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github
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uoa1184615/EquationFreeGit-master
|
patchSmooth1.m
|
.m
|
EquationFreeGit-master/Patch/patchSmooth1.m
| 376 |
utf_8
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8eeed72eb5476e704fc56d7f0f700ded
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% legacy interface patchSmooth1() auto-invokes new patchSys1()
function dudt=patchSmooth1(t,u,patches)
global smOOthCount
if isempty(smOOthCount), smOOthCount=1;
else smOOthCount=smOOthCount+1; end
l2=log2(smOOthCount);
if abs(l2-round(l2))<1e-9
warning('Use new patchSys1 instead of old patchSmooth1')
end
if nargin<3, global patches, end
dudt=patchSys1(t,u,patches);
|
github
|
uoa1184615/EquationFreeGit-master
|
configPatches1.m
|
.m
|
EquationFreeGit-master/Patch/configPatches1.m
| 24,742 |
utf_8
|
7290399fac4ce9faab19acbc25ff17ed
|
% configPatches1() creates a data struct of the design of
% 1D patches for later use by the patch functions such as
% patchSys1(). AJR, Nov 2017 -- 23 Mar 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{configPatches1()}: configure spatial
patches in 1D}
\label{sec:configPatches1}
\localtableofcontents
Makes the struct~\verb|patches| for use by the patch\slash
gap-tooth time derivative\slash step function
\verb|patchSys1()|. \cref{sec:configPatches1eg} lists an
example of its use.
\begin{matlab}
%}
function patches = configPatches1(fun,Xlim,Dom ...
,nPatch,ordCC,dx,nSubP,varargin)
version = '2023-03-23';
%{
\end{matlab}
\paragraph{Input}
If invoked with no input arguments, then executes an example
of simulating Burgers' \pde---see \cref{sec:configPatches1eg}
for the example code.
\begin{itemize}
\item \verb|fun| is the name of the user function,
\verb|fun(t,u,patches)| or \verb|fun(t,u)| or
\verb|fun(t,u,patches,...)|, that computes time derivatives
(or time-steps) of quantities on the 1D micro-grid within
all the 1D~patches.
\item \verb|Xlim| give the macro-space spatial domain of the
computation, namely the interval $[ \verb|Xlim(1)|,
\verb|Xlim(2)|]$.
\item \verb|Dom| sets the type of macroscale conditions for
the patches, and reflects the type of microscale boundary
conditions of the problem. If \verb|Dom| is \verb|NaN| or
\verb|[]|, then the field~\verb|u| is macro-periodic in the
1D spatial domain, and resolved on equi-spaced patches. If
\verb|Dom| is a character string, then that specifies the
\verb|.type| of the following structure, with
\verb|.bcOffset| set to the default zero. Otherwise
\verb|Dom| is a structure with the following components.
\begin{itemize}
\item \verb|.type|, string, of either \verb|'periodic'| (the
default), \verb|'equispace'|, \verb|'chebyshev'|,
\verb|'usergiven'|. For all cases except \verb|'periodic'|,
users \emph{must} code into \verb|fun| the micro-grid
boundary conditions that apply at the left(right) edge of
the leftmost(rightmost) patches.
\item \verb|.bcOffset|, optional one or two element array,
in the cases of \verb|'equispace'| or \verb|'chebyshev'|
the patches are placed so the left\slash right macroscale
boundaries are aligned to the left\slash right edges of the
corresponding extreme patches, but offset by \verb|bcOffset|
of the sub-patch micro-grid spacing. For example, use
\verb|bcOffset=0| when applying Dirichlet boundary values on
the extreme edge micro-grid points, whereas use
\verb|bcOffset=0.5| when applying Neumann boundary conditions
halfway between the extreme edge micro-grid points.
\item \verb|.X|, optional array, in the case~\verb|'usergiven'|
it specifies the locations of the centres of the
\verb|nPatch| patches---the user is responsible it makes
sense.
\end{itemize}
\item \verb|nPatch| is the number of equi-spaced spatial
patches.
\item \verb|ordCC|, must be~$\geq -1$, is the `order' of
interpolation across empty space of the macroscale patch
values to the edge of the patches for inter-patch coupling:
where \verb|ordCC| of~$0$ or~$-1$ gives spectral
interpolation; and \verb|ordCC| being odd specifies
staggered spatial grids.
\item \verb|dx| (real) is usually the sub-patch micro-grid
spacing in~\(x\).
However, if \verb|Dom| is~\verb|NaN| (as for pre-2023), then
\verb|dx| actually is \verb|ratio|, namely the ratio of
(depending upon \verb|EdgyInt|) either the half-width or
full-width of a patch to the equi-spacing of the patch
mid-points---adjusted a little when $\verb|nEdge|>1$. So
either $\verb|ratio|=\tfrac12$ means the patches abut and
$\verb|ratio|=1$ is overlapping patches as in holistic
discretisation, or $\verb|ratio|=1$ means the patches abut.
Small~\verb|ratio| should greatly reduce computational time.
\item \verb|nSubP| is the number of equi-spaced microscale
lattice points in each patch. If not using \verb|EdgyInt|,
then $\verb|nSubP/nEdge|$ must be odd integer so that there
is/are centre-patch lattice point(s). So for the defaults
of $\verb|nEdge|=1$ and not \verb|EdgyInt|, then
\verb|nSubP| must be odd.
\item \verb|'nEdge'|, \emph{optional}, default=1, the number
of edge values set by interpolation at the edge regions of
each patch. The default is one (suitable for microscale
lattices with only nearest neighbour interactions).
\item \verb|EdgyInt|, true/false, \emph{optional},
default=false. If true, then interpolate to left\slash
right edge-values from right\slash left next-to-edge values.
If false or omitted, then interpolate from centre-patch
values.
\item \verb|nEnsem|, \emph{optional-experimental},
default one, but if more, then an ensemble over this
number of realisations.
\item \verb|hetCoeffs|, \emph{optional}, default empty.
Supply a 1D or 2D array of microscale heterogeneous
coefficients to be used by the given microscale \verb|fun|
in each patch. Say the given array~\verb|cs| is of size
$m_x\times n_c$, where $n_c$~is the number of different sets
of coefficients. The coefficients are to be the same for
each and every patch; however, macroscale variations are
catered for by the $n_c$~coefficients being $n_c$~parameters
in some macroscale formula.
\begin{itemize}
\item If $\verb|nEnsem|=1$, then the array of coefficients
is just tiled across the patch size to fill up each patch,
starting from the first point in each patch. Best accuracy
usually obtained when the periodicity of the coefficients
is a factor of \verb|nSubP-2*nEdge| for \verb|EdgyInt|, or
a factor of \verb|(nSubP-nEdge)/2| for not \verb|EdgyInt|.
\item If $\verb|nEnsem|>1$ (value immaterial), then reset
$\verb|nEnsem|:=m_x$ and construct an ensemble of all
$m_x$~phase-shifts of the coefficients. In this scenario,
the inter-patch coupling couples different members in the
ensemble. When \verb|EdgyInt| is true, and when the
coefficients are diffusivities\slash elasticities, then this
coupling cunningly preserves symmetry.
\end{itemize}
\item \verb|nCore|, \emph{optional-experimental}, default
one, but if more, and only for non-EdgyInt, then
interpolates from an average over the core of a patch, a
core of size ??. Then edge values are set according to
interpolation of the averages?? or so that average at edges
is the interpolant??
\item \verb|'parallel'|, true/false, \emph{optional},
default=false. If false, then all patch computations are on
the user's main \textsc{cpu}---although a user may well
separately invoke, say, a \textsc{gpu} to accelerate
sub-patch computations.
If true, and it requires that you have \Matlab's Parallel
Computing Toolbox, then it will distribute the patches over
multiple \textsc{cpu}s\slash cores. In \Matlab, only one
array dimension can be split in the distribution, so it
chooses the one space dimension~$x$. A user may
correspondingly distribute arrays with property
\verb|patches.codist|, or simply use formulas invoking the
preset distributed arrays \verb|patches.x|. If a user has
not yet established a parallel pool, then a `local' pool is
started.
\end{itemize}
\paragraph{Output} The struct \verb|patches| is created and
set with the following components. If no output variable is
provided for \verb|patches|, then make the struct available
as a global variable.\footnote{When using \texttt{spmd}
parallel computing, it is generally best to avoid global
variables, and so instead prefer using an explicit output
variable.}
\begin{matlab}
%}
if nargout==0, global patches, end
patches.version = version;
%{
\end{matlab}
\begin{itemize}
\item \verb|.fun| is the name of the user's function
\verb|fun(t,u,patches)| or \verb|fun(t,u)| or
\verb|fun(t,u,patches,...)|, that computes the time
derivatives (or steps) on the patchy lattice.
\item \verb|.ordCC| is the specified order of inter-patch
coupling.
\item \verb|.periodic|: either true, for interpolation on
the macro-periodic domain; or false, for general
interpolation by divided differences over non-periodic
domain or unevenly distributed patches.
\item \verb|.stag| is true for interpolation using only odd
neighbouring patches as for staggered grids, and false for
the usual case of all neighbour coupling.
\item \verb|.Cwtsr| and \verb|.Cwtsl|, only for
macro-periodic conditions, are the $\verb|ordCC|$-vector of
weights for the inter-patch interpolation onto the right and
left edges (respectively) with patch:macroscale ratio as
specified or as derived from~\verb|dx|.
\item \verb|.x| (4D) is $\verb|nSubP| \times1 \times1
\times \verb|nPatch|$ array of the regular spatial
locations~$x_{iI}$ of the $i$th~microscale grid point in
the $I$th~patch.
\item \verb|.ratio|, only for macro-periodic conditions, is
the size ratio of every patch.
\item \verb|.nEdge| is, for each patch, the number of edge
values set by interpolation at the edge regions of each
patch.
\item \verb|.le|, \verb|.ri| determine inter-patch coupling
of members in an ensemble. Each a column vector of
length~\verb|nEnsem|.
\item \verb|.cs| either
\begin{itemize}
\item \verb|[]| 0D, or
\item if $\verb|nEnsem|=1$, $(\verb|nSubP(1)|-1)\times
n_c$ 2D array of microscale heterogeneous coefficients, or
\item if $\verb|nEnsem|>1$, $(\verb|nSubP(1)|-1) \times
n_c\times m_x$ 3D array of $m_x$~ensemble of phase-shifts
of the microscale
heterogeneous coefficients.
\end{itemize}
\item \verb|.parallel|, logical: true if patches are
distributed over multiple \textsc{cpu}s\slash cores for the
Parallel Computing Toolbox, otherwise false (the default is
to activate the \emph{local} pool).
\item \verb|.codist|, \emph{optional}, describes the
particular parallel distribution of arrays over the active
parallel pool.
\end{itemize}
\subsection{If no arguments, then execute an example}
\label{sec:configPatches1eg}
\begin{matlab}
%}
if nargin==0
disp('With no arguments, simulate example of Burgers PDE')
%{
\end{matlab}
The code here shows one way to get started: a user's script
may have the following three steps (``\into'' denotes
function recursion).
\begin{enumerate}\def\itemsep{-1.5ex}
\item configPatches1
\item ode15s integrator \into patchSys1 \into user's PDE
\item process results
\end{enumerate}
Establish global patch data struct to point to and interface
with a function coding Burgers' \pde: to be solved on
$2\pi$-periodic domain, with eight patches, spectral
interpolation couples the patches, with micro-grid
spacing~$0.06$, and with seven microscale points forming
each patch.
\begin{matlab}
%}
global patches
patches = configPatches1(@BurgersPDE, [0 2*pi], ...
'periodic', 8, 0, 0.06, 7);
%{
\end{matlab}
Set some initial condition, with some microscale randomness.
\begin{matlab}
%}
u0=0.3*(1+sin(patches.x))+0.1*randn(size(patches.x));
%{
\end{matlab}
Simulate in time using a standard stiff integrator and the
interface function \verb|patchSys1()|
(\cref{sec:patchSys1}).
\begin{matlab}
%}
if ~exist('OCTAVE_VERSION','builtin')
[ts,us] = ode15s( @patchSys1,[0 0.5],u0(:));
else % octave version
[ts,us] = odeOcts(@patchSys1,[0 0.5],u0(:));
end
%{
\end{matlab}
Plot the simulation using only the microscale values
interior to the patches: either set $x$-edges to \verb|nan|
to leave the gaps; or use \verb|patchEdgyInt1| to
re-interpolate correct patch edge values and thereby join
the patches. \cref{fig:config1Burgers} illustrates an
example simulation in time generated by the patch scheme
applied to Burgers'~\pde.
\begin{matlab}
%}
figure(1),clf
if 1, patches.x([1 end],:,:,:)=nan; us=us.';
else us=reshape(patchEdgyInt1(us.'),[],length(ts));
end
mesh(ts,patches.x(:),us)
view(60,40), colormap(0.7*hsv)
title('Burgers PDE: patches in space, continuous time')
xlabel('time $t$'), ylabel('space $x$'), zlabel('$u(x,t)$')
%{
\end{matlab}
\begin{figure}
\centering \caption{\label{fig:config1Burgers}field
$u(x,t)$ of the patch scheme applied to Burgers'~\pde.}
\includegraphics[scale=0.85]{configPatches1}
\end{figure}
Upon finishing execution of the example, optionally save
the graph to be shown in \cref{fig:config1Burgers}, then
exit this function.
\begin{matlab}
%}
ifOurCf2eps(mfilename)
return
end%if nargin==0
%{
\end{matlab}
\IfFileExists{../Patch/BurgersPDE.m}{\input{../Patch/BurgersPDE.m}}{}
\IfFileExists{../Patch/odeOcts.m}{\input{../Patch/odeOcts.m}}{}
\begin{devMan}
\subsection{Parse input arguments and defaults}
\begin{matlab}
%}
p = inputParser;
fnValidation = @(f) isa(f, 'function_handle'); %test for fn name
addRequired(p,'fun',fnValidation);
addRequired(p,'Xlim',@isnumeric);
%addRequired(p,'Dom'); % nothing yet decided
addRequired(p,'nPatch',@isnumeric);
addRequired(p,'ordCC',@isnumeric);
addRequired(p,'dx',@isnumeric);
addRequired(p,'nSubP',@isnumeric);
addParameter(p,'nEdge',1,@isnumeric);
addParameter(p,'EdgyInt',false,@islogical);
addParameter(p,'nEnsem',1,@isnumeric);
addParameter(p,'hetCoeffs',[],@isnumeric);
addParameter(p,'parallel',false,@islogical);
addParameter(p,'nCore',1,@isnumeric);
parse(p,fun,Xlim,nPatch,ordCC,dx,nSubP,varargin{:});
%{
\end{matlab}
Set the optional parameters.
\begin{matlab}
%}
patches.nEdge = p.Results.nEdge;
patches.EdgyInt = p.Results.EdgyInt;
patches.nEnsem = p.Results.nEnsem;
cs = p.Results.hetCoeffs;
patches.parallel = p.Results.parallel;
patches.nCore = p.Results.nCore;
%{
\end{matlab}
Check parameters.
\begin{matlab}
%}
assert(Xlim(1)<Xlim(2) ...
,'two entries of Xlim must be ordered increasing')
assert((mod(ordCC,2)==0)|(patches.nEdge==1) ...
,'Cannot yet have nEdge>1 and staggered patch grids')
assert(3*patches.nEdge<=nSubP ...
,'too many edge values requested')
assert(rem(nSubP,patches.nEdge)==0 ...
,'nSubP must be integer multiple of nEdge')
if ~patches.EdgyInt, assert(rem(nSubP/patches.nEdge,2)==1 ...
,'for non-edgyInt, nSubP/nEdge must be odd integer')
end
if (patches.nEnsem>1)&(patches.nEdge>1)
warning('not yet tested when both nEnsem and nEdge non-one')
end
if patches.nCore>1
warning('nCore>1 not yet tested in this version')
end
%{
\end{matlab}
For compatibility with pre-2023 functions, if parameter
\verb|Dom| is \verb|Nan|, then we set the \verb|ratio| to
be the value of the so-called \verb|dx| parameter.
\begin{matlab}
%}
if ~isstruct(Dom), pre2023=isnan(Dom);
else pre2023=false; end
if pre2023, ratio=dx; dx=nan; end
%{
\end{matlab}
Default macroscale conditions are periodic with evenly
spaced patches.
\begin{matlab}
%}
if isempty(Dom), Dom=struct('type','periodic'); end
if (~isstruct(Dom))&isnan(Dom), Dom=struct('type','periodic'); end
%{
\end{matlab}
If \verb|Dom| is a string, then just set type to that
string, and then get corresponding defaults for others
fields.
\begin{matlab}
%}
if ischar(Dom), Dom=struct('type',Dom); end
%{
\end{matlab}
Check what is and is not specified, and provide default of
Dirichlet boundaries if no \verb|bcOffset| specified when
needed.
\begin{matlab}
%}
patches.periodic=false;
switch Dom.type
case 'periodic'
patches.periodic=true;
if isfield(Dom,'bcOffset')
warning('bcOffset not available for Dom.type = periodic'), end
if isfield(Dom,'X')
warning('X not available for Dom.type = periodic'), end
case {'equispace','chebyshev'}
if ~isfield(Dom,'bcOffset'), Dom.bcOffset=[0;0]; end
if length(Dom.bcOffset)==1
Dom.bcOffset=repmat(Dom.bcOffset,2,1); end
if isfield(Dom,'X')
warning('X not available for Dom.type = equispace or chebyshev')
end
case 'usergiven'
if isfield(Dom,'bcOffset')
warning('bcOffset not available for usergiven Dom.type'), end
assert(isfield(Dom,'X'),'X required for Dom.type = usergiven')
otherwise
error([Dom.type ' is unknown Dom.type'])
end%switch Dom.type
%{
\end{matlab}
\subsection{The code to make patches and interpolation}
First, store the pointer to the time derivative function in
the struct.
\begin{matlab}
%}
patches.fun=fun;
%{
\end{matlab}
Second, store the order of interpolation that is to provide
the values for the inter-patch coupling conditions. Spectral
coupling is \verb|ordCC| of~$0$ and~$-1$.
\begin{matlab}
%}
assert((ordCC>=-1) & (floor(ordCC)==ordCC), ...
'ordCC out of allowed range integer>=-1')
%{
\end{matlab}
For odd~\verb|ordCC|, interpolate based upon odd
neighbouring patches as is useful for staggered grids.
\begin{matlab}
%}
patches.stag=mod(ordCC,2);
ordCC=ordCC+patches.stag;
patches.ordCC=ordCC;
%{
\end{matlab}
Check for staggered grid and periodic case.
\begin{matlab}
%}
if patches.stag, assert(mod(nPatch,2)==0, ...
'Require an even number of patches for staggered grid')
end
%{
\end{matlab}
Third, set the centre of the patches in the macroscale grid
of patches, depending upon \verb|Dom.type|.
\begin{matlab}
%}
switch Dom.type
%{
\end{matlab}
%: case periodic
The periodic case is evenly spaced within the spatial domain.
Store the size ratio in \verb|patches|.
\begin{matlab}
%}
case 'periodic'
X=linspace(Xlim(1),Xlim(2),nPatch+1);
DX=X(2)-X(1);
X=X(1:nPatch)+diff(X)/2;
pEI=patches.EdgyInt;% abbreviation
pnE=patches.nEdge; % abbreviation
if pre2023, dx = ratio*DX/(nSubP-pnE*(1+pEI))*(2-pEI);
else ratio = dx/DX*(nSubP-pnE*(1+pEI))/(2-pEI); end
patches.ratio=ratio;
%{
\end{matlab}
In the case of macro-periodicity, precompute the weightings
to interpolate field values for coupling.
\todo{Might sometime extend to coupling via derivative values.}
\begin{matlab}
%}
if ordCC>0
[Cwtsr,Cwtsl] = patchCwts(ratio,ordCC,patches.stag);
patches.Cwtsr = Cwtsr; patches.Cwtsl = Cwtsl;
end
%{
\end{matlab}
%: case equispace
The equi-spaced case is also evenly spaced but with the
extreme edges aligned with the spatial domain boundaries,
modified by the offset.
%\todo{This warning needs refinement for multi-edges??}
\begin{matlab}
%}
case 'equispace'
X=linspace(Xlim(1)+((nSubP-1)/2-Dom.bcOffset(1))*dx ...
,Xlim(2)-((nSubP-1)/2-Dom.bcOffset(2))*dx ,nPatch);
DX=diff(X(1:2));
width=(1+patches.EdgyInt)/2*(nSubP-1-patches.EdgyInt)*dx;
if DX<width*0.999999
warning('too many equispace patches (double overlapping)')
end
%{
\end{matlab}
%: case chebyshev
The Chebyshev case is spaced according to the Chebyshev
distribution in order to reduce macro-interpolation errors,
\(X_i \propto -\cos(i\pi/N)\), but with the extreme edges
aligned with the spatial domain boundaries, modified by the
offset, and modified by possible `boundary layers'.
\footnote{ However, maybe overlapping patches near a
boundary should be viewed as some sort of spatial analogue
of the `christmas tree' of projective integration and its
projection to a slow manifold. Here maybe the overlapping
patches allow for a `christmas tree' approach to the
boundary layers. Needs to be explored??}
\begin{matlab}
%}
case 'chebyshev'
halfWidth=dx*(nSubP-1)/2;
X1 = Xlim(1)+halfWidth-Dom.bcOffset(1)*dx;
X2 = Xlim(2)-halfWidth+Dom.bcOffset(2)*dx;
% X = (X1+X2)/2-(X2-X1)/2*cos(linspace(0,pi,nPatch));
%{
\end{matlab}
Search for total width of `boundary layers' so that in the
interior the patches are non-overlapping Chebyshev. But
the width for assessing overlap of patches is the following
variable \verb|width|. We need to find~\verb|b|, the number of patches `glued' together at the boundaries.
\begin{matlab}
%}
pEI=patches.EdgyInt;% abbreviation
pnE=patches.nEdge; % abbreviation
width=(1+pEI)/2*(nSubP-pnE-pEI*pnE)*dx;
for b=0:2:nPatch-2
DXmin=(X2-X1-b*width)/2*( 1-cos(pi/(nPatch-b-1)) );
if DXmin>width, break, end
end%for
if DXmin<width*0.999999
warning('too many Chebyshev patches (mid-domain overlap)')
end
%{
\end{matlab}
Assign the centre-patch coordinates.
\begin{matlab}
%}
X = [ X1+(0:b/2-1)*width ...
(X1+X2)/2-(X2-X1-b*width)/2*cos(linspace(0,pi,nPatch-b)) ...
X2+(1-b/2:0)*width ];
%{
\end{matlab}
%: case usergiven
The user-given case is entirely up to a user to specify, we
just force it to have the correct shape of a row.
\begin{matlab}
%}
case 'usergiven'
X = reshape(Dom.X,1,[]);
end%switch Dom.type
%{
\end{matlab}
Fourth, construct the microscale grid in each patch, centred
about the given mid-points~\verb|X|. Reshape the grid to be
4D to suit dimensions (micro,Vars,Ens,macro).
\begin{matlab}
%}
xs = dx*( (1:nSubP)-mean(1:nSubP) );
patches.x = reshape( xs'+X ,nSubP,1,1,nPatch);
%{
\end{matlab}
\subsection{Set ensemble inter-patch communication}
For \verb|EdgyInt| or centre interpolation respectively,
\begin{itemize}
\item the right-edge\slash centre realisations
\verb|1:nEnsem| are to interpolate to left-edge~\verb|le|,
and
\item the left-edge\slash centre realisations
\verb|1:nEnsem| are to interpolate to~\verb|re|.
