- ter array rather than an integer. The routine that prints the tree will also have to be modified to reflect this. + +6. Compile and run the program that calculated the inner product of two sparse vectors using the data supplied. This should produce the answer 19. + +# 26.14 Bibliography + +Duff I.S., Erismon A.M., Reid J.K., Direct Methods for Sparse Matrices , Oxford Science Publications, 1986. + +• Authoritative coverage of this area. Relatively old, but well regarded. Code segments and examples are a mixture of Fortran 77 and Algol 60 (which of course do not support pointers) and therefore the implementa- tion of linked lists is done using the existing features of these languages. The onus is on the programmer to correctly implement linked lists using fixed size arrays rather than using the features provided by pointers in a language. It is remarkable how elegant these solutions are, given the lack of dynamic data structures in these two languages. + +Hopkins T., Phillips C., Numerical Methods in Practice, Using the NAG Library . Addison-Wesley, 1988. + +• Good adjunct to the NAG library documentation for the less numerate user. + +Schneider G.M., Bruell S.C., Advanced Programming and Problem Solving with Pascal, Wiley, 1981. + +The book is aimed at computer science students and follows the curricu- lum guidelines laid down in Communications of the ACM , August 1985, Course CS2. The book is very good for the complete beginner as the ex- amples are very clearly laid out and well explained. There is a coverage of data structures, abstract data types and their implementation, algorithms for sorting and searching, the principles of software development as they relate to the specification, design, implementation and verification of pro- grams in an orderly and disciplined fashion — their words. + +Vowels R.A., Algorithms and Data Structures in $F$ and Fortran, Unicomp, 1998. + +• The only book currently that uses Fortran 90/95 and F. Visit the Fortran web site for more details. They are the publishers. + +• http://www.fortran.com/fortran/market.html Wirth N., Algorithms $^+$ Data Structures $=$ Programs , Prentice-Hall, 1976. • An early but illuminating book on the subject. Well worth a read. Pascal is used. Wirth N., Algorithms $^+$ Data Structures , Prentice-Hall, 1986. • This is the Modula 2 version. Closer to Fortran than the Pascal version. + +# ISO TR 15580 IEEE Arithmetic + +“'Can you do addition?' the White Queen asked. 'What's one and one and one and one and one and one and one and one and one and one?'” + +'I don't know,' said Alice. 'I lost count.' + +Lewis Carroll, Through the Looking Glass and What Alice Found There + +# Aims + +The aims of this chapter are to look in more depth at arithmetic and in particular at the support that Fortran provides for the IEEE 754 standard. There is a coverage of: + +• Hardware support for arithmetic. • Integer formats. • Floating point formats: single and double. • Special values: denormal, infinity and not a number — NAN. • Exceptions and flags: divide by zero, inexact, invalid, overflow, under- flow. + +# 27 ISO TR 15580 — IEEE Arithmetic + +The literature contains details of the IEEE 754 standard and the bibliography con- tains details of a number of printed and on-line sources. + +# 27.1 History + +When we use programming languages to do arithmetic two major concerns are the ability to develop reliable and portable numerical software. Arithmetic is done in hardware and there are a number of things to consider: + +• The range of hardware available both now and in the past. • The evolution of hardware. + +There has been a very considerable change in arithmetic units since the first com- puters. The following is a list of hardware and computing systems that the authors have some used or have heard of. It is not exhaustive or definitive, but rather re- flects the authors' age and experience: + +• CDC • Cray • IBM • ICL • Fujitsu • DEC • Compaq • Gateway • Sun • Silicon Graphics • Hewlett Packard • Data General • Honeywell • Elliot • Mostek • National Semiconductors • Intel + +• Zilog • Motorola • Signetics • Amdahl • Texas Instruments • Cyrix + +Some of the operating systems include: + +• NOS • NOS/BE • Kronos • UNIX • VMS • Dos • Windows 3.x • Windows 95 • Windows 98 • Windows NT • Windows 2000 • MVS • VM • CP/M • Macintosh • OS/2 + +Again the list is not exhaustive or definitive. The intention is simply to provide some idea of the wide range of hardware, computer manufacturers and operating systems that have been around in the past 50 years. + +To cope with the anarchy in this area Doctor Robert Stewart (acting on behalf of the IEEE) convened a meeting which led to the birth of IEEE 754. + +The first draft, which was prepared by William Kahan, Jerome Coonen and Harold Stone, was called the KCS draft and eventually adopted as IEEE 754. A fascinat- ing account of the development of this standard can be found in An Interview with the Old Man of Floating Point , and the bibliography provides a web address for this interview. Kahan went on to get the ACM Turing Award in 1989 for his work in this area. + +This has become a de facto standard amongst arithmetic units in modern hardware. Note that it is not possible to describe precisely the answers a program will give, and the authors of the standard knew this. This goal is virtually impossible to achieve when one considers floating point arithmetic. Reasons for this include: + +• The conversions of numbers between decimal and binary formats. • The use of elementary library functions. • Results of calculations may be in hardware inaccessible to the program- mer. • Intermediate results in subexpressions or arguments to procedures. + +The bibliography contains details of a paper that addresses this issue in much greater depth — Differences Among IEEE 754 Implementations. + +Fortran is one of a small number of languages that provides access to IEEE arith- metic, and it achieves this via TR1880 which is an integral part of Fortran 2003. The C standard (C9X) addresses this issue and Java offers limited IEEE arithmetic support. More information can be found in the references at the end of the chapter. + +# 27.2 IEEE 754 Specifications + +The standard specifies a number of things including: + +• Single precision floating point format. • Double precision floating point format. • Two classes of extended floating point formats. • Accuracy requirements on the following floating point operations: • Add. • Subtract. • Multiply. • Divide. • Square root. • Remainder. • Round numbers in floating point format to integer values. + +• Convert between different floating point formats. • Convert between floating point and integer format. • Compare. • Base conversion, i.e., when converting between decimal and binary float- ing point formats and vice versa. • Exception handling for: • Divide by zero. • Overflow. • Underflow. • Invalid operation. • Inexact. • Rounding directions. • Rounding precisions. + +We will look briefly at each of these requirements. + +# 27.2.1 Single precision floating point format + +This is a 32-bit quantity made up of a sign bit, 8-bit biased exponent and 23-bit mantissa. The standard also specifies that certain of the bit patterns are set aside and do not represent normal numbers. This means that valid numbers are in the range $3.40282347\mathrm{E}+38$ to 1.17549435E-38 and the precision is between 6 and 9 digits depending on the numbers. + +The special bit patterns provide the following: + +• $+0$ • -0 • subnormal numbers in the range 1.17549421E-38 to 1.40129846E-45 • $^+$ infinity • - infinity • quiet NaN (Not a Number) • signalling NaN + +One of the first systems that the authors worked with that had special bit patterns set aside was the CDC 6000 range of computers that had negative indefinite and infinity. Thus the ideas are not new, as this was in the late 1970s. + +The support of positive and negative zero means that certain problems can be han- dled correctly including: + +• The evaluation of the log function which has a discontinuity at zero. • The equation $\sqrt{\mathscr{V}}_{z}=\mathscr{V}_{\sqrt{z}}$ can be solved when $z=-1$ . + +See also the Kahan paper Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit for more details. + +Subnormals, which permit gradual underflow, fill the gap between 0 and the small- est normal number. + +Simply stated underflow occurs when the result of an arithmetic operation is so small that it is subject to a larger than normal rounding error when stored. The ex- istence of subnormals means that greater precision is available with these small numbers than with normal numbers. The key features of gradual underflow are: + +• When underflow does occur there should never be a loss of accuracy any greater than that from ordinary roundoff. • The operations of addition, subtraction, comparison and remainder are al- ways exact. • Algorithms written to take advantage of subnormal numbers have smaller error bounds than other systems. • If $x$ and $y$ are within a factor of 2 then x-y is error free, which is used in a number of algorithms that increase the precision at critical regions. + +The combination of positive and negative zero and subnormal numbers means that when $x$ and $y$ are small and $x$ -y has been flushed to zero the evaluation of + +• $\mathcal{V}_{(x-y)}$ + +can be flagged and located. + +Certain arithmetic operations cause problems including: + +$$ +\begin{array}{l}{\bullet\quad\ 0\ ^{*}\infty}\\ {\bullet\quad\ 0\ /\ 0}\\ {\bullet\quad\ {\sqrt{x}}\ \,\mathrm{when}\ x<0}\end{array} +$$ + +and the support for $\mathrm{NaN}$ handles these cases. The support for positive and negative infinity allows the handling of • $x\mathrm{~/~}0$ when $x$ is nonzero and of either sign + +and the outcome of this means that we write our programs to take the appropriate action. In some cases this would mean recalculating using another approach. + +For more information see the references in the bibliography. + +# 27.2.2 Double precision floating point format + +This is a 64-bit quantity made up of a sign bit, 11-bit biased exponent and 52-bit mantissa. As with single precision the standard specifies that certain of the bit pat- terns are set aside and do not represent normal numbers. This means we have valid numbers in the range 1.7976931348623157 E 308 to 2.2250738585072014 E-308 and precision between 15 and 17 digits depending on the numbers. + +As with single precision there are bit patterns set aside for the same special condi- tions. + +Note that this does not mean that the hardware has to handle the manipulation of this 64-bit quantity in an identical fashion. The Sparc and Intel family handle the above as two 32-bit quantities but the order of the two component parts is reversed — so-called big endian and little endian. + +# 27.2.3 Two classes of extended floating point formats + +These formats are not mandatory. A number of variants of double extended exist including: + +• Sun — four 32-bit words, one sign bit, 15-bit biased exponent and 112-bit mantissa, numbers in the range 3.362E-4932 to 1.189E4932, 33–36 digits of significance. • Intel — 10 bytes — one sign bit, 15-bit biased exponent, 63-bit mantissa, numbers in the range 3.362E-4932 to 1.189E4932, 18–21 digits of signifi- cance. • PowerPC — as Sun. + +# 27.2.4 Accuracy requirements + +Remainder and compare must be exact. The rest should return the exact result if possible. If not, there are well-defined rounding rules to apply. + +# 27.2.5 Base conversion — Converting between decimal and binary floating point formats and vice versa + +These results should be exact if possible; if not the results must differ by toler- ances that depend on the rounding mode. + +# Chapter 27 + +# 27.2.6 Exception handling + +It must be possible to signal to the user the occurrence of the following conditions or exceptions: + +• Divide by zero. • Overflow. • Underflow. • Invalid operation. • Inexact. + +The ability to detect the above is a big step forward in our ability to write robust and portable code. These operations do occur in calculations and it is essential to have user programmer control over what action to take. + +# 27.2.7 Rounding directions + +Four rounding directions are available: + +• Nearest — the default. • Down. • Up. • Chop. + +Access to directed rounding can be used to implement interval arithmetic, for ex- ample. + +# 27.2.8 Rounding precisions + +The only mandatory part here is that machines that perform computations in ex- tended mode let the programmer control the precision via a control word. This means that if software is being developed on machines that support extended modes those machines can be switched to a mode that would enable the software to run on a system that didn't support extended modes. This area looks like a can of worms. Look at the Kahan paper for more information — Lecture Notes on the Status of IEEE 754 . + +# 27.3 Resumé + +The above has provided a quick tour of IEEE 754. We'll now look at what Fortran has to offer to support it. + +# 27.4 ISO TR 15580 + +Fortran provides access to the facilities via the USE statement. The current stan- dard does not have the concept of an intrinsic module. TR 15580 introduces this concept. Three modules are provided: + +• ieee_features • ieee_exceptions • ieee_arithmetic + +The first thing to consider is the degree of conformance to the IEEE standard. It is possible that not all of the features are supported. Thus the first thing to do is to run one or more test programs to determine the degree of support for a particular system. + +# 27.4.1 IEEE_FEATURES module + +This module defines a derived type, IEEE FEATURES TYPE, and up to 11 con- stants of that type representing IEEE features: + +• IEEE_DATATYPE — whether any IEEE data types are available. • IEEE_DENORMAL — whether IEEE denormal values are available. • IEEE_DIVIDE — whether division has the accuracy required by IEEE. • IEEE_HALTING — whether control of halting is supported. • IEEE INEXACT FLAG — whether the inexact exception is supported. • IEEE_INF — whether IEEE positive and negative infinities are available. • IEEE INVALID FLAG — whether the invalid exception is supported. • IEEE_NAN — whether IEEE NaNs are available. • IEEE_ROUNDING — whether all IEEE rounding modes are available. • IEEE_SQRT — whether SQRT conforms to the IEEE standard. • IEEE UNDERFLOW FLAG — whether underflow is supported. + +# 27.4.2 IEEE_EXCEPTIONS module + +This module provides data types, constants and generic procedures for