\end{itemize}
\verb|re| and \verb|li| are `transposes' of each other as
\verb|re(li)=le(ri)| are both \verb|1:nEnsem|.
Alternatively, one may use the statement
\begin{verbatim}
c=hankel(c(1:nSubP-1),c([nSubP 1:nSubP-2]));
\end{verbatim}
to \emph{correspondingly} generates all phase shifted copies
of microscale heterogeneity (see \verb|homoDiffEdgy1| of
\cref{sec:homoDiffEdgy1}).
The default is nothing shifty. This setting reduces the
number of if-statements in function \verb|patchEdgeInt1()|.
\begin{matlab}
%}
nE = patches.nEnsem;
patches.le = 1:nE;
patches.ri = 1:nE;
%{
\end{matlab}
However, if heterogeneous coefficients are supplied via
\verb|hetCoeffs|, then do some non-trivial replications.
First, get microscale periods, patch size, and replicate
many times in order to subsequently sub-sample: \verb|nSubP|
times should be enough. If \verb|cs| is more then 2D, then
the higher-dimensions are reshaped into the 2nd dimension.
\begin{matlab}
%}
if ~isempty(cs)
[mx,nc] = size(cs);
nx = nSubP(1);
cs = repmat(cs,nSubP,1);
%{
\end{matlab}
If only one member of the ensemble is required, then
sub-sample to patch size, and store coefficients in
\verb|patches| as is.
\begin{matlab}
%}
if nE==1, patches.cs = cs(1:nx-1,:); else
%{
\end{matlab}
But for $\verb|nEnsem|>1$ an ensemble of
$m_x$~phase-shifts of the coefficients is constructed from
the over-supply. Here code phase-shifts over the
periods---the phase shifts are like Hankel-matrices.
\begin{matlab}
%}
patches.nEnsem = mx;
patches.cs = nan(nx-1,nc,mx);
for i = 1:mx
is = (i:i+nx-2);
patches.cs(:,:,i) = cs(is,:);
end
patches.cs = reshape(patches.cs,nx-1,nc,[]);
%{
\end{matlab}
Further, set a cunning left\slash right realisation of
inter-patch coupling. The aim is to preserve symmetry in
the system when also invoking \verb|EdgyInt|. What this
coupling does without \verb|EdgyInt| is unknown. Use
auto-replication.
\begin{matlab}
%}
patches.le = mod((0:mx-1)'+mod(nx-2,mx),mx)+1;
patches.ri = mod((0:mx-1)'-mod(nx-2,mx),mx)+1;
%{
\end{matlab}
Issue warning if the ensemble is likely to be affected by
lack of scale separation. Need to justify this and the
arbitrary threshold more carefully??
\begin{matlab}
%}
if ratio*patches.nEnsem>0.9, warning( ...
'Probably poor scale separation in ensemble of coupled phase-shifts')
scaleSeparationParameter = ratio*patches.nEnsem
end
%{
\end{matlab}
End the two if-statements.
\begin{matlab}
%}
end%if-else nEnsem>1
end%if not-empty(cs)
%{
\end{matlab}
\paragraph{If parallel code} then first assume this is not
within an \verb|spmd|-environment, and so we invoke
\verb|spmd...end| (which starts a parallel pool if not
already started). At this point, the global \verb|patches|
is copied for each worker processor and so it becomes
\emph{composite} when we distribute any one of the fields.
Hereafter, {\em all fields in the global variable
\verb|patches| must only be referenced within an
\verb|spmd|-environment.}%
\footnote{If subsequently outside spmd, then one must use
functions like \texttt{getfield(patches\{1\},'a')}.}
\begin{matlab}
%}
if patches.parallel
% theparpool=gcp()
spmd
%{
\end{matlab}
Second, choose to slice parallel workers in the spatial
direction.
\begin{matlab}
%}
pari = 1;
patches.codist=codistributor1d(3+pari);
%{
\end{matlab}
\verb|patches.codist.Dimension| is the index that is split
among workers. Then distribute the coordinate direction
among the workers: the function must be invoked inside an
\verb|spmd|-group in order for this to work---so we do not
need \verb|parallel| in argument list.
\begin{matlab}
%}
switch pari
case 1, patches.x=codistributed(patches.x,patches.codist);
otherwise
error('should never have bad index for parallel distribution')
end%switch
end%spmd
%{
\end{matlab}
If not parallel, then clean out \verb|patches.codist| if it
exists. May not need, but safer.
\begin{matlab}
%}
else% not parallel
if isfield(patches,'codist'), rmfield(patches,'codist'); end
end%if-parallel
%{
\end{matlab}
\paragraph{Fin}
\begin{matlab}
%}
end% function
%{
\end{matlab}
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
chanDispSpmd.m
|
.m
|
EquationFreeGit-master/Patch/chanDispSpmd.m
| 12,849 |
utf_8
|
7f77310d0654cfb42d486fbc11f9a69e
|
% chanDispSpmd simulates 2D shear dispersion in a long thin
% channel with 1D patches as a Proof of Principle example of
% parallel computing with spmd. AJR, Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{chanDispSpmd}: simulation of a 1D shear
dispersion via simulation on small patches across a channel}
\label{sec:chanDispSpmd}
\localtableofcontents
Simulate 1D shear dispersion along long thin channel,
dispersion that is emergent from micro-scale dynamics in 2D
space. Use 1D patches as a Proof of Principle example of
parallel computing with \verb|spmd|. In this shear
dispersion, although the micro-scale diffusivities are
one-ish, the shear causes an effective longitudinal
`diffusivity' of the order of~$\Pe^2$\,---which is typically
much larger than the micro-scale diffusivity
\cite[e.g.]{Taylor53}.
The spatial domain is the channel (large) $L$-periodic
in~$x$ and $|y|<1$\,. Seek to predict a concentration
field~$c(x,y,t)$ satisfying the linear
advection-diffusion~\pde
\begin{equation}
\D tc = -\Pe u(y)\D xc+\D x{}\Big[\kappa_x(y)\D xc\Big]
+\D y{}\Big[\kappa_y(y)\D yc\Big].
\label{eq:pdeChanDisp}
\end{equation}
where \Pe\ denotes a Peclet number, parabolic advection
velocity $u(y)=\tfrac32(1-y^2)$ with noise, and parabolic
diffusivity $\kappa_x(y)=\kappa_y(y)=(1-y^2)$ with noise.
The noise is to be multiplicative and log-normal to ensure
advection and diffusion are all positive, and to be periodic
in~$x$.
For a microscale computation we discretise in space with
$x$-spacing~$\delta x$, and $n_y$~points over $|y|<1$ with
spacing $\delta y:=2/n_y$ at $y_j:=-1+(j-\tfrac12)\delta
y$\,, $j=1:n_y$\,. Our microscale discretisation of
\pde~\eqref{eq:pdeChanDisp} is then
\begin{align}&
\D t{c_{ij}}=-\Pe u(y_j)\frac{c_{i+1,j}-c_{i-1,j}}{2\delta x}
+\frac{d_{i,j+1/2}-d_{i,j-1/2}}{\delta y}
+\frac{D_{i+1/2,j}-D_{i-1/2,j}}{\delta x}\,,
\nonumber\\&\quad
d_{ij}:=\kappa_y(y_j)\frac{c_{i,j+1/2}-c_{i,j-1/2}}{\delta y}\,,\quad
D_{ij}:=\kappa_x(y_j)\frac{c_{i+1/2,j}-c_{i-1/2,j}}{\delta x}\,.
\label{eq:ddeChanDisp}
\end{align}
These are coded in \cref{sec:chanDispMicro} for the computation.
Choose one of four cases:
\begin{itemize}
\item \verb|theCase=1| is corresponding code without
parallelisation (in this toy problem it is much the quickest
because there is no expensive interprocessor communication);
\item \verb|theCase=2| illustrates that \verb|RK2mesoPatch|
invokes \verb|spmd| computation if parallel has been
configured.
\item \verb|theCase=3| shows how users explicitly invoke
\verb|spmd|-blocks around the time integration.
\item \verb|theCase=4| invokes projective integration for
long-time simulation via short bursts of the
micro-computation, bursts done within \verb|spmd|-blocks for
parallel computing.
\end{itemize}
First, clear all to remove any existing globals, old
composites, etc---although a parallel pool persists. Then
choose the case.
\begin{matlab}
%}
clear all
theCase = 1
%{
\end{matlab}
The micro-scale \pde\ is evaluated at positions~$y_j$ across
the channel, $|y|<1$\,. The even indexed points are the
collocation points for the \pde, whereas the odd indexed
points are the half-grid points for specification of
$y$-diffusivities.
\begin{matlab}
%}
ny = 7
y = linspace(-1,1,2*ny+1);
yj = y(2:2:end);
%{
\end{matlab}
Set micro-scale advection (array~1) and diffusivity
(array~2) with (roughly) parabolic shape
\cite[e.g.]{Watt94b, MacKenzie03}. Here modify the parabola
by a heterogeneous log-normal factor with specified period
along the channel: modify the strength of the heterogeneity
by the coefficient of~\verb|randn| from zero to perhaps one:
coefficient~$0.3$ appears a good moderate value. Remember
that \verb|configPatches1| reshapes \verb|cHetr| to~2D.
\begin{matlab}
%}
mPeriod = 4
cHetr = shiftdim([3/2 1],-1).*(1-y.^2) ...
.*exp(0.3*randn([mPeriod 2*ny+1 2]));
%{
\end{matlab}
Configure the patch scheme with some arbitrary choices of
domain, patches, size ratios. Choose some random order of
interpolation to see the alternatives. Set \verb|patches|
information to be global so the info can be used for
Cases~1--2 without being explicitly passed as arguments.
Choose the parallel option if not Case~1, which invokes
\verb|spmd|-block internally, so that field variables become
\emph{distributed} across cpus.
\begin{matlab}
%}
if theCase<=2, global patches, end
nPatch=15
nSubP=2+mPeriod
ratio=0.2+0.2*(theCase<4)
Len=nPatch/ratio
ordCC=2*randi([0 3])
disp('**** Setting configPatches1')
patches = configPatches1(@chanDispMicro, [0 Len], nan ...
, nPatch, ordCC, ratio, nSubP, 'EdgyInt',true ...
,'hetCoeffs',cHetr ,'parallel',(theCase>1) );
%{
\end{matlab}
When using parallel then additional parameters to
\verb|patches| should be set within a \verb|spmd| block
(because \verb|patches| is a co-distributed structure).
\begin{matlab}
%}
Peclet = 10
if theCase==1, patches.Pe = Peclet;
else spmd, patches.Pe = Peclet; end
end
%{
\end{matlab}
\subsection{Simulate heterogeneous advection-diffusion}
Set initial conditions of a simulation as shown in
\cref{fig:chanDispSpmdt0}.
\begin{matlab}
%}
disp('**** Set initial condition and test dc0dt =')
if theCase==1
%{
\end{matlab}
Without parallel processing, invoke the usual operations.
\begin{matlab}
%}
c0 = 10*exp(-(ratio*patches.x-2.5).^2/2) +0*yj;
c0 = c0.*(1+0.2*rand(size(c0)));
dc0dt = patchSys1(0,c0);
%{
\end{matlab}
With parallel, we must use an \verb|spmd|-block for
computations: there is no difference in cases~2--4 here.
Also, we must sometimes use \verb|patches.codist| to
explicitly code how to distribute new arrays over the cpus.
Now \verb|patchSys1| does not invoke \verb|spmd| so
higher level code must, as here. Even if \verb|patches| is
global, inside \verb|spmd|-block we \emph{must} pass it
explicitly as a parameter to \verb|patchSys1|.
\begin{matlab}
%}
else, spmd
c0 = 10*exp(-(ratio*patches.x-2.5).^2/2) +0*yj;
c0 = c0.*(1+0.2*rand(size(c0),patches.codist));
dc0dt = patchSys1(0,c0,patches)
end%spmd
end%if theCase
%{
\end{matlab}
\begin{figure}
\centering \caption{\label{fig:chanDispSpmdt0}initial
field~$u(x,y,0)$ of the patch scheme applied to a
heterogeneous advection-diffusion~\pde.
\cref{fig:chanDispSpmdtFin} plots the roughly smooth field
values at time $t=4$. In this example the patches are
relatively large, ratio~$0.4$, for visibility.}
\includegraphics[scale=0.8]{chanDispSpmdt0}
\end{figure}
Integrate in time, either via the automatic \verb|ode23| or
via \verb|RK2mesoPatch| which reduces communication between
patches. By default, \verb|RK2mesoPatch| does ten
micro-steps for each specified meso-step in~\verb|ts|. For
stability: with noise up to~$0.3$, need micro-steps less
than~$0.005$; with noise~$1$, need micro-steps less
than~$0.0015$.
\begin{matlab}
%}
warning('Integrating system in time, wait patiently')
ts=4*linspace(0,1);
%{
\end{matlab}
Go to the selected case.
\begin{matlab}
%}
switch theCase
%{
\end{matlab}
\begin{enumerate}
\item For non-parallel, we could use \verb|RK2mesoPatch| as
indicated below, but instead choose to use standard
\verb|ode23| as here \verb|patchSys1| accesses patch
information via global \verb|patches|. For post-processing,
reshape each and every row of the computed solution to the
correct array size---namely that of the initial condition.
\begin{matlab}
%}
case 1
% [cs,uerrs] = RK2mesoPatch(ts,c0);
[ts,cs] = ode23(@patchSys1,ts,c0(:));
cs=reshape(cs,[length(ts) size(c0)]);
%{
\end{matlab}
\item In the second case, \verb|RK2mesoPatch| detects a
parallel patch code has been requested, but has only one cpu
worker, so it auto-initiates an \verb|spmd|-block for the
integration. Both this and the next case return
\emph{composite} results, so just keep one version of the
results.
\begin{matlab}
%}
case 2
cs = RK2mesoPatch(ts,c0);
cs = cs{1};
%{
\end{matlab}
\item In this third case, a user could merge this explicit
\verb|spmd|-block with the previous one that sets the
initial conditions.
\begin{matlab}
%}
case 3,spmd
cs = RK2mesoPatch(ts,c0,[],patches);
end%spmd
cs = cs{1};
%{
\end{matlab}
\item In this fourth case, use Projective Integration (PI)
over long times (\verb|PIRK4| also works). Currently the PI
is done serially, with parallel \verb|spmd|-blocks only
invoked inside function \verb|aBurst()| (\cref{secmBfPI}) to
compute each burst of the micro-scale simulation. For a
Peclet number of ten, the macro-scale time-step needs to be
less than about~$0.5$ (which here is very little
projection)---presumably the mean advection in a macro-step
needs to be less than about the patch spacing. The function
\verb|microBurst()| here interfaces to \verb|aBurst()|
(\cref{secCHS1mBfPI}) in order to provide shaped initial
states, and to provide the patch information.
\begin{matlab}
%}
case 4
microBurst = @(tb0,xb0,bT) ...
aBurst(tb0 ,reshape(xb0,size(c0)) ,patches);
ts = 0:0.7:5
cs = PIRK2(microBurst,ts,gather(c0(:)));
cs = reshape(cs,[length(ts) size(c0)]);
%{
\end{matlab}
\end{enumerate}
End the four cases.
\begin{matlab}
%}
end%switch theCase
%{
\end{matlab}
\subsection{Plot the solution}
Optionally set to save some plots to file.
\begin{matlab}
%}
if 0, global OurCf2eps, OurCf2eps=true, end
%{
\end{matlab}
\paragraph{Animate the computed solution field over time}
\begin{matlab}
%}
figure(1), clf, colormap(0.8*hsv)
%{
\end{matlab}
First get the $x$-coordinates and omit the patch-edge
values from the plot (because they are not here
interpolated).
\begin{matlab}
%}
if theCase==1, x = patches.x;
else, spmd
x = gather( patches.x );
end%spmd
x = x{1};
end
x([1 end],:,:,:) = nan;
%{
\end{matlab}
For every time step draw the concentration values as a set
of surfaces on 2D patches, with a short pause to display
animation.
\begin{matlab}
%}
nTimes = length(ts)
for l = 1:nTimes
%{
\end{matlab}
At each time, squeeze sub-patch data into a 3D array,
permute to get all the $x$-variation in the first two
dimensions, and reshape into $x$-variation for each and
every~$(y)$.
\begin{matlab}
%}
c = reshape( permute( squeeze( ...
cs(l,:,:,:,:) ) ,[1 3 2]) ,numel(x),ny);
%{
\end{matlab}
Draw surface of each patch, to show both micro-scale and
macro-scale variation in space.
\begin{matlab}
%}
if l==1
hp = surf(x(:),yj,c');
axis([0 Len -1 1 0 max(c(:))])
axis equal
xlabel('space $x$'), ylabel('$y$'); zlabel('$c(x,y,t)$')
ifOurCf2eps([mfilename 't0'])
legend(['time = ' num2str(ts(l),'%4.2f')] ...
,'Location','north')
disp('**** pausing, press blank to animate')
pause
else
hp.ZData = c';
legend(['time = ' num2str(ts(l),'%4.2f')])
pause(0.1)
end
%{
\end{matlab}
\begin{figure}
\centering \caption{\label{fig:chanDispSpmdtFin}final
field~$c(x,y,4)$ of the patch scheme applied to a
heterogeneous advection-diffusion
\pde~\eqref{eq:pdeChanDisp} with heterogeneous factor
log-normal, here distributed $\exp[\mathcal N(0,1)]$. }
\includegraphics[scale=0.8]{chanDispSpmdtFin}
\end{figure}
Finish the animation loop, and optionally save the final
plot to file, \cref{fig:chanDispSpmdtFin}.
\begin{matlab}
%}
end%for over time
ifOurCf2eps([mfilename 'tFin'])
%{
\end{matlab}
\paragraph{Macro-scale view}
Plot a macro-scale mesh of the predictions: at each of a
selection of times, for every patch, plot the patch-mean
value at the mean-$x$.
\begin{matlab}
%}
figure(2), clf, colormap(0.8*hsv)
X = squeeze(mean(x(2:end-1,:,:,:)));
C = squeeze(mean(mean(cs(:,2:end-1,:,:,:),2),3));
j = 1:ceil(nTimes/30):nTimes;
mesh(X,ts(j),C(j,:));
xlabel('space $x$'),ylabel('time $t$'),zlabel('$C(X,t)$')
zlim([-0.1 11])
ifOurCf2eps([mfilename 'Macro'])
%{
\end{matlab}
\begin{figure}
\centering \caption{\label{fig:chanDispSpmdMacro}macro-scale
view of heterogeneous advection-diffusion~\pde\ along a
(periodic) channel obtained via the patch scheme. }
\includegraphics[scale=0.8]{chanDispSpmdMacro}
\end{figure}
\subsection{\texttt{microBurst} function for Projective Integration}
\label{secCHS1mBfPI}
Projective Integration stability appears to require bursts
longer than~$0.2$. Each burst is done in parallel
processing. Here use \verb|RK2mesoPatch| to take take
meso-steps, each with default ten micro-steps so the
micro-scale step is~$0.0033$. With macro-step~$0.5$,
these parameters usually give stable projective integration.
\begin{matlab}
%}
function [tbs,xbs] = aBurst(tb0,xb0,patches)
normx=max(abs(xb0(:)));
disp(['* aBurst t=' num2str(tb0) ' |x|=' num2str(normx)])
assert(normx<20,'solution exploding')
tbs = tb0+(0:0.033:0.2);
spmd
xb0 = codistributed(xb0,patches.codist);
xbs = RK2mesoPatch(tbs,xb0,[],patches);
end%spmd
xbs=reshape(xbs{1},length(tbs),[]);
end%function
%{
\end{matlab}
Fin.
\input{../Patch/chanDispMicro.m}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
Combescure2022.m
|
.m
|
EquationFreeGit-master/Patch/Combescure2022.m
| 20,113 |
utf_8
|
1b2e5e650fdd07081db07d4a2c4c942a
|
% For an example nonlinear elasticity in 1D, simulate and
% use MatCont to continue parametrised equilibria. An
% example of working via patches in space. Adapted from the
% example Figure 3(a) of Combescure(2022). AJR Nov 2022 --
% 13 Mar 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{Combescure2022}: simulation and
continuation of a 1D example nonlinear elasticity, via
patches}
\label{sec:Combescure2022}
Here we explore a nonlinear 1D elasticity problem with
complicated microstructure. Executes a simulation.
\emph{But the main aim is to show how one may use the
MatCont continuation toolbox \cite[]{Govaerts2019} together
with the Patch Scheme toolbox} \cite[]{Maclean2020a} in
order to explore parameter space by continuing branches of
equilibria, etc.
\begin{figure}
\centering
\caption{\label{fig:toyElas}1D arrangement of non-linear
springs with connections to (a) next-to-neighbour node
\protect\cite[Fig.~3(a)]{Combescure2022}. The blue box is
one micro-cell of one period, width~\(2b\), containing an
odd and an even~\(i\).}
\setlength{\unitlength}{0.01\linewidth}
\begin{picture}(100,31)
\put(0,0){\framebox(100,31){}}
\put(0,0){\includegraphics[width=\linewidth]{Figs/toyElas}}
\put(36,4){\color{blue}\framebox(27,23){cell}}
\end{picture}
\end{figure}
\cref{fig:toyElas} shows the microscale elasticity---adapted
from Fig.~3(a) by \cite{Combescure2022}. Let the spatial
microscale lattice be at rest at points~\(x_i\), with
constant spacing~\(b\). With displacement
variables~\(u_i(t)\), simulate the microscale lattice toy
elasticity system with 2-periodicity: for \(p=1,2\)
(respectively black and red in \cref{fig:toyElas}) and for
every~\(i\),
\begin{align}
&\epsilon^p_i:=\frac1{pb}(u_{i+p/2}-u_{i-p/2}),
&&\sigma^p_i:=w'_p(\epsilon^p_i),
\nonumber\\
&\DD t{u_{i}}= \sum_{p=1}^2\frac1{pb}(\sigma^p_{i+p/2}-\sigma^p_{i-p/2}),
&&w'_p(\epsilon):=\epsilon-M_p\epsilon^3+\epsilon^5.
\label{eq:heteroNLE}
\end{align}
The system has a microscale heterogeneity via the two
different functions~\(w'_p(\epsilon)\)
\cite[\S4]{Combescure2022}:
\begin{itemize}
\item microscale `instability' (structure) arises with
\(M_1:=2\) and \(M_2:=1\)
(\cref{fig:Comb22diffuSvis2b,fig:Comb22cpl}(b)); and
\item large scale `instability' (structure) arises with
\(M_1:=-1\) and \(M_2:=3\)
(\cref{fig:Comb22diffuLvis1,fig:Comb22cpl}(a)).
\end{itemize}
\paragraph{Microscale case} Set \(M_1:=2\) and \(M_2:=1\)\,.
We fix the boundary conditions \(u(0)=0\) and parametrise
solutions by~\(u(L)\). There are equilibria \(u\approx
u(L)x/L\), but under large compression (large
negative~\(u(L)\)) interesting structures develop.
\cref{fig:Comb22diffuSvis2b} shows boundary layers with
microscale variations develop for \(u(L)<-13\). This figure
plots a strain~\(\epsilon\) as the strain is nearly constant
across the interior, so the boundary layers show up clearly.
As~\(u(L)\) decreases further, \cref{fig:Comb22diffuSvis2b}
shows the family of equilibria form complicated folds.
\cref{tblMicro} lists that MatCont also reports some branch
points and neutral saddle equilibria in this same regime
(see \cref{fig:Comb22cpl}(b)). I have not yet followed any
of the branches.
\begin{SCfigure}
\centering
\caption{\label{fig:Comb22diffuSvis2b}the case of microscale
`instability' appears as fluctuations close to both
boundaries. As the system is physically compressed, the
equilibrium curve has complicated folds, as shown here (and
\cref{fig:Comb22cpl}(b)).}
\includegraphics[scale=0.8]{Comb22diffuSvis2b}
\end{SCfigure}
\begin{SCtable}
\centering\caption{\label{tblMicro}Interesting equilibria
for the cases of small scale instability: \(M_1:=2\),
\(M_2:=1\) (\cref{fig:Comb22diffuSvis2b,fig:Comb22cpl}(b)).
The rightmost column gives the \(-u(L)\)~parameter values
for corresponding critical points in the three-patch code
(\cref{fig:Comb22diffuSvis2N3}).}
\begin{tabular}{@{}rp{12.1em}r@{}}
\hline
$-u(L)$&MatCont description &\text{Patch}\\\hline
14.684 & Branch point &14.599\\
14.702 & Limit point &14.610\\
14.612 & Neutral Saddle Equilibrium &-\\
14.063 & Neutral Saddle Equilibrium &-\\
13.972 & Limit point &13.817\\
13.988 & Branch point &13.828\\
17.184 & Branch point &17.197\\
- & Limit point &17.227\\
17.183 & Neutral Saddle Equilibrium &17.211\\
%15.034 & Neutral Saddle Equilibrium \\
%15.024 & Limit point \\
%15.032 & Branch point \\
%17.987 & Branch point \\
%17.993 & Limit point \\
%17.987 & Neutral Saddle Equilibrium \\
\hline
\end{tabular}
\end{SCtable}
The previous paragraph's discussion is for a full domain
simulation, albeit done through an imposed computational
framework of physically abutting patches.
\cref{fig:Comb22diffuSvis2N3} shows the corresponding
MatCont continuation for the patch scheme with \(N=3\)
patches in the domain. Just three patches may well be
reasonable as the structures in this problem are the two
boundary layers, and a constant interior.
\cref{fig:Comb22diffuSvis2N3} shows the patch scheme
reasonably resolves these. \cref{tblMicro} also lists the
special points, as reported by MatCont, in the equilibria of
the patch scheme. The locations of these special points
reasonably match those found by the full domain simulation.
Importantly, MatCont is about \emph{ten times quicker to
execute on the patches} than on the full domain code. This
speed-up indicates that on larger scale problems the patch
scheme could be very useful in continuation explorations.
\begin{SCfigure}
\centering
\caption{\label{fig:Comb22diffuSvis2N3}using just three
patches, the case of microscale instability appears as
fluctuations close to both boundaries. As the system is
physically compressed, the equilibrium curve has complicated
folds, as shown, and that approximately match
\cref{fig:Comb22diffuSvis2b}. But it is computed ten times
quicker.}
\includegraphics[scale=0.8]{Comb22diffuSvis2N3}
\end{SCfigure}
\paragraph{Large scale case} Set \(M_1:=-1\) and
\(M_2:=3\)\,. We fix the boundary conditions \(u(0)=0\) and
parametrise solutions by~\(u(L)\). There are equilibria
\(u\approx u(L)x/L\), but under large compression (large
negative~\(u(L)\)) interesting structures develop.
\cref{fig:Comb22diffuLvis1} shows an interior region of
higher magnitude strain develops. Again, this figure plots a
strain~\(\epsilon\) as the strain is nearly constant across
the domain, so the interior structure shows up clearly.
As~\(u(L)\) decreases further, \cref{fig:Comb22diffuLvis1}
shows the family of equilibria form complicated folds.
\cref{tblLarge} lists that MatCont also reports some branch
points and neutral saddle equilibria in this regime (see
\cref{fig:Comb22cpl}(a)). I have not yet followed any of the
branches.
\begin{SCfigure}
\centering
\caption{\label{fig:Comb22diffuLvis1}the case of large scale
`instability'. Spatial structure appears in the middle of
the domain. As the system is physically compressed, the
equilibrium curve has complicated folds, as shown here and
in \cref{fig:Comb22cpl}(a).}
\includegraphics[scale=0.8]{Comb22diffuLvis1}
\end{SCfigure}
\begin{SCtable}
\centering\caption{\label{tblLarge}Interesting equilibria
for the cases of large scale instability: \(M_1:=-1\),
\(M_2:=3\) (\cref{fig:Comb22diffuLvis1,fig:Comb22cpl}(a)).}
\begin{tabular}{@{}rp{12.1em}r@{}}
\hline
$-u(L)$&MatCont description \\\hline
21.295 & Limit point \\
18.783 & Branch point \\
18.762 & Neutral Saddle Equilibrium \\
18.761 & Neutral Saddle Equilibrium \\
18.761 & Limit point \\
18.934 & Branch point \\
19.393 & Branch point \\
19.928 & Branch point \\
20.490 & Branch point \\
21.055 & Branch point \\
21.627 & Branch point \\
\hline
\end{tabular}
% these are from N=3 patches
%21.469 & Branch point \\
%23.342 & Neutral Saddle Equilibrium \\
%23.462 & Branch point \\
%29.95 & Hopf \\
\end{SCtable}
The patch scheme with \(N=3\) patches does not make
reasonable predictions here. I suspect this failure is
because the nontrivial interior structure here occupies too
much of the domain to fit into one `small' patch. Here the
patch scheme may be useful if the physical domain is larger.
\subsection{Configure heterogeneous toy elasticity systems}
\label{sec:chtes}
Set some physical parameters. Each cell is of
width~\(dx:=2b\) as I choose to store~\(u_i\) for odd~\(i\)
in \verb|u((i+1)/2,1,:)| and for even~\(i\) in
\verb|u(i/2,2,:)|, that is, the the physical displacements
form the array
\begin{equation*}
\verb|u|=\begin{bmatrix} u_1&u_2\\ u_3&u_4\\ u_5&u_6\\ \vdots&\vdots \end{bmatrix}.
\end{equation*}
Then corresponding velocities are adjoined as 3rd and 4th column.
\begin{matlab}
%}
clear all
global b M vis
b = 1 % separation of lattice points
N = 42 % # lattice steps in L
L = b*N % length of domain
%{
\end{matlab}
The nonlinear coefficients of stress-strain are in
array~\verb|M|, chosen by~\verb|theCase|.
\begin{matlab}
%}
theCase = 2
switch theCase
case 1, M = [0 0 0 0] % linear spring coefficients
case 2, M = [ 2 1 1 1] % micro scale instability??
case 3, M = [-1 3 1 1] % large scale instability??
end% switch
vis = 0.1 % does not appear to affect the equilibria
tEnd = 25
%{
\end{matlab}
Patch parameters: here \verb|nSubP| is the number of cells.
\begin{matlab}
%}
edgyInt = true
nSubP = 6, nPatch = 5 % gives full-domain on N=42, dx=2
%nSubP = 6, nPatch = 3 % patches for some crude comparison
%{
\end{matlab}
Establish the global data struct~\verb|patches| for the
microscale heterogeneous lattice elasticity
system~\cref{eq:heteroNLE}. Solved with \verb|nPatch|
patches, and interpolation (as high-order as possible) to
provide the edge-values of the inter-patch coupling
conditions.
\begin{matlab}
%}
global patches
configPatches1(@heteroNLE,[0 L],'equispace',nPatch ...
,0,2*b,nSubP,'EdgyInt',edgyInt);
xx = patches.x+[-1 1]*b/2; % staggered sub-cell positions
%{
\end{matlab}
\subsection{Simulate in time}
Set the initial displacement and velocity of a simulation.
Integrate some time using standard integrator.
\begin{matlab}
%}
u0 = [ sin(pi/L*xx) -0*0.14*cos(pi/L*xx) ];
tic
[ts,ust] = ode23(@patchSys1, tEnd*linspace(0,1,41), u0(:) ...
,[],patches,0);
cpuIntegrateTime = toc
%{
\end{matlab}
\paragraph{Plot space-time surface of the simulation} To see
the edge values of the patches, interpolate and then adjoin
a row of \verb|nan|s between patches. Because of the
odd/even storage we need to do a lot of permuting and
reshaping. First, array of sub-cell coordinates in a
column for each patch, separating patches also by an extra
row of nans.
\begin{matlab}
%}
xs = reshape( permute( xx ,[2 1 3 4]), 2*nSubP,nPatch);
xs(end+1,:) = nan;
%{
\end{matlab}
Interpolate patch edge values, at all times simultaneously
by including time data into the 2nd dimension, and 2nd
reshaping it into the 3rd dimension.
\begin{matlab}
%}
uvs = reshape( permute( reshape(ust ...
,length(ts),nSubP,4,1,nPatch) ,[2 3 1 4 5]) ,nSubP,[],1,nPatch);
uvs = reshape( patchEdgeInt1(uvs) ,nSubP,4,[],nPatch);
%{
\end{matlab}
Extract displacement field, merge the 1st two columns,
permute the time variations to the 3rd, separate patches by
NaNs, and merge spatial data into the 1st column.
\begin{matlab}
%}
us = reshape( permute( uvs(:,1:2,:,:) ...
,[2 1 4 3]) ,2*nSubP,nPatch,[]);
us(end+1,:,:) = nan;
us = reshape(us,[],length(ts));
%{
\end{matlab}
Plot space-time surface of displacements over the macroscale
duration of the simulation.
\begin{matlab}
%}
figure(1), clf()
mesh(ts,xs(:),us)
view(60,40), colormap(0.8*jet), axis tight
xlabel('time $t$'), ylabel('space $x$'), zlabel('$u(x,t)$')
%{
\end{matlab}
Ditto for the velocity.
\begin{matlab}
%}
vs = reshape( permute( uvs(:,3:4,:,:) ...
,[2 1 4 3]) ,2*nSubP,nPatch,[]);
vs(end+1,:,:) = nan;
vs = reshape(vs,[],length(ts));
figure(2), clf()
mesh(ts,xs(:),vs)
view(60,40), colormap(0.8*jet), axis tight
xlabel('time $t$'), ylabel('space $x$'), zlabel('$v(x,t)$')
drawnow
%{
\end{matlab}
\subsection{MatCont continuation}
First, use \verb|fsolve| to find an equilibrium at some
starting compressive displacement---a compression that
differs depending upon the case of nonlinearity.
\begin{matlab}
%}
muL0 = 12+6*(theCase==3)
u0 = [ -muL0*xx/L 0*xx ];
u0([1 end],:,:,:)=nan;
patches.i = find(~isnan(u0));
nVars=length(patches.i)
ueq=fsolve(@(v) dudtSys(0,v,muL0),u0(patches.i));
%{
\end{matlab}
Start search for equilibria at other compression parameters.
Starting from zero, need 1000+ to find both the large-scale
and small-scale instability cases. But need less points
when starting from parameter~\(12\) or so.
\begin{matlab}
%}
disp('Searching for equilibria, may take 1000+ secs')
[uv0,vec0]=init_EP_EP(@matContSys,ueq,muL0,[1]);
opt=contset; % initialise MatCont options
opt=contset(opt,'Singularities',true); %to report branch points, p.24
opt=contset(opt,'MaxNumPoints',400); % restricts how far matcont goes
opt=contset(opt,'Backward',true); % strangely, needs to go backwards??
[uv,vec,s,h,f]=cont(@equilibrium, uv0, [], opt); %MatCont continuation
%{
\end{matlab}
\paragraph{Post-process the report}
\begin{matlab}
%}
disp('List of interesting critical points')
muLs=uv(nVars+1,:);
for j=1:numel(s)
disp([num2str(muLs(s(j).index),5) ' & ' s(j).msg ' \\'])
end
%{
\end{matlab}
Find a range of parameter and corresponding indices where
all the critical points occur.
\begin{matlab}
%}
p1=muLs(end); pe=muLs(1);
if numel(s)>3, for j=2:numel(s)-1
p1=min(p1,muLs(s(j).index));
pe=max(pe,muLs(s(j).index));
end, end
pMid=(p1+pe)/2, pWid=abs(pe-p1)
iPars=find(abs(muLs(:)-pMid)<pWid);%include some either side
%{
\end{matlab}
Choose an `evenly spaced' subset of the range so we only
plot up to sixty of the parameter values reported in the
range.
\begin{matlab}
%}
nPars=numel(iPars)
dP=ceil((nPars-1)/60)
iP=1:dP:nPars;
muLP=muLs(iPars(iP));
%{
\end{matlab}
Interpolate patch edge values, at all parameters
simultaneously by including parameter-wise data into the 2nd
dimension, and 2nd reshaping it into the 3rd dimension.
\begin{matlab}
%}
uvs=nan(numel(iP),numel(u0));
uvs(:,patches.i)=uv(1:nVars,iPars(iP))';
uvs = reshape( permute( reshape(uvs ...
,length(muLP),nSubP,4,1,nPatch) ,[2 3 1 4 5]) ,nSubP,[],1,nPatch);
uvs = reshape( patchEdgeInt1(uvs) ,nSubP,4,[],nPatch);
%{
\end{matlab}
Extract displacement field, merge the 1st two columns,
permute the parameter variations to the 3rd, separate
patches by NaNs, and merge spatial data into the 1st column.
\begin{matlab}
%}
us = reshape( permute( uvs(:,1:2,:,:) ...
,[2 1 4 3]) ,2*nSubP,nPatch,[]);
us(end+1,:,:) = nan;
us = reshape(us,[],length(muLP));
%{
\end{matlab}
Plot space-time surface of displacements over the macroscale
duration of the simulation.
\begin{matlab}
%}
figure(4), clf()
mesh(muLP,xs(:),us)
view(60,40), colormap(0.8*jet), axis tight
xlabel('-u(L)'), ylabel('space $x$'), zlabel('$u(x)$')
%{
\end{matlab}
Plot space-time surface of strain, differences in
displacements, over the parameter variation.
\begin{matlab}
%}
figure(5), clf()
mesh(muLP,xs(1:end-1),diff(us))
view(45,20), colormap(0.8*jet), axis tight
xlabel('$-u(L)$'), ylabel('space $x$'), zlabel('strain $\delta u(x)$')
ifOurCf2eps(['Comb22diffu' num2str(theCase)],[12 9])%optionally save
%{
\end{matlab}
\begin{figure}
\centering
\caption{\label{fig:Comb22cpl}cross-sections through
\cref{fig:Comb22diffuLvis1,fig:Comb22diffuSvis2b}: (a)~large
scale case, at the mid-point in space of
\cref{fig:Comb22diffuLvis1}; (b)~microscale case, in a
boundary layer of \cref{fig:Comb22diffuSvis2b}. These
cross-sections are labelled with the various critical
points.}
\begin{tabular}{@{}cc@{}}
(a) large scale case & (b) microscale case\\
\includegraphics[scale=0.75]{Figs/Comb22cpl3}&
\includegraphics[scale=0.75]{Figs/Comb22cpl2}
\end{tabular}
\end{figure}
\paragraph{Labelled parameter plot} Get the labelled 2D
plots of \cref{fig:Comb22cpl} via MatCont's \verb|cpl|
function. In high-D problems it is unlikely that any one
variable is a good thing to plot, so I show how to plot
something else, here a strain. I use all the computed points
so reform~\verb|uvs| (possibly better to have merged the
critical points into the list of plotted parameters??).
\begin{matlab}
%}
uvs = nan(numel(muLs),numel(u0));
uvs(:,patches.i) = uv(1:nVars,:)';
uvs = reshape( uvs ,[],nSubP,4,nPatch);
%{
\end{matlab}
As a function of the parameter, plot the strain in the
middle of the domain (the middle of the middle patch),
unless it is the microscale case when we plot a strain near
the middle of the left boundary layer.
\begin{matlab}
%}
if theCase==2, thePatch=1;
else thePatch=(nPatch+1)/2;
end%if
figure(7),clf
du = diff( uvs(:,nSubP/2,1:2,thePatch) ,1,3);
cpl([muLs;du'],[],s);
xlabel('$-u(L)$')
if thePatch==1, ylabel('boundary layer strain')
else ylabel('mid-domain strain')
end
ifOurCf2eps(['Comb22cpl' num2str(theCase)],[9 7])%optionally save
%{
\end{matlab}
\subsection{\texttt{matContSys}: basic function for MatCont analysis}
This is the simple `odefile' of the patch scheme wrapped
around the microcode.
\begin{matlab}
%}
function out = matContSys%(t,coordinates,flag,y,z)
out{1} = [];%@init;
out{2} = @dudtSys;
out{3} = [];%@jacobian;
out{4} = [];%@jacobianp;
out{5} = [];%@hessians;
out{6} = [];%@hessiansp;
out{7} = [];
out{8} = [];
out{9} = [];
end% function matContSys
%{
\end{matlab}
\subsection{\texttt{dudtSys()}: wraps around the patch wrapper}
This function adjoins \verb|patches| to the argument list,
places the variables within the patch structure, and then
extracts their time derivatives to return. Used by both
MatCont and \verb|fsolve|.
\begin{matlab}
%}
function ut = dudtSys(t,u,p)
global patches
%{
\end{matlab}
The 4 here is the number of variables in each micro-cell,
that is, notionally `at' each \(x\)-grid point.
\begin{matlab}
%}
U=nan(1,4)+patches.x;
U(patches.i)=u(:);
Ut=patchSys1(t,U,patches,p);
ut=Ut(patches.i);
end
%{
\end{matlab}
\subsection{\texttt{heteroNLE()}: forced heterogeneous elasticity}
\label{sec:heteroNLE}
This function codes the lattice heterogeneous example
elasticity inside the patches. Computes the time derivative
at each point in the interior of a patch, output
in~\verb|uvt|.
\begin{matlab}
%}
function uvt = heteroNLE(t,uv,patches,muL)
if nargin<4, muL=0; end% default end displacement is zero
global b M vis
%{
\end{matlab}
Separate state vector into displacement and velocity fields:
\(u_{ijI}\)~is the displacement at the \(j\)th~point in the
\(i\)th 2-cell in the \(I\)th~patch; similarly for
velocity~\(v_{ijI}\). That is, physically neighbouring
points have different~\(j\), whereas physical
next-to-neighbours have \(i\)~different by one.
\begin{matlab}
%}
u=uv(:,1:2,:,:); v=uv(:,3:4,:,:); % separate u and v=du/dt
%{
\end{matlab}
Provide boundary conditions, here fixed displacement and
velocity in the left/right sub-cells of the
leftmost/rightmost patches.
\begin{matlab}
%}
u(1,:,:,1)=0;
v(1,:,:,1)=0;
u(end,:,:,end)=-muL;
v(end,:,:,end)=0;
%{
\end{matlab}
Compute the two different strain fields, and also a first
derivative for some optional viscosity.
\begin{matlab}
%}
eps2 = diff(u)/(2*b);
eps1 = [u(:,2,:,:)-u(:,1,:,:) u([2:end 1],1,:,:)-u(:,2,:,:)]/b;
eps1(end,2,:,:)=nan; % as this value is fake
vx1 = [v(:,2,:,:)-v(:,1,:,:) v([2:end 1],1,:,:)-v(:,2,:,:)]/b;
vx1(end,2,:,:)=nan; % as this value is fake
%{
\end{matlab}
Set corresponding nonlinear stresses
\begin{matlab}
%}
sig2 = eps2-M(2)*eps2.^3+M(4)*eps2.^5;
sig1 = eps1-M(1)*eps1.^3+M(3)*eps1.^5;
%{
\end{matlab}
Preallocate output array, and fill in time derivatives of
displacement and velocity, from velocity and gradient of
stresses, respectively.
\begin{matlab}
%}
uvt = nan+uv; % preallocate output array
i=2:size(uv,1)-1;
% rate of change of position
uvt(i,1:2,:,:) = v(i,:,:,:);
% rate of change of velocity +some artificial viscosity??
uvt(i,3:4,:,:) = diff(sig2) ...
+[ sig1(i,1,:,:)-sig1(i-1,2,:,:) diff(sig1(i,:,:,:),1,2)] ...
+vis*[ vx1(i,1,:,:)-vx1(i-1,2,:,:) diff(vx1(i,:,:,:),1,2) ];
end% function heteroNLE
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroDiff.m
|
.m
|
EquationFreeGit-master/Patch/heteroDiff.m
| 1,322 |
utf_8
|
e25f9b6b97e32d8fd2d51c533139ff0f
|
% Computes the time derivatives of heterogeneous diffusion
% in 1D on patches. Used by homogenisationExample.m,
% homoDiffEdgy1.m Optionally becomes Burgers PDE with
% heterogeneous advection. AJR, Apr 2019 -- Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{heteroDiff()}: heterogeneous diffusion}
\label{sec:heteroDiff}
This function codes the lattice heterogeneous diffusion
inside the patches. For 2D input arrays~\verb|u|
and~\verb|x| (via edge-value interpolation of
\verb|patchSys1|, \cref{sec:patchSys1}), computes the
time derivative~\cref{eq:HomogenisationExample} at each
point in the interior of a patch, output in~\verb|ut|. The
column vector of diffusivities~\(c_i\), and possibly
Burgers' advection coefficients~\(b_i\), have previously
been stored in struct~\verb|patches.cs|.
\begin{matlab}
%}
function ut = heteroDiff(t,u,patches)
dx = diff(patches.x(2:3)); % space step
i = 2:size(u,1)-1; % interior points in a patch
ut = nan+u; % preallocate output array
ut(i,:,:,:) = diff(patches.cs(:,1,:).*diff(u))/dx^2;
% possibly include heterogeneous Burgers' advection
if size(patches.cs,2)>1 % check for advection coeffs
buu = patches.cs(:,2,:).*u.^2;
ut(i,:) = ut(i,:)-(buu(i+1,:)-buu(i-1,:))/(dx*2);
end
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
burgersMap.m
|
.m
|
EquationFreeGit-master/Patch/burgersMap.m
| 715 |
utf_8
|
cf2b2e562653643533e6ba812a3c7deb
|
% Microscale Euler step of the Burgers PDE on a lattice in
% x. Used by BurgersExample.m AJR, 4 Apr 2019 -- Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{burgersMap()}: discretise the PDE microscale}
\label{sec:burgersMap}
This function codes the microscale Euler integration map of
the lattice differential equations inside the patches. Only
the patch-interior values are mapped (\verb|patchSys1()|
overrides the edge-values anyway).
\begin{matlab}
%}
function u = burgersMap(t,u,patches)
u = squeeze(u);
dx = diff(patches.x(2:3));
dt = dx^2/2;
i = 2:size(u,1)-1;
u(i,:) = u(i,:) +dt*( diff(u,2)/dx^2 ...