IEEE ex- ceptions: + +TYPE IEEE_STATUS_TYPE + +Variables of this type can hold a floatingpoint status value. SUBROUTINE IEEE_GET_STATUS(STATUS_VALUE) + +TYPE(IEEE_STATUS_TYPE),INTENT(OUT) :: STATUS_VALUE Stores the current floatingpoint status into the STATUS_VALUE argument. SUBROUTINE IEEE_SET_STATUS(STATUS_VALUE)TYPE(IEEE_STATUS_TYPE),INTENT(IN) :: STATUS_VALUE Sets the current floatingpoint status from the STATUS_VALUE argument. TYPE IEEE_FLAG_TYPE Values of this type specify individual IEEE exception flags; constants for these are available as follows: TYPE(IEEE_FLAG_TYPE),PARAMETER :: IEEE DIVIDE BY ZERO TYPE(IEEE_FLAG_TYPE),PARAMETER :: IEEE_INEXACT TYPE(IEEE_FLAG_TYPE),PARAMETER :: IEEE_INVALID TYPE(IEEE_FLAG_TYPE),PARAMETER :: IEEE_OVERFLOW TYPE(IEEE_FLAG_TYPE),PARAMETER :: IEEE_UNDERFLOW In addition, two array constants are available for indicating common combinations of flags: TYPE(IEEE_FLAG_TYPE),PARAMETER :: & IEEE_USUAL(3) $=$ (/& IEEE_DIVIDE_BY_ZERO,& IEEE_INVALID, & IEEE_OVERFLOW $/$ ), & IEEE_ALL(5) $=$ (/& IEEE_DIVIDE_BY_ZERO,& IEEE_INEXACT, & IEEE_INVALID,& IEEE_OVERFLOW, & IEEE_UNDERFLOW /) LOGICAL FUNCTION IEEE SUPPORT FLAG(FLAG,X) TYPE(IEEE_FLAG_TYPE),INTENT(IN) :: FLAG REAL(kind),INTENT(IN),OPTIONAL :: X + +Returns TRUE if detection of the specified IEEE exception is supported for the REAL kind of X (if X is present), or for all REAL kinds (if X is absent). LOGICAL FUNCTION IEEE SUPPORT HALTING(FLAG) TYPE(IEEE_FLAG_TYPE),INTENT(IN) :: FLAG Returns TRUE if IEEE SET HALTING MODE can be used to change whether the processor terminates the program on receiving the specified exception. ELEMENTAL SUBROUTINE & IEEE_GET_FLAG(FLAG,FLAG_VALUE) TYPE(IEEE_FLAG_TYPE),INTENT(IN) :: FLAG LOGICAL,INTENT(OUT) :: FLAG_VALUE Sets (each element of) FLAG_VALUE to TRUE if the corresponding exception specified by FLAG is signalling, and to FALSE otherwise. ELEMENTAL SUBROUTINE & IEEE_GET_HALTING_MODE(FLAG,HALTING) TYPE(IEEE_FLAG_TYPE),INTENT(IN) :: FLAG LOGICAL,INTENT(OUT) :: HALTING Sets (each element of) HALTING to TRUE if the corresponding exception speci- fied by FLAG is signalling, and to FALSE otherwise. ELEMENTAL SUBROUTINE IEEE_SET_FLAG(FLAG,FLAG_VALUE) TYPE(IEEE_FLAG_TYPE),INTENT(OUT) :: FLAG LOGICAL,INTENT(IN) :: FLAG_VALUE Sets the exception flag specified by (each element of) FLAG to signalling or quiet according to the corresponding element of FLAG_VALUE. ELEMENTAL SUBROUTINE & IEEE_SET_HALTING_MODE(FLAG,HALTING)TYPE(IEEE_FLAG_TYPE),INTENT(OUT) :: FLAG LOGICAL,INTENT(IN) :: HALTING Sets the halting mode for each exception specified by FLAG to the value of the corresponding element of HALTING (TRUE $=$ halt). 27.4.3 IEEE_ARITHMETIC module These are given below: + +# 27.4.3.1 IEEE data type selection + +INTEGER FUNCTION SELECTED_REAL_KIND(P,R) + +INTEGER(kind1),OPTIONAL :: P + +INTEGER(kind2),OPTIONAL :: R + +The same as the SELECTED REAL KIND intrinsic, but only returns information about the IEEE kinds of reals. 27.4.3.2 General support enquiry functions LOGICAL FUNCTION IEEE SUPPORT DATA TYPE(X) REAL(kind),OPTIONAL :: X Whether IEEE arithmetic is supported for the same kind of REAL as $\mathrm{X}$ (or for all REAL kinds if X is absent). LOGICAL FUNCTION IEEE SUPPORT DENORMAL(X) REAL(kind),OPTIONAL :: X Whether IEEE denormal values are supported for the same kind of REAL as X (or for all REAL kinds if X is absent). LOGICAL FUNCTION IEEE SUPPORT DIVIDE(X) REAL(kind),OPTIONAL :: X Whether division is carried out to the accuracy specified by the IEEE standard for the same kind of REAL as X (or for all REAL kinds if X is absent). LOGICAL FUNCTION IEEE_SUPPORT_INF(X) REAL(kind),OPTIONAL :: X Whether IEEE infinite values are supported for the same kind of REAL as $\mathrm{X}$ (or for all REAL kinds if X is absent). LOGICAL FUNCTION IEEE SUPPORT NAN(X) REAL(kind),OPTIONAL :: X Whether IEEE NaN (Not-a-Number) values are supported for the same kind of REAL as X (or for all REAL kinds if X is absent). LOGICAL FUNCTION IEEE SUPPORT SQR T(X) REAL(kind),OPTIONAL :: X Whether SQRT conforms to the IEEE standard for the same kind of REAL as X (or for all REAL kinds if X is absent). + +LOGICAL FUNCTION IEEE SUPPORT STANDARD(X) REAL(kind),OPTIONAL :: X Whether all the IEEE facilities specified by the TR are supported for the same kind of REAL as X (or for all REAL kinds if X is absent). + +# 27.4.3.3 Rounding modes + +TYPE IEEE_ROUND_TYPE Values of this type specify the IEEE rounding mode. TYPE (IEEE_ROUND_TYPE) , & PARAMETER :: IEEE_DOWN TYPE (IEEE_ROUND_TYPE) , & PARAMETER :: IEEE_NEAREST TYPE (IEEE_ROUND_TYPE) , PARAMETER :: IEEE_TO_ZERO TYPE (IEEE_ROUND_TYPE) , PARAMETER :: IEEE_UP LOGICAL FUNCTION IEEE SUPPORT ROUNDING(ROUND_VALUE,X) TYPE(IEEE_ROUND_TYPE),INTENT(IN) :: ROUND_VALUE REAL(kind),OPTIONAL :: X Whether the specified IEEE rounding mode is supported for the same kind of REAL as X (or for all REAL kinds if X is absent). SUBROUTINE IEEE_GET_ROUNDING_MODE(ROUND_VALUE)TYPE(IEEE_ROUND_TYPE),INTENT(OUT) :: ROUND_VALUE Sets the ROUND_VALUE argument to the current IEEE rounding mode. SUBROUTINE IEEE_SET_ROUNDING_MODE(ROUND_VALUE)TYPE (IEEE_ROUND_TYPE) , INTENT(IN) :: ROUND_VALUE Sets the current IEEE rounding mode to that specified by ROUND_VALUE. + +# 27.4.3.4 Number classification + +TYPE IEEE_CLASS_TYPE Values of this type indicate the IEEE class of a number. TYPE (IEEE_CLASS_TYPE) , & PARAMETER :: IEEE NEGATIVE DENORMAL TYPE (IEEE_CLASS_TYPE) , PARAMETER :: IEEE NEGATIVE IN F + +TYPE (IEEE_CLASS_TYPE) , PARAMETER :: IEEE NEGATIVE NORMAL TYPE (IEEE_CLASS_TYPE) , PARAMETER :: IEEE_NEGATIVE_ZERO TYPE (IEEE_CLASS_TYPE) , PARAMETER :: IEEE POSITIVE DENORMAL TYPE (IEEE_CLASS_TYPE) , PARAMETER :: IEEE POSITIVE IN F TYPE(IEEE_CLASS_TYPE) , PARAMETER :: IEEE_POSITIVE_NORMAL TYPE(IEEE_CLASS_TYPE) , PARAMETER :: IEEE_POSITIVE_ZERO TYPE(IEEE_CLASS_TYPE) , PARAMETER :: IEEE_QUIET_NAN TYPE(IEEE_CLASS_TYPE) , PARAMETER :: IEEE signalling NAN ELEMENTAL TYPE(IEEE_CLASS_TYPE) FUNCTION IEEE_CLASS(X) REAL(kind),INTENT(IN) :: X Returns the appropriate value of IEEE_CLASS_TYPE for the number X, which may be of any IEEE kind. In addition to ISO/IEC TR 15580:1998(E), the module IEEE_ARITHMETIC de- fines the $"="$ and $"/="$ operators for the IEEE_CLASS_TYPE. These may be used to test the return value of the IEEE_CLASS function, e.g., USE,INTRINSIC :: IEEE_ARITHMETIC, ONLY: IEEE_CLASS, & IEEE_QUIET_NAN, OPERATOR( $\left(\stackrel{\footnotesize\mathrm{==}}{}\right)$ ) + +IF (IEEE_CLASS(X) $==$ IEEE_QUIET_NAN) THEN + +ELEMENTAL REAL(kind) FUNCTION IEEE_VALUE(X,CLASS) REAL(kind),INTENT(IN) :: X TYPE(IEEE_CLASS_TYPE),INTENT(IN) :: CLASS Returns a sample value of the specified class for the same kind of real as $\mathrm{X}$ , which may be of any IEEE kind. ELEMENTAL LOGICAL FUNCTION IEEE_IS_FINITE(X) REAL(kind),INTENT(IN) :: X Returns TRUE if $\mathrm{X}$ is not infinite or NaN. ELEMENTAL LOGICAL FUNCTION IEEE_IS_NAN(X) REAL(kind),INTENT(IN) :: X + +Returns TRUE if X is either a signalling or quiet NaN. ELEMENTAL LOGICAL FUNCTION IEEE_IS_NEGATIVE(X) REAL(kind),INTENT(IN) :: X Returns TRUE if X is negative, including negative zero. ELEMENTAL LOGICAL FUNCTION IEEE_IS_NORMAL(X) REAL(kind),INTENT(IN) :: X Returns TRUE if $\mathrm{X}$ is not an infinity, NaN, or denormal. ELEMENTAL LOGICAL FUNCTION IEEE_UNORDERED(X,Y) REAL(kind),INTENT(IN) :: X,Y Returns TRUE if X is a NaN or if Y is a NaN. 27.4.3.5 Arithmetic operations ELEMENTAL REAL(kind) FUNCTION IEEE_COPY_SIGN(X,Y) REAL (kind) , INTENT(IN) :: X,Y Returns X with the sign of Y, even for NaNs and infinities. ELEMENTAL REAL (kind) FUNCTION IEEE_LOGB(X) REAL (kind) , INTENT(IN) :: X Returns the unbiased exponent as a REAL value: If X is zero, IEEE DIVIDE BY ZERO signals and the result is –infinity if IEEE infinities are supported for that kind, and –HUGE(X) if not. If X is infinite, the result is +infinity. If X is a NaN, the result is a quiet NaN (the same one if $\mathrm{X}$ is a quiet NaN); other- wise the result is EXPONENT(X)-1. ELEMENTAL REAL (kind) FUNCTION IEEE_NEXT_AFTER(X,Y) REAL (kind) , INTENT(IN) :: X,Y The same as NEAREST(X,1.0_kind) for $\mathrm{Y}{>}\mathrm{X}$ and NEAREST(X,-1.0_kind) for $\mathrm{Y}{<}\mathrm{X}$ ; if $\mathrm{Y}{=}\mathrm{X}$ , the result is X, if either X or Y are NaNs the result is one of these NaNs. ELEMENTAL REAL (kind) FUNCTION IEEE_REM(X,Y) REAL (kind) , INTENT(IN) :: X,Y + +# Chapter 27 + +$\mathrm{{X}{\cdot}\mathrm{{Y}^{*}\mathrm{{N}}}}$ exactly, where $_\mathrm{N}$ is the integer nearest to the exact value X/Y. If the result is zero, it has the same sign as X. This function is not affected by the rounding mode. + +ELEMENTAL REAL (kind) FUNCTION IEEE_RINT(X) + +REAL (kind) , INTENT(IN) :: X + +Round to an integer according to the current rounding mode. + +ELEMENTAL REAL (kind) FUNCTION IEEE_SCALB(X,I) + +REAL (kind1) , INTENT(IN) :: X + +INTEGER (kind2) , INTENT(IN) :: I + +The same as SCALE(X,I). + +# 27.5 Summary + +Support for the above is relatively limited at the time of writing this book. There is always a time lag between the formal publication of a standard and the implemen- tation in production compilers. As compiler support improves examples will be added to our web site. Our home page is: + +• http://www.kcl.ac.uk/fortran + +# 27.6 Bibliography + +Hauser J.R., Handling Floating Point Exceptions in Numeric Programs, ACM Transaction on Programming Languages and Systems , Vol. 18, No. 2, March 1996, pp. 139–174. + +• The paper looks at a number of techniques for handling floating point ex- ceptions in numeric code. One of the conclusions is for better structured support for floating point exception handling in new programming lan- guages, or of course better standards for existing languages. + +IEEE, IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std 754-1985 , Institute of Electrical and Electronic Engineers Inc. + +• The formal definition of IEEE 754. Knuth D., Seminumerical Algorithms , Addison-Wesley, 1969. + +• There is a coverage of floating point arithmetic, multiple precision arith- metic, radix conversion and rational arithmetic. + +Sun, Numerical Computation Guide , SunPro. + +• Very good coverage of the numeric formats for IEEE Standard 754 for Binary Floating-Point Arithmetic. All SunPro compiler products support the features of the IEEE 754 standard. + +# 27.6.1 Web-based sources + +http://validgh.com/goldberg/addendum.html + +• Differences Among IEEE 754 Implementations. The material in this paper will eventually be included in the Sun Numerical Computation Guide as an addendum to Appendix D, David Goldberg's What Every Computer Scientist Should Know about Floating Point Arithmetic. + +http://docs.sun.com/ + +• Follow the links to the Floating Point and Common Tools AnswerBook . The Numerical Computation Guide can be browsed on-line or down- loaded as a pdf file. The last time we checked it was about 260 pages. Good source of information if you have Sun equipment. + +http://www.validgh.com/ + +• This web site contains technical and business information relating to the validgh professional consulting practice of David G. Hough. Contains links to the Goldberg paper and the above addendum by Doug Priest. + +http://babbage.cs.qc.edu/courses/cs341/IEEE-754references.html + +• Brief coverage of IEEE arithmetic with pointers to further sources. There is also a coverage of the storage layout and ranges of floating point num- bers. Computer Science 341 is an introduction to the design of a com- puter's hardware, particularly the CPU and memory systems. + +http://www.nag.co.uk/nagware/NP/TR.html + +• NAG provide coverage of TR 15580 and TR 15581. The first is the sup- port Fortran has for IEEE arithmetic. + +http://www.cs.berkeley.edu/\~wkahan/ + +• Willam Kahan home page. + +http://www.cs.berkeley.edu/\~wkahan/ieee754status/754story.html + +• An Interview with the Old Man of Floating Point. Reminiscences elicited from William Kahan by Charles Severance, which appeared in an issue of IEEE Computer - March 1998. + +http://www.cs.berkeley.edu/\~wkahan/ieee754status/ieee754.ps + +• Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic. Well worth a read. + +http://www.stewart.cs.sdsu.edu/cs575/labs/l3floatpt.html + +• CS 575 Supercomputing — Lab 3: Floating Point Arithmetic. CS 575 is an interdisciplinary course to introduce students in the sciences and engi- neering to advanced computing techniques using the supercomputers at the San Diego Supercomputer Center (SDSC). + +http://www.mathcom.com/nafaq/index.html + +• FAQ: Numerical Analysis and Associated Fields Resource Guide . A sum- mary of Internet resources for a number of fields related to numerical analysis. + +http://www.math.psu.edu/dna/disasters/ariadne.html + +• The Explosion of the Ariane 5 : A 64-bit floating point number relating to the horizontal velocity of the rocket with respect to the platform was con- verted to a 16-bit signed integer. The number was larger than 32,768, the largest integer storeable in a 16-bit signed integer, and thus the conversion failed. + +# 27.6.2 Hardware sources + +Osbourne A., Kane G., 4-bit and 8-bit Microprocessor Handbook, + +• Good source of information on 4-bit and 8-bit microprocessors. Osbourne A., Kane G., 16-Bit Microprocessor Handbook, Osbourne/McGraw-Hill, 1981. • Ditto 16-bit microprocessors. Intel, 386 DX Microprocessor Hardware Reference Manual , Intel. • The first Intel offering with 32-bit addressing. Intel, 80386 System Software Writer's Guide, Intel. • Developer's guide to the above. http://www.intel.com/ • Intel's home page. http://developer.intel.com/design/pentiumiii/ • Details of the Pentium III processor. + +http://www.cyrix.com/ + +• Cyrix home page. + +Bhandarkar D.P., Alpha Implementations and Architecture: Complete Reference and Guide, Digital Press, 1996. + +• Looks at some of the trade-offs and design philosophy behind the alpha chip. The author worked with VAX, MicroVAX and VAX vectors as well as the Prism. Also looks at the GEM compiler technology that DEC/Compaq use. + +http://www.digital.com/alphaserver/workstations/ + +• Home page for the Compaq/DEC Alpha systems. + +http://www.sgi.com/ + +• Silicon Graphics home page. + +http://www.sun.com/ + +• Sun home page. + +http://www.ibm.com/ + +• IBM home page. + +# 27.6.3 Operating Systems + +Deitel H.M., An Introduction to Operating Systems , Addison-Wesley, 1990. + +• The revised first edition includes case studies of UNIX, VMS, CP/M, MVS and VM. The second edition adds OS/2 and the Macintosh operat- ing systems. There is a coverage of hardware, software, firmware, process