-20*u(i,:).*(u(i+1,:)-u(i-1,:))/(2*dx) );
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
hyperDiffHetero.m
|
.m
|
EquationFreeGit-master/Patch/hyperDiffHetero.m
| 5,091 |
utf_8
|
59b2e77ff7d0ff83e8b8355ac6ea570b
|
% Simulate a heterogeneous version of hyper-diffusion PDE in
% 1D on patches as an example application with pairs of edge
% points needing to be interpolated between patches in
% space. AJR, 12 Apr 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{hyperDiffHetero}: simulate a heterogeneous
hyper-diffusion PDE in 1D on patches}
\label{sec:hyperDiffHetero}
\localtableofcontents
\cref{fig:hyperDiffHeteroU} shows an example simulation in
time generated by the patch scheme applied to a
heterogeneous version of the hyper-diffusion \pde. That such
simulations makes valid predictions was established by
\cite{Bunder2013b} who proved that the scheme is accurate
when the number of points in a patch is tied to a multiple
of the periodicity of the pattern.
\begin{figure}
\centering \caption{\label{fig:hyperDiffHeteroU}
hyper-diffusing field~\(u(x,t)\) in the patch scheme applied
to microscale heterogeneous hyper-diffusion
(\cref{sec:hyperDiffHetero}). The log-time axis shows:
\(t<10^{-2}\), rapid decay of sub-patch micro-structure;
\(10^{-2}<t<1\), meso-time quasi-equilibrium; and
\(1<t<10^2\), slow decay of macroscale structures.}
\includegraphics[scale=0.9]{hyperDiffHeteroUxt}
\end{figure}%
We aim to simulate the heterogeneous hyper-diffusion \pde
\begin{equation}
u_t= -D[c_1(x)Du] \quad\text{where operator }
D := \partial_x( c_2(x) \partial_x ),
\label{eq:hyperDiffHetero}
\end{equation}
for microscale periodic coefficients~\(c_l(x)\), and
boundary conditions of \(u=u_x=0\) at \(x=0,L\). In this 1D
space, the macroscale, homogenised, effective
hyper-diffusion should be some unknown `average' of these
coefficients, but we use the patch scheme to provide a
computational homogenisation. We discretise the \pde\ to a
lattice of values~\(u_i(t)\), with lattice spacing~\(dx\),
and governed by
\begin{equation*}
\dot u_i = -D[c_{i1}D u_i] \quad\text{where operator }
D := \delta( c_{i2}\delta )/dx^2
\end{equation*}
in terms of centred difference operator \(\delta u_i :=
u_{i+1/2} - u_{i-1/2}\).
Set the desired microscale periodicity, and correspondingly
choose random microscale diffusion coefficients (with
some subscripts shifted by a half).
\begin{matlab}
%}
clear all
basename = mfilename
%global OurCf2eps, OurCf2eps=true %optional to save plots
nGap = 3 % controls size of gap between patches
nPtsPeriod = 5
dx = 0.5/nGap/nPtsPeriod
%{
\end{matlab}
Create some random heterogeneous coefficients, log-uniform.
\begin{matlab}
%}
csVar = 1
cs = 0.2*exp( -csVar/2+csVar.*rand(nPtsPeriod,2) )
%{
\end{matlab}
Establish global data struct~\verb|patches| for
heterogeneous hyper-diffusion on a finite domain with, on
average, one patch per unit length. Use seven patches, and
use high-order interpolation with \(\verb|ordCC|=0\).
\begin{matlab}
%}
nPatch = 7
nSubP = 2*nPtsPeriod+4 % or +2 for not-edgyInt
Len = nPatch;
ordCC = 0;
dom.type = 'equispace';
dom.bcOffset = 0.5 % for BC type
patches = configPatches1(@hyperDiffPDE,[0 Len],dom ...
,nPatch,ordCC,dx,nSubP,'EdgyInt',true,'nEdge',2 ...
,'hetCoeffs',cs);
xs=squeeze(patches.x);
%{
\end{matlab}
\paragraph{Simulate in time}
Set an initial condition, and here integrate forward in time
using a standard method for stiff systems---because of the
simplicity of linear problems this method works quite
efficiently here. Integrate the interface \verb|patchSys1|
(\cref{sec:patchSys1}) to the microscale differential
equations.
\begin{matlab}
%}
u0 = sin(2*pi/Len*patches.x).*rand(nSubP,1,1,nPatch);
tic
[ts,us] = ode15s(@patchSys1, [0 100], u0(:) ,[],patches);
simulateTime = toc
us = reshape(us,length(ts),numel(patches.x(:)),[]);
%{
\end{matlab}
Plot the simulation in \cref{fig:hyperDiffHeteroU}, using
log-axis for time so we can see a little of both micro- and
macro-dynamics.
\begin{matlab}
%}
figure(1),clf
xs([1:2 end-1:end],:) = nan;
t0=min(find(ts>1e-5));
mesh(ts(t0:3:end),xs(:),us(t0:3:end,:)'), view(55,50)
colormap(0.7*hsv)
xlabel('time $t$'), ylabel('space $x$'), zlabel('$u(x,t)$')
ca=gca; ca.XScale='log'; ca.XLim=ts([t0 end]);
ifOurCf2eps([basename 'Uxt'])
%{
\end{matlab}
Fin.
\subsection{Heterogeneous hyper-diffusion PDE inside patches}
As a microscale discretisation of hyper-diffusion
\pde~\cref{eq:hyperDiffHetero} \(u_t= -D[c_1(x)Du] \), where
heterogeneous operator \(D = \partial_x( c_2(x) \partial_x
)\).
\begin{matlab}
%}
function ut=hyperDiffPDE(t,u,patches)
dx=diff(patches.x(1:2)); % microscale spacing
%{
\end{matlab}
Code Dirichlet boundary conditions of zero function and
derivative at left-end of left-patch, and right-end of
right-patch. For slightly simpler coding, squeeze out the
two singleton dimensions.
\begin{matlab}
%}
u = squeeze(u);
if ~patches.periodic % discretise BC u=u_x=0
u(1:2,1)=0;
u(end-1:end,end)=0;
end%if
%{
\end{matlab}
Here code straightforward centred discretisation in space.
\begin{matlab}
%}
ut = nan+u; % preallocate output array
v = patches.cs(2:end,1).*diff(patches.cs(:,2).*diff(u))/dx^2;
ut(3:end-2,:) = -diff(patches.cs(2:end-1,2).*diff(v))/dx^2 ;
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
rotFilmMicro.m
|
.m
|
EquationFreeGit-master/Patch/rotFilmMicro.m
| 3,955 |
utf_8
|
09ccd41412bd78572f9d4d444e4228c8
|
% rotFilmMicro() computes the time derivatives of a 2D
% shallow water flow on a rotating heterogeneous substrate
% on 2D patches in space. AJR, Dec 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{rotFilmMicro()}: 2D shallow water flow
on a rotating heterogeneous substrate}
\label{sec:rotFilmMicro}
This function codes the heterogeneous shallow water
flow~\eqref{eqs:spinddt} inside 2D patches. The \pde{}s are
discretised on the multiscale lattice in terms of evolving
variables~$h_{ijIJ}$, $u_{ijIJ}$ and~$v_{ijIJ}$. For 6D
input array~\verb|huv| (via edge-value interpolation of
\verb|patchEdgeInt2()|, \cref{sec:patchSys2}), computes
the time derivatives~\eqref{eqs:spinddt} at each point in
the interior of a patch, output in~\verb|huvt|. The
heterogeneous bed drag and diffusivities,~$b_{ij}$
and~$\nu_{ij}$, have previously been merged and stored in
the array~\verb|patches.cs| (2D${}\times3$): herein
\verb|patches| is named~\verb|p|.
\begin{matlab}
%}
function huvt = rotFilmMicro(t,huv,p)
[nx,ny,~]=size(huv); % micro-grid points in patches
i = 2:nx-1; % x interior points in a patch
j = 2:ny-1; % y interior points in a patch
dx = diff(p.x(2:3)); % x space step
dy = diff(p.y(2:3)); % y space step
huvt = nan+huv; % preallocate output array
%{
\end{matlab}
Set indices of fields in the arrays. Need to store different
diffusivity values for the $x,y$-directions as they are
evaluated at different points in space.
\begin{matlab}
%}
h=1; u=2; v=3;
b=1; nux=2; nuy=3;
%{
\end{matlab}
Use a staggered micro-grid so that $\verb|h(i,j)| =h_{ij}$,
$\verb|u(i,j)| =u_{i+1/2,j}$, and $\verb|v(i,j)|
=v_{i,j+1/2}$. We need the following to interpolate some
quantities to other points on the staggered micro-grid. But
the first two statements fill-in two needed corner values
because they are not (currently) interpolated by
\verb|patchEdgeInt2()|.
\begin{matlab}
%}
huv(1,ny,u,:,:,:) = huv(2,ny,u,:,:,:)+huv(1,ny-1,u,:,:,:) ...
-huv(2,ny-1,u,:,:,:);
huv(nx,1,v,:,:,:) = huv(nx,2,v,:,:,:)+huv(nx-1,1,v,:,:,:) ...
-huv(nx-1,2,v,:,:,:);
v4u = (huv(i,j-1,v,:,:,:)+huv(i+1,j,v,:,:,:) ...
+huv(i,j,v,:,:,:)+huv(i+1,j-1,v,:,:,:))/4;
u4v = (huv(i,j+1,u,:,:,:)+huv(i-1,j,u,:,:,:) ...
+huv(i,j,u,:,:,:)+huv(i-1,j+1,u,:,:,:))/4;
h2u = (huv(2:nx,:,h,:,:,:)+huv(1:nx-1,:,h,:,:,:))/2;
h2v = (huv(:,2:ny,h,:,:,:)+huv(:,1:ny-1,h,:,:,:))/2;
%{
\end{matlab}
Evaluate conservation of mass \pde~\eqref{eq:spindhdt}
(needing averages of~$h$ at half-grid points):
\begin{matlab}
%}
huvt(i,j,h,:,:,:) = ...
- (h2u(i,j ,:,:,:,:).*huv(i ,j,u,:,:,:) ...
-h2u(i-1,j,:,:,:,:).*huv(i-1,j,u,:,:,:) )/dx ...
- (h2v(i,j ,:,:,:,:).*huv(i,j ,v,:,:,:) ...
-h2v(i,j-1,:,:,:,:).*huv(i,j-1,v,:,:,:))/dy ;
%{
\end{matlab}
Evaluate the $x$-direction momentum
\pde~\eqref{eq:spindvdt} (needing to interpolate
component~$v$ to $u$-points):
\begin{matlab}
%}
huvt(i,j,u,:,:,:) = ...
- p.cs(i,j,b).*huv(i,j,u,:,:,:) + p.f.*v4u ...
- huv(i,j,u,:,:,:).*(huv(i+1,j,u,:,:,:)-huv(i-1,j,u,:,:,:))/(2*dx) ...
- v4u.*(huv(i,j+1,u,:,:,:)-huv(i,j-1,u,:,:,:))/(2*dy) ...
- p.g*(huv(i+1,j,h,:,:,:)-huv(i,j,h,:,:,:))/dx ...
+ diff(p.cs(:,j,nux).*diff(huv(:,j,u,:,:,:),[],1),[],1)/dx^2 ...
+ diff(p.cs(i,:,nuy).*diff(huv(i,:,u,:,:,:),[],2),[],2)/dy^2 ;
%{
\end{matlab}
Evaluate the $y$-direction momentum
\pde~\eqref{eq:spindvdt} (needing to interpolate
component~$u$ to $v$-points):
\begin{matlab}
%}
huvt(i,j,v,:,:,:) = ...
- p.cs(i,j,b).*huv(i,j,v,:,:,:) - p.f.*u4v ...
- u4v.*(huv(i+1,j,v,:,:,:)-huv(i-1,j,v,:,:,:))/(2*dx) ...
- huv(i,j,v,:,:,:).*(huv(i,j+1,v,:,:,:)-huv(i,j-1,v,:,:,:))/(2*dy) ...
- p.g*(huv(i,j+1,h,:,:,:)-huv(i,j,h,:,:,:))/dy ...
+ diff(p.cs(:,j,nux).*diff(huv(:,j,v,:,:,:),[],1),[],1)/dx^2 ...
+ diff(p.cs(i,:,nuy).*diff(huv(i,:,v,:,:,:),[],2),[],2)/dy^2 ;
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroBurstF.m
|
.m
|
EquationFreeGit-master/Patch/heteroBurstF.m
| 732 |
utf_8
|
e1c0b0563873325bb9e37ed4ee71f983
|
% Simulates a burst of the system linked to by the
% configuration of patches. Used by ??.m
% AJR, 4 Apr 2019 -- 21 Oct 2022
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{heteroBurstF()}: a burst of
heterogeneous diffusion}
\label{sec:heteroBurstF}
This code integrates in time the derivatives computed by
\verb|heteroDiff| from within the patch coupling of
\verb|patchSys1|. Try~\verb|ode23|, although \verb|ode45|
may give smoother results. Sample every period of the
microscale time fluctuations (or, at least, close to the
period).
\begin{matlab}
%}
function [ts, ucts] = heteroBurstF(ti, ui, bT)
global microTimePeriod
[ts,ucts] = ode45( @patchSys1,ti+(0:microTimePeriod:bT),ui(:));
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
waterWavePDE.m
|
.m
|
EquationFreeGit-master/Patch/waterWavePDE.m
| 2,385 |
utf_8
|
ca0bad5d58464a63a5c783c3f1be9605
|
% Codes a nonlinear water wave PDE on a staggered 1D grid
% inside patches in space. Used by waterWaveExample.m
% AJR, 4 Apr 2019 -- Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{waterWavePDE()}: water wave PDE}
\label{sec:waterWavePDE}
This function codes the staggered lattice equation inside
the patches for the nonlinear wave-like \pde\
system~\cref{eqs:patch:N}. Also, regularise the absolute
value appearing the the \pde{}s via the one-line
function~\verb|rabs()|.
\begin{matlab}
%}
function Ut = waterWavePDE(t,U,patches)
rabs = @(u) sqrt(1e-4 + u.^2);
%{
\end{matlab}
As before, set the micro-grid spacing, reserve space for
time derivatives, and index the patch-interior points of the
micro-grid.
\begin{matlab}
%}
dx = diff(patches.x(2:3));
U = squeeze(U);
Ut = nan(size(U)); ht = Ut;
i = 2:size(U,1)-1;
%{
\end{matlab}
Need to estimate~\(h\) at all the \(u\)-points, so
into~\verb|V| use averages, and linear extrapolation to
patch-edges.
\begin{matlab}
%}
ii = i(2:end-1);
V = Ut;
V(ii,:) = (U(ii+1,:)+U(ii-1,:))/2;
V(1:2,:) = 2*U(2:3,:)-V(3:4,:);
V(end-1:end,:) = 2*U(end-2:end-1,:)-V(end-3:end-2,:);
%{
\end{matlab}
Then estimate \(\D x{(hu)}\) from~\(u\) and the
interpolated~\(h\) at the neighbouring micro-grid points.
\begin{matlab}
%}
ht(i,:) = -(U(i+1,:).*V(i+1,:)-U(i-1,:).*V(i+1,:))/(2*dx);
%{
\end{matlab}
Correspondingly estimate the terms in the momentum \pde:
\(u\)-values in~\(\verb|U|_i\) and~\(\verb|V|_{i\pm1}\); and
\(h\)-values in~\(\verb|V|_i\) and~\(\verb|U|_{i\pm1}\).
\begin{matlab}
%}
Ut(i,:) = -0.985*(U(i+1,:)-U(i-1,:))/(2*dx) ...
-0.003*U(i,:).*rabs(U(i,:)./V(i,:)) ...
-1.045*U(i,:).*(V(i+1,:)-V(i-1,:))/(2*dx) ...
+0.26*rabs(V(i,:).*U(i,:)).*(V(i+1,:)-2*U(i,:)+V(i-1,:))/dx^2/2;
%{
\end{matlab}
where the mysterious division by two in the second
derivative is due to using the averaged values of~\(u\) in
the estimate:
\begin{eqnarray*}
u_{xx}&\approx&\frac1{4\delta^2}(u_{i-2}-2u_i+u_{i+2})
\\&=&\frac1{4\delta^2}(u_{i-2}+u_i-4u_i+u_i+u_{i+2})
\\&=&\frac1{2\delta^2}\left(\frac{u_{i-2}+u_i}2-2u_i+\frac{u_i+u_{i+2}}2\right)
\\&=&\frac1{2\delta^2}\left(\bar u_{i-1}-2u_i+\bar u_{i+1}\right).
\end{eqnarray*}
Then overwrite the unwanted~\(\dot u_{ij}\) with the
corresponding wanted~\(\dot h_{ij}\).
\begin{matlab}
%}
Ut(patches.hPts) = ht(patches.hPts);
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
configPatches2.m
|
.m
|
EquationFreeGit-master/Patch/configPatches2.m
| 31,145 |
utf_8
|
fe35a8aef7255a762724332dc96dcbe2
|
% configPatches2() creates a data struct of the design of 2D
% patches for later use by the patch functions such as
% patchSys2(). AJR, Nov 2018 -- 12 Apr 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{configPatches2()}: configures spatial
patches in 2D}
\label{sec:configPatches2}
\localtableofcontents
Makes the struct~\verb|patches| for use by the patch\slash
gap-tooth time derivative\slash step function
\verb|patchSys2()|. \cref{sec:configPatches2eg} lists an
example of its use.
\begin{matlab}
%}
function patches = configPatches2(fun,Xlim,Dom ...
,nPatch,ordCC,dx,nSubP,varargin)
version = '2023-04-12';
%{
\end{matlab}
\paragraph{Input}
If invoked with no input arguments, then executes an example
of simulating a nonlinear diffusion \pde\ relevant to the
lubrication flow of a thin layer of fluid---see
\cref{sec:configPatches2eg} for an example code.
\begin{itemize}
\item \verb|fun| is the name of the user function,
\verb|fun(t,u,patches)| or \verb|fun(t,u)| or
\verb|fun(t,u,patches,...)|, that computes time-derivatives
(or time-steps) of quantities on the 2D micro-grid within
all the 2D~patches.
\item \verb|Xlim| array/vector giving the rectangular
macro-space domain of the computation, namely
$[\verb|Xlim(1)|, \verb|Xlim(2)|] \times [\verb|Xlim(3)|,
\verb|Xlim(4)|]$. If \verb|Xlim| has two elements, then the
domain is the square domain of the same interval in both
directions.
\item \verb|Dom| sets the type of macroscale conditions for
the patches, and reflects the type of microscale boundary
conditions of the problem. If \verb|Dom| is \verb|NaN| or
\verb|[]|, then the field~\verb|u| is doubly macro-periodic
in the 2D spatial domain, and resolved on equi-spaced
patches. If \verb|Dom| is a character string, then that
specifies the \verb|.type| of the following structure, with
\verb|.bcOffset| set to the default zero. Otherwise
\verb|Dom| is a structure with the following components.
\begin{itemize}
\item \verb|.type|, string, of either \verb|'periodic'| (the
default), \verb|'equispace'|, \verb|'chebyshev'|,
\verb|'usergiven'|. For all cases except \verb|'periodic'|,
users \emph{must} code into \verb|fun| the micro-grid
boundary conditions that apply at the left\slash right\slash
bottom\slash top edges of the leftmost\slash rightmost\slash
bottommost\slash topmost patches, respectively.
\item \verb|.bcOffset|, optional one, two or four element
vector/array, in the cases of \verb|'equispace'| or
\verb|'chebyshev'| the patches are placed so the left\slash
right\slash top\slash bottom macroscale boundaries are
aligned to the left\slash right\slash top\slash bottom edges
of the corresponding extreme patches, but offset by
\verb|.bcOffset| of the sub-patch micro-grid spacing. For
example, use \verb|bcOffset=0| when the micro-code applies
Dirichlet boundary values on the extreme edge micro-grid
points, whereas use \verb|bcOffset=0.5| when the microcode
applies Neumann boundary conditions halfway between the
extreme edge micro-grid points. Similarly for the top and
bottom edges.
If \verb|.bcOffset| is a scalar, then apply the same offset
to all boundaries. If two elements, then apply the first
offset to both \(x\)-boundaries, and the second offset to
both \(y\)-boundaries. If four elements, then apply the
first two offsets to the respective \(x\)-boundaries, and
the last two offsets to the respective \(y\)-boundaries.
\item \verb|.X|, optional vector/array with \verb|nPatch(1)|
elements, in the case \verb|'usergiven'| it specifies the
\(x\)-locations of the centres of the patches---the user is
responsible the locations makes sense.
\item \verb|.Y|, optional vector/array with \verb|nPatch(2)|
elements, in the case \verb|'usergiven'| it specifies the
\(y\)-locations of the centres of the patches---the user is
responsible the locations makes sense.
\end{itemize}
\item \verb|nPatch| sets the number of equi-spaced spatial
patches: if scalar, then use the same number of patches in
both directions, otherwise \verb|nPatch(1:2)| gives the
number of patches~($\geq1$) in each direction.
\item \verb|ordCC| is the `order' of interpolation for
inter-patch coupling across empty space of the macroscale
patch values to the edge-values of the patches: currently
must be~$0,2,4,\ldots$; where $0$~gives spectral
interpolation.
\item \verb|dx| (real---scalar or two element) is usually
the sub-patch micro-grid spacing in~\(x\) and~\(y\). If
scalar, then use the same \verb|dx| in both directions,
otherwise \verb|dx(1:2)| gives the spacing in each of the
two directions.
However, if \verb|Dom| is~\verb|NaN| (as for pre-2023), then
\verb|dx| actually is \verb|ratio| (scalar or two element),
namely the ratio of (depending upon \verb|EdgyInt|) either
the half-width or full-width of a patch to the equi-spacing
of the patch mid-points---adjusted a little when $\verb|nEdge|>1$. So
either $\verb|ratio|=\tfrac12$ means the patches abut and
$\verb|ratio|=1$ is overlapping patches as in holistic
discretisation, or $\verb|ratio|=1$ means the patches abut.
Small~\verb|ratio| should greatly reduce computational time.
\item \verb|nSubP| is the number of equi-spaced microscale
lattice points in each patch: if scalar, then use the same
number in both directions, otherwise \verb|nSubP(1:2)| gives
the number in each direction. If not using \verb|EdgyInt|,
then $\verb|nSubP./nEdge|$ must be odd integer(s) so that there
is/are centre-patch lattice lines. So for the defaults
of $\verb|nEdge|=1$ and not \verb|EdgyInt|, then
\verb|nSubP| must be odd.
\item \verb|'nEdge'|, \emph{optional} (integer---scalar or two element), default=1, the width of edge values set by interpolation at the
edge regions of each patch. If two elements, then respectively the width in \(x,y\)-directions. The default is one (suitable
for microscale lattices with only nearest neighbour
interactions).
\item \verb|EdgyInt|, true/false, \emph{optional},
default=false. If true, then interpolate to left\slash
right\slash top\slash bottom edge-values from right\slash
left\slash bottom\slash top next-to-edge values. If false
or omitted, then interpolate from centre cross-patch lines.
\item \verb|nEnsem|, \emph{optional-experimental},
default one, but if more, then an ensemble over this
number of realisations.
\item \verb|hetCoeffs|, \emph{optional}, default empty.
Supply a 2D or 3D array of microscale heterogeneous
coefficients to be used by the given microscale \verb|fun|
in each patch. Say the given array~\verb|cs| is of size
$m_x\times m_y\times n_c$, where $n_c$~is the number of
different sets of coefficients. For example, in
heterogeneous diffusion, $n_c=2$ for the diffusivities in
the \emph{two} different spatial directions (or $n_c=3$ for
the diffusivity tensor). The coefficients are to be the same
for each and every patch; however, macroscale variations are
catered for by the $n_c$~coefficients being $n_c$~parameters
in some macroscale formula.
\begin{itemize}
\item If $\verb|nEnsem|=1$, then the array of coefficients
is just tiled across the patch size to fill up each patch,
starting from the $(1,1)$-point in each patch. Best accuracy
usually obtained when the periodicity of the coefficients
is a factor of \verb|nSubP-2*nEdge| for \verb|EdgyInt|, or
a factor of \verb|(nSubP-nEdge)/2| for not \verb|EdgyInt|.