management, process concepts, asynchronous concurrent processes, con- current programming, deadlock and indefinite postponement, storage man- agement, real storage, virtual storage, processor management, distributed computing, disk performance optimisation, file and database systems, per- formance, coprocessors, risc, data flow, analytic modelling, networks, se- curity and it concludes with case studies of the these operating systems. The book is well written and an easy read. + +# 27.6.4 Java and IEEE 754 + +http://www.cs.berkeley.edu/\~darcy/Borneo/ + +• Borneo Language Homepage : Borneo is a dialect of the Java language de- signed to have true support for the IEEE 754 floating point standard. The status of arithmetic in Java is fluid. At the time of writing this book Sun had withdrawn from the formal language standardisation process. Sun + +have a publication at their web site that addresses changes to the Java lan- guage specification for JDK Release 1.2 floating point arithmetic. Their home Java page is + +• http://www.java.sun.com/ + +# 27.6.5 C and IEEE 754 + +http://wwwold.dkuug.dk/JTC1/SC22/WG14/ + +The official home of JTC1/SC22/WG14 - C. The C programming lan- guage standard ISO/IEC 9899 was adopted by ISO in 1990. ANSI then re- placed their first standard X3.159 by the ANSI/ISO 9899 standard identi- cal to ISO/IEC 9899:1990. + +# ISO TR 15581 Allocatable Enhancements + +# Aim + +The aim of this chapter is to provide a small number of examples illustrating some of the features introduced with ISO TR 15581: + +• Allocatable dummy arrays. • Allocatable function results. • Allocatable structure components. + +# Chapter 28 + +# 28 ISO TR 15581 Allocatable Enhancements + +In this chapter we provide three examples that illustrate the features introduced by TR 15581. The facilities mean that we do not have to use pointers and this has several efficiency benefits as the compiler does not have to worry about aliasing and whether it can deallocate temporaries or not. There is also the issue of contigu- ous memory allocation for allocatable arrays, which can't be guaranteed when using pointers and sections and strides other than unity. + +# 28.1 Allocatable dummy array example + +In the Quicksort example the actual array allocation took place in the main pro- gram. In this example we do the allocation in the Read_Data subroutine: + +PROGRAM ch2801 IMPLICIT NONE INTEGER :: How_Many CHARACTER (LEN $\scriptstyle{\dot{=}20}$ ) :: File_Name REAL , ALLOCATABLE , DIMENSION(:) :: Raw_Data integer , dimension(8) :: timing + +INTERFACE SUBROUTINE Read_Data(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER (LEN $\mathbf{\dot{\rho}}=\mathbf{\dot{\rho}}$ ) , INTENT(IN) :: File_Name INTEGER , INTENT(IN) :: How_Many REAL , INTENT(OUT) , ALLOCATABLE , & DIMENSION(:) :: Raw_Data END SUBROUTINE Read_Data END INTERFACE + +INTERFACE SUBROUTINE Sort_Data(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many REAL , INTENT(INOUT) , DIMENSION(:) :: Raw_Data END SUBROUTINE Sort_Data END INTERFACE + +INTERFACE SUBROUTINE Print_Data(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many + +REAL , INTENT(IN) , DIMENSION(:) :: Raw_Data END SUBROUTINE Print_Data END INTERFACE PRINT \* , ' How many data items are there?' READ \* , How_Many PRINT \* , ' What is the file name?' READ '(A)',File_Name call date_and_time(values $=$ timing) print \* , ' initial' print \* , timing(6),timing(7),timing(8) CALL Read_Data(File_Name,Raw_Data,How_Many) call date_and_time(values $=$ timing) print \* , ' read and allocate' print \* , timing(6),timing(7),timing(8) CALL Sort_Data(Raw_Data,How_Many) call date_and_time(values $=$ timing) print \* , ' sort' print \* , timing(6),timing(7),timing(8) CALL Print_Data(Raw_Data,How_Many) call date_and_time(values $=$ timing) print \* , ' print' print \* , timing(6),timing(7),timing(8) PRINT $\star$ , ' ' PRINT \*, ' Data written to file SORTED.DAT' + +END PROGRAM ch2801 + +SUBROUTINE Read_Data(File_Name,Raw_Data,How_Many) IMPLICIT NONE CHARACTER ( $\tt L E N\mathrm{=\star}$ ) , INTENT(IN) :: File_Name INTEGER , INTENT(IN) :: How_Many REAL , INTENT(OUT) , ALLOCATABLE , & DIMENSION(:) :: Raw_Data + +INTEGER :: I ALLOCATE(Raw_Data(1:How_Many))OPEN(FILE $_{\parallel}=$ File_Name,UNIT $\vDash\vDash$ ) DO $\scriptstyle{\mathbb{T}}=1$ ,How_Many READ (UNIT $\vDash\vDash$ , $\scriptstyle{\mathrm{FT}}={\star}$ ) Raw_Data(I) ENDDO + +END SUBROUTINE Read_Data + +SUBROUTINE Sort_Data(Raw_Data,How_Many) IMPLICIT NONE INTEGER , INTENT(IN) :: How_Many REAL , INTENT(INOUT) , DIMENSION(:) :: Raw_Data + +CALL QuickSort(1,How_Many) + +CONTAINS + +RECURSIVE SUBROUTINE QuickSort(L,R) IMPLICIT NONE INTEGER , INTENT(IN) :: L,R INTEGER :: I,J,tt REAL :: V,T + +$\dot{\bot}\!=\!\bot$ $\dot{\daleth}\,{=}\,\underline{{\boldsymbol{\Upsilon}}}$ $\mathbb{V}=$ raw_data( int(( $\cdot\,\mathtt{L}\,\mathtt{+r}$ )/2) ) do do while (raw_data(i) < v ) $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ enddo do while (v < raw_data(j) ) $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ enddo if ( $\dot{\Sigma}\!<\!=\!\dot{\Pi}$ ) then t $=$ raw_data(i) raw_data(i) $=$ raw_data(j) raw_data $(\,\dot{\,}\,)=\,\!\mathsf{C}$ $\dot{\bot}\!=\!\dot{\bot}\!+\!\bot$ $\dot{\bar{\jmath}}=\dot{\bar{\jmath}}-\it{1}$ endif if ( $\dot{\perp}{>}\dot{\supset}$ ) exit enddo + +if (i + + #include int main() { float a,b,sum; sum $\imath\!=\!0\cdot0$ ; cout $<$ " Type in two numbers " ; cin $>$ a > b ; sum $=$ a + b ; cout $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!<\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$ return (0); } + +# Cobol: ISO/IEC 1989:1985 + +# Fortran 90: ISO/IEC 1539:1990 + +PROGRAM Example IMPLICIT NONE REAL :: A REAL :: B REAL :: Sum $=\!0$ .0 PRINT \* , ' Type in two numbers' READ \* , A,B $\mathtt{S u m}\mathtt{\ =\ \texttt{A}\ +\ \texttt{B}}$ PRINT \* , A ,' + ', B , ' = ', Sum END PROGRAM Example + +# Appendix B — Sample Program Examples + +# ICON: No standard. + +Lisp + +# Logo: No standard. + +# Modula 2: ISO/IEC Draft 10514 + +MODULE Example; FROM InOut IMPORT Write,WriteLn,WriteString; FROM InOut IMPORT ReadReal,WriteReal; VAR A,B : REAL; + +BEGIN Sum : $\begin{array}{r l}{\mathbf{\sigma}=}&{{}0\mathbf{\sigma}.\;0}\end{array}$ ;WriteString(" Type in two numbers"); WriteLn; ReadReal(A); ReadReal(B); Sum : $=$ A + B; WriteReal(A,10); WriteString(" $^+$ "); WriteReal(B,10);WriteString(" = "); WriteReal(Sum,10); WriteLn; END Example. + +# Oberon: No standard. + +# Pascal: Pascal — ISO 7185:1990; Extended Pascal — ISO/IEC 10206: 1991 + +PROGRAM Example(INPUT,OUTPUT); + +BEGIN WRITELN(' Type in two numbers'); READLN(A,B); Sum : $\begin{array}{r l r l}{\mathrm{~\boldmath~\omega~}}&{{}=}&{\mathrm{~\boldmath~\delta~}_{\mathrm{~\tiny~\mathbb~A~}}\,\mathrm{~\boldmath~+~}\,\mathrm{~\boldmath~\delta~}_{\mathrm{~\tiny~\mathbb~B~}}}\end{array}$ ;WRITELN(A, ' + ' , B ,' = ' , Sum) END. + +Postscript: Prolog: ISO/IEC Draft 13211-1 SQL: ISO 9075:1992(E) Simula: No international standard, but a Swedish one does exist. Smalltalk: Snobol: + +# C ASCII Character Set + +![](images/0bc6d2a8ca7f6fd61f19e724f04dc2fca66baba6fb446ac9f5fe336d980fef21.jpg) + +# D Intrinsic Functions and Procedures + +The following abbreviations and typographic conventions are used in this appen- dix. + +# Argument type and result type: + +I Integer R Real C Complex N Numeric (any of integer, real, complex) L Logical P Pointer T Target DP Double precision Char Character, length $=1$ . S Character + +# Class + +E Elemental function I Inquiry function T Transformation al function S Subroutine + +See Chapter 14 for more information on these classifications. + +# Arguments in italics + +ALL(Mask,Dim)are optional arguments, i.e., Dim may be omitted in the example above. + +# Double precision + +Before Fortran 90 if you required real variables to have greater precision than the default real then the only option available was to declare them as double precision. With the introduction of kind types with Fortran 90 the use of double precision + +# 536 Appendix D — Intrinsic Functions and Procedures + +declarations is not recommended, and instead real entities with a kind type offering more than the default precision should be used. + +# Kind optional argument + +There are several functions that have an optional argument Kind, e.g., AINT(A,Kind). If Kind is absent the result is the same kind type as the first argu- ment, in this case A. If Kind is present the result has the kind type specified by this argument. + +# Result type + +When the result type is the same as the argument type then the result is not just the same type as the argument but also the same kind. + +# Miscellaneous rules + +When the argument is Back it is of logical type. + +When the argument is Count_Rate, Count_Max, Dim, Kind, Len, Order, N_Copies, Shape, Shift, Values it is of integer type. + +When the argument is Mask it is of logical type. + +When the argument is Target it is of pointer or target type. + +# ABS(A) + +Yields the absolute value unless A is complex; see below. + +Argument: A Type: N Result: As argument Class: E + +Note : If A is complex $(x,y)$ then the functions returns $\sqrt{\phantom{x}x^{2}+y^{2}}$ + +Example : R1 $=$ ABS(A) + +# ACHAR(I) + +Returns character in the ASCII character set. + +Argument: I Type: I Result: Char Class: E + +Example : $\mathrm{C=}$ ACHAR(I) + +# ACOS(X) + +Arccosine (inverse cosine). + +Argument: X Type: R Result: As argument Class: E + +Example : $\mathrm{Y{=}A C O S(X)}$ + +Adjust string left, removing leading blanks and inserting trailing blanks. + +Argument: String Type: S Result: As argument Class: E + +Example: $\mathrm{S=}$ ADJUSTL(S) + +ADJUSTR(String) + +Adjust string right, removing trailing blanks and inserting leading blanks. + +Argument: String Type: S Result: As argument Class: E + +Example: $\mathrm{S=}$ ADJUSTR(S) + +AIMAG(Z) Imaginary part of complex argument. + +Argument: Z Type: C Result: As argument Class: E + +Example : Y=AIMAG(Z) AINT(A, Kind ) Truncation. + +Argument: A Type: R Result: As A Class: E Argument: Kind Type: I + +Example: $\mathrm{Y{=}A l N T(Z)}$ and when $\mathrm{Z=}0.3\ \mathrm{Y=}0$ , when $_{\textrm{Z=}2.73\ \mathrm{Y=}2.0}$ , when $Z{=}{-}2.73$ $\mathrm{Y}{=}{-}2.0$ + +ALL(Mask,Dim)Determines whether all values are true in Mask along dimension Dim. + +Argument: Mask Type: L + +Appendix D — Intrinsic Functions and Procedures Result: L Class: T + +Note : Dim must be a scalar in the range $1\leq D i m\leq n$ where $n$ is the rank of Mask. The result is scalar if Dim is absent or Mask has rank 1. Otherwise it works on the dimension Dim of Mask and the result is an array of rank $_{n-1}$ . + +Example : T=ALL(M) + +ALLOCATED(Array) + +Returns true if array is allocated. + +Argument: Array Type: Any Result: L Class: I + +Note : Array must be declared with the ALLOCATABLE attribute. + +Example : IF (ALLOCATED(Array) ) THEN ... + +# ANINT(A, Kind) + +Rounds reals, i.e., returns nearest whole number. + +Argument: A Type: R Result: As A Class: E + +Example : $Z=$ ANINT(A), if $\mathrm{A}=5.63\;\mathrm{Z}=6$ , if $\mathrm{A}{=}{5.7}\ \mathrm{Z}={-6.0}$ + +ANY(Mask,Dim) + +Determines whether any value is true in Mask along dimension Dim. + +Argument: Mask Type: L Result: L Class: T + +Note : Mask must be an array. The result is a scalar if Dim is absent or if Mask is of rank 1. Otherwise it works on the dimension Dim of Mask and the result is an array of rank $_{n-1}$ . + +Example : T=ANY(A) + +ASIN(X) + +Arcsine. + +Argument: X Type: R Result: As argument Class: E + +Example : $Z=$ ASIN(X) + +ASSOCIATED(Pointer, Target) + +Returns the association status of the pointer. + +Argument: Pointer Type: P Result: L Class: I + +# Note : + +1. If Target is absent then the result is true if POINTER is associated with a target, otherwise false. 2. If Target is present and is a target, the result is true if Pointer is currently associ- ated with Target and false if it is not. 3. If Target is present and is a pointer, the result is true if both Pointer and Target are currently associated with the same target, and is false otherwise. If either Pointer or Target is disassociated the result is false. + +Example : $\mathrm{T=}$ ASSOCIATED(P) + +ATAN(X) Arctangent. + +Argument: X Type: R Result: As argument Class: E + +Example : $Z=$ ATAN(X) + +ATAN2(Y,X)Arctangent of Y / X. + +Argument: Y Type: R Result: As arguments Class: E + +Example: Z=ATAN2(Y,X) + +BIT_SIZE(I) + +Returns the number of bits, as defined by the numeric model for integer numbers in Chapter 8. + +Argument: I Type: I Result: As argument Class: I + +Example : N_Bits $\scriptstyle,\equiv$ SIZE(I) + +Appendix D — Intrinsic Functions and Procedures + +# BTEST(I,Pos) + +Returns true if the bit is set in the integer argument at the position given by the second argument. + +Argument: I Type: I Result: L Class: E + +Example : T=BTEST(I,Pos) + +CEILING(A, Kind ) + +Returns the smallest integer greater than or equal to the argument. + +Argument: A Type: R Result: I Class: E + +# Note: + +If kind is present the result has the kind type parameter Kind. + +Otherwise the result is of type default integer. + +Example: $\mathrm{I}=$ CEILING(A) If $\mathrm{A}{=}12.21$ then $\mathrm{I}{=}13$ , if $\mathrm{A}{=}3.16$ then $\mathrm{I}{=}{-}3$ + +CHAR(I, Kind ) + +Returns the character in a given position in the processor collating sequence asso- ciated with the specified kind type parameter. Normally ASCII. + +Argument: I Type: I Result: CHAR Class: E + +Example : $\scriptstyle{\mathrm{C=}}$ CHAR(65) and for the ASCII character set $\mathrm{C}{=}\mathrm{'}\mathrm{A'}$ . + +# CMPLX(X, Y,Kind ) + +Converts to complex from integer, real and complex. + +Argument: X Type: N Result: C Class: E + +Note : 1. If $\mathrm{X}$ is complex and Y is absent it is as if $\mathrm{Y}$ were present with the value AIMAG(X). + +2. If X is not complex and Y is absent, it is as if Y were present with the value 0. + +Example : $Z=$ CMPLX(X,Y) + +# CONJG(Z) + +Conjugate of a complex argument. + +Argument: Z Type: C Result: As Z Class: E + +Example : Z1 $=$ CONJG(Z) COS(X) Cosine. + +Argument: X Type: R, C Result: As argument Class: E + +Note : The arguments of all trigonometric functions should be in radians, not de- grees. + +Example : A=COS(X) + +COSH(X) Hyperbolic cosine. + +Argument: X Type: R Result: As argument Class: E + +Example : $Z=$ COSH(X) + +COUNT(Mask,Dim)Returns the number of true elements in Mask along dimension Dim. + +Argument: Mask Type: L Result: I Class: T + +Note : Dim must be a scalar