\item If $\verb|nEnsem|>1$ (value immaterial), then reset
$\verb|nEnsem|:=m_x\cdot m_y$ and construct an ensemble of
all $m_x\cdot m_y$ phase-shifts of the coefficients. In
this scenario, the inter-patch coupling couples different
members in the ensemble. When \verb|EdgyInt| is true, and
when the coefficients are diffusivities\slash elasticities
in~$x$ and~$y$ directions, respectively, then this
coupling cunningly preserves symmetry.
\end{itemize}
\item \verb|'parallel'|, true/false, \emph{optional},
default=false. If false, then all patch computations are on
the user's main \textsc{cpu}---although a user may well
separately invoke, say, a \textsc{gpu} to accelerate
sub-patch computations.
If true, and it requires that you have \Matlab's Parallel
Computing Toolbox, then it will distribute the patches over
multiple \textsc{cpu}s\slash cores. In \Matlab, only one
array dimension can be split in the distribution, so it
chooses the one space dimension~$x,y$ corresponding to the
highest~\verb|\nPatch| (if a tie, then chooses the rightmost
of~$x,y$). A user may correspondingly distribute arrays
with property \verb|patches.codist|, or simply use formulas
invoking the preset distributed arrays \verb|patches.x|, and
\verb|patches.y|. If a user has not yet established a
parallel pool, then a `local' pool is started.
\end{itemize}
\paragraph{Output} The struct \verb|patches| is created and
set with the following components. If no output variable is
provided for \verb|patches|, then make the struct available
as a global variable.\footnote{When using \texttt{spmd}
parallel computing, it is generally best to avoid global
variables, and so instead prefer using an explicit output
variable.}
\begin{matlab}
%}
if nargout==0, global patches, end
patches.version = version;
%{
\end{matlab}
\begin{itemize}
\item \verb|.fun| is the name of the user's function
\verb|fun(t,u,patches)| or \verb|fun(t,u)| or
\verb|fun(t,u,patches,...)|, that computes the time
derivatives (or steps) on the patchy lattice.
\item \verb|.ordCC| is the specified order of inter-patch
coupling.
\item \verb|.periodic|: either true, for interpolation on
the macro-periodic domain; or false, for general
interpolation by divided differences over non-periodic
domain or unevenly distributed patches.
\item \verb|.stag| is true for interpolation using only odd
neighbouring patches as for staggered grids, and false for
the usual case of all neighbour coupling---not yet
implemented.
\item \verb|.Cwtsr| and \verb|.Cwtsl|, only for
macro-periodic conditions, are the
$\verb|ordCC|\times 2$-array of weights for the inter-patch
interpolation onto the right\slash top and left\slash bottom
edges (respectively) with patch:macroscale ratio as
specified or as derived from~\verb|dx|.
\item \verb|.x| (6D) is $\verb|nSubP(1)| \times1 \times1
\times1 \times \verb|nPatch(1)| \times1$ array of the
regular spatial locations~$x_{iI}$ of the microscale grid
points in every patch.
\item \verb|.y| (6D) is $1 \times \verb|nSubP(2)| \times1
\times1 \times1 \times \verb|nPatch(2)|$ array of the
regular spatial locations~$y_{jJ}$ of the microscale grid
points in every patch.
\item \verb|.ratio| $1\times 2$, only for
macro-periodic conditions, are the size ratios of
every patch.
\item \verb|.nEdge| $1\times 2$, is the width of edge
values set by interpolation at the edge regions of each
patch, in the \(x,y\)-directions respectively.
\item \verb|.le|, \verb|.ri|, \verb|.bo|, \verb|.to|
determine inter-patch coupling of members in an ensemble.
Each a column vector of length~\verb|nEnsem|.
\item \verb|.cs| either
\begin{itemize}
\item \verb|[]| 0D, or
\item if $\verb|nEnsem|=1$, $(\verb|nSubP(1)|-1)\times
(\verb|nSubP(2)|-1)\times n_c$ 3D array of microscale
heterogeneous coefficients, or
\item if $\verb|nEnsem|>1$, $(\verb|nSubP(1)|-1)\times
(\verb|nSubP(2)|-1)\times n_c\times m_xm_y$ 4D array of
$m_xm_y$~ensemble of phase-shifts of the microscale
heterogeneous coefficients.
\end{itemize}
\item \verb|.parallel|, logical: true if patches are
distributed over multiple \textsc{cpu}s\slash cores for the
Parallel Computing Toolbox, otherwise false (the default is
to activate the \emph{local} pool).
\item \verb|.codist|, \emph{optional}, describes the
particular parallel distribution of arrays over the active
parallel pool.
\end{itemize}
\subsection{If no arguments, then execute an example}
\label{sec:configPatches2eg}
\begin{matlab}
%}
if nargin==0
disp('With no arguments, simulate example of nonlinear diffusion')
%{
\end{matlab}
The code here shows one way to get started: a user's script
may have the following three steps (``\into'' denotes
function recursion).
\begin{enumerate}\def\itemsep{-1.5ex}
\item configPatches2
\item ode23 integrator \into patchSys2 \into user's PDE
\item process results
\end{enumerate}
Establish global patch data struct to interface with a
function coding a nonlinear `diffusion' \pde: to be solved
on $6\times4$-periodic domain, with $9\times7$ patches,
spectral interpolation~($0$) couples the patches, with
$5\times5$ points forming the micro-grid in each patch, and
a sub-patch micro-grid spacing of~\(0.12\) (relatively large
for visualisation). \cite{Roberts2011a} established that
this scheme is consistent with the \pde\ (as the patch
spacing decreases).
\begin{matlab}
%}
global patches
patches = configPatches2(@nonDiffPDE,[-3 3 -2 2] ...
,'periodic', [9 7], 0, 0.12, 5 ,'EdgyInt',false);
%{
\end{matlab}
Set an initial condition of a perturbed-Gaussian using
auto-replication of the spatial grid.
\begin{matlab}
%}
u0 = exp(-patches.x.^2-patches.y.^2);
u0 = u0.*(0.9+0.1*rand(size(u0)));
%{
\end{matlab}
Initiate a plot of the simulation using only the microscale
values interior to the patches: optionally set $x$~and
$y$-edges to \verb|nan| to leave the gaps between patches.
\begin{matlab}
%}
figure(1), clf, colormap(0.8*hsv)
x = squeeze(patches.x); y = squeeze(patches.y);
if 1, x([1 end],:) = nan; y([1 end],:) = nan; end
%{
\end{matlab}
Start by showing the initial conditions of
\cref{fig:configPatches2ic} while the simulation computes.
\begin{matlab}
%}
u = reshape(permute(squeeze(u0) ...
,[1 3 2 4]), [numel(x) numel(y)]);
hsurf = mesh(x(:),y(:),u');
axis([-3 3 -3 3 -0.03 1]), view(60,40)
legend('time = 0.00','Location','north')
xlabel('space $x$'), ylabel('space $y$'), zlabel('$u(x,y)$')
colormap(hsv)
ifOurCf2eps([mfilename 'ic'])
%{
\end{matlab}
\begin{figure}
\centering \caption{\label{fig:configPatches2ic}initial
field~$u(x,y,t)$ at time $t=0$ of the patch scheme applied
to a nonlinear `diffusion'~\pde: \cref{fig:configPatches2t3}
plots the computed field at time $t=3$.}
\includegraphics[scale=0.9]{configPatches2ic}
\end{figure}
Integrate in time to $t=4$ using standard functions. In
\Matlab\ \verb|ode15s| would be natural as the patch scheme
is naturally stiff, but \verb|ode23| is quicker \cite
[Fig.~4] {Maclean2020a}. Ask for output at non-uniform
times because the diffusion slows.
\begin{matlab}
%}
disp('Wait to simulate nonlinear diffusion h_t=(h^3)_xx+(h^3)_yy')
drawnow
if ~exist('OCTAVE_VERSION','builtin')
[ts,us] = ode23(@patchSys2,linspace(0,2).^2,u0(:));
else % octave version is quite slow for me
lsode_options('absolute tolerance',1e-4);
lsode_options('relative tolerance',1e-4);
[ts,us] = odeOcts(@patchSys2,[0 1],u0(:));
end
%{
\end{matlab}
Animate the computed simulation to end with
\cref{fig:configPatches2t3}. Use \verb|patchEdgeInt2| to
interpolate patch-edge values.
\begin{matlab}
%}
for i = 1:length(ts)
u = patchEdgeInt2(us(i,:));
u = reshape(permute(squeeze(u) ...
,[1 3 2 4]), [numel(x) numel(y)]);
set(hsurf,'ZData', u');
legend(['time = ' num2str(ts(i),'%4.2f')])
pause(0.1)
end
ifOurCf2eps([mfilename 't3'])
%{
\end{matlab}
\begin{figure}
\centering
\caption{\label{fig:configPatches2t3}field~$u(x,y,t)$ at
time $t=3$ of the patch scheme applied to a nonlinear
`diffusion'~\pde\ with initial condition in
\cref{fig:configPatches2ic}.}
\includegraphics[scale=0.9]{configPatches2t3}
\end{figure}
Upon finishing execution of the example, exit this function.
\begin{matlab}
%}
return
end%if no arguments
%{
\end{matlab}
\IfFileExists{../Patch/nonDiffPDE.m}{\input{../Patch/nonDiffPDE.m}}{}
\begin{devMan}
\subsection{Parse input arguments and defaults}
\begin{matlab}
%}
p = inputParser;
fnValidation = @(f) isa(f, 'function_handle');%test for fn name
addRequired(p,'fun',fnValidation);
addRequired(p,'Xlim',@isnumeric);
%addRequired(p,'Dom'); % nothing yet decided
addRequired(p,'nPatch',@isnumeric);
addRequired(p,'ordCC',@isnumeric);
addRequired(p,'dx',@isnumeric);
addRequired(p,'nSubP',@isnumeric);
addParameter(p,'nEdge',1,@isnumeric);
addParameter(p,'EdgyInt',false,@islogical);
addParameter(p,'nEnsem',1,@isnumeric);
addParameter(p,'hetCoeffs',[],@isnumeric);
addParameter(p,'parallel',false,@islogical);
%addParameter(p,'nCore',1,@isnumeric); % not yet implemented
parse(p,fun,Xlim,nPatch,ordCC,dx,nSubP,varargin{:});
%{
\end{matlab}
Set the optional parameters.
\begin{matlab}
%}
patches.nEdge = p.Results.nEdge;
if numel(patches.nEdge)==1
patches.nEdge = repmat(patches.nEdge,1,2);
end
patches.EdgyInt = p.Results.EdgyInt;
patches.nEnsem = p.Results.nEnsem;
cs = p.Results.hetCoeffs;
patches.parallel = p.Results.parallel;
%patches.nCore = p.Results.nCore;
%{
\end{matlab}
Initially duplicate parameters for both space dimensions as
needed.
\begin{matlab}
%}
if numel(Xlim)==2, Xlim = repmat(Xlim,1,2); end
if numel(nPatch)==1, nPatch = repmat(nPatch,1,2); end
if numel(dx)==1, dx = repmat(dx,1,2); end
if numel(nSubP)==1, nSubP = repmat(nSubP,1,2); end
%{
\end{matlab}
Check parameters.
\begin{matlab}
%}
assert(Xlim(1)<Xlim(2) ...
,'first pair of Xlim must be ordered increasing')
assert(Xlim(3)<Xlim(4) ...
,'second pair of Xlim must be ordered increasing')
assert((mod(ordCC,2)==0)|all(patches.nEdge==1) ...
,'Cannot yet have nEdge>1 and staggered patch grids')
assert(all(3*patches.nEdge<=nSubP) ...
,'too many edge values requested')
assert(all(rem(nSubP,patches.nEdge)==0) ...
,'nSubP must be integer multiple of nEdge')
if ~patches.EdgyInt, assert(all(rem(nSubP./patches.nEdge,2)==1) ...
,'for non-edgyInt, nSubP./nEdge must be odd integer')
end
if (patches.nEnsem>1)&all(patches.nEdge>1)
warning('not yet tested when both nEnsem and nEdge non-one')
end
%if patches.nCore>1
% warning('nCore>1 not yet tested in this version')
% end
%{
\end{matlab}
For compatibility with pre-2023 functions, if parameter
\verb|Dom| is \verb|Nan|, then we set the \verb|ratio| to
be the value of the so-called \verb|dx| vector.
\begin{matlab}
%}
if ~isstruct(Dom), pre2023=isnan(Dom);
else pre2023=false; end
if pre2023, ratio=dx; dx=nan; end
%{
\end{matlab}
Default macroscale conditions are periodic with evenly
spaced patches.
\begin{matlab}
%}
if isempty(Dom), Dom=struct('type','periodic'); end
if (~isstruct(Dom))&isnan(Dom), Dom=struct('type','periodic'); end
%{
\end{matlab}
If \verb|Dom| is a string, then just set type to that
string, and subsequently set corresponding defaults for
others fields.
\begin{matlab}
%}
if ischar(Dom), Dom=struct('type',Dom); end
%{
\end{matlab}
We allow different macroscale domain conditions in the
different directions. But for the moment do not allow
periodic to be mixed with the others (as the interpolation
mechanism is different code)---hence why we choose
\verb|periodic| be seven characters, whereas the others are
eight characters. The different conditions are coded in
different rows of \verb|Dom.type|, so we duplicate the
string if only one row specified.
\begin{matlab}
%}
if size(Dom.type,1)==1, Dom.type=repmat(Dom.type,2,1); end
%{
\end{matlab}
Check what is and is not specified, and provide default of
zero
(Dirichlet boundaries) if no \verb|bcOffset| specified when
needed. Do so for both directions independently.
\begin{matlab}
%}
patches.periodic=false;
for p=1:2
switch Dom.type(p,:)
case 'periodic'
patches.periodic=true;
if isfield(Dom,'bcOffset')
warning('bcOffset not available for Dom.type = periodic'), end
msg=' not available for Dom.type = periodic';
if isfield(Dom,'X'), warning(['X' msg]), end
if isfield(Dom,'Y'), warning(['Y' msg]), end
case {'equispace','chebyshev'}
if ~isfield(Dom,'bcOffset'), Dom.bcOffset=zeros(2,2); end
% for mixed with usergiven, following should still work
if numel(Dom.bcOffset)==1
Dom.bcOffset=repmat(Dom.bcOffset,2,2); end
if numel(Dom.bcOffset)==2
Dom.bcOffset=repmat(Dom.bcOffset(:)',2,1); end
msg=' not available for Dom.type = equispace or chebyshev';
if (p==1)& isfield(Dom,'X'), warning(['X' msg]), end
if (p==2)& isfield(Dom,'Y'), warning(['Y' msg]), end
case 'usergiven'
% if isfield(Dom,'bcOffset')
% warning('bcOffset not available for usergiven Dom.type'), end
msg=' required for Dom.type = usergiven';
if p==1, assert(isfield(Dom,'X'),['X' msg]), end
if p==2, assert(isfield(Dom,'Y'),['Y' msg]), end
otherwise
error([Dom.type ' is unknown Dom.type'])
end%switch Dom.type
end%for p
%{
\end{matlab}
\subsection{The code to make patches}
First, store the pointer to the time derivative function in
the struct.
\begin{matlab}
%}
patches.fun = fun;
%{
\end{matlab}
Second, store the order of interpolation that is to provide
the values for the inter-patch coupling conditions. Spectral
coupling is \verb|ordCC| of~$0$ or (not yet??)~$-1$.
\todo{Perhaps implement staggered spectral coupling.}
\begin{matlab}
%}
assert((ordCC>=-1) & (floor(ordCC)==ordCC), ...
'ordCC out of allowed range integer>=-1')
%{
\end{matlab}
For odd~\verb|ordCC| do interpolation based upon odd
neighbouring patches as is useful for staggered grids.
\begin{matlab}
%}
patches.stag = mod(ordCC,2);
assert(patches.stag==0,'staggered not yet implemented??')
ordCC = ordCC+patches.stag;
patches.ordCC = ordCC;
%{
\end{matlab}
Check for staggered grid and periodic case.
\begin{matlab}
%}
if patches.stag, assert(all(mod(nPatch,2)==0), ...
'Require an even number of patches for staggered grid')
end
%{
\end{matlab}
\paragraph{Set the macro-distribution of patches}
Third, set the centre of the patches in the macroscale grid
of patches. Loop over the coordinate directions, setting
the distribution into~\verb|Q| and finally assigning to
array of corresponding direction.
\begin{matlab}
%}
for q=1:2
qq=2*q-1;
%{
\end{matlab}
Distribution depends upon \verb|Dom.type|:
\begin{matlab}
%}
switch Dom.type(q,:)
%{
\end{matlab}
%: case periodic
The periodic case is evenly spaced within the spatial domain.
Store the size ratio in \verb|patches|.
\begin{matlab}
%}
case 'periodic'
Q=linspace(Xlim(qq),Xlim(qq+1),nPatch(q)+1);
DQ=Q(2)-Q(1);
Q=Q(1:nPatch(q))+diff(Q)/2;
pEI=patches.EdgyInt; % abbreviation
pnE=patches.nEdge(q);% abbreviation
if pre2023, dx(q) = ratio(q)*DQ/(nSubP(q)-pnE*(1+pEI))*(2-pEI);
else ratio(q) = dx(q)/DQ*(nSubP(q)-pnE*(1+pEI))/(2-pEI);
end
patches.ratio=ratio;
%{
\end{matlab}
%: case equispace
The equi-spaced case is also evenly spaced but with the
extreme edges aligned with the spatial domain boundaries,
modified by the offset.
\begin{matlab}
%}
case 'equispace'
Q=linspace(Xlim(qq)+((nSubP(q)-1)/2-Dom.bcOffset(qq))*dx(q) ...
,Xlim(qq+1)-((nSubP(q)-1)/2-Dom.bcOffset(qq+1))*dx(q) ...
,nPatch(q));
DQ=diff(Q(1:2));
width=(1+patches.EdgyInt)/2*(nSubP(q)-1-patches.EdgyInt)*dx;
if DQ<width*0.999999
warning('too many equispace patches (double overlapping)')
end
%{
\end{matlab}
%: case chebyshev
The Chebyshev case is spaced according to the Chebyshev
distribution in order to reduce macro-interpolation errors,
\(Q_i \propto -\cos(i\pi/N)\), but with the extreme edges
aligned with the spatial domain boundaries, modified by the
offset, and modified by possible `boundary layers'.
\footnote{ However, maybe overlapping patches near a
boundary should be viewed as some sort of spatially analogue
of the `christmas tree' of projective integration and its
integration to a slow manifold. Here maybe the overlapping
patches allow for a `christmas tree' approach to the
boundary layers. Needs to be explored??}
\begin{matlab}
%}
case 'chebyshev'
halfWidth=dx(q)*(nSubP(q)-1)/2;
Q1 = Xlim(1)+halfWidth-Dom.bcOffset(qq)*dx(q);
Q2 = Xlim(2)-halfWidth+Dom.bcOffset(qq+1)*dx(q);
% Q = (Q1+Q2)/2-(Q2-Q1)/2*cos(linspace(0,pi,nPatch));
%{
\end{matlab}
Search for total width of `boundary layers' so that in the
interior the patches are non-overlapping Chebyshev. But
the width for assessing overlap of patches is the following
variable \verb|width|.
\begin{matlab}
%}
pEI=patches.EdgyInt; % abbreviation
pnE=patches.nEdge(q);% abbreviation
width=(1+pEI)/2*(nSubP(q)-pnE*(1+pEI))*dx(q);
for b=0:2:nPatch(q)-2
DQmin=(Q2-Q1-b*width)/2*( 1-cos(pi/(nPatch(q)-b-1)) );
if DQmin>width, break, end
end%for
if DQmin<width*0.999999
warning('too many Chebyshev patches (mid-domain overlap)')
end
%{
\end{matlab}
Assign the centre-patch coordinates.
\begin{matlab}
%}
Q =[ Q1+(0:b/2-1)*width ...
(Q1+Q2)/2-(Q2-Q1-b*width)/2*cos(linspace(0,pi,nPatch(q)-b)) ...
Q2+(1-b/2:0)*width ];
%{
\end{matlab}
%: case usergiven
The user-given case is entirely up to a user to specify, we
just force it to have the correct shape of a row.
\begin{matlab}
%}
case 'usergiven'
if q==1, Q = reshape(Dom.X,1,[]);
else Q = reshape(Dom.Y,1,[]);
end%if
end%switch Dom.type
%{
\end{matlab}
Assign \(Q\)-coordinates to the correct spatial direction.
At this stage they are all rows.
\begin{matlab}
%}
if q==1, X=Q; end
if q==2, Y=Q; end
end%for q
%{
\end{matlab}
\paragraph{Construct the micro-grids}
Fourth, construct the microscale grid in each patch, centred
about the given mid-points~\verb|X,Y|. Reshape the grid to be
6D to suit dimensions (micro,Vars,Ens,macro).
\begin{matlab}
%}
xs = dx(1)*( (1:nSubP(1))-mean(1:nSubP(1)) );
patches.x = reshape( xs'+X ...
,nSubP(1),1,1,1,nPatch(1),1);
ys = dx(2)*( (1:nSubP(2))-mean(1:nSubP(2)) );
patches.y = reshape( ys'+Y ...
,1,nSubP(2),1,1,1,nPatch(2));
%{
\end{matlab}
\paragraph{Pre-compute weights for macro-periodic}
In the case of macro-periodicity, precompute the weightings
to interpolate field values for coupling. \todo{Might sometime
extend to coupling via derivative values.}
\begin{matlab}
%}
if patches.periodic
ratio = reshape(ratio,1,2); % force to be row vector
patches.ratio=ratio;
if ordCC>0
[Cwtsr,Cwtsl] = patchCwts(ratio,ordCC,patches.stag);
patches.Cwtsr = Cwtsr; patches.Cwtsl = Cwtsl;
end%if
end%if patches.periodic
%{
\end{matlab}
\subsection{Set ensemble inter-patch communication}
For \verb|EdgyInt| or centre interpolation respectively,
\begin{itemize}
\item the right-edge\slash centre realisations
\verb|1:nEnsem| are to interpolate to left-edge~\verb|le|,
and
\item the left-edge\slash centre realisations
\verb|1:nEnsem| are to interpolate to~\verb|re|.
\end{itemize}
\verb|re| and \verb|li| are `transposes' of each other as
\verb|re(li)=le(ri)| are both \verb|1:nEnsem|. Similarly for
bottom-edge\slash centre interpolation to top-edge
via~\verb|to|, and top-edge\slash centre interpolation to
bottom-edge via~\verb|bo|.
The default is nothing shifty. This setting reduces the
number of if-statements in function \verb|patchEdgeInt2()|.
\begin{matlab}
%}
nE = patches.nEnsem;
patches.le = 1:nE; patches.ri = 1:nE;
patches.bo = 1:nE; patches.to = 1:nE;
%{
\end{matlab}
However, if heterogeneous coefficients are supplied via
\verb|hetCoeffs|, then do some non-trivial replications.