in the range $1\leq D i m\leq n$ , where $n$ is the rank of Mask. The result is scalar if Dim is absent or Mask has rank 1. Otherwise it works on the dimension Dim of Mask and the result is an array of rank $_{n-1}$ . + +Example : N=COUNT(A) CPU_TIME(Time) + +Returns the processor time. + +Argument: Time Type: R Result: N/A Class : S + +# Appendix D — Intrinsic Functions and Procedures + +Example: CALL CPU_TIME(Time) + +# CSHIFT(Array,Shift, Dim ) + +Circular shift on a rank 1 array or rank 1 sections of higher-rank arrays. + +Argument: Array Type: Any Result: As Array Class: T + +Note : Array must be an array, Shift must be a scalar if Array has rank 1, otherwise it is an array of rank $_{n-1}$ , where $n$ is the rank of Array. Dim must be a scalar with a value in the range $1\leq D i m\leq n$ . + +Example : Array $=$ CSHIFT(Array,10) + +DATE_AND_TIME( Date,Time,Zone,Values ) + +Returns the current date and time (compatible with ISO 8601:1988). + +Argument: Date Type: S Result: N/A Class: S + +Time and Zone are of type S. + +# Note: + +1. Date is optional and must be scalar and 8 characters long in order to return the complete value of the form CCYYMMDD, where CC is the century, YY is the year, MM is the month and DD is the day. It is INTENT(OUT). + +2. Time is optional and must be scalar and 10 characters long in order to return the complete value of the form hhmmss.sss where hh is the hour, mm is the minutes and ss.sss is the seconds and milliseconds. It is INTENT(OUT). + +3. Zone is optional and must be scalar and must be 5 characters long in order to re- turn the complete value of the form hhmm where hh and mm are the time differences with respect to Coordinated Universal Time in hours and minutes. It is INTENT(OUT). + +4. Values is optional and a rank 1 array of size 8. It is INTENT(OUT). The values returned are as follows: Values $(1)=$ the year Values(2) $=$ the month Values(3) $=$ the day Values(4) $=$ the time with respect to Coordinated Universal Time in minutes. Values(5) $=$ the hour (24 hour clock) Values(6) $=$ the minutes Values(7) $=$ the seconds Values $(8)=$ the milliseconds in the range 0–999. + +Example : CALL DATE_TIME(D,T,Z,V) + +# DBLE(A) + +Converts to double precision from integer, real, and complex + +Argument: A Type: N Result: DP Class: E + +Example : D=DBLE(A) + +# DIGITS(X) + +Returns the number of significant digits of the argument as defined in the numeric models for integer and reals in Chapter 8. + +Argument: X Type: I,R Result: I Class: I + +Example : I=DIGITS(X) + +# DIM(X,Y) + +Returns first argument minus minimum of the two arguments: X -MIN(X,Y). + +Argument: X Type: I Result: As arguments Class: E + +Example : Z=DIM(X,Y) + +DOT_PRODUCT(Vector_1,Vector_2) + +Performs the mathematical dot product of two rank 1 arrays. + +Argument: Vector_1 Type: Nt Result: As arguments Class: T + +Vector_2 is as Vector_1. + +Note: 1. If Vector_1 is of type integer or real the result has the value SUM(Vector_1\*Vector_2). 2. If Vector_1 is complex the result has the value SUM(CONJG(Vector_1)\*Vector_2). 3. If Vector_1 is logical the result has the value ANY(Vector_1 .AND. Vector_2). + +Example : $\mathbf{A}\!\!=\!\!$ DOT_PRODUCT(X,Y) + +# DPROD(X,Y) + +Double precision product of two reals. + +Appendix D — Intrinsic Functions and Procedures + +Argument: X Type: R Result: DP Class: E + +Example : D=DPROD(X,Y) + +EOSHIFT(Array,Shift, Boundary,Dim ) + +End of shift of a rank 1 array or rank 1 section of a higher-rank array. + +Argument: Array Type: Any Result: As Array Class: T + +Boundary is as Array. + +Note : Array must be an array, Shift must be a scalar if Array has rank 1, otherwise it is an array of rank $_{n-1}$ , where $n$ is the rank of Array. Boundary must be scalar if Array has rank 1, otherwise it must be either scalar or of rank $n$ -1. Dim must be a scalar with a value in the range $1\leq D i m\leq n$ . + +Example : $\mathbf{A}\!\!=$ EOSHIFT(A,Shift) + +EPSILON(X) + +Smallest difference between two reals of that kind. See Chapter 8 and real numeric model. + +Argument: X Type: R Result: As argument Class: I + +Example : Tiny $\risingdotseq$ EPSILON(X) + +EXP(X) Exponential, $\mathrm{e}^{\mathrm{X}}$ + +Argument: X Type: R, C Result: As argument Class: E + +Example : Y=EXP(X) + +# EXPONENT(X) + +Returns the exponent component of the argument. See Chapter 8 and the real nu- meric model. + +Argument: X Type: R Result: I Class: E + +Example : I=EXPONENT(X) + +FLOOR(A, Kind) . + +Returns the greatest integer less than or equal to the argument + +Argument: A Type: R Result: I Class: E + +# Note: + +If kind is present the result has the kind type parameter Kind, otherwise the result is of type default integer. + +Example: $\mathrm{I}{=}$ FLOOR(A) and when $\mathrm{A}{=}5.2\ \mathrm{I}$ has the value 5, when $\mathrm{A}{=}9.7\ \mathrm{I}$ has the value -10 + +FRACTION(X) + +Returns the fractional part of the real numeric model of the argument See Chapter 8 and the real numeric model. + +Argument: X Type: R Result: As X Class: E + +Example : $\mathrm{F}{=}$ FRACTION(X) + +# HUGE(X) + +Returns the largest number for the kind type of the argument. See Chapter 8 and the real and integer numeric models. + +Argument: X Type: I,R Result: As argument Class: I + +Example : H=HUGE(X) + +# IACHAR(C) + +Returns the position of the character argument in the ASCII collating sequence. + +Argument: C Type: Char Result: I Class: E + +Example : I=IACHAR('A') returns the value 65. + +# IAND(I,J) + +Performs a logical AND on the arguments. + +Argument: I Type: I Result: As arguments Class: E + +Appendix D — Intrinsic Functions and Procedures + +Example : K=IAND(I,J) + +IBCLR(I,Pos) + +Clears one bit of the argument to zero. + +Argument: I Type: I Result: As I Class: E + +Note : $0\leq P o s0.0$ ) returns the minimum of the positive elements of C. + +$B=\!\!\left(\!\!\begin{array}{c c c}{{1}}&{{3}}&{{5}}\\ {{2}}&{{4}}&{{6}}\end{array}\!\!\right)$ + +MINVAL(B,DIM $=\!1$ ) returns (1,3,5). + +MINVAL(B,DIM $\scriptstyle{=2}$ ) returns (1,2). + +# MOD(A,B) + +Returns the remainder when first argument divided by second. + +Argument: A Type: I, R Result: As arguments Class: E + +Note : If $\mathrm{B}{=}0$ the result is processor dependent. For $\textrm{\textbf{B}}\sqrt{\textrm{0}}$ the result is A - $\mathrm{INT(A/B)\astB}$ . + +Example : R=MOD(A,B) If $\mathrm{A}{=}8$ and $_{\mathrm{B}=5}$ then $\scriptstyle{\mathrm{R}}=3$ If $\mathrm{A}{=}{8}$ and $_{\mathrm{B}=5}$ then $\mathrm{R}{=}3$ + +If $\mathrm{A}{=}8$ and $\mathrm{B}{=}{-}5$ then $\scriptstyle{\mathrm{R}}=3$ If $\mathrm{A}{=}\mathrm{8}$ and $\mathrm{B}{=}{-}5$ then $\mathrm{R}{=}3$ + +MODULO(A,B) + +Returns the modulo of the arguments. + +Argument: A Type: I,R Result: As A Class: E + +Note : + + 1. If $\mathrm{B}{=}0$ then the result is processor dependent. + + 2. Integer A The result is R where $\mathrm{A=Q\astB+R}$ and $\mathrm{Q}$ is integer for $_{\mathrm{B}>0}$ , $0\le\mathrm{R}<\mathrm{B}$ for $\mathrm{B}<0$ , $\mathrm{B}<\mathrm{R}\leq0$ 3. Real A The result is A - FLOOR(A/B) \* B. + +Example : $\mathbf{R}{=}$ MODULO(A,B) If $\mathrm{A}{=}8$ and $_{\mathrm{B}=5}$ then $\scriptstyle{\mathrm{R}}=3$ If $\mathrm{A}{=}\mathrm{8}$ and $_{\mathrm{B}=5}$ then $\scriptstyle{\mathrm{R}}=2$ If $\mathrm{A}{=}8$ and $\mathrm{B}{=}{-}5$ then $\scriptstyle{\mathrm{R}}=-2$ If $\mathrm{A}{=}\mathrm{8}$ and $\mathrm{B}{=}{-}5$ then $\mathrm{R}{=}3$ + +MVBITS(From,F_Pos,Len,To,T_Pos) + +Copies a sequence of bits from one data object to another. + +Argument: From Type: I Result: N/A Class: S + +All arguments are of integer type. + +Note : From must be INTENT(IN). F_Pos must be INTENT(IN), $\mathrm{F}\_\mathrm{Pos}\geq0$ , F_Pos+Len $\leq$ BIT_SIZE(From). Len must be INTENT(IN), Len $\ge0$ . To must be INTENT(INOUT). T_Pos must be INTENT(IN), T_Pos $\ge0$ , T_Pos $^+$ Len $\leq$ BIT_SIZE(To). + +Example : CALL MVBITS(F,FP,L,T,TP) + +# NEAREST(X,Next) + +Returns the nearest different number. See Chapter 8 and the real numeric model. Argument: X Type: R + +Appendix D — Intrinsic Functions and Procedures Result: As X Class: E + +Next is of type R. Example : N=NEAREST(X,Next) + +NINT(A, Kind ) Yields nearest integer. + +Argument: A Type: RI Result: I Class: E + +Note : + + 1. $\mathrm{A}>0$ , the result is $\mathrm{INT}(\mathrm{A}{+}0.5)$ . + + 2. $\mathrm{A}\le0$ , the result is INT(A-0.5). + +Example : I=NINT(X) + +NOT(I) Returns the logical complement of the argument. + +Argument: I Type: I Result: As I Class: E + +Example : I=NOT(I) + +NULL( Mold) Returns a disassociated pointer. + +Argument: Mold Type: P Result: As argument Class: T + +Note: If the argument Mold is present the result is the same as Mold. Otherwise it is determined by context. Example: REAL , POINTER :: $\mathrm{P}{=}>$ NULL() PACK(Array,Mask, Vector ) Packs an array into an array of rank 1, under the control of a mask. + +Argument: Array Type: Any Result: As Array Class: T + +# Note : + +1. Array must be an array. + + 2. Mask be conformable with Array. + + 3. Vector must have rank 1 and have at least as many elements as there are TRUE elements in Mask. 4. If Mask is scalar with the value TRUE. Vector must have at least as many ele- ments as there are in Array. + + 5. The result is an array of rank 1. + + 6. If Vector is present the result size is that of Vector. + + 7. If Vector is not present the result size is $t.$ , the number of TRUE elements in Mask, unless Mask is scalar with a value TRUE in which case the result size is the size of Array. + +Example : R=PACK(A,M) + +# PRECISION(X) + +Returns the decimal precision of the argument. See Chapter 8 and numeric models. + +Argument: X Type: R, C Result: I Class: I + +Example : $\mathrm{I}{=}$ PRECISION(X) + +# PRESENT(A) + +Returns whether an optional argument is present. + +Argument: A Type: Any Result: L Class: I + +Note : A must be an optional argument of the procedure in which the PRESENT function reference appears. + +Example : IF (PRESENT(X)) THEN ... + +PRODUCT(Array, Dim,Mask ) + +The product of all of the elements of Array along the dimension Dim correspond- ing to the TRUE elements of Mask. + +Argument: Array Type: N Result: As Array Class: T + +# Note : + +1. Array must be an array. + + $2.~1\leq~\operatorname{Dim}~\leq\mathfrak{n}$ where n is the rank of Array. + + 3. Mask must be conformable with Array. + +4. Result is scalar if Dim is absent, or Array has rank 1, otherwise the result is an array of rank $_{n-1}$ . + +# Example : + +1. PRODUCT((/1,2,3/)) the result is 6. + + 2. PRODUCT(C,Mask $\mathrm{=}\mathrm{C}>0.0$ ) forms the product of the positive elements of C. + + 3. If $B=\!\!\left(\!\!\begin{array}{c c c}{{1}}&{{3}}&{{5}}\\ {{2}}&{{4}}&{{6}}\end{array}\!\!\right)$ + +PRODUCT(B,DIM=1) is (2,12,30) and PRODUCT(B,DIM $\scriptstyle{\stackrel{=}{2}}$ ) is (15,48) + +# RADIX(X) + +Returns the base of the numeric argument. See Chapter 8 and numeric models. + +Argument: X Type: I,R Result: I Class: I + +Example : Base=RADIX(X) + +RANDOM_NUMBER(X) + +Returns one pseudorandom number or an array of pseudorandom numbers from the uniform distribution over the range $0\leq x<1$ + +Argument: X Type: R Result: N/A Class: S + +Note : X is INTENT(OUT). + +Example : CALL RANDOM_NUMBER(X) + +RANDOM_SEED(Size,Put,Get) + +Restarts (seeds) or queries the pseudorandom generator used by RAN- DOM_NUMBER. + +Argument: Size Type: I Result: N/A Class: S + +All arguments are of integer type. + +Note : 1. Size is INTENT(OUT). It is set to the number $\mathbf{N}$ of integers that the processor uses to hold the value of the seed. + +2. Put is INTENT(IN). It is an array of rank 1 and size $\geq\mathrm{N}$ . It is used by the pro- cessor to set the seed value. + +3. Get is INTENT(OUT). It is an array of rank 1 and size $\geq\mathrm{N}$ . It is set by the pro- cessor to the current value of the seed. + +Example : CALL RANDOM_SEED + +# RANGE(X) + +Returns the decimal exponent range of the real argument. See Chapter 8 and the numeric model representing the argument. + +Argument: X Type: N Result: I Class: I + +Example : I=RANGE(N) + +REAL(A, Kind ) + +Converts to real from integer, real or complex. + +Argument: A Type: N Result: R Class: E + +Example : $X=$ REAL(A) + +REPEAT(String,N_Copies) + +Concatenates several copies of a string. + +Argument: String Type: S Result: S Class: T + +Example : New_S $\leftleftarrows$ REPEAT(S,10) + +RESHAPE(Source,Shape,Pad,Order) + +Constructs an array of a specified shape from the elements of a given array. + +Argument: Source Type: Any Result: As Source Class: T + +Note : 1. Source must be an array. If Pad is absent or of size zero the size of Source must + +be $\geq$ PRODUCT(Shape). + + 2. Shape must be a rank 1 array and $0\leq\mathrm{size}<8$ . + + 3. Pad must be an array. + + 4. Order must have the same shape as Shape and its value must be a permutation of $\left(1,2,\ldots\,,n\right)$ where $n$ is the size of Shape. If absent it is as if it were present with the value $\scriptstyle(1,2,\ldots,n)$ . 