First, get microscale periods, patch size, and replicate
many times in order to subsequently sub-sample: \verb|nSubP|
times should be enough. If \verb|cs| is more then 3D, then
the higher-dimensions are reshaped into the 3rd dimension.
\begin{matlab}
%}
if ~isempty(cs)
[mx,my,nc] = size(cs);
nx = nSubP(1); ny = nSubP(2);
cs = repmat(cs,nSubP);
%{
\end{matlab}
If only one member of the ensemble is required, then
sub-sample to patch size, and store coefficients in
\verb|patches| as is.
\begin{matlab}
%}
if nE==1, patches.cs = cs(1:nx-1,1:ny-1,:); else
%{
\end{matlab}
But for $\verb|nEnsem|>1$ an ensemble of
$m_xm_y$~phase-shifts of the coefficients is constructed
from the over-supply. Here code phase-shifts over the
periods---the phase shifts are like Hankel-matrices.
\begin{matlab}
%}
patches.nEnsem = mx*my;
patches.cs = nan(nx-1,ny-1,nc,mx,my);
for j = 1:my
js = (j:j+ny-2);
for i = 1:mx
is = (i:i+nx-2);
patches.cs(:,:,:,i,j) = cs(is,js,:);
end
end
patches.cs = reshape(patches.cs,nx-1,ny-1,nc,[]);
%{
\end{matlab}
Further, set a cunning left\slash right\slash bottom\slash
top realisation of inter-patch coupling. The aim is to
preserve symmetry in the system when also invoking
\verb|EdgyInt|. What this coupling does without
\verb|EdgyInt| is unknown. Use auto-replication.
\begin{matlab}
%}
le = mod((0:mx-1)+mod(nx-2,mx),mx)+1;
patches.le = reshape( le'+mx*(0:my-1) ,[],1);
ri = mod((0:mx-1)-mod(nx-2,mx),mx)+1;
patches.ri = reshape( ri'+mx*(0:my-1) ,[],1);
bo = mod((0:my-1)+mod(ny-2,my),my)+1;
patches.bo = reshape( (1:mx)'+mx*(bo-1) ,[],1);
to = mod((0:my-1)-mod(ny-2,my),my)+1;
patches.to = reshape( (1:mx)'+mx*(to-1) ,[],1);
%{
\end{matlab}
Issue warning if the ensemble is likely to be affected by
lack of scale separation. \todo{Maybe need to justify this
and the arbitrary threshold more carefully??}
\begin{matlab}
%}
if prod(ratio)*patches.nEnsem>0.9, warning( ...
'Probably poor scale separation in ensemble of coupled phase-shifts')
scaleSeparationParameter = ratio*patches.nEnsem
end
%{
\end{matlab}
End the two if-statements.
\begin{matlab}
%}
end%if-else nEnsem>1
end%if not-empty(cs)
%{
\end{matlab}
\paragraph{If parallel code} then first assume this is not
within an \verb|spmd|-environment, and so we invoke
\verb|spmd...end| (which starts a parallel pool if not
already started). At this point, the global \verb|patches|
is copied for each worker processor and so it becomes
\emph{composite} when we distribute any one of the fields.
Hereafter, {\em all fields in the global variable
\verb|patches| must only be referenced within an
\verb|spmd|-environment.}%
\footnote{If subsequently outside spmd, then one must use
functions like \texttt{getfield(patches\{1\},'a')}.}
\begin{matlab}
%}
if patches.parallel
% theparpool=gcp()
spmd
%{
\end{matlab}
Second, decide which dimension is to be sliced among
parallel workers (for the moment, do not consider slicing
the ensemble). Choose the direction of most patches, biased
towards the last.
\begin{matlab}
%}
[~,pari]=max(nPatch+0.01*(1:2));
patches.codist=codistributor1d(4+pari);
%{
\end{matlab}
\verb|patches.codist.Dimension| is the index that is split
among workers. Then distribute the appropriate coordinate
direction among the workers: the function must be invoked
inside an \verb|spmd|-group in order for this to work---so
we do not need \verb|parallel| in argument list.
\begin{matlab}
%}
switch pari
case 1, patches.x=codistributed(patches.x,patches.codist);
case 2, patches.y=codistributed(patches.y,patches.codist);
otherwise
error('should never have bad index for parallel distribution')
end%switch
end%spmd
%{
\end{matlab}
If not parallel, then clean out \verb|patches.codist| if it exists.
May not need, but safer.
\begin{matlab}
%}
else% not parallel
if isfield(patches,'codist'), rmfield(patches,'codist'); end
end%if-parallel
%{
\end{matlab}
\paragraph{Fin}
\begin{matlab}
%}
end% function
%{
\end{matlab}
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
SwiftHohenbergPattern.m
|
.m
|
EquationFreeGit-master/Patch/SwiftHohenbergPattern.m
| 6,925 |
utf_8
|
8c112fc501c1e2dc9975cdcca02022a3
|
% Simulate Swift--Hohenberg PDE in 1D on patches as an
% example application of patches in space with pairs of edge
% points needing to be interpolated between patches. AJR,
% 28 Mar 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{SwiftHohenbergPattern}: patterns of the
Swift--Hohenberg PDE in 1D on patches}
\label{sec:SwiftHohenbergPattern}
\localtableofcontents
\cref{fig:SwiftHohenbergPatternUxt} shows an example
simulation in time generated by the patch scheme applied to
the patterns arising from the Swift--Hohenberg \pde. That
such simulations of patterns makes valid predictions was
established by \cite{Bunder2013b} who proved that the scheme
is accurate when the number of points in a patch is just
more than a multiple of the periodicity of the pattern.
\begin{figure}
\centering \caption{\label{fig:SwiftHohenbergPatternUxt}the
pattern forming field~\(u(x,t)\) in the patch (gap-tooth)
scheme applied to a microscale discretisation of the
Swift--Hohenberg \pde\ (\cref{sec:SwiftHohenbergPattern}).
Physically we see the rapid decay of much microstructure,
but also the meso-time growth of sub-patch-scale patterns,
wavenumber~\(k_0\), that are modulated over the inter-patch
distances and over long times.}
\includegraphics[scale=0.9]{Figs/SwiftHohenbergPatternUxt}
\end{figure}%
Consider a lattice of values~\(u_i(t)\), with lattice
spacing~\(dx\), and governed by a microscale centred
discretisation of the Swift--Hohenberg \pde
\begin{equation}
\partial_tu = -(1+\partial_x^2/k_0^2)^2u+\Ra u-u^3,
\label{eq:SwiftHohenbergPattern}
\end{equation}
with boundary conditions of \(u=u_x=0\) at \(x=0,L\). For
\Ra\ just above critical, say \(\Ra=0.1\), the system
rapidly evolves to spatial quasi-periodic solutions with
period\({} \approx 0.166\) when wavenumber parameter \(k_0 =
38\). On medium times these spatial oscillations grow to
near equilibrium amplitude of~\(\sqrt{\Ra}\), and over very
long times the phases of the oscillations evolve in space to
adapt to the boundaries.
Set the desired microscale periodicity of the emergent pattern.
\begin{matlab}
%}
clear all, close all
%global OurCf2eps, OurCf2eps=true %optional to save plots
Ra = 0.1 % Ra>0 leads to patterns
nGap = 3
%waveLength = 0.496688741721854 /nGap %for nPatch==5
waveLength = 0.497630331753555 /nGap %for nPatch==7
%waveLength = 0.5 /nGap %for periodic case
nPtsPeriod = 10
dx = waveLength/nPtsPeriod
k0 = 2*pi/waveLength
%{
\end{matlab}
Establish global data struct~\verb|patches| for
the Swift--Hohenberg \pde\ on some length domain. Use
seven patches. Quartic (fourth-order) interpolation
\(\verb|ordCC|=4\) provides values for the inter-patch
coupling conditions.
\begin{matlab}
%}
nPatch = 7
nSubP = 2*nPtsPeriod+4
%nSubP = 2*nGap*nPtsPeriod+4 % full-domain
Len = nPatch;
ordCC = 4;
dom.type='equispace';
dom.bcOffset=0.5
patches = configPatches1(@SwiftHohenbergPDE,[0 Len],dom ...
,nPatch,ordCC,dx,nSubP,'EdgyInt',true,'nEdge',2);
xs=squeeze(patches.x);
%{
\end{matlab}
\begin{figure}
\centering
\caption{\label{fig:SwiftHohenbergPatternEquilib} an
equilibrium of the Swift--Hohenberg \pde\ on seven patches
in 1D~space. In the sub-patch patterns, there is a small
phase shift in the patterns from patch to patch. And the
amplitude of the pattern has to go to `zero' at the
boundaries. }
\def\extraAxisOptions{small,mark size=1pt,width=12cm,height=4cm}
\inputFigs{SwiftHohenbergPatternEquilib}
\end{figure}
\subsubsection{Find equilibrium with fsolve}
Start the search from some guess.
\begin{matlab}
%}
fprintf('\n**** Find equilibrium with fsolve\n')
u = 0.4*sin(k0*patches.x);
%{
\end{matlab}
But set the pairs of patch-edge values to \verb|Nan| in
order to use \verb|patches.i| to index the interior
sub-patch points as they are the variables.
\begin{matlab}
%}
u([1:2 end-1:end],:) = nan;
patches.i = find(~isnan(u));
%{
\end{matlab}
Seek the equilibrium, and report the norm of the residual,
via the generic patch system wrapper \verb|theRes|
(\cref{sec:theRes}).
\begin{matlab}
%}
tic
[u(patches.i),res] = fsolve(@(v) theRes(v,patches,k0,Ra) ...
,u(patches.i) ,optimoptions('fsolve','Display','off'));
solveTime = toc
normRes = norm(res)
assert(normRes<1e-6,'**** fsolve solution not accurate')
%{
\end{matlab}
\paragraph{Plot the equilibrium} see
\cref{fig:SwiftHohenbergPatternEquilib}.
\begin{matlab}
%}
figure(1),clf
subplot(2,1,1)
plot(xs,squeeze(u),'.-')
xlabel('space $x$'),ylabel('equilibrium $u(x)$')
ifOurCf2tex([mfilename 'Equilib'])%optionally save
%{
\end{matlab}
\subsubsection{Simulate in time}
Set an initial condition, and here integrate forward in time
using a standard method for stiff systems---because of the
simplicity of linear problems this method works quite
efficiently here. Integrate the interface \verb|patchSys1|
(\cref{sec:patchSys1}) to the microscale differential
equations.
\begin{matlab}
%}
fprintf('\n**** Simulate in time\n')
u0 = 0*patches.x+0.1*randn(nSubP,1,1,nPatch);
tic
[ts,us] = ode15s(@patchSys1, [0 40], u0(:) ,[],patches,k0,Ra);
simulateTime = toc
us = reshape(us,length(ts),numel(patches.x(:)),[]);
%{
\end{matlab}
Plot the simulation in \cref{fig:SwiftHohenbergPatternUxt}.
\begin{matlab}
%}
figure(2),clf
xs([1:2 end-1:end],:) = nan;
mesh(ts(1:3:end),xs(:),us(1:3:end,:)'), view(65,60)
colormap(0.7*hsv)
xlabel('time $t$'), ylabel('space $x$'), zlabel('$u(x,t)$')
ifOurCf2eps([mfilename 'Uxt'])
%{
\end{matlab}
Fin.
\subsection{The Swift--Hohenberg PDE and BCs inside patches}
As a microscale discretisation of Swift--Hohenberg \pde\
\(u_t= -(1+\partial_{x}^2/k_0^2)^2u +\Ra u -u^3\), here code
straightforward centred discretisation in space.
\begin{matlab}
%}
function ut=SwiftHohenbergPDE(t,u,patches,k0,Ra)
dx=diff(patches.x(1:2)); % microscale spacing
i=3:size(u,1)-2; % interior points in patches
%{
\end{matlab}
Code Dirichlet boundary conditions of zero function and
derivative, \(u=u_x=0\), at the left-end of the
leftmost-patch, and the right-end of the rightmost-patch.
For slightly simpler coding, squeeze out the two singleton
dimensions.
\begin{matlab}
%}
u = squeeze(u);
u(1:2,1)=0;
u(end-1:end,end)=0;
%{
\end{matlab}
Here code straightforward centred discretisation in space.
\begin{matlab}
%}
ut=nan+u; % preallocate output array
v = u(2:end-1,:)+diff(u,2)/dx^2/k0^2;
ut(i,:) = -( v(2:end-1,:)+diff(v,2)/dx^2/k0^2 ) ...
+Ra*u(i,:) -u(i,:).^3;
end
%{
\end{matlab}
\subsection{\texttt{theRes()}: wrapper function to zero for equilibria}
\label{sec:theRes}
This functions converts a vector of values into the interior
values of the patches, then evaluates the time derivative of
the system at time zero, and returns the vector of
patch-interior time derivatives.
\begin{matlab}
%}
function f=theRes(u,patches,k0,Ra)
v=nan(size(patches.x));
v(patches.i) = u;
f = patchSys1(0,v(:),patches,k0,Ra);
f = f(patches.i);
end%function theRes
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
patchEdgeInt2.m
|
.m
|
EquationFreeGit-master/Patch/patchEdgeInt2.m
| 21,894 |
utf_8
|
69c7eba8d39f1ca697e955100def73e9
|
% patchEdgeInt2() provides the interpolation across 2D space
% for 2D patches of simulations of a lattice system such as
% a PDE discretisation. AJR, Nov 2018 -- 12 Apr 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{patchEdgeInt2()}: sets 2D patch
edge values from 2D macroscale interpolation}
\label{sec:patchEdgeInt2}
Couples 2D patches across 2D space by computing their edge
values via macroscale interpolation. Research
\cite[]{Roberts2011a, Bunder2019d} indicates the patch
centre-values are sensible macroscale variables, and
macroscale interpolation of these determine patch-edge
values. However, for computational homogenisation in
multi-D, interpolating patch next-to-edge values appears
better \cite[]{Bunder2020a}. This function is primarily
used by \verb|patchSys2()| but is also useful for user
graphics. \footnote{Script \texttt{patchEdgeInt2test.m}
verifies this code.}
Communicate patch-design variables via a second argument
(optional, except required for parallel computing of
\verb|spmd|), or otherwise via the global
struct~\verb|patches|.
\begin{matlab}
%}
function u = patchEdgeInt2(u,patches)
if nargin<2, global patches, end
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|u| is a vector\slash array of length
$\verb|prod(nSubP)| \cdot \verb|nVars| \cdot \verb|nEnsem|
\cdot \verb|prod(nPatch)|$ where there are $\verb|nVars|
\cdot \verb|nEnsem|$ field values at each of the points in
the $\verb|nSubP1| \cdot \verb|nSubP2| \cdot \verb|nPatch1|
\cdot \verb|nPatch2|$ multiscale spatial grid on the
$\verb|nPatch1| \cdot \verb|nPatch2|$ array of patches.
\item \verb|patches| a struct set by \verb|configPatches2()|
which includes the following information.
\begin{itemize}
\item \verb|.x| is $\verb|nSubP1| \times1 \times1 \times1
\times \verb|nPatch1| \times1 $ array of the spatial
locations~$x_{iI}$ of the microscale grid points in every
patch. Currently it \emph{must} be an equi-spaced lattice on
the microscale index~$i$, but may be variable spaced in
macroscale index~$I$.
\item \verb|.y| is similarly $1 \times \verb|nSubP2| \times1
\times1 \times1 \times \verb|nPatch2|$ array of the spatial
locations~$y_{jJ}$ of the microscale grid points in every
patch. Currently it \emph{must} be an equi-spaced lattice on
the microscale index~$j$, but may be variable spaced in
macroscale index~$J$.
\item \verb|.ordCC| is order of interpolation, currently
only $\{0,2,4,\ldots\}$
\item \verb|.periodic| indicates whether macroscale is
periodic domain, or alternatively that the macroscale has
left, right, top and bottom boundaries so interpolation is
via divided differences.
\item \verb|.stag| in $\{0,1\}$ is one for staggered grid
(alternating) interpolation. Currently must be zero.
\item \verb|.Cwtsr| and \verb|.Cwtsl| are the coupling
coefficients for finite width interpolation in both the
$x,y$-directions---when invoking a periodic domain.
\item \verb|.EdgyInt|, true/false, for determining
patch-edge values by interpolation: true, from opposite-edge
next-to-edge values (often preserves symmetry); false, from
centre cross-patch values (near original scheme).
\item \verb|.nEdge|, two elements, the width of edge
values set by interpolation at the \(x,y\)-edge regions,
respectively, of each patch (default is one for both
\(x,y\)-edges).
\item \verb|.nEnsem| the number of realisations in the
ensemble.
\item \verb|.parallel| whether serial or parallel.
\end{itemize}
\end{itemize}
\paragraph{Output}
\begin{itemize}
\item \verb|u| is 6D array, $\verb|nSubP1| \cdot
\verb|nSubP2| \cdot \verb|nVars| \cdot \verb|nEnsem| \cdot
\verb|nPatch1| \cdot \verb|nPatch2|$, of the fields with
edge values set by interpolation.
\end{itemize}
\begin{devMan}
Test for reality of the field values, and define a function
accordingly. Could be problematic if some variables are
real and some are complex, or if variables are of quite
different sizes.
\begin{matlab}
%}
if max(abs(imag(u(:))))<1e-9*max(abs(u(:)))
uclean=@(u) real(u);
else uclean=@(u) u;
end
%{
\end{matlab}
Determine the sizes of things. Any error arising in the
reshape indicates~\verb|u| has the wrong size.
\begin{matlab}
%}
[~,ny,~,~,~,Ny] = size(patches.y);
[nx,~,~,~,Nx,~] = size(patches.x);
nEnsem = patches.nEnsem;
nVars = round(numel(u)/numel(patches.x)/numel(patches.y)/nEnsem);
assert(numel(u) == nx*ny*Nx*Ny*nVars*nEnsem ...
,'patchEdgeInt2: input u has wrong size for parameters')
u = reshape(u,[nx ny nVars nEnsem Nx Ny ]);
%{
\end{matlab}
For the moment assume the physical domain is either
macroscale periodic or macroscale rectangle so that the
coupling formulas are simplest. These index vectors point
to patches and, if periodic, their four immediate neighbours.
\begin{matlab}
%}
I=1:Nx; Ip=mod(I,Nx)+1; Im=mod(I-2,Nx)+1;
J=1:Ny; Jp=mod(J,Ny)+1; Jm=mod(J-2,Ny)+1;
%{
\end{matlab}
\paragraph{Implement multiple width edges by folding}
Subsample~\(x,y\) coordinates, noting it is only differences
that count \emph{and} the microgrid~\(x,y\) spacing must be
uniform.
\begin{matlab}
%}
%x = patches.x;
%if patches.nEdge(1)>1
% m = patches.nEdge(1);
% x = x(1:m:nx,:,:,:,:,:);
% nx = nx/m;
% u = reshape(u,m,nx,ny,nVars,nEnsem,Nx,Ny);
% nVars = nVars*m;
% u = reshape( permute(u,[2:3 1 4:7]) ...
% ,nx,ny,nVars,nEnsem,Nx,Ny);
%end%if patches.nEdge(1)
%y = patches.y;
%if patches.nEdge(2)>1
% m = patches.nEdge(2);
% y = y(:,1:m:ny,:,:,:,:);
% ny = ny/m;
% u = reshape(u,nx,m,ny,nVars,nEnsem,Nx,Ny);
% nVars = nVars*m;
% u = reshape( permute(u,[1 3 2 4:7]) ...
% ,nx,ny,nVars,nEnsem,Nx,Ny);
%end%if patches.nEdge(2)
x = patches.x;
y = patches.y;
if mean(patches.nEdge)>1
mx = patches.nEdge(1);
my = patches.nEdge(2);
x = x(1:mx:nx,:,:,:,:,:);
y = y(:,1:my:ny,:,:,:,:);
nx = nx/mx;
ny = ny/my;
u = reshape(u,mx,nx,my,ny,nVars,nEnsem,Nx,Ny);
nVars = nVars*mx*my;
u = reshape( permute(u,[2 4 1 3 5:8]) ...
,nx,ny,nVars,nEnsem,Nx,Ny);
end%if patches.nEdge
%{
\end{matlab}
The centre of each patch (as \verb|nx| and~\verb|ny| are
odd for centre-patch interpolation) is at indices
\begin{matlab}
%}
i0 = round((nx+1)/2);
j0 = round((ny+1)/2);
%{
\end{matlab}
\subsection{Periodic macroscale interpolation schemes}
\begin{matlab}
%}
if patches.periodic
%{
\end{matlab}
Get the size ratios of the patches.
\begin{matlab}
%}
rx = patches.ratio(1);
ry = patches.ratio(2);
%{
\end{matlab}
\subsubsection{Lagrange interpolation gives patch-edge values}
Compute centred differences of the mid-patch values for the
macro-interpolation, of all fields. Here the domain is
macro-periodic.
\begin{matlab}
%}
ordCC = patches.ordCC;
if ordCC>0 % then finite-width polynomial interpolation
%{
\end{matlab}
Interpolate the three directions in succession, in this way
we naturally fill-in corner values. Start with
\(x\)-direction, and give most documentation for that case
as the \(y\)-direction is essentially the same.
\paragraph{\(x\)-normal edge values} The patch-edge values
are either interpolated from the next-to-edge values, or
from the centre-cross values (not the patch-centre value
itself as that seems to have worse properties in general).
Have not yet implemented core averages.
\begin{matlab}
%}
if patches.EdgyInt % interpolate next-to-face values
U = u([2 nx-1],2:(ny-1),:,:,I,J);
else % interpolate centre-cross values
U = u(i0,2:(ny-1),:,:,I,J);
end;%if patches.EdgyInt
%{
\end{matlab}
Just in case any last array dimension(s) are one, we force a
padding of the sizes, then adjoin the extra dimension for
the subsequent array of differences.
\begin{matlab}
%}
szUO=size(U); szUO=[szUO ones(1,6-length(szUO)) ordCC];
%{
\end{matlab}
Use finite difference formulas for the interpolation, so
store finite differences ($\mu\delta, \delta^2, \mu\delta^3,
\delta^4, \ldots$) in these arrays. When parallel, in order
to preserve the distributed array structure we use an index
at the end for the differences.
\begin{matlab}
%}
if ~patches.parallel, dmu = zeros(szUO); % 7D
else dmu = zeros(szUO,patches.codist); % 7D
end%if patches.parallel
%{
\end{matlab}
First compute differences $\mu\delta$ and $\delta^2$.
\begin{matlab}
%}
if patches.stag % use only odd numbered neighbours
error('polynomial interpolation not yet for staggered patch coupling')
% dmux(:,:,:,:,I,:,1) = (Ux(:,:,:,:,Ip,:)+Ux(:,:,:,:,Im,:))/2; % \mu
% dmux(:,:,:,:,I,:,2) = (Ux(:,:,:,:,Ip,:)-Ux(:,:,:,:,Im,:)); % \delta
% Ip = Ip(Ip); Im = Im(Im); % increase shifts to \pm2
% dmuy(:,:,:,:,:,J,1) = (Ux(:,:,:,:,:,Jp)+Ux(:,:,:,:,:,Jm))/2; % \mu
% dmuy(:,:,:,:,:,J,2) = (Ux(:,:,:,:,:,Jp)-Ux(:,:,:,:,:,Jm)); % \delta
% Jp = Jp(Jp); Jm = Jm(Jm); % increase shifts to \pm2
else %disp('starting standard interpolation')
dmu(:,:,:,:,I,:,1) = (U(:,:,:,:,Ip,:) ...
-U(:,:,:,:,Im,:))/2; %\mu\delta
dmu(:,:,:,:,I,:,2) = (U(:,:,:,:,Ip,:) ...
-2*U(:,:,:,:,I,:) +U(:,:,:,:,Im,:)); %\delta^2
end% if patches.stag
%{
\end{matlab}
Recursively take $\delta^2$ of these to form successively
higher order centred differences in space.
\begin{matlab}
%}
for k = 3:ordCC
dmu(:,:,:,:,I,:,k) = dmu(:,:,:,:,Ip,:,k-2) ...