5. The result is an array of shape, Shape. + +Example : RESHAPE((/1,2,3,4,5,6/),(/2,3/)) has the value $\left(\begin{array}{l l l}{{1}}&{{3}}&{{5}}\\ {{2}}&{{4}}&{{6}}\end{array}\right)$ RESHAPE( (/1,2,3,4,5,6/) , (/2,4/) , (/0,0/) , (/2,1/) ) has the value $\left(\begin{array}{l l l l}{{1}}&{{2}}&{{3}}&{{4}}\\ {{5}}&{{6}}&{{0}}&{{0}}\end{array}\right)$ + +# RRSPACING(X) + +Returns the reciprocal of the relative spacing of model numbers near the argument value. See Chapter 8 and the real numeric model. + +Argument: X Type: R Result: As X Class: E + +Example : $Z=$ RRSPACING(X) + +# SCALE(X,I) + +Returns $X*b^{\operatorname{I}}$ where b is the base in the model representation of $X$ . See Chapter 8 and the real numeric model. + +Argument: X Type: R Result: As X Class: E + +I is of integer type. + +Example : Z=SCALE(X,I) + +# SCAN(String,Set,Back) + +Scans a string for any one of the characters in a set of characters. + +Argument: String Type: S Result: I Class: E + +Note : 1. The default is to scan from the left, and will only be from the right when Back is present and has the value TRUE. 2. Zero is returned if the scan fails. + +Example : W=SCAN(String,Set) + +# SELECTED_INT_KIND(R) + +Returns a value of the kind type parameter of an integer data type that represents all integer values $n$ with $-10^{R}\hat{<}n\hat{<}10^{R}$ + +Argument: R Type: I + +# Result: I Class: T + +R must be scalar. If a kind type parameter is not available then the value $^{-1}$ is returned. + +Example : I=SELECTED_INT_KIND(2) + +SELECTED_REAL_KIND( P,R ) + +Returns a value of the kind type parameter of a real data type with decimal preci- sion of at least $P$ digits and a decimal exponent range of at least $R$ . + +Argument: P and R Type: I Result: I Class: T + +Note : + + 1. $P$ and $R$ must be scalar. + + 2. The value $^{-1}$ is returned if the precision is not available, the value $^{-2}$ if the ex- ponent range is not available, and $^{-3}$ if neither is available. + +Example : I=SELECTED_REAL_KIND(P,R) + +# SET_EXPONENT(X,I) + +Returns the model number whose fractional part is the fractional part of the model representation of X and whose exponent part is I. + +Argument: X Type: R Result: As X Class: E + +I is of integer type. + +Example : Exp_Par $\leftrightharpoons$ SET_EXPONENT(X,I) + +# SHAPE(Source) + +Returns the shape of the array argument or scalar. + +Argument: Source Type: Any Result: I Class: I + +Note : 1. Source may be array valued or scalar. It must not be a pointer that is disassoci- ated or an allocatable array that is not allocated. It must not be an assumed-size array. + +2. The result is an array of rank 1 whose size is equal to the rank of Source. + +Example : $\mathrm{S}{=}$ SHAPE(A(2:5,-1:1)) yields $\scriptstyle{\mathrm{S}=(4,3)}$ + +Appendix D — Intrinsic Functions and Procedures + +# SIGN(A,B) + +Absolute value of A times the sign of B. + +Argument: A Type: I, R Result: As A Class: E + +# Note: + +In the special case where B is zero normally the result would have the value ABS(A), but if B is one of the real kind types and the processor is able to distin- guish between plus zero and minus zero then the result is ABS(A) if B is plus zero and the result is –ABS(A) if B is minus zero. + +B is as A. + +Example : A=SIGN(A,B) + +SIN(X) + +Sine. + +Argument: X Type: R, C Result: As argument Class: E + +Note : The argument is in radians. + +Example : Z=SIN(X) + +SINH(X) Hyperbolic sine. + +Argument: X Type: R Result: As argument Class: E + +Example : Z=SINH(X) + +SIZE(Array, Dim ) + +Returns the extent of an array along a specified dimension or the total number of elements in an array. + +Argument: Array Type: Any Result: I Class: I + +# Note : + +1. Array must be an array. It must not be a pointer that is disassociated or an allocatable array that is not allocated. If Array is an assumed-size array Dim must be present with a value less than the rank of Array. + +2. Dim must be scalar and in the range $1\leq\operatorname{Dim}\leq n$ where $n$ is the rank of Array. + + 3. Result is equal to the extent of dimension Dim of Array, or if Dim is absent, the total number of elements of Array. + +Example : A=SIZE(Array) + +SPACING(X) + +Returns the absolute spacing of model numbers near the argument value. See Chapter 8 and the real numeric model. + +Argument: X Type: R Result: As X Class: E + +Example : S=SPACING(X) + +SPREAD(Source,Dim,N_Copies) + +Creates an array with an additional dimension, replicating the values in the original array. + +Argument: Source Type: Any Result: As Source Class: T + +# Note : + +1. Source may be array valued or scalar, with rank less than 7. + + 2. Dim must be scalar and in the range $1\,\leq\,{\mathrm{Diim}}\,\leq\,n{+}1$ where $n$ is the rank of Source. + + 3. N_Copies must be scalar. + + 4. The result is an array of rank $_{n+1}$ . + +# Example : + +If A is the array (2,3,4) then SPREAD(A,DIM $=\!1$ ,NCOPIES $^{=3}$ ) then the result is th $\mathrm{\textbf{emdash}}\left(\!\!\begin{array}{c c c}{{2}}&{{3}}&{{4}}\\ {{2}}&{{3}}&{{4}}\\ {{2}}&{{3}}&{{4}}\end{array}\!\!\right)$ + +# SQRT(X) + +Square root. + +Argument: X Type: R, C Result: As argument Class: E + +Example $\mathbf{A}\!\!=$ SQRT(B) + +# Appendix D — Intrinsic Functions and Procedures + +SUM(Array, Dim,Mask ) + +Returns the sum of all elements of Array along the dimension Dim corresponding to the true elements of Mask. + +Argument: Array Type: N Result: As Array Class: T + +# Note : + +1. Array must be an array. + + 2. $1\leq\operatorname{Dim}\leq n$ where $n$ is the rank of Array. + + 3. Mask must be conformable with Array. + + 4. Result is scalar if Dim is absent, or Array has rank 1, otherwise the result is an array of rank $_{n-1}$ . + +# Example : + +$\it.\mathrm{\bfSUM}((1,\!2,\!3/))$ the result is 6. + + $2.\mathrm{\DeltaSUM(C,Maxk=C>0.0)}$ forms the arithmetic sum of the positive elements of C. + + 3. If $B=\!\!\left(\!\!\begin{array}{c c c}{{1}}&{{3}}&{{5}}\\ {{2}}&{{4}}&{{6}}\end{array}\!\!\right) + +$ $\mathrm{SUM}(\mathrm{B,Dim}{=}1)$ is (3,7,11) + + $\mathrm{SUM}(\mathrm{B},\mathrm{Dim}{=}2)$ is (9,12) + +SYSTEM_CLOCK( Count,Count_Rate,Count_Max ) + +Returns integer data from a real time clock. + +Argument: Count Type: I Result: N/A Class: S + +# Note : + +1. Count is INTENT(OUT) and is set to a processor dependent value based on the current value of the processor clock or to -HUGE(0) if there is no clock. $0\leq$ Count + + $\leq$ Count_Max. + + 2. Count_Rate is INTENT(OUT) and it is set to the number of processor clock counts per second, or zero if there is no clock. 3. Count_max is INTENT(OUT) and is set to the maximum value that Count can have or to zero if there is no clock. + +Example : CALL SYSTEM_CLOCK(C,R,M) + +# TAN(X) + +Tangent. + +Argument: X Type: R + +Result: As argument Class: E + +Note : X must be in radians. + +Example : Y=TAN(X) + +TANH(X) Hyperbolic tangent. + +Argument: X Type: R Result: As argument Class: E + +Example : Y=TANH(X) + +# TINY(X) + +Returns the smallest positive number in the model representing numbers of the same type and kind type parameter as the argument. + +Argument: X Type: R Result: As X Class: I + +Example : $\mathrm{T}\mathrm{=}$ TINY(X) + +TRANSFER(Source,Mold, Size ) + +Returns a result with a physical representation identical to that of Source, but inter- preted with the type and type parameters of Mold. + +Argument: Source Type: Any Result: As Mold Class: T + +Warning : A thorough understanding of the implementation specific internal repre- sentation of the data types involved is necessary for successful use of this function. Consult the documentation that accompanies the compiler that you work with be- fore using this function. + +TRANSPOSE(Matrix) + +Transposes an array of rank 2. + +Argument: Matrix Type: Any Result: As argument Class: T + +Note : Matrix must be of rank 2. If its shape is $(n,m)$ then the resultant matrix has shape $(m,n)$ . + +Appendix D — Intrinsic Functions and Procedures + +Example : For $A=\!\!{\left(\begin{array}{l l l}{1}&{2}&{3}\\ {4}&{5}&{6}\\ {7}&{8}&{9}\end{array}\right)}{\mathrm{TRANSOE(A)~yields}}{\left(\begin{array}{l l l}{1}&{4}&{7}\\ {2}&{5}&{8}\\ {3}&{6}&{9}\end{array}\right)}$ + +# TRIM(String) + +Returns the argument with trailing blanks removed. + +Argument: String Type: S Result: As String Class: T + +Note : String must be a scalar. + +Example : T_S=TRIM(S) + +UBOUND(Array, Dim ) + +Returns all the upper bounds of an array or a specified upper bound. + +Argument: Array Type: Any Result: I Class: I + +# Note : + +$1\!\le\!D i m\!\le n$ , where $n$ is the rank of Array. The result is scalar if Dim is present oth- erwise the result is an array of rank 1 and size $n$ . Result is a scalar if Dim is present otherwise is an array of rank 1, and size $n$ . + +Example : Z=UBOUND(A) + +# UNPACK(Vector,Mask,Field) + +Unpacks an array of rank 1 into an array under the control of a mask. + +Argument: Vector Type: Any Result: As Vector Class: T + +# Note : + +1. Vector must have rank 1. Its size must be at least $t$ , where $t$ is the number of true elements in Mask. + + 2. Mask must be array valued. + + 3. Field must be conformable with Mask. Result is an array with the same shape as Mask. + +# Example : + +$$ +\begin{array}{r}{\mathrm{With}\;V e c t o r=(1,2,3)\;\mathrm{~and}\;\;M a s k=\left(\begin{array}{c c c}{F}&{T}&{F}\\ {T}&{F}&{F}\\ {F}&{F}&{T}\end{array}\right)\;\mathrm{and}\;\;F i e l d=\left(\begin{array}{c c c}{1}&{0}&{0}\\ {0}&{1}&{0}\\ {0}&{0}&{1}\end{array}\right)}\end{array} +$$ + +$$ +{\mathrm{The~result~is~}}{\left(\begin{array}{l l l}{1}&{2}&{0}\\ {1}&{1}&{0}\\ {0}&{0}&{3}\end{array}\right)} +$$ + +VERIFY(String,Set, Back ) + +Verify that a set of characters contains all the characters in a string by identifying the position of the first character in a string of characters that does not appear in a given set of characters. + +Argument: String Type: S Result: I Class: E + +# Note : + +1. The default is to scan from the left, and will only be from the right when Back is present and has the value TRUE. + +2. The value of the result is zero if each character in String is in Set, or if String has zero length. + +Example : $\mathrm{I}{=}$ VERIFY(String,Set) + +# E English and Latin Texts + +YET IF HE SHOULD GIVE UP WHAT HE HAS BEGUN, AND AGREE TO MAKE US OR OUR KINGDOM SUBJECT TO THE KING OF ENGLAND OR THE ENGLISH, WE SHOULD EXERT OURSELVES AT ONCE TO DRIVE HIM OUT AS OUR ENEMY AND A SUBVERTER OF HIS OWN RIGHTSAND OURS, AND MAKE SOME OTHER MAN WHO WAS ABLE TO DE- FEND US OUR KING; FOR, AS LONG AS BUT A HUNDRED OF USREMAIN ALIVE, NEVER WILL WE ON ANY CONDITIONS BE BROUGHT UNDER ENGLISH RULE. IT IS IN TRUTH NOT FOR GLORY, NOR RICHES, NOR HONOURS THAT WE ARE FIGHTING, BUT FOR FREEDOM - FOR THAT ALONE, WHICH NO HONEST MAN GIVES UP BUT WITH LIFE IT-SELF. + +QUEM SI AB INCEPTIS DIESISTERET, REGI ANGLORUM AUT ANGLICIS NOS AUT REGNUM NOSTRUM VOLENS SUBICERE, TANQUAM INIMICUM NOSTRUM ET SUI NOSTRIQUE JURIS SUBUERSOREM STATIM EXPELLERE NITEREMUR ET ALIUM REGEM NOSTRUM QUI AD DEFENSIONEM NOSTRAM SUFFICERET FACEREMUS. QUIA QUANDIU CENTUM EX NOBIS VIUI REMANSERINT, NUCQUAM ANGLORUM DOMINIO ALIQUATENUS VOLUMUS SUBIUGARI. NON ENIM PROPTER GLORIAM, DIUICIAS AUT HONORES PUGNAMUS SET PROPTER LIBERATEM SOLUMMODO QUAM NEMO BONUS NISI SIMUL CUM VITA AMITTIT. + +from 'The Declaration of Arbroath' c.1320. The English translation is by Sir James Fergusson. + +# F Coded Text Extract + +OH YABY NSFOUN, YAN DUBZY LZ DBUYLTUBFAJ BYYBOHNX GPDA FNUZNDYOLH YABY YAN SBF LZ B GOHTMN FULWOHDN DLWNUNX YAN GFBDN LZ BH NHYOUN DOYJ, BHX YAN SBF LZ YAN NSFOUN OYGNMZ BH NHYOUN FULWOHDN. OH YAN DLPUGN LZ YOSN, YANGN NKYNHGOWN SBFG VNUN ZLPHX GLSNALV VBHYOHT, BHX GL YAN DLMMNTN LZ DBUYLTUBFANUG NWLMWNX B SBF LZ YAN NSFOUN YABY VBG YAN GBSN GDBMN BG YAN NSFOUN BHX YABY DLOHDOXNX VOYA OY FLOHY ZLU FLOHY. MNGG BYYNHYOWN YL YAN GYPXJ LZ DBUYLTUBFAJ, GPDDNNXOHT TNHNUBYOLHG DBSN YL RPXTN B SBF LZ GPDA SBTHOYPXN DPSENUGLSN, BHX, HLY VOYALPY OUUNWNUNHDN, YANJ BEBHXLHNX OY YL YAN UOTLPUG LZ GPH BHX UBOH. OH YAN VNGYNUH XNGNUYG, YBYYNUNX ZUBTSNHYG LZ YAN SBF BUN GYOMM YL EN ZLPHX, GANMYNUOHT BH LDDBGOLHBM ENBGY LU ENTTBU; OH YAN VALMN HBYOLH, HL LYANU UNMOD OG MNZY LZ YAN XOGDOFMOHN LZ TNLTUBFAJ. + +# G Formal syntax + +# Statement ordering + +FORMAT statements may appear anywhere between the USE statement and the CONTAINS statement. + +The following table summarises the usage of the various statements within individ- ual scoping units. + +![](images/3c279cf4b28903aa07c7c82a2ad852585073cbf6fb2da0352a5112ccca45d317.jpg) + +1 Misc Dec (Miscellaneous declaration) are PARAMETER statements, IMPLICIT statements, type dec- laration statements and specification statements. + +# Syntax summary of some frequently used Fortran constructs + +The following provides simple syntactical definitions of some of the more fre- quently used parts of Fortran 95. + +# Main program + +PROGRAM [ program-name ] [ specification-construct ] ... [ executable-construct ] ... [CONTAINS [ internal procedure ] ... ] END [ PROGRAM [ program-name ] ] + +# Subprogram + +procedure heading + +[ specification-construct ] ... [ executable-construct ] ... [CONTAINS [ internal procedure ] ... ] + +# Module + +[ specification-construct ] ... [CONTAINS subprogram [ subprogram ] ... ] END [ MODULE [ module-name ] + +# Internal procedure + +procedure heading + +[ specification construct ] ... [ executable construct ] ... + +procedure ending + +# Procedure heading + +[ RECURSIVE ] [ type specification ] FUNCTION function-name & ( [ dummy argument list ] ) [ RESULT ( result name + + ) ] + + [ RECURSIVE ] SUBROUTINE subroutine name & [ ( [ dummy argument list ] ) ] + +# Procedure ending + +END [ FUNCTION [ function name ] ] END [ SUBROUTINE [ subroutine name ] ] + +# Specification construct + +derived type definition interface block specification statement + +# Derived type definition + +TYPE [[ , access specification ] :: ] type name [ PRIVATE ] [ SEQUENCE ] [ type specification [[ , POINTER ] :: ] component specification list ] + +END TYPE [ type name ] + +# Interface block + +INTERFACE [ generic specification ] [ procedure heading [ specification construct ] ... procedure ending ] ... [ MODULE PROCEDURE module procedure name list ] ... END INTERFACE + +# Specification statement + +ALLOCATABLE [ :: ] allocatable array list DIMENSION array dimension list EXTERNAL external name list FORMAT ( [ format specification list ] ) IMPLICIT implicit specification INTENT ( intent specification ) :: dummy argument name list INTRINSIC intrinsic procedure name list OPTIONAL [ :: ] optional object list PARAMETER ( named constant definition list ) POINTER [ :: ] pointer name list PUBLIC [ [ :: ] module entity name list ] PRIVATE[ [ :: ] module entity name list ] SAVE[ [ :: ] saved object list ] TARGET [ :: ] target name list USE module name [ , rename list ] USE module name , ONLY : [ access list ] type specification [ [ , attribute specification ] ... :: & + +# Type specification + +INTEGER [ ( [ KIND $=$ ] kind parameter ) ] REAL[ ( [ KIND $=$ ] kind parameter ) ] COMPLEX[ ( [ KIND $=$ ] kind parameter ) ] CHARACTER[ ( [ KIND $\mathrm{=}$ ] kind parameter ) ] CHARACTER[ ( [ KIND $\mathrm{=}$ ] kind parameter ) ] & [ LEN $\left.\stackrel{\cdot}{=}\right.