-2*dmu(:,:,:,:,I,:,k-2) +dmu(:,:,:,:,Im,:,k-2);
end
%{
\end{matlab}
Interpolate macro-values to be Dirichlet edge values for
each patch \cite[]{Roberts06d, Bunder2013b}, using weights
computed in \verb|configPatches2()|. Here interpolate to
specified order.
For the case where next-to-edge values interpolate to the
opposite edge-values: when we have an ensemble of
configurations, different configurations might be coupled to
each other, as specified by \verb|patches.le|,
\verb|patches.ri|, \verb|patches.to| and \verb|patches.bo|.
\begin{matlab}
%}
k=1+patches.EdgyInt; % use centre or two edges
u(nx,2:(ny-1),:,patches.ri,I,:) ...
= U(1,:,:,:,:,:)*(1-patches.stag) ...
+sum( shiftdim(patches.Cwtsr(:,1),-6).*dmu(1,:,:,:,:,:,:) ,7);
u(1 ,2:(ny-1),:,patches.le,I,:,:) ...
= U(k,:,:,:,:,:)*(1-patches.stag) ...
+sum( shiftdim(patches.Cwtsl(:,1),-6).*dmu(k,:,:,:,:,:,:) ,7);
%{
\end{matlab}
\paragraph{\(y\)-normal edge values} Interpolate from either
the next-to-edge values, or the centre-cross-line values.
\begin{matlab}
%}
if patches.EdgyInt % interpolate next-to-face values
U = u(:,[2 ny-1],:,:,I,J);
else % interpolate centre-cross values
U = u(:,j0,:,:,I,J);
end;%if patches.EdgyInt
%{
\end{matlab}
Adjoin extra dimension for the array of differences.
\begin{matlab}
%}
szUO=size(U); szUO=[szUO ones(1,6-length(szUO)) ordCC];
%{
\end{matlab}
Store finite differences ($\mu\delta, \delta^2, \mu\delta^3,
\delta^4, \ldots$) in this array.
\begin{matlab}
%}
if ~patches.parallel, dmu = zeros(szUO); % 7D
else dmu = zeros(szUO,patches.codist); % 7D
end%if patches.parallel
%{
\end{matlab}
First compute differences $\mu\delta$ and $\delta^2$.
\begin{matlab}
%}
if patches.stag % use only odd numbered neighbours
error('polynomial interpolation not yet for staggered patch coupling')
else %disp('starting standard interpolation')
dmu(:,:,:,:,:,J,1) = (U(:,:,:,:,:,Jp) ...
-U(:,:,:,:,:,Jm))/2; %\mu\delta
dmu(:,:,:,:,:,J,2) = (U(:,:,:,:,:,Jp) ...
-2*U(:,:,:,:,:,J) +U(:,:,:,:,:,Jm)); %\delta^2
end% if stag
%{
\end{matlab}
Recursively take $\delta^2$.
\begin{matlab}
%}
for k = 3:ordCC
dmu(:,:,:,:,:,J,k) = dmu(:,:,:,:,:,Jp,k-2) ...
-2*dmu(:,:,:,:,:,J,k-2) +dmu(:,:,:,:,:,Jm,k-2);
end
%{
\end{matlab}
Interpolate macro-values using the weights pre-computed by
\verb|configPatches2()|. An ensemble of configurations may
have cross-coupling.
\begin{matlab}
%}
k = 1+patches.EdgyInt; % use centre or two edges
u(:,ny,:,patches.to,:,J) ...
= U(:,1,:,:,:,:)*(1-patches.stag) ...
+sum( shiftdim(patches.Cwtsr(:,2),-6).*dmu(:,1,:,:,:,:,:) ,7);
u(:,1 ,:,patches.bo,:,J) ...
= U(:,k,:,:,:,:)*(1-patches.stag) ...
+sum( shiftdim(patches.Cwtsl(:,2),-6).*dmu(:,k,:,:,:,:,:) ,7);
%{
\end{matlab}
\subsubsection{Case of spectral interpolation}
Assumes the domain is macro-periodic.
\begin{matlab}
%}
else% patches.ordCC<=0, spectral interpolation
%{
\end{matlab}
We interpolate in terms of the patch index, $j$~say, not
directly in space. As the macroscale fields are $N$-periodic
in the patch index~$I$, the macroscale Fourier transform
writes the centre-patch values as $U_I=\sum_{k}C_ke^{ik2\pi
I/N}$. Then the edge-patch values $U_{I\pm r}
=\sum_{k}C_ke^{ik2\pi/N(I\pm r)} =\sum_{k}C'_ke^{ik2\pi
I/N}$ where $C'_k=C_ke^{ikr2\pi/N}$. For $N$~patches we
resolve `wavenumbers' $|k|<N/2$, so set row vector
$\verb|ks|=k2\pi/N$ for `wavenumbers' $\mathcode`\,="213B
k=(0,1, \ldots, k_{\max}, -k_{\max}, \ldots, -1)$ for
odd~$N$, and $\mathcode`\,="213B k=(0,1, \ldots, k_{\max},
\pm(k_{\max}+1) -k_{\max}, \ldots, -1)$ for even~$N$.
Deal with staggered grid by doubling the number of fields
and halving the number of patches (\verb|configPatches2|
tests there are an even number of patches). Then the
patch-ratio is effectively halved. The patch edges are near
the middle of the gaps and swapped.
\begin{matlab}
%}
if patches.stag % transform by doubling the number of fields
error('staggered grid not yet implemented??')
v=nan(size(u)); % currently to restore the shape of u
u=cat(3,u(:,1:2:nPatch,:),u(:,2:2:nPatch,:));
stagShift=reshape(0.5*[ones(nVars,1);-ones(nVars,1)],1,1,[]);
iV=[nVars+1:2*nVars 1:nVars]; % scatter interp to alternate field
r=r/2; % ratio effectively halved
nPatch=nPatch/2; % halve the number of patches
nVars=nVars*2; % double the number of fields
else % the values for standard spectral
stagShift = 0;
iV = 1:nVars;
end%if patches.stag
%{
\end{matlab}
Interpolate the two directions in succession, in this way we
naturally fill-in edge-corner values. Start with
\(x\)-direction, and give most documentation for that case
as the other is essentially the same. Need these indices of
patch interior.
\begin{matlab}
%}
ix = 2:nx-1; iy = 2:ny-1;
%{
\end{matlab}
\paragraph{\(x\)-normal edge values} Now set wavenumbers
into a vector at the correct dimension. In the case of
even~$N$ these compute the $+$-case for the highest
wavenumber zig-zag mode, $\mathcode`\,="213B k=(0,1, \ldots,
k_{\max}, +(k_{\max}+1) -k_{\max}, \ldots, -1)$.
\begin{matlab}
%}
kMax = floor((Nx-1)/2);
kr = shiftdim( rx*2*pi/Nx*(mod((0:Nx-1)+kMax,Nx)-kMax) ,-3);
%{
\end{matlab}
Compute the Fourier transform of the centre-cross values.
Unless doing patch-edgy interpolation when FT the
next-to-edge values. If there are an even number of points,
then if complex, treat as positive wavenumber, but if real,
treat as cosine. When using an ensemble of configurations,
different configurations might be coupled to each other, as
specified by \verb|patches.le|, \verb|patches.ri|,
\verb|patches.to| and \verb|patches.bo|.
\begin{matlab}
%}
if ~patches.EdgyInt
Cm = fft( u(i0,iy,:,:,:,:) ,[],5);
Cp = Cm;
else
Cm = fft( u( 2,iy ,:,patches.le,:,:) ,[],5);
Cp = fft( u(nx-1,iy ,:,patches.ri,:,:) ,[],5);
end%if ~patches.EdgyInt
%{
\end{matlab}
Now invert the Fourier transforms to complete interpolation.
Enforce reality when appropriate.
\begin{matlab}
%}
u(nx,iy,:,:,:,:) = uclean( ifft( ...
Cm.*exp(1i*(stagShift+kr)) ,[],5) );
u( 1,iy,:,:,:,:) = uclean( ifft( ...
Cp.*exp(1i*(stagShift-kr)) ,[],5) );
%{
\end{matlab}
\paragraph{\(y\)-normal edge values} Set wavenumbers into a
vector.
\begin{matlab}
%}
kMax = floor((Ny-1)/2);
kr = shiftdim( ry*2*pi/Ny*(mod((0:Ny-1)+kMax,Ny)-kMax) ,-4);
%{
\end{matlab}
Compute the Fourier transform of the patch values on the
centre-lines for all the fields.
\begin{matlab}
%}
if ~patches.EdgyInt
Cm = fft( u(:,j0,:,:,:,:) ,[],6);
Cp = Cm;
else
Cm = fft( u(:,2 ,:,patches.bo,:,:) ,[],6);
Cp = fft( u(:,ny-1 ,:,patches.to,:,:) ,[],6);
end%if ~patches.EdgyInt
%{
\end{matlab}
Invert the Fourier transforms to complete interpolation.
\begin{matlab}
%}
u(:,ny,:,:,:,:) = uclean( ifft( ...
Cm.*exp(1i*(stagShift+kr)) ,[],6) );
u(:, 1,:,:,:,:) = uclean( ifft( ...
Cp.*exp(1i*(stagShift-kr)) ,[],6) );
%{
\end{matlab}
\begin{matlab}
%}
end% if ordCC>0 else, so spectral
%{
\end{matlab}
\subsection{Non-periodic macroscale interpolation}
\begin{matlab}
%}
else% patches.periodic false
assert(~patches.stag, ...
'not yet implemented staggered grids for non-periodic')
%{
\end{matlab}
Determine the order of interpolation~\verb|px| and~\verb|py|
(potentially different in the different directions!), and
hence size of the (forward) divided difference tables
in~\verb|F|~(7D) for interpolating to left/right, and
top/bottom edges. Because of the product-form of the patch
grid, and because we are doing \emph{only} either edgy
interpolation or cross-patch interpolation (\emph{not} just
the centre patch value), the interpolations are all 1D
interpolations.
\begin{matlab}
%}
if patches.ordCC<1
px = Nx-1; py = Ny-1;
else px = min(patches.ordCC,Nx-1);
py = min(patches.ordCC,Ny-1);
end
ix=2:nx-1; iy=2:ny-1; % indices of edge 'interior' (ix n/a)
%{
\end{matlab}
\subsubsection{\(x\)-direction values}
Set function values in first `column' of the tables for
every variable and across ensemble. For~\verb|EdgyInt|, the
`reversal' of the next-to-edge values are because their
values are to interpolate to the opposite edge of each
patch. \todo{Have no plans to implement core averaging as
yet.}
\begin{matlab}
%}
F = nan(patches.EdgyInt+1,ny-2,nVars,nEnsem,Nx,Ny,px+1);
if patches.EdgyInt % interpolate next-to-edge values
F(:,:,:,:,:,:,1) = u([nx-1 2],iy,:,:,:,:);
X = x([nx-1 2],:,:,:,:,:);
else % interpolate mid-patch cross-patch values
F(:,:,:,:,:,:,1) = u(i0,iy,:,:,:,:);
X = x(i0,:,:,:,:,:);
end%if patches.EdgyInt
%{
\end{matlab}
\paragraph{Form tables of divided differences} Compute
tables of (forward) divided differences
\cite[e.g.,][]{DividedDifferences} for every variable, and
across ensemble, and for left/right edges. Recursively find
all divided differences.
\begin{matlab}
%}
for q = 1:px
i = 1:Nx-q;
F(:,:,:,:,i,:,q+1) ...
= (F(:,:,:,:,i+1 ,:,q)-F(:,:,:,:,i,:,q)) ...
./(X(:,:,:,:,i+q,:) -X(:,:,:,:,i,:));
end
%{
\end{matlab}
\paragraph{Interpolate with divided differences} Now
interpolate to find the edge-values on left/right edges
at~\verb|Xedge| for every interior~\verb|Y|.
\begin{matlab}
%}
Xedge = x([1 nx],:,:,:,:,:);
%{
\end{matlab}
Code Horner's recursive evaluation of the interpolation
polynomials. Indices~\verb|i| are those of the left edge of
each interpolation stencil, because the table is of forward
differences. This alternative: the case of order~\(p_x\)
and~\(p_y\) interpolation across the domain, asymmetric near
the boundaries of the rectangular domain.
\begin{matlab}
%}
i = max(1,min(1:Nx,Nx-ceil(px/2))-floor(px/2));
Uedge = F(:,:,:,:,i,:,px+1);
for q = px:-1:1
Uedge = F(:,:,:,:,i,:,q)+(Xedge-X(:,:,:,:,i+q-1,:)).*Uedge;
end
%{
\end{matlab}
Finally, insert edge values into the array of field values,
using the required ensemble shifts.
\begin{matlab}
%}
u(1 ,iy,:,patches.le,:,:) = Uedge(1,:,:,:,:,:);
u(nx,iy,:,patches.ri,:,:) = Uedge(2,:,:,:,:,:);
%{
\end{matlab}
\subsubsection{\(y\)-direction values}
Set function values in first `column' of the tables for
every variable and across ensemble.
\begin{matlab}
%}
F = nan(nx,patches.EdgyInt+1,nVars,nEnsem,Nx,Ny,py+1);
if patches.EdgyInt % interpolate next-to-edge values
F(:,:,:,:,:,:,1) = u(:,[ny-1 2],:,:,:,:);
Y = y(:,[ny-1 2],:,:,:,:);
else % interpolate mid-patch cross-patch values
F(:,:,:,:,:,:,1) = u(:,j0,:,:,:,:);
Y = y(:,j0,:,:,:,:);
end;
%{
\end{matlab}
Form tables of divided differences.
\begin{matlab}
%}
for q = 1:py
j = 1:Ny-q;
F(:,:,:,:,:,j,q+1) ...
= (F(:,:,:,:,:,j+1 ,q)-F(:,:,:,:,:,j,q)) ...
./(Y(:,:,:,:,:,j+q) -Y(:,:,:,:,:,j));
end
%{
\end{matlab}
Interpolate to find the edge-values on top/bottom
edges~\verb|Yedge| for every~\(x\).
\begin{matlab}
%}
Yedge = y(:,[1 ny],:,:,:,:);
%{
\end{matlab}
Code Horner's recursive evaluation of the interpolation
polynomials. Indices~\verb|j| are those of the bottom edge
of each interpolation stencil, because the table is of
forward differences.
\begin{matlab}
%}
j = max(1,min(1:Ny,Ny-ceil(py/2))-floor(py/2));
Uedge = F(:,:,:,:,:,j,py+1);
for q = py:-1:1
Uedge = F(:,:,:,:,:,j,q)+(Yedge-Y(:,:,:,:,:,j+q-1)).*Uedge;
end
%{
\end{matlab}
Finally, insert edge values into the array of field values,
using the required ensemble shifts.
\begin{matlab}
%}
u(:,1 ,:,patches.bo,:,:) = Uedge(:,1,:,:,:,:);
u(:,ny,:,patches.to,:,:) = Uedge(:,2,:,:,:,:);
%{
\end{matlab}
\subsubsection{Optional NaNs for safety}
We want a user to set outer edge values on the extreme
patches according to the microscale boundary conditions that
hold at the extremes of the domain. Consequently, unless testing, override
their computed interpolation values with~\verb|NaN|.
\begin{matlab}
%}
if isfield(patches,'intTest')&&patches.intTest
else % usual case
u( 1,:,:,:, 1,:) = nan;
u(nx,:,:,:,Nx,:) = nan;
u(:, 1,:,:,:, 1) = nan;
u(:,ny,:,:,:,Ny) = nan;
end%if
%{
\end{matlab}
End of the non-periodic interpolation code.
\begin{matlab}
%}
end%if patches.periodic else
%{
\end{matlab}
\paragraph{Unfold multiple edges} No need to restore~\(x,y\).
\begin{matlab}
%}
if mean(patches.nEdge)>1
nVars = nVars/(mx*my);
u = reshape( u ,nx,ny,mx,my,nVars,nEnsem,Nx,Ny);
nx = nx*mx;
ny = ny*my;
u = reshape( permute(u,[3 1 4 2 5:8]) ...
,nx,ny,nVars,nEnsem,Nx,Ny);
end%if patches.nEdge
%{
\end{matlab}
Fin, returning the 6D array of field values with
interpolated edges.
\begin{matlab}
%}
end% function patchEdgeInt2
%{
\end{matlab}
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
waveFirst.m
|
.m
|
EquationFreeGit-master/Patch/waveFirst.m
| 987 |
utf_8
|
665ff1a1681b400ba148a4f1f8e7723e
|
% Computes the time derivatives of a 1D, heterogeneous,
% first-order, wave PDE in 1D on patches. AJR, 17 Dec 2019
% -- Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{waveFirst()}: first-order wave PDE}
\label{sec:waveFirst}
This function codes a lattice, first-order, heterogeneous,
wave \pde\ inside patches. Optionally adds some viscous
dissipation. For 2D input arrays~\verb|u| and~\verb|x| (via
edge-value interpolation of \verb|patchSys1|,
\cref{sec:patchSys1}), computes the time
derivative~\cref{eq:waveEdgy1} at each point in the interior
of a patch, output in~\verb|ut|.
\begin{matlab}
%}
function ut = waveFirst(t,u,patches)
u=squeeze(u);
dx = diff(patches.x(2:3)); % space step
i = 2:size(u,1)-1; % interior points in a patch
ut = nan+u; % preallocate output array
ut(i,:) = -(patches.cs(i).*u(i+1,:) ...
-patches.cs(i-1).*u(i-1,:))/(2*dx) ...
+patches.nu*diff(u,2)/dx^2;
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroWave3.m
|
.m
|
EquationFreeGit-master/Patch/heteroWave3.m
| 1,956 |
utf_8
|
69fefa988c006f6c2501b1435e78cb1c
|
% Computes the time derivatives of heterogeneous waves
% in 3D on patches. AJR, Aug--Sep 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{heteroWave3()}: heterogeneous Waves}
\label{sec:heteroWave3}
This function codes the lattice heterogeneous waves
inside the patches. The wave \pde\ is
\begin{equation*}
u_t=v,\quad v_t=\grad(C\divv u)
\end{equation*}
for diagonal matrix~\(C\) which has microscale variations.
For 8D input arrays~\verb|u|, \verb|x|, \verb|y|,
and~\verb|z| (via edge-value interpolation of
\verb|patchSys3|, \cref{sec:patchSys3}), computes the
time derivative at each point in the interior of a patch,
output in~\verb|ut|. The three 3D array of heterogeneous
coefficients,~$c^x_{ijk}$, $c^y_{ijk}$ and~$c^z_{ijk}$, have
previously been stored in~\verb|patches.cs| (4D).
Supply patch information as a third argument (required by
parallel computation), or otherwise by a global variable.
\begin{matlab}
%}
function ut = heteroWave3(t,u,patches)
if nargin<3, global patches, end
%{
\end{matlab}
Microscale space-steps, and interior point indices.
\begin{matlab}
%}
dx = diff(patches.x(2:3)); % x micro-scale step
dy = diff(patches.y(2:3)); % y micro-scale step
dz = diff(patches.z(2:3)); % z micro-scale step
i = 2:size(u,1)-1; % x interior points in a patch
j = 2:size(u,2)-1; % y interior points in a patch
k = 2:size(u,3)-1; % z interior points in a patch
%{
\end{matlab}
Reserve storage and then assign interior patch values to the
heterogeneous diffusion time derivatives. Using \verb|nan+u|
appears quicker than \verb|nan(size(u),patches.codist)|
\begin{matlab}
%}
ut = nan+u; % preallocate output array
ut(i,j,k,1,:) = u(i,j,k,2,:);
ut(i,j,k,2,:) ...
=diff(patches.cs(:,j,k,1,:).*diff(u(:,j,k,1,:),1),1)/dx^2 ...
+diff(patches.cs(i,:,k,2,:).*diff(u(i,:,k,1,:),1,2),1,2)/dy^2 ...
+diff(patches.cs(i,j,:,3,:).*diff(u(i,j,:,1,:),1,3),1,3)/dz^2;
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
SwiftHohenbergHetero.m
|
.m
|
EquationFreeGit-master/Patch/SwiftHohenbergHetero.m
| 13,941 |
utf_8
|
b586d4ee2d187e24411e6fdf3390a7bd
|
% Simulate a heterogeneous version of Swift--Hohenberg PDE
% in 1D on patches as an example application with pairs of
% edge points needing to be interpolated between patches in
% space. AJR, 28 Mar 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{SwiftHohenbergHetero}: patterns of a
heterogeneous Swift--Hohenberg PDE in 1D on patches}
\label{sec:SwiftHohenbergHetero}
\localtableofcontents
\cref{fig:SwiftHohenbergHeteroU} shows an example simulation
in time generated by the patch scheme applied to the
patterns arising from a heterogeneous version of the
Swift--Hohenberg \pde. That such simulations of patterns
makes valid predictions was established by
\cite{Bunder2013b} who proved that the scheme is accurate
when the number of points in a patch is tied to a multiple
of the periodicity of the pattern.
\begin{figure}
\centering \caption{\label{fig:SwiftHohenbergHeteroU} the
field~\(u(x,t)\) in the patch (gap-tooth) scheme applied to
microscale heterogeneous Swift--Hohenberg \pde\
(\cref{sec:SwiftHohenbergHetero}). The heterogeneous
coefficients are approximately uniform over~\([0.9,1.1]\).
This heterogeneity has no noticeable affect on the
simulation.}
\includegraphics[scale=0.85]{r26479SwiftHohenbergHeteroUxt}
\end{figure}%
Consider a lattice of values~\(u_i(t)\), with lattice
spacing~\(dx\), arising from a microscale discretisation of
the pattern forming, heterogeneous, Swift--Hohenberg \pde
\begin{equation}
\partial_t u=-D[c_1(x)Du] +\Ra u-u^3,
\quad D:=1+\partial_x[c_2(x)\partial_x\cdot]/k_0^2,
\label{eq:SwiftHohenbergHetero}
\end{equation}
where \(c_\ell(x)\) have period~\(2\pi/k_0\).
Coefficients~\(c_\ell\) are chosen iid random, nearly
uniform, with mean near one. With mean one, the
periodicity of~\(c_\ell\) approximately matches the
periodicity of the resultant spatial pattern.
The current patch scheme coding preserves symmetry in the
case of periodic patches (for every order of interpolation).
For equispace and chebyshev options, the coupling currently
fails symmetry.
Consider the spectrum in the symmetric cases of periodic
patches (based upon only the cases \(N=5,7\)). There are
\(2N\)~small eigenvalues, separated by a gap from the rest.
In the homogeneous case, these occur as \(N\)~pairs. With
small heterogeneity, they appear to split into
\(N-1\)~pairs, and two distinct. With stronger heterogeneity
(say~\(0.5\)), they \emph{often} appear to also split into
two clusters, each of~\(N\) eigenvalues, with one
small-valued cluster, and one meso-valued cluster---curious.
Further analysis with sparse approximation of the invariant
spaces suggests the following:
\begin{itemize}
\item for homogeneous, the \(2N\)~modes are local
oscillations in each patch, with two modes each
corresponding to phase shifts of the possible oscillations;
\item for heterogeneous \begin{itemize}
\item \(N\)~eigenmodes appear to be one phase `locking' to
the heterogeneity; and
\item \(N\)~eigenmodes appear to be other phase `locking' to
the heterogeneity. Unless it is something to do with the
coupling, but then it only appears with heterogeneity.
\end{itemize}
\end{itemize}
Consider the spectrum with BCs of \(u=u_{xx}=0\) at ends.
Non-symmetric so some eigenvalues are complex! For small or
zero heterogeneity find \(2N-2\) eigenvalues are small.