$ ] length parameter )LOGICAL[ ( [ KIND $=$ ] kind parameter ) ] TYPE ( type name ) + +# Attribute specification + +ALLOCATABLE DIMENSION ( array specification ) EXTERNAL INTENT ( intent specification ) INTRINSIC OPTIONAL PARAMETER POINTER PRIVATE PUBLIC SAVE TARGET + +# Executable construct + +action statement case construct do construct if construct where construct + +# Action statement + +ALLOCATE ( allocation list ) [ ,STAT $^{\ast}{=}$ scalar integer variable ] ) CALL subroutinename [ ( [ actual argument specification list] ) ] CLOSE ( close specification list ) CYCLE [ do construct name ] DEALLOCATE( name list ) [ , STAT $=$ scalar integer variable ] ) ENDFILE external file unit + +EXIT [ do construct name ] GOTO label IF ( scalar logical expression ) action statement INQUIRE ( inquire specification list ) [ output item list ] NULLIFY ( pointer object list ) OPEN ( connect specification list ) PRINT format [ , output item list ] READ (i/o control specification list ) [ input item list ] READ format [ , output item list ] RETURN [ scalar integer expression ] REWIND ( position specification list ) STOP [ access code ] WHERE ( array logical expression ) array assignment expression WRITE ( i/o control specification list ) [ output item list ] pointer variable $=>$ target expression variable $=$ expression + +# H Compiler Options + +In this appendix we look at what compiler options there are that can help us at both compile time and run time. + +# CVF + +/check or /check:all + +Equivalent to: /check:(arg temp created, bounds, flawed_pentium, format, power, output conversion, overflow, underflow). + +/debug + +If you specify /debug:full, /debug, $/Z\mathrm{i}$ , or /Z7, the compiler produces symbol table information needed for full symbolic debugging of unoptimised code and global symbol information needed for linking. This is the default for a debug configura- tion in the visual development environment. + +/exe[:file] + +The /exe or /Fe option specifies the name of the executable program (EXE) or dy- namic-link library (DLL) file being created. To request that a DLL be created instead of an executable program, specify the /dll option. + +/fltconsistency + +The /fltconsistency or $/\mathrm{Op}$ option enables improved floating point consistency on ia32 systems. floating point operations are not reordered and the result of each floating point operation is stored into the target variable rather than being kept in the floating point processor for use in a subsequent calculation. This option is ig- nored on ia64 systems + +/fpe:0 + +The /fpe:level option controls floating point exception handling at run time for the main program. This includes whether exceptional floating point values are allowed and how precisely run time exceptions are reported. The /fpe:level option specifies how the compiler should handle the following floating point exceptions: + +When floating point calculations result in a divide by zero, overflow, or invalid op- eration. When floating point calculations result in an underflow. When a denormalised number or other exceptional number (positive infinity, negative infin- ity, or a NaN) is present in an arithmetic expression + +/list + +The /list or /Fs option creates a listing of the source file with compile time infor- mation appended. To name the source listing file, specify file. + +# Appendix H — Compiler Options + +/map + +/map[:file], /nomap, or /Fmfile The /map or $/\mathrm{Fm}$ option controls whether or not a link map is created. To name the map file, specify file. + +/nooptimize + +/show + +The /show option specifies what information is included in a listing. In the visual development environment, specify the Source Listing Options in the Listing File Compiler Option Category. + +/traceback + +The /traceback option requests that the compiler generate extra information in the object file that allows the display of source file traceback information at run time when a severe error occurs. + +/warn:all + +The /warn option instructs the compiler to generate diagnostic messages for de- fined classes of additional checking that can be performed at compile time. + +# Intel + +/check:all + +Determines whether several run time conditions are checked.keyword: all, none, + + [no]arg temp created, [no]bounds, [no]format, [no]output conversion + +/debug:full + +Determines the type of debugging information generated by the compiler in the ob- ject file. keyword: minimal, partial, full, none. + +/exe:%1intel + +/[no]fltconsistency + +Determines whether improved floating point consistency is used. + +/fpe:0 + +Specifies floating point exception handling at run time for the main program; $\mathrm{~n~}=$ 0, 1, or 3. 0 - floating underflow results in zero; all other floating point exceptions abort execution + +/[no]map[:name] + +Determines whether the compiler generates a link map (optionally, named name ). + +/[no]stand[:keyword] + +Tells the compiler to issue warnings for nonstandard Fortran language elements. keyword: f90, f95, none. + +/[no]traceback + +Specifies whether the compiler should generate extra information in the object file that allows the display of source file traceback information at run time when a se- vere error occurs. + +/warn:all + +/map + +# Lahey + +-chk + +Specify -chk to generate a fatal run time error message when substring and array subscripts are out of range, when non common variables are accessed before they are initialised, when array expression shapes do not match, and when procedure ar- guments do not match in type, attributes, size, or shape. + +-f95Specify -f95 to generate warnings when the compiler encounters nonstandard For- tran 95 code. + +-gSpecify -g to instruct the compiler to generate an expanded symbol table and other information for the debugger. + +-info Specify -info to display informational messages at compile time. Informational messages include such things as the level of loop unrolling performed, variables declared but never used, divisions changed to multiplication by reciprocal, etc. + +-lst Specify -lst to generate a listing file that contains the source program, compiler op- tions, date and time of compilation, and any compiler diagnostics. + +-OUT Default: the name of the first object or source file. + +-map Default: create a map file with same name as output file + +-trap diou + +The -trap option specifies how each of four numeric data processor (NDP) excep- tions will be handled at execution time of your program. + +-trace The -trace option causes a call traceback with procedure names and line numbers to be generated with run time error messages. + +-fullwarn provides the maximum level of warning and informational messages. + +Specify -xref to generate cross-reference information. This information is shown in the listing file in addition to the information that the -lst option would provide. + +# NAG + +-C + +ompile code with all possible run time checks. all array calls do none present pointer + +-float-store + +gnu C based systems. Do not store floating point variables in machines with float- ing point registers wider than 64 bits. + +-glineInclude line number information in run time error messages. + +-ieee $=$ stop + +Enables all IEEE arithmetic facilities except for nonstop arithmetic. Execution is terminated on floating overflow, divide by zero or invalid operand. + +-o output + +-strict95 + +# Salford + +/CHECKMATE + +A synonym for /FULL_UNDEF. /FULL_UNDEF implies /UNDEF which in turn implies /CHECK. + +/DEBUG + +causes FTN95 to generate symbolic information and to activate the symbolic debugger when fatal errors occur. /DEBUG is included in both /CHECK and /UNDEF, which are normally preferred. /DEBUG can be used on its own in order to allow the debugger to be used on "dirty" code, which intentionally violates some of the rules of Fortran. + +/FULL_DEBUG + +Outputs full debugging information including PARAMETERs + +/FULL_UNDEF + +Like /UNDEF but also INTENT(OUT) arguments are initialised as undefined on entry to a procedure (see /INHIBIT_CHECK to switch this off) and by default character variables are initialised as undefined. The reading of an undefined value precipitates an error condition. + +/link Generate executable. + +/list + +/map + +Control the output of a listing file. + +/xref Generates a crossreference of all variables in a program and subprograms. + +The use of the Win32 /UNDERFLOW option ensures that the first occurrence of underflow in an arithmetical computation is treated as a failure and is not ignored as would otherwise be the case. A large number of occurrences of underflow dur- ing execution can result in long execution times because of the way in which the underflow condition is treated. If an underflow is trapped, the message + +ERROR: Floating point arithmetic underflow + +is output and the interactive debugger is entered. If underflows occur during pro- gram execution and the /UNDERFLOW option is not used, a message is output at the end of the run specifying the number of underflows that have occurred. + +# Index + +# A + +A edit descriptor 186, 244 ABS function 205, 206, 211, 542 Accuracy 86, 102, 426, 476, 478–479 ACHAR function 211, 250, 542 ACOS function 205, 211, 542 ActiveX 49-50 Actual argument 299, 310, 313, 315, 346, 367, 425, 427 Ada 2, 9, 43-44, 56-58, 60, 64, 536 Addition 78, 80, 126, 133-135, 137, 139, 141, 143, 145, 147, 149, 260, 517 Addition operator 80, 437 Adjustable array 310–311, 520 ADJUSTL function 211, 250, 543 ADJUSTR function 211, 250, 543 AIMAG function 211, 543 AINT function 211, 542, 543 Algol 60 16, 38-41, 470 Algol 68 2, 40-41, 56, 536 + +Algorithm 28, 33-34, 105, 108, 221–222, 268, 340, 369, 397, 470–471, 478, 488, 511, 514, 518–519 ALL function 211, 541, 543 ALLOCATABLE arrays 119, 133 ALLOCATABLE attribute 52, 120, 149, 310, 315, 361, 520, 544 ALLOCATABLE dummy array 493–494 ALLOCATE statement 121, 149, 270, 288, 292, 424 Alternate RETURN 367–368 ANINT function 211, 544 ANY function 211, 544 APL 2, 41, 60 Argument 210, 212, 295, 359, 370, 390, 392–393, 398, 447, 450–457, 541–573 actual 299, 310, 313, 315, 346, 367, 425, 427, 520 allocatable dummy array argument 493, 494 array 206, 313, 316, 317, 319, 320, 367, 554, 556, 567 assumed-length 315, 316, 521 + +assumed-shape 310, 313, 315, 316, 319, 320, 336, 367, 521 assumed-size 310, 367, 521, 567, 568 character 315 dummy 52, 220, 299, 310, 311, 313, 315–316, 336, 346, 367, 424–427, 442– 445, 520–523 keyword and optional 425, 427 PRESENT intrinsic function 210, 426–427, 563 rank 2 and higher array arguments 316 Arithmetic 5, 77–79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 116, 366, 473–475, 477, 479, 481, 483, 485, 487–491 Arithmetic assignment statement 68, 70, 78, 102, 213, 271 Arithmetic IF 366 Array constructor 140, 148 Array element 134, 148 Array element ordering 133–134, 138,142, 148, 170, 438, 520 Array functions 211 Array section 138 Arrays 107–149 adjustable 310–311, 520 allocatable 51, 119, 133, 270, 310 321, 425, 494, 520 assumed-shape 310, 313, 315–316 319–320, 336, 367, 521 assumed-size 310, 367, 521 automatic 310,322,424–425 bounds 134, 149, 310–311, 313 315, 317, 321, 478, 503, 516, 521 constructor 52, 133, 136, 140–141 148, 502, 520 + +conformable 134–135, 147, 149, 522 deferred shape 149, 310, 321, 361, 424, 523 element 133, 135, 170, 206, 246 446, 520 explicit shape 310–311, 313, 316–317, 367, 520–521, 523 extent 13–135, 149, 310, 313 315, 367, 524 rank 134–135, 138, 140–146, 148, 149, 310, 313, 315–317 361, 367, 424, 438, 444, 470, 503 520, 523, 527 section 133, 138, 140, 144–145 148, 168–169, 189, 438, 494, 520–521, 528 shape 134,142,542,565 size 110, 114, 119–120, 133–134 149, 270, 310, 313, 424–425, 524 528 stride 144, 147, 494, 528 whole array manipulation 135 ASCII character set 5, 242, 542, 546, 552 ASIN function 205, 211, 544 ASSIGN and assigned GOTO statements 367 Assigned FORMAT statements 367 Assignment statement 68, 70, 78, 102, 104, 112, 213, 270–271 ASSOCIATED function 273 Assumed length dummy argument 315–316, 521 Assumed-shape arrays 310, 313, 315–316, 319–320, 336, 367, 521 Assumed-size 310, 367, 521 ATAN function 205, 211, 522, 545 + +ATAN2 function 205, 211, 256, 545 Attribute specification 579 Automatic arrays 310, 322, 424–425 + +# B + +Basic 2, 41, 59–60, 536 BIT_SIZE function 210, 545 Blanks 179–180, 185, 232 Blanks, nulls and zeros 185 Block IF statement 225–226, 237 BTEST function 98–101, 211, 546 + +# C + +C 2, 41–42, 49–50, 53, 56, 58–59 $\mathrm{C++}$ 2,48–50,56,58,60,508,537 CALL statement 295, 299, 307, 528 CASE statement 223, 229–230, 367–368 CEILING function 51, 211, 546 CHAR function 211, 249–250, 546 Character argument 315 Character functions 247 Character I/O 241–252 Character operators 244 CHARACTER statement 66–67, 241–252 Character string 250 CLOSE statement 151, 163, 176, 188, 195 + +CMPLX function 211, 254–256, 546 Co-array Fortran 508 Cobol 2, 37–38, 40, 60, 537 Collating sequence 248 Comments 66, 70–71, 73 Common mistakes 162 Compilation unit 522 Compilers used 6–7 Complex 89, 253–256, 297, 302, 305, 478, 541 Computational functions 211 Concatenate 565 Conformable 134 CONJG function 211, 255, 547 CONTAINS statement 219, 304, 576 Continuation character & 71 Control Structures 223–225, 227, 229, 231, 233, 235, 237, 239 Converting from Fortran 77 366–367, 369, 371, 373, 375, 377, 379, 381, 383, 385, 387,389, 391, 393, 395, 397,399, 401, 403, 405, 407, 409 COS function 204–205, 211, 254, 424 547 COSH function 211, 547 COUNT function 211 547 CSHIFT function 211, 548 CYCLE and EXIT 234, 515 CYCLE statement 223, 234 + +# D + +Data description statements 65 + +Data processing statements 70 Data structures 511,514 Data type 3–4, 41, 46–47, 51–52, 63, 65, 69, 82, 92–93, 226, 242 244, 250, 254, 256, 264–268, 270, 288, 346–347, 362, 410, 412, 427–428, 441, 446, 470, 481, 516, 521–522 DATE_AND_TIME subroutine 211, 326, 328, 334, 337, 352, 355–356, 455–456, 458–459,495, 548 DBLE function 211, 549 DEALLOCATE statement 270, 326, 328, 428, 579 Debugging 260, 512–513, 516, 581–582, 585 Declaration 370, 398, 448, 517, 574 Decremented features 366 Default kind 102, 523 Deferred-shape arrays 149, 310, 321, 361, 424, 523 Deleted features 366 Derived type definition 578 DIGITS function 102, 210, 549 DIM function 146, 211, 549, 557–558 DIMENSION attribute 3, 110, 112, 115, 119, 126, 129 additional form 126 Division 78, 230 DO construct 130 + +DO loop 100, 113, 116, 118–119, 127–128, 130,137, 140, 169, 201, 284, 313, 325 DO statement 113, 126, 128–129 DO WHILE construct 212–215, 217, 224, 232–233, 236, 238, 278, 281–283, 317, 408–409, 414,416 DOT_PRODUCT function 136 141, 207, 211, 549 DOUBLE PRECISION 367, 369–370, 377 DPROD function 211, 549 Dummy argument 299, 523 assumed-length 315, 316, 521 assumed-shape 310, 313, 315, 316, 319, 320, 336, 367, 521 assumed-size 310, 367, 521, 567, 568 character 315 explicit shape array310–311, 313, 316–317, 367, 520–521, 523 Dummy procedure argument 425 Dynamic data structures 412, $^{470+}$ + +E edit descriptor 162 Editors 19–20 Efficiency 511,516 Elemental 206–207, 218, 411–412, 443–445, 541 Elemental functions 206, 218 Elemental subroutine 411–412, 445 Elements of a programming language 64 + +ELSE block 215–216, 228 ELSE IF 223, 229, 237–238 ELSEWHERE block 147END DO statement 201, 230, 234 END FUNCTION statement 212–213, 220 END PROGRAM statement 67, 71 END SELECT statement 230–231, 238 Entity 523 Entity oriented declaration 125–126 EOSHIFT function 211, 550 EPSILON function 94, 96, 210, 550 Errors when reading 192 Evaluation and testing 31 Exception handling 50, 477, 480, 488, 581–582 EXIT statement 223, 230, 234, 515 EXP function 205, 211, 235–236, 239, 470, 550 Explicit interface 356, 368, 503, 523 Explicit-shape array 310–310, 520–521, 523 EXPONENT function 210, 550 Exponentiation operator 78, 80 Expressions 135, 259 Extent 134 External 370, 387, 392–393, 398, 576 + +# F + +F edit descriptor 151, 155 File name 164, 179, 193, 198, 316 FILE $\scriptstyle{\varepsilon=}$ specifier 163–164, 173, 197 Files 195–202 unformatted 174–176, 191, 199 Fixed fields on input 180 FLOOR function 51, 211, 551 FMT $\leftrightharpoons$ specifier 163–165 FORALL statement and construct 51, 133, 147–148 FORM $\circeq$ specifier 175,191 FORMAT statement 150–193 Formatted data 199 Fortran character set 72, 242, 251, 504 FRACTION function 210, 551 Functions 94, 203–205, 207, 209, 211, 213, 215, 217, 219, 221, 370, 398, 450–451, 455, 458, 478, 541–573 arguments 219 array 211 computational 211 elemental 206, 218, 411–412, 443–444 generic 205–206, 210 inquiry 210, 524 internal 218 intrinsic 203–212, 541–573 pure 51, 203, 2128, 442–443 recursive 203, 214–215, 219 transfer and conversion 210 transformation al 203, 206 user defined 203–212 + +# Index + +# G + +Gaussian elimination 341, 356–357 Generic 205, 411, 427 Generic functions 205 Generic procedures 411, 427 Global 378 Good programming guidelines 73 GOTO statement 224, 237, 240, 367, 511, 515 + +# H + +High Performance Fortran 51, 147, 509 Host association 53, 504, 524 HPF 7, 51, 147, 509 HUGE function 94, 96, 102, 210, 551 + +# I + +I edit descriptor 162,180 IACHAR function 211, 250, 551 IAND function 211, 551 IBCLR function 211, 552 IBITS function 211, 552 IBSET function 211, 552 ICHAR function 211, 249, 552 ICON 2, 46, 538 IEEE 5, 52–53, 57, 61, 94, 105, 473–477, 479–492, 504, 584 IEOR function 211, 552 + +IF statement 225–226, 228–229, 258, 366, 515 IMPLICIT NONE statement 66 Implied DO loops 168 INDEX function 247–248, 251 I/O 3–4, 12, 16, 66–67, 71, 122, 198, 200, 294, 337, 443, 503 INQUIRE statement 580 Inquiry functions 210 INT function 98 Integer data type 69, 71 Integer kind type 88–90 Intent 299 INTENT attribute 295, 299, 307, 443, 445 Interface 301, 304, 306, 336, 508, 576, 578 Interface block 301, 304, 306, 336, 576, 578 Internal functions 218 Internal procedure 577 Internal subroutine 325 Intrinsic functions 203–212, 541–573 IOR function 211, 553 ISHFT function 211, 554 ISHFTC function 211, 554 ISO TR 15580 472–492 ISO TR 15581 493–499 + +# K + +Keyword and optional argument 425, 427 Kind 88, 96–98, 542–544, 546, 551, 553, 556, 562, 565, 576 KIND function 90 Kind type parameter 90–91, 344, 361 523, 525 + +# L + +L edit descriptor 261 LaTeX 45 LBOUND function 210, 554 LEN function 210, 248, 250, 555 LEN_TRIM function 211, 248, 250, 555 LGE function 206, 211, 249–250, 555 LGT function 206, 211, 249–250, 555 Linked list 270 LINPACK 357,363 Lisp 2, 39–40, 56, 60, 538 LLE function 206, 211, 249–250, 555 LLT function 206, 211, 249–250, 556 Local variables 295, 298–300, 303, 306, 334–336, 359, 419, 443 Local variables and the SAVE attribute 295, 300 LOG function 205, 207–208, 211, 254 556 LOG10 function 205, 211, 556 + +Logical expression 223, 225 LOGICAL function 211, 556 Logical operators 257, 261 Logical variable 257 Logo 2, 44, 538 Lower bound 52, 144, 235, 315, 503, 554 + +# M + +Main program 296, 576 Maintenance 30, 32 Mantissa 92–93, 95, 101, 477, 479 Mask 146–148 Masked array assignment 146 MATMUL function 136, 211, 321–322 556 MAX function 52, 211, 503, 557 MAXEXPONENT function 210, 557 MAXLOC function 51, 211, 359, 361 557 MAXVAL function 211, 359, 361, 421 558 MERGE function 211, 559 MIN function 52, 211, 503, 559 MINEXPONENT function 210, 559 MINLOC function 51, 211, 559 MINVAL function 211, 560 MOD function 205, 208–212, 217, 220, 560 + +# Index + +Modula 2 2, 29, 44, 47, 53, 58,61, 268, 323, 471, 512, 538 + +Module 5, 51, 54–56, 301, 341–343, 345–347, 349, 351, 353, 355, 357, 359, 361, 363, 411, 501–503, 505, 513, 576–577 PUBLIC and PRIVATE attributes 446–448, 578–579 USE 342–344, 347, 368 USE ONLY 390–393, 395 Modules and packaging 411 Modules containing procedures 342, 349, 353 Modules for derived data types 346 Modules for global data 342 MODULO function 211, 561 MPI 508 Multiple statements 71 MVBITS function 211, 561 + +# N + +NEAREST function 210, 561 Nested user defined types 266 Nesting 122 Netlib 518 Networking 17 NINT function 211, 562 NOT function 211, 562 NULL function 51, 210, 275–276, 562 NULLIFY statement 270,279 + +# O + +Oberon 2 2, 46–47, 512 Object oriented programming 29, 48, 52, 502, 504 OPEN statement 163, 188, 193, 197 OPENMP 508 Operator and assignment overloading 411–412, 436 Operator hierarchy 259 Operators 78,80 Optional arguments 336, 425–427 Overflow 87, 153, 156–158, 473 477, 480, 581, 584 + +# P + +PACK function 211, 438–439, 562 Parameters 114,369,397 Pascal 2, 34, 41, 43–44, 56–57, 59–61, 222, 268, 470–471, 512, 538 PL/1 2,40 POINTER attribute 270–271, 526 Pointers 269–294 arrays of pointers 283 assignment statement 270–271 association status 270 Postscript 45, 56, 539 Precision 86, 96–98, 326, 330, 343–344, 357–359, 361–362, 418–419, 422, 424, 426, 517 + +PRECISION function 94, 96, 102, 210 563 PRESENT function 563 Pretty print 517 PRINT statement 67, 70–71, 152–153, 164, 177 PRIVATE attribute 446–448, 578–579 PRIVATE statement 578 PRODUCT function 211, 563 Program testing 516 Program unit 115, 148, 213–214, 299–300, 336, 342, 344–345 362, 425, 427, 441, 513, 527 Programming languages 2, 39, 65 Programming style 511, 513–514, 519 Prolog 2, 45, 56, 59, 370, 397, 539 PUBLIC attribute 446–448, 578–579 PUBLIC statement 578 PURE keyword 51, 203, 2128, 442–443 PVM 509 + +# R + +RADIX function 210, 564 RANDOM_NUMBER subroutine 211, 314, 564 RANDOM_SEED subroutine 211, 564 Range and precision of numbers 88, 90–91, 94, 525 RANGE function 210, 565 + +Rank 134–135, 138, 140–146, 148, 149, 310, 313, 315–317 361, 367, 424, 438, 444, 470, 503 520, 523, 52 Rank 2 and higher arrays as parameters 316 READ statement 67, 70, 187–188, 198, 200, 243, 279, 293, 367, 505 ADVANCE $\scriptstyle{\varepsilon=}$ 278–279, 293, 441 IOSTAT $\leftleftarrows$ 200–201, 278–279, 281, 283, 293, 505 Reading 179–181, 183, 185, 187, 189, 191, 193, 280 Reading in Data 179–181, 183, 185, 187, 189, 191, 193 REAL function 205, 211, 565 Real kind type 91, 94, 101, 221, 256, 568 Reals 155, 158, 181–182 Recursion 51, 214, 216–217, 222, 337, 511, 515 Recursive 214–216, 332, 337, 340 Recursive functions 214 Recursive subroutines 332 Referencing a subroutine 299 Relational expression 527 Relational operator 226, 527 REPEAT function 211, 250, 565 REPEAT UNTIL loop 232 Repetition 165–166 RESHAPE function 142, 148, 520 RETURN statement 219 + +# Index + +REWIND 179, 188–189, 193–194, 580 Rounding and truncation 81 RRSPACING function 210, 566 + +# S + +SAVE attribute 295, 300 SCALE function 210, 488, 566 SCAN function 211, 250, 566 Scope 39, 295, 300 Scope of variables 295,300 Scoping unit 527 SELECT CASE statement 230–231, 238, 408–409 SELECTED IN T KIND function 90 SELECTED_REAL_KIND function 90–92, 95, 210, 256, 567 SET_EXPONENT function 210, 567 Shape 134,142,542,565 SHAPE function 142,148,520 Shared DO termination and 366 SIGN function 51 Simula 2, 40, 46, 48, 56, 60, 417, 539 SIN function 204–207, 211, 254, 568 Singly linked list 278 SINH function 211, 568 SIZE function 210, 568 Skipping spaces and lines 187 Smalltalk 2, 47, 56, 58, 539 Snobol 2, 40, 46, 58, 60, 539 + +SPACING function 210, 569 Specification construct 577 Specification statement 578 SPREAD function 211, 569 SQL 2, 43, 45, 57, 539 SQRT function 205, 211, 219, 221, 228, 256, 569 Standardisation 42 Stepwise refinement 29 STOP statement 201 Stride 144, 147, 528 Structured programming 42,511,515 Subprogram 577 Subroutine 204, 295–297, 299–301, 303, 305, 307, 309– 311, 313, 315, 317, 319, 321, 323, 325, 327, 329, 331, 333, 335, 337, 339, 370, 387, 392–393, 398, 424, 439, 541 actual arguments 299 assumed-shape arrays 315, 320 automatic arrays 310, 322 contained 325–326 dummy arguments 298–299, 313, 315–316, 336 interface blocks 295, 301, 304, 306–307, 313, 336, 339 internal 325 INTENT attribute 295, 299, 307 KEYWORD and optional arguments 425 local variables and the SAVE attribute 295 rank 2 and higher arrays as arguments 316 Substring 250,553 + +Subtraction operator 78 SUM function 207 Supplying your own functions 212 SYSTEM_CLOCK subroutine 211, 570 Systems Analysis and Design 33–34 + +# T + +TAN function 204–205, 211, 424, 570 TANH function 211, 571 Target 541–542, 545 Target attribute 270–271, 336 TeX 45, 59 TINY function 210, 571 Transfer and conversion functions 210 TRANSFER function 100, 211, 571 Transformation al functions 206 TRANSPOSE function 211, 321–322, 571 Trees 440 TRIM function 211, 251, 572 Triplet 144, 147–148, 528 Truncation 82, 543 Type declaration 70–71 Type definition 264 + +Unformatted files 174, 176, 191, 199, 516 UNPACK function 211, 572 Upper bound 144, 235, 315, 572 USE 342–344, 347, 368, 576, 578 USE ONLY 390–393, 395, 578 User defined functions 296 User defined types 263–268 + +# V + +Variables 68–70, 114, 122–124, 128, 481, 503 status – undefined 70, 101, 270, 273 Vector 107, 110, 438, 549, 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'horizontal'] + invalid_chars: True +Classification Results: is_text_pdf=True, results={'by_image_area': True, 'by_text_len': True, 'by_avg_words': True, 'by_img_num': True, 'by_text_layout': True, 'by_img_narrow_strips': True, 'by_invalid_chars': True} +Error occurred: list indices must be integers or slices, not str +GPU 0 - Batch size: 1, VRAM usage: 0.00GB / 44.34GB +Processing PDF: input/e0f88b7f2e5a62dec8f0a1825a312092.pdf +Metadata: { + "is_needs_password": 0, + "is_encrypted": false, + "total_page": 236, + "page_width_pts": 482, + "page_height_pts": 680, + "image_info_per_page": [ + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [ + [ + 56, + 309, + 383, + 528, + 45 + ] + ], + [], + [ + [ + 141, + 269, + 382, + 475, + 53 + ] + ], + [ + [ + 102, + 354, + 317, + 592, + 58 + ] + ], + [], + [], + [ + [ + 108, + 57, + 380, + 390, + 68 + ] + ], + [ + [ + 102, + 421, + 318, + 590, + 73 + ] + ], + [], + [], + [ + [ + 171, + 258, + 386, + 591, + 84 + ] + ], + [], + [], + [ 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"PDF 1.5", + "title": "Titelei 1..4", + "author": "", + "subject": "", + "keywords": "", + "creator": "3B2 Total Publishing System 7.51d/W", + "producer": "Acrobat Distiller 5.0 (Windows)", + "creationDate": "D:20040701185454-04'00'", + "modDate": "D:20040701185713-04'00'", + "trapped": "", + "encryption": null + } +} +Classify parameters: total_page=236, page_width=482, page_height=680, img_sz_list=[[], [], [], [], []]..., text_len_list=[244, 585, 79, 1849, 1439]..., img_num_list=[0, 0, 0, 0, 0]..., text_layout_list=['horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal']..., invalid_chars=True +Detailed Classify Inputs for input/e0f88b7f2e5a62dec8f0a1825a312092.pdf: + total_page: 236 + page_width: 482 + page_height: 680 + img_sz_list (length 50): [[], [], [], [], [], [], [], [], [], [], [[56, 309, 383, 528, 45]], [], [[141, 269, 382, 475, 53]], [[102, 354, 317, 592, 58]], [], [], [[108, 57, 380, 390, 68]], [[102, 421, 318, 590, 73]], [], [], [[171, 258, 386, 591, 84]], 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'by_text_layout': True, 'by_img_narrow_strips': True, 'by_invalid_chars': True} +Error occurred: list indices must be integers or slices, not str +GPU 0 - Batch size: 2, VRAM usage: 0.00GB / 44.34GB +Processing PDF: input/e5cf785df9dc15bbc59962defb61e981.pdf +Metadata: { + "is_needs_password": 0, + "is_encrypted": false, + "total_page": 228, + "page_width_pts": 540, + "page_height_pts": 666, + "image_info_per_page": [ + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [ + [ + 249, + 382, + 342, + 542, + 32 + ] + ], + [], + [], + [], + [], + [], + [], + [], + [], + [ + [ + 42, + 280, + 210, + 448, + 60 + ] + ], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [ + [ + 332, + 386, + 362, + 413, + 999 + ] + ], + [], + [ + [ + 235, + 145, + 280, + 190, + 999 + ] + ], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [] + ], + "text_len_per_page": [ + 73, + 926, + 235, + 1243, + 4096, + 4555, + 1681, + 0, + 2408, + 2284, 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"producer": "Acrobat Distiller Server 5.0.5 (Sparc Solaris)", + "creationDate": "D:20070813164847Z", + "modDate": "D:20070813221529+05'30'", + "trapped": "", + "encryption": null + } +} +Classify parameters: total_page=228, page_width=540, page_height=666, img_sz_list=[[], [], [], [], []]..., text_len_list=[73, 926, 235, 1243, 4096]..., img_num_list=[0, 0, 0, 0, 0]..., text_layout_list=['horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal']..., invalid_chars=True +Detailed Classify Inputs for input/e5cf785df9dc15bbc59962defb61e981.pdf: + total_page: 228 + page_width: 540 + page_height: 666 + img_sz_list (length 50): [[], [], [], [], [], [], [], [], [], [], [[249, 382, 342, 542, 32]], [], [], [], [], [], [], [], [], [[42, 280, 210, 448, 60]], [], [], [], [], [], [], [], [], [], [], [[332, 386, 362, 413, 999]], [], [[235, 145, 280, 190, 999]], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []] + text_len_list (length 228): [73, 926, 235, 1243, 4096, 4555, 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"D:20040701185454-04'00'", + "modDate": "D:20040701185713-04'00'", + "trapped": "", + "encryption": null + } +} +Classify parameters: total_page=236, page_width=482, page_height=680, img_sz_list=[[], [], [], [], []]..., text_len_list=[244, 585, 79, 1849, 1439]..., img_num_list=[0, 0, 0, 0, 0]..., text_layout_list=['horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal']..., invalid_chars=True +Detailed Classify Inputs for input/e0f88b7f2e5a62dec8f0a1825a312092.pdf: + total_page: 236 + page_width: 482 + page_height: 680 + img_sz_list (length 50): [[], [], [], [], [], [], [], [], [], [], [[56, 309, 383, 528, 45]], [], [[141, 269, 382, 475, 53]], [[102, 354, 317, 592, 58]], [], [], [[108, 57, 380, 390, 68]], [[102, 421, 318, 590, 73]], [], [], [[171, 258, 386, 591, 84]], [], [], [[102, 416, 260, 591, 94]], [[167, 427, 383, 593, 98]], [], [[139, 410, 381, 559, 106]], [], [], [], [[79, 454, 384, 549, 118]], [], [], [[102, 59, 339, 287, 126]], [], [[102, 58, 317, 345, 133]], [], 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'horizontal', 'horizontal'] + invalid_chars: True +Classification Results: is_text_pdf=True, results={'by_image_area': True, 'by_text_len': True, 'by_avg_words': True, 'by_img_num': True, 'by_text_layout': True, 'by_img_narrow_strips': True, 'by_invalid_chars': True} +Error occurred: list indices must be integers or slices, not str +GPU 1 - Batch size: 3, VRAM usage: 0.00GB / 44.34GB +Processing PDF: input/f2f24088a4ac947b8f3b29ace552cb90.pdf +Metadata: { + "is_needs_password": 0, + "is_encrypted": false, + "total_page": 599, + "page_width_pts": 441, + "page_height_pts": 666, + "image_info_per_page": [ + [], + [ + [ + 52, + 606, + 162, + 636, + 13 + ] + ], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [] + ], + "text_len_per_page": [ + 41, + 124, + 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'horizontal', 'horizontal', 'horizontal'] + invalid_chars: True +Classification Results: is_text_pdf=True, results={'by_image_area': True, 'by_text_len': True, 'by_avg_words': True, 'by_img_num': True, 'by_text_layout': True, 'by_img_narrow_strips': True, 'by_invalid_chars': True} +Error occurred: list indices must be integers or slices, not str +Processing PDF: input/dcc863d5a46cff540ccca1430f488931.pdf +Metadata: { + "is_needs_password": 0, + "is_encrypted": false, + "total_page": 582, + "page_width_pts": 415, + "page_height_pts": 637, + "image_info_per_page": [ + [ + [ + 0, + 0, + 449, + 682, + 1752 + ] + ], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [] + ], + "text_len_per_page": [ + 0, + 91, + 74, + 48, + 1713, + 2044, + 2946, + 2419, + 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'horizontal']..., invalid_chars=True +Detailed Classify Inputs for input/dcc863d5a46cff540ccca1430f488931.pdf: + total_page: 582 + page_width: 415 + page_height: 637 + img_sz_list (length 50): [[[0, 0, 449, 682, 1752]], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []] + text_len_list (length 582): [0, 91, 74, 48, 1713, 2044, 2946, 2419, 3591, 4475, 4392, 3988, 4915, 4844, 4074, 2470, 1901, 2052, 2380, 2341, 2155, 2262, 1830, 1906, 1115, 2302, 127, 2491, 1854, 1296, 2005, 2195, 1934, 2678, 2860, 2593, 2767, 2287, 2452, 1577, 2452, 2709, 2530, 1708, 2711, 2469, 2679, 2148, 3021, 1226, 1621, 2420, 1421, 2981, 1845, 1862, 2208, 2775, 2605, 2755, 2329, 2214, 1643, 2573, 2596, 2430, 2694, 2811, 2584, 2775, 2901, 2765, 2728, 2792, 2304, 279, 1209, 2892, 1979, 2772, 150, 1109, 2025, 2667, 1533, 1922, 2565, 2657, 2479, 2862, 2624, 347, 1887, 1179, 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'horizontal', 'horizontal']..., invalid_chars=True +Detailed Classify Inputs for input/e5cf785df9dc15bbc59962defb61e981.pdf: + total_page: 228 + page_width: 540 + page_height: 666 + img_sz_list (length 50): [[], [], [], [], [], [], [], [], [], [], [[249, 382, 342, 542, 32]], [], [], [], [], [], [], [], [], [[42, 280, 210, 448, 60]], [], [], [], [], [], [], [], [], [], [], [[332, 386, 362, 413, 999]], [], [[235, 145, 280, 190, 999]], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []] + text_len_list (length 228): [73, 926, 235, 1243, 4096, 4555, 1681, 0, 2408, 2284, 1774, 2816, 1243, 2321, 1608, 2663, 1638, 3059, 2118, 2516, 2130, 1681, 1965, 2377, 2787, 2693, 2812, 2462, 2298, 2480, 2448, 2158, 1678, 2464, 1794, 1962, 1927, 2624, 3240, 2442, 2128, 2404, 2574, 2359, 1749, 1312, 1840, 1635, 1906, 2343, 2315, 3018, 2024, 2101, 1560, 2002, 1500, 1424, 2563, 3201, 1406, 1255, 1426, 2966, 2073, 2090, 1753, 2801, 2166, 2865, 2431, 2137, 2014, 1875, 2016, 2097, 2037, 1820, 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'horizontal', 'horizontal']..., invalid_chars=True +Detailed Classify Inputs for input/b8ff2f69a03b82d2f2ba8100ab287148.pdf: + total_page: 397 + page_width: 453 + page_height: 684 + img_sz_list (length 50): [[[0, 0, 600, 450, 1193]], [[0, 0, 308, 497, 3]], [], [], [], [[35, 597, 170, 641, 17]], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []] + text_len_list (length 397): [0, 0, 44, 1792, 9, 122, 1068, 49, 962, 1654, 433, 9, 2730, 3967, 3381, 2898, 3316, 3670, 2487, 3221, 3050, 3758, 2221, 1962, 2157, 2158, 1891, 2504, 1800, 3585, 2168, 1537, 3474, 521, 2979, 3353, 3073, 2227, 2513, 1483, 2858, 2732, 2632, 2680, 2633, 2286, 1995, 1853, 2485, 2362, 3945, 3559, 3622, 3414, 1910, 2960, 2148, 3468, 2263, 1954, 2388, 3952, 1738, 3895, 3851, 2302, 2936, 2843, 2269, 3317, 3395, 2743, 2819, 2044, 2557, 3640, 3604, 1891, 2985, 2378, 2104, 3173, 2779, 1177, 2834, 1569, 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+Detailed Classify Inputs for input/e0f88b7f2e5a62dec8f0a1825a312092.pdf: + total_page: 236 + page_width: 482 + page_height: 680 + img_sz_list (length 50): [[], [], [], [], [], [], [], [], [], [], [[56, 309, 383, 528, 45]], [], [[141, 269, 382, 475, 53]], [[102, 354, 317, 592, 58]], [], [], [[108, 57, 380, 390, 68]], [[102, 421, 318, 590, 73]], [], [], [[171, 258, 386, 591, 84]], [], [], [[102, 416, 260, 591, 94]], [[167, 427, 383, 593, 98]], [], [[139, 410, 381, 559, 106]], [], [], [], [[79, 454, 384, 549, 118]], [], [], [[102, 59, 339, 287, 126]], [], [[102, 58, 317, 345, 133]], [], [[102, 59, 288, 580, 141]], [], [[102, 404, 318, 561, 150], [102, 58, 319, 386, 148]], [], [[102, 58, 343, 217, 158]], [], [[103, 56, 376, 280, 164]], [[110, 56, 383, 314, 168]], [[102, 416, 374, 548, 172]], [[165, 59, 381, 266, 176]], [[102, 413, 318, 587, 180]], [[164, 62, 380, 342, 184]], [[102, 59, 261, 160, 189], [102, 462, 232, 592, 188]]] + text_len_list (length 236): [244, 585, 79, 1849, 1439, 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'by_img_num': True, 'by_text_layout': True, 'by_img_narrow_strips': True, 'by_invalid_chars': True} +Error occurred: list indices must be integers or slices, not str +Processing PDF: input/dcc863d5a46cff540ccca1430f488931.pdf +Metadata: { + "is_needs_password": 0, + "is_encrypted": false, + "total_page": 582, + "page_width_pts": 415, + "page_height_pts": 637, + "image_info_per_page": [ + [ + [ + 0, + 0, + 449, + 682, + 1752 + ] + ], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [] + ], + "text_len_per_page": [ + 0, + 91, + 74, + 48, + 1713, + 2044, + 2946, + 2419, + 3591, + 4475, + 4392, + 3988, + 4915, + 4844, + 4074, + 2470, + 1901, + 2052, + 2380, + 2341, + 2155, + 2262, + 1830, + 1906, + 1115, + 2302, + 127, + 2491, + 1854, + 1296, + 2005, + 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'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal'] + invalid_chars: True +Classification Results: is_text_pdf=True, results={'by_image_area': True, 'by_text_len': True, 'by_avg_words': True, 'by_img_num': True, 'by_text_layout': True, 'by_img_narrow_strips': True, 'by_invalid_chars': True} +Error occurred: list indices must be integers or slices, not str +GPU 3 - Batch size: 2, VRAM usage: 0.00GB / 44.34GB +Processing PDF: input/e5cf785df9dc15bbc59962defb61e981.pdf +Metadata: { + "is_needs_password": 0, + "is_encrypted": false, + "total_page": 228, + "page_width_pts": 540, + "page_height_pts": 666, + "image_info_per_page": [ + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [ + [ + 249, + 382, + 342, + 542, + 32 + ] + ], + [], + [], + [], + [], + [], + [], + [], + [], + [ + [ + 42, + 280, + 210, + 448, + 60 + ] + ], + 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50): [[], [], [], [], [], [], [], [], [], [], [[249, 382, 342, 542, 32]], [], [], [], [], [], [], [], [], [[42, 280, 210, 448, 60]], [], [], [], [], [], [], [], [], [], [], [[332, 386, 362, 413, 999]], [], [[235, 145, 280, 190, 999]], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []] + text_len_list (length 228): [73, 926, 235, 1243, 4096, 4555, 1681, 0, 2408, 2284, 1774, 2816, 1243, 2321, 1608, 2663, 1638, 3059, 2118, 2516, 2130, 1681, 1965, 2377, 2787, 2693, 2812, 2462, 2298, 2480, 2448, 2158, 1678, 2464, 1794, 1962, 1927, 2624, 3240, 2442, 2128, 2404, 2574, 2359, 1749, 1312, 1840, 1635, 1906, 2343, 2315, 3018, 2024, 2101, 1560, 2002, 1500, 1424, 2563, 3201, 1406, 1255, 1426, 2966, 2073, 2090, 1753, 2801, 2166, 2865, 2431, 2137, 2014, 1875, 2016, 2097, 2037, 1820, 1383, 1474, 1760, 1149, 1337, 2112, 2939, 739, 1563, 2154, 1059, 1545, 1723, 2070, 1118, 1276, 2136, 1889, 1180, 1072, 1187, 1519, 2284, 1794, 1773, 2862, 2009, 1353, 1786, 2392, 2397, 2483, 1572, 1522, 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'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal'] + invalid_chars: True +Classification Results: is_text_pdf=True, results={'by_image_area': True, 'by_text_len': True, 'by_avg_words': True, 'by_img_num': True, 'by_text_layout': True, 'by_img_narrow_strips': True, 'by_invalid_chars': True} +Error occurred: list indices must be integers or slices, not str +Processing PDF: input/b8ff2f69a03b82d2f2ba8100ab287148.pdf +Metadata: { + "is_needs_password": 0, + "is_encrypted": false, + "total_page": 397, + "page_width_pts": 453, + "page_height_pts": 684, + "image_info_per_page": [ + [ + [ + 0, + 0, + 600, + 450, + 1193 + ] + ], + [ + [ + 0, + 0, + 308, + 497, + 3 + ] + ], + [], + [], + [], + [ + [ + 35, + 597, + 170, + 641, + 17 + ] + ], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + [], + 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2518, + 2622, + 3080, + 3164, + 2665, + 2692, + 2901, + 2541, + 2238, + 3123, + 2970, + 2840, + 3300, + 3581, + 672, + 2547, + 3830, + 4003, + 3455, + 3383, + 9, + 1824, + 2532, + 2552, + 2206, + 2397, + 2151, + 2507, + 2337, + 2536, + 2660, + 2333, + 2583, + 2345 + ], + "text_layout_per_page": [ + "unknow", + "unknow", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal", + "horizontal" + ], + "text_language": "en", + "imgs_per_page": [ + 1, + 1, + 0, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 2, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 2, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 3, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 0, + 8, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 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600, 450, 1193]], [[0, 0, 308, 497, 3]], [], [], [], [[35, 597, 170, 641, 17]], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []] + text_len_list (length 397): [0, 0, 44, 1792, 9, 122, 1068, 49, 962, 1654, 433, 9, 2730, 3967, 3381, 2898, 3316, 3670, 2487, 3221, 3050, 3758, 2221, 1962, 2157, 2158, 1891, 2504, 1800, 3585, 2168, 1537, 3474, 521, 2979, 3353, 3073, 2227, 2513, 1483, 2858, 2732, 2632, 2680, 2633, 2286, 1995, 1853, 2485, 2362, 3945, 3559, 3622, 3414, 1910, 2960, 2148, 3468, 2263, 1954, 2388, 3952, 1738, 3895, 3851, 2302, 2936, 2843, 2269, 3317, 3395, 2743, 2819, 2044, 2557, 3640, 3604, 1891, 2985, 2378, 2104, 3173, 2779, 1177, 2834, 1569, 2438, 3155, 2564, 3134, 2813, 3510, 3219, 3150, 3852, 3170, 1524, 2414, 3002, 3305, 3105, 3387, 2606, 2765, 3282, 3040, 3426, 3350, 3130, 2006, 2310, 2228, 2303, 2334, 2480, 887, 2483, 2720, 2119, 3060, 2102, 2278, 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2400, 1447, 2976, 3610, 2424, 2271, 2424, 3016, 2743, 1154, 4178, 2380, 3294, 2505, 3492, 2521, 2544, 2529, 1922, 2469, 3186, 1966, 3040, 2006, 2612, 4038, 3622, 3896, 3089, 2874, 3689, 3144, 2782, 3470, 3164, 2777, 2535, 2290, 2039, 2540, 1918, 2832, 3733, 2885, 2522, 2033, 3841, 3125, 3316, 3373, 2979, 2701, 3229, 1854, 2061, 2272, 2524, 2186, 1720, 2447, 2275, 2361, 1973, 2833, 2858, 2730, 2639, 2801, 3812, 4227, 3255, 2839, 2723, 2518, 2622, 3080, 3164, 2665, 2692, 2901, 2541, 2238, 3123, 2970, 2840, 3300, 3581, 672, 2547, 3830, 4003, 3455, 3383, 9, 1824, 2532, 2552, 2206, 2397, 2151, 2507, 2337, 2536, 2660, 2333, 2583, 2345] + img_num_list (length 397): [1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal', 'horizontal'] + invalid_chars: True +Classification Results: is_text_pdf=True, results={'by_image_area': True, 'by_text_len': True, 'by_avg_words': True, 'by_img_num': True, 'by_text_layout': True, 'by_img_narrow_strips': True, 'by_invalid_chars': True} +Error occurred: list indices must be integers or slices, not str +GPU 3 - Batch size: 3, VRAM usage: 0.00GB / 44.34GB +Processed 20 PDFs. 0 succeeded, 0 timed out, 20 failed. diff --git a/processing_results.jsonl b/processing_results.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/processing_summary.txt b/processing_summary.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2bcafb6021da8c5b43c568d6a29286b45cdf5bb --- /dev/null +++ b/processing_summary.txt @@ -0,0 +1,26 @@ +Total PDFs processed: 20 +Successful: 0 +Timed out: 0 +Failed: 20 + +Failed PDFs: + - input/f2f24088a4ac947b8f3b29ace552cb90.pdf: Error: list indices must be integers or slices, not str + - input/e0f88b7f2e5a62dec8f0a1825a312092.pdf: Error: list indices must be integers or slices, not str + - input/dcc863d5a46cff540ccca1430f488931.pdf: Error: list indices must be integers or slices, not str + - input/e5cf785df9dc15bbc59962defb61e981.pdf: Error: list indices must be integers or slices, not str + - input/b8ff2f69a03b82d2f2ba8100ab287148.pdf: Error: list indices must be integers or slices, not str + - input/f2f24088a4ac947b8f3b29ace552cb90.pdf: Error: list indices must be integers or slices, not str + - input/e0f88b7f2e5a62dec8f0a1825a312092.pdf: Error: list indices must be integers or slices, not str + - input/dcc863d5a46cff540ccca1430f488931.pdf: Error: list indices must be integers or slices, not str + - input/e5cf785df9dc15bbc59962defb61e981.pdf: Error: list indices must be integers or slices, not str + - input/b8ff2f69a03b82d2f2ba8100ab287148.pdf: Error: list indices must be integers or slices, not str + - input/f2f24088a4ac947b8f3b29ace552cb90.pdf: Error: list indices must be integers or slices, not str + - input/e0f88b7f2e5a62dec8f0a1825a312092.pdf: Error: list indices must be integers or slices, not str + - input/dcc863d5a46cff540ccca1430f488931.pdf: Error: list indices must be integers or slices, not str + - input/e5cf785df9dc15bbc59962defb61e981.pdf: Error: list indices must be integers or slices, not str + - input/b8ff2f69a03b82d2f2ba8100ab287148.pdf: Error: list indices must be integers or slices, not str + - input/f2f24088a4ac947b8f3b29ace552cb90.pdf: Error: list indices must be integers or slices, not str + - input/e0f88b7f2e5a62dec8f0a1825a312092.pdf: Error: list indices must be integers or slices, not str + - input/dcc863d5a46cff540ccca1430f488931.pdf: Error: list indices must be integers or slices, not str + - input/e5cf785df9dc15bbc59962defb61e981.pdf: Error: list indices must be integers or slices, not str + - input/b8ff2f69a03b82d2f2ba8100ab287148.pdf: Error: list indices must be integers or slices, not str