Effectively, two modes in each of \(N-2\) interior patches,
and one mode each in the two end patches. With increasing
heterogeneity (say above~\(0.3\)), the gap decreases as a
couple (or some) of the small eigenvalues become larger in
magnitude.
Consider the spectrum with BCs of \(u=u_{x}=0\) at ends.
Non-symmetric so some eigenvalues are complex! For small or
zero heterogeneity find \(2N-4\) eigenvalues are small.
Effectively, two modes in each of \(N-2\) interior patches.
With increasing heterogeneity (say above~\(0.4\)), half
\((N-2)\) of the small eigenvalues become larger in
magnitude (presumably some phase `locking' to the
heterogeneity): effectively forms two clusters of modes.
Set the desired microscale periodicity of the patterns,
here~\(0.062\), and on the microscale lattice of
spacing~\(0.0062\), correspondingly choose random microscale
material coefficients. The wavenumber of this microscale
patterns is \(k_0\approx 101\).
\begin{matlab}
%}
clear all
%global OurCf2eps, OurCf2eps=true %optional to save plots
basename = ['r' num2str(floor(1e5*rem(now,1))) mfilename]
Ra = 0.1 % Ra>0 leads to patterns
nGap = 8 % controls size of gap between patches
waveLength = 0.496688741721854 /nGap %for nPatch==5
%waveLength = 0.497630331753555 /nGap %for nPatch==7
%waveLength = 0.5 /nGap %for periodic case
nPtsPeriod = 10
dx = waveLength/nPtsPeriod
k0 = 2*pi/waveLength
%{
\end{matlab}
Create some random heterogeneous coefficients.
\begin{matlab}
%}
heteroVar = 0.99*[1 1] % must be <2
cl = 1./(1-heteroVar/2+heteroVar.*rand(nPtsPeriod,2));
cRange = quantile(cl,0:0.5:1)
%{
\end{matlab}
Establish global data struct~\verb|patches| for
heterogeneous Swift--Hohenberg \pde\ with, on average, one
patch per units length. Use seven patches to start with.
Quartic (fourth-order) interpolation \(\verb|ordCC|=4\)
provides values for the inter-patch coupling conditions.
Or use as high-order as possible with \(\verb|ordCC|=0\).
\begin{matlab}
%}
nPatch = 5
nSubP = 2*nPtsPeriod+4 % +2 for not-edgyInt
%nSubP = 2*nGap*nPtsPeriod+4 % approx full-domain
Len = nPatch;
ordCC = 0;
dom.type='equispace';
dom.bcOffset=0.5
patches = configPatches1(@heteroSwiftHohenbergPDE,[0 Len],dom ...
,nPatch,ordCC,dx,nSubP,'EdgyInt',true,'nEdge',2 ...
,'hetCoeffs',cl);
xs=squeeze(patches.x);
%{
\end{matlab}
\subsubsection{Explore the Jacobian}
Finds that with periodic patches, everything is symmetric.
However, for equispace or chebyshev, the patch coupling is
not symmetric---is this to be expected?
\begin{matlab}
%}
fprintf('\n**** Explore the Jacobian\n')
u0 = 0*patches.x;
u0([1:2 end-1:end],:) = nan;
patches.i = find(~isnan(u0));
nVars = numel(patches.i)
Jac = nan(nVars);
for j=1:nVars
Jac(:,j)=theRes((1:nVars)==j,patches,k0,0,0);
end
%{
\end{matlab}
Check on the symmetry of the Jacobian
\begin{matlab}
%}
nonSymmetric = norm(Jac-Jac')
Jac(abs(Jac)<1e-12)=0;
antiJac = Jac-Jac';
antiJac(abs(antiJac)<1e-12)=0;
figure(6),clf
spy(Jac,'.'),hold on, spy(antiJac,'rx'),hold off
if nonSymmetric>5e-9, warning('failed symmetry'),
else Jac = (Jac+Jac')/2; %tweak to symmetry
end
%{
\end{matlab}
Compute eigenvalues and eigenvectors.
\begin{matlab}
%}
figure(5),clf
[evec,mEval] = eig(-Jac ,'vector');
[~,j]=sort(real(mEval));
mEval=mEval(j); evec=evec(:,j);
loglog(real(mEval),'.')
ylabel('$-\Re\lambda$')
ifOurCf2tex([basename 'Eval'])%optionally save
%{
\end{matlab}
\begin{SCfigure}
\centering
\caption{\label{fig:SwiftHohenbergHeteroEval} eigenvalues of
the patch scheme on the heterogeneous Swift--Hohenberg \pde\
(linearised). With \(N=5\) patches and \bc{}s of
\(u=u_x=0\) at \(x\in\{0,5\}\), there are \(2(N-2)=6\) small
eigenvalues, \(|\lambda|<0.001\), corresponding to six slow
modes in the interior.}
\def\extraAxisOptions{mark size=1pt}
\inputFigs{r26479SwiftHohenbergHeteroEval}
\end{SCfigure}
Explore sparse approximations of all the slowest together
(lots of iterations required), or separately of the two
clusters of the slowest (few iterations needed). First
ascertain whether one or two clusters of small eigenvalues.
\begin{matlab}
%}
logGaps=diff(log10(real(mEval)));
[~,j]=sort(-logGaps);
%someLogGaps=[logGaps(j(1:5)) j(1:5)]
if logGaps(j(2))<0.4*logGaps(j(1)), nSlow=j(1)
else nSlow=min( sort(j(1:2)) , 3*nPatch)
end
log10Gap=logGaps(nSlow)
smallEvals=-mEval(1:nSlow(end)+2)
%{
\end{matlab}
Second, make eigenvectors all real, sparsely approximate
cluster modes via an algorithm developed from
\cite{ZhenfangHu2014}, and plot.
\cref{fig:SwiftHohenbergHeteroEvec} shows that each pair of
basis vectors are phase-shifted by~\(90^\circ\).
\begin{matlab}
%}
js=find(imag(mEval)>0);
evec(:,js)=imag(evec(:,js));
evec=real(evec);
if numel(nSlow)==1, S = spcart(evec(:,1:nSlow));
else S = spcart(evec(:,1:nSlow(1)));
S = [S spcart(evec(:,nSlow(1)+1:nSlow(2))) ];
end;
figure(3),clf
vStep=ceil(max(abs(S(:)))*10+1)/10
for j=1:nSlow(end)
u0(patches.i)=S(:,j);
plot(xs,vStep*(j-1)+squeeze(u0),'.-'),hold on
end
hold off, xlabel('space $x$')
ifOurCf2tex([basename 'Evec'])%optionally save
%{
\end{matlab}
\begin{figure}
\centering
\caption{\label{fig:SwiftHohenbergHeteroEvec} sparse
approximations of the eigenvectors of the six slow modes of
\cref{fig:SwiftHohenbergHeteroEval}. Plotted are sparse
basis vectors for the invariant space spanned by the six
slow eigenvectors: each basis vector shifted vertically to
separate. Thus a fair approximation is that there are
effectively two modes for each of the \(N-2=3\) interior
patches.}
\def\extraAxisOptions{small, mark size=1pt, width=13cm, height=7cm}
\inputFigs{r26479SwiftHohenbergHeteroEvec}
\end{figure}
Reorganise the eigenvectors to maybe clarify.
\begin{matlab}
%}
[i,j]=find(abs(S)>vStep/2);
j=find([1;diff(j)]);
[i,k]=sort(i(j));
figure(4)
for p=1:2
clf,subplot(2,1,1)
for j=p:2:numel(k)
u0(patches.i)=S(:,k(j));
plot(xs,squeeze(u0),'.-'),hold on
end% for j
hold off, xlabel('space $x$')
ifOurCf2tex([basename 'Evec' num2str(p)])%optionally save
end%for p
%{
\end{matlab}
\begin{figure}
\centering
\caption{\label{fig:SwiftHohenbergHeteroEvec2} sparse basis
approximations for the invariant subspace of the six slow
modes of \cref{fig:SwiftHohenbergHeteroEval}. A replot of
\cref{fig:SwiftHohenbergHeteroEvec} but with three of the
basis vectors superimposed in each of the two panels.}
\def\extraAxisOptions{small, mark size=1pt, width=13cm, height=3cm}
\inputFigs{r26479SwiftHohenbergHeteroEvec1}
\inputFigs{r26479SwiftHohenbergHeteroEvec2}
\end{figure}
\subsubsection{Find an equilibrium with fsolve}
Start the search from some guess.
\begin{matlab}
%}
fprintf('\n**** Find equilibrium with fsolve\n')
u = 0.4*sin(2*pi/waveLength*patches.x);
%{
\end{matlab}
But set the pairs of patch-edge values to \verb|Nan| in
order to use \verb|patches.i| to index the interior
sub-patch points as they are the variables.
\begin{matlab}
%}
u([1:2 end-1:end],:) = nan;
patches.i = find(~isnan(u));
%{
\end{matlab}
Seek the equilibrium, and report the norm of the residual,
via the generic patch system wrapper \verb|theRes|
(\cref{sec:theResSWhetero}).
\begin{matlab}
%}
tic
[u(patches.i),res] = fsolve(@(v) theRes(v,patches,k0,Ra,1) ...
,u(patches.i) ,optimoptions('fsolve','Display','off'));
solveTime = toc
normRes = norm(res)
if normRes>1e-7, warning('residual large: bad equilibrium'),end
%{
\end{matlab}
\paragraph{Plot the equilibrium} see
\cref{fig:SwiftHohenbergHeteroEquilib}.
\begin{matlab}
%}
figure(1),clf
subplot(2,1,1)
plot(xs,squeeze(u),'.-')
xlabel('space $x$'),ylabel('equilibrium $u(x)$')
ifOurCf2tex([basename 'Equilib'])%optionally save
%{
\end{matlab}
\begin{figure}
\centering
\caption{\label{fig:SwiftHohenbergHeteroEquilib} an
equilibrium of the heterogeneous Swift--Hohenberg \pde\
determined by the patch scheme}
\def\extraAxisOptions{small, mark size=1pt, width=13cm, height=4cm}
\inputFigs{r26479SwiftHohenbergHeteroEquilib}
\end{figure}
\subsubsection{Simulate in time}
Set an initial condition, and here integrate forward in time
using a standard method for stiff systems---because of the
simplicity of linear problems this method works quite
efficiently here. Integrate the interface \verb|patchSys1|
(\cref{sec:patchSys1}) to the microscale differential
equations.
\begin{matlab}
%}
fprintf('\n**** Simulate in time\n')
u0 = 0*sin(2*pi/waveLength*patches.x)+0.1*randn(nSubP,1,1,nPatch);
tic
[ts,us] = ode15s(@patchSys1, [0 40], u0(:) ,[],patches,k0,Ra,1);
simulateTime = toc
us = reshape(us,length(ts),numel(patches.x(:)),[]);
%{
\end{matlab}
Plot the simulation in \cref{fig:SwiftHohenbergHeteroU}.
\begin{matlab}
%}
figure(2),clf
xs([1:2 end-1:end],:) = nan;
mesh(ts(1:3:end),xs(:),us(1:3:end,:)'), view(65,60)
colormap(0.7*hsv)
xlabel('time $t$'), ylabel('space $x$'), zlabel('$u(x,t)$')
ifOurCf2eps([basename 'Uxt'])
%{
\end{matlab}
Fin.
\subsection{Heterogeneous SwiftHohenberg PDE+BCs inside patches}
As a microscale discretisation of Swift--Hohenberg \pde\
\(u_t= -D[c_1(x)Du] +\Ra u -u^3\), where heterogeneous
operator \(D = 1 +\partial_x( c_2(x) \partial_x )/k_0^2\).
\begin{matlab}
%}
function ut=heteroSwiftHohenbergPDE(t,u,patches,k0,Ra,cubic)
dx=diff(patches.x(1:2)); % microscale spacing
i=3:size(u,1)-2; % interior points in patches
%{
\end{matlab}
Code a couple of different boundary conditions of zero
function and derivative(s) at left-end of left-patch, and
right-end of right-patch. For slightly simpler coding,
squeeze out the two singleton dimensions.
\begin{matlab}
%}
u = squeeze(u);
if ~patches.periodic
switch 1
case 1 % these are u=u_x=0
u(1:2,1)=0;
u(end-1:end,end)=0;
case 2 % these are u=u_{xx}=0
u(1:2,1) = [-u(3,1); 0];
u(end-1:end,end) = [0; -u(end-2,end)];
end% case
end%if
%{
\end{matlab}
Here code straightforward centred discretisation in space.
\begin{matlab}
%}
ut = nan+u; % preallocate output array
v = u(2:end-1,:)+diff(patches.cs(: ,2).*diff(u))/dx^2/k0^2;
v = v.*patches.cs(2:end,1);
v = v(2:end-1,:)+diff(patches.cs(2:end-1,2).*diff(v))/dx^2/k0^2;
ut(i,:) = -v +Ra*u(i,:) -cubic*u(i,:).^3;
end
%{
\end{matlab}
\subsection{\texttt{theRes()}: a wrapper function}
\label{sec:theResSWhetero}
This functions converts a vector of values into the interior
values of the patches, then evaluates the time derivative of
the system at time zero, and returns the vector of
patch-interior time derivatives.
\begin{matlab}
%}
function f=theRes(u,patches,k0,Ra,cubic)
v=nan(size(patches.x));
v(patches.i) = u;
f = patchSys1(0,v(:),patches,k0,Ra,cubic);
f = f(patches.i);
end%function theRes
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
patchSys2.m
|
.m
|
EquationFreeGit-master/Patch/patchSys2.m
| 3,739 |
utf_8
|
89dfc4cb2288733f30c09a385da6fe21
|
% patchSys2() Provides an interface to time integrators
% for the dynamics on patches in 2D coupled across space.
% The system must be a lattice system such as PDE
% discretisations. AJR, Nov 2018 -- 12 Apr 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{patchSys2()}: interface 2D space to time integrators}
\label{sec:patchSys2}
To simulate in time with 2D spatial patches we often need to
interface a users time derivative function with time
integration routines such as \verb|ode23| or~\verb|PIRK2|.
This function provides an interface. Communicate
patch-design variables (\cref{sec:configPatches2}) either
via the global struct~\verb|patches| or via an optional
third argument. \verb|patches| is required for the parallel
computing of \verb|spmd|, or if parameters are to be passed
though to the user microscale function.
\begin{matlab}
%}
function dudt = patchSys2(t,u,patches,varargin)
if nargin<3, global patches, end
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|u| is a vector\slash array of length
$\verb|prod(nSubP)| \cdot \verb|nVars| \cdot \verb|nEnsem|
\cdot \verb|prod(nPatch)|$ where there are $\verb|nVars|
\cdot \verb|nEnsem|$ field values at each of the points in
the $\verb|nSubP(1)| \times \verb|nSubP(2)| \times
\verb|nPatch(1)| \times \verb|nPatch(2)|$ grid.
\item \verb|t| is the current time to be passed to the
user's time derivative function.
\item \verb|patches| a struct set by \verb|configPatches2()|
with the following information used here.
\begin{itemize}
\item \verb|.fun| is the name of the user's function
\verb|fun(t,u,patches,...)| that computes the time
derivatives on the patchy lattice. The array~\verb|u| has
size $\verb|nSubP(1)| \times \verb|nSubP(2)| \times
\verb|nVars| \times \verb|nEsem| \times \verb|nPatch(1)|
\times \verb|nPatch(2)|$. Time derivatives must be computed
into the same sized array, although herein the patch
edge-values are overwritten by zeros.
\item \verb|.x| is $\verb|nSubP(1)| \times1 \times1 \times1
\verb|nPatch(1)| \times1$ array of the spatial
locations~$x_{i}$ of the microscale $(i,j)$-grid points
in every patch. Currently it \emph{must} be an equi-spaced
lattice on both macro- and micro-scales.
\item \verb|.y| is similarly $1 \times \verb|nSubP(2)|
\times1 \times1 \times1 \times \verb|nPatch(2)|$ array of
the spatial locations~$y_{j}$ of the microscale
$(i,j)$-grid points in every patch. Currently it
\emph{must} be an equi-spaced lattice on both macro- and
micro-scales.
\end{itemize}
\item \verb|varargin|, optional, is arbitrary list of
parameters to be passed onto the users time-derivative
function as specified in configPatches2.
\end{itemize}
\paragraph{Output}
\begin{itemize}
\item \verb|dudt| is a vector\slash array of of time
derivatives, but with patch edge-values set to zero. It is
of total length $\verb|prod(nSubP)| \cdot \verb|nVars|
\cdot \verb|nEnsem| \cdot \verb|prod(nPatch)|$ and the same
dimensions as~\verb|u|.
\end{itemize}
\begin{devMan}
Reshape the fields~\verb|u| as a 6D-array, and sets the edge
values from macroscale interpolation of centre-patch values.
\cref{sec:patchEdgeInt2} describes \verb|patchEdgeInt2()|.
\begin{matlab}
%}
sizeu = size(u);
u = patchEdgeInt2(u,patches);
%{
\end{matlab}
Ask the user function for the time derivatives computed in
the array, overwrite its edge values with the dummy value of
zero (as \verb|ode15s| chokes on NaNs), then return to the
user\slash integrator as same sized array as input.
\begin{matlab}
%}
dudt = patches.fun(t,u,patches,varargin{:});
m = patches.nEdge(1);
dudt([1:m end-m+1:end],:,:) = 0;
m = patches.nEdge(2);
dudt(:,[1:m end-m+1:end],:) = 0;
dudt = reshape(dudt,sizeu);
%{
\end{matlab}
Fin.
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
patchSmooth2.m
|
.m
|
EquationFreeGit-master/Patch/patchSmooth2.m
| 376 |
utf_8
|
f8dd4fe1b147b1fab1ccfaee915fee3e
|
% legacy interface patchSmooth2() auto-invokes new patchSys2()
function dudt=patchSmooth2(t,u,patches)
global smOOthCount
if isempty(smOOthCount), smOOthCount=1;
else smOOthCount=smOOthCount+1; end
l2=log2(smOOthCount);
if abs(l2-round(l2))<1e-9
warning('Use new patchSys2 instead of old patchSmooth2')
end
if nargin<3, global patches, end
dudt=patchSys2(t,u,patches);
|
github
|
uoa1184615/EquationFreeGit-master
|
wavePDE.m
|
.m
|
EquationFreeGit-master/Patch/wavePDE.m
| 897 |
utf_8
|
7602308553e43185da7a08cacefca52c
|
% Microscale discretisation of the 2D ideal wave PDE inside
% 2D patches in space. Used by the example wave2D.m
% AJR, 4 Apr 2019 -- Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{wavePDE()}: Example of simple wave PDE inside patches}
As a microscale discretisation of \(u_{tt}=\delsq(u)\), so
code \(\dot u_{ijkl}=v_{ijkl}\) and \(\dot v_{ijkl}
=\frac1{\delta x^2} (u_{i+1,j,k,l} -2u_{i,j,k,l}
+u_{i-1,j,k,l}) + \frac1{\delta y^2} (u_{i,j+1,k,l}
-2u_{i,j,k,l} +u_{i,j-1,k,l})\).
\begin{matlab}
%}
function uvt = wavePDE(t,uv,patches)
dx = diff(patches.x(1:2));
dy = diff(patches.y(1:2)); % microscale spacing
i = 2:size(uv,1)-1;
j = 2:size(uv,2)-1; % interior patch-points
uvt = nan+uv; % preallocate storage
uvt(i,j,1,:) = uv(i,j,2,:);
uvt(i,j,2,:) = diff(uv(:,j,1,:),2,1)/dx^2 ...
+diff(uv(i,:,1,:),2,2)/dy^2;
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
twoscaleDiffEquil2Errs.m
|
.m
|
EquationFreeGit-master/Patch/twoscaleDiffEquil2Errs.m
| 12,228 |
utf_8
|
6ac1bdbb99ab92212ecd4b99b2944c73
|
% Explore errors in the steady state of twoscale
% heterogeneous diffusion in 2D on patches as an example,
% inspired by section 5.3.1 of Freese et al., 2211.13731.
% AJR, 31 Jan 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{twoscaleDiffEquil2Errs}: errors in
equilibria of a 2D twoscale heterogeneous diffusion via
small patches}
\label{sec:twoscaleDiffEquil2Errs}
\begin{figure}
\centering\caption{\label{fig:twoscaleDiffEquil2Errsus} For
various numbers of patches as indicated on the colorbar,
plot the equilibrium of the multiscale diffusion problem of
Freese with Dirichlet zero-value boundary conditions
(\cref{sec:twoscaleDiffEquil2Errs}). We only compare
solutions only in these 25~common patches.}
\includegraphics[scale=0.8]{Figs/twoscaleDiffEquil2Errsus}
\end{figure}%
Here we find the steady state~\(u(x,y)\) to the
heterogeneous \pde\ (inspired by Freese et al.\footnote{
\protect \url{http://arxiv.org/abs/2211.13731}} \S5.3.1)
\begin{equation*}
u_t=A(x,y)\grad\grad u+f,
\end{equation*}
on domain \([-1,1]^2\) with Dirichlet BCs, for coefficient
`diffusion' matrix, varying with some microscale
period~\(\epsilon\) (here \(\epsilon\approx 0.24, 0.12,
0.06, 0.03\)), of
\begin{equation*}
A:=\begin{bmatrix} 2& a\\a & 2 \end{bmatrix}
\quad \text{with } a:=\sin(\pi x/\epsilon)\sin(\pi y/\epsilon),
\end{equation*}
and for forcing \(f:=10(x+y+\cos\pi x)\) (for which the
solution has magnitude up to one).\footnote{Freese et al.\
had forcing \(f:=(x+\cos3\pi x)y^3\), but here we want
smoother forcing so we get meaningful results in a minute or
two computation.\footnote{Except in the `usergiven' case,
for $N=65$, that is $152\,100$ unknowns, it takes an hour to
compute the Jacobian, then chokes.} For the same reason we
do not invoke their smaller \(\epsilon\approx 0.01\).}
\begin{figure}
\centering\caption{\label{fig:twoscaleDiffEquil2Errs} For
various numbers of patches as indicated on the colorbar,
plot the equilibrium of the multiscale diffusion problem of
Freese with Dirichlet zero-value boundary conditions
(\cref{sec:twoscaleDiffEquil2Errs}). We only compare
solutions only in these 25~common patches.}
\includegraphics[scale=0.8]{Figs/twoscaleDiffEquil2Errs}
\end{figure}%
Here we explore the errors for increasing number~\(N\) of
patches (in both directions). Find mean-abs errors to be the
following (for different orders of interpolation and patch
distribution):
\begin{equation*}
\begin{array}{rcccc}
N&5&9&17&33%&65
\\\hline
\text{equispace, 2nd-order} &6\E2 &3\E2 &1\E2 &3\E3
\\
\text{equispace, 4th-order} &3\E2 &8\E3 &7\E4 &7\E5
\\
\text{chebyshev, 4th-order} &1\E2 &2\E2 &6\E3 &2\E3
\\
\text{usergiven, 4th-order} &1\E2 &2\E2 &4\E3 &\text{n/a}
\\
\text{equispace, 6th-order} &3\E2 &1\E3 &1\E4 &2\E5
\\\hline
\end{array}
\end{equation*}
\paragraph{Script start} Clear, and initiate global patches.
Choose the type of patch distribution to be either
`equispace', `chebyshev', or `usergiven'. Also set order of
interpolation (fourth-order is good start).
\begin{matlab}
%}
clear all
global patches
%global OurCf2eps, OurCf2eps=true %option to save plot
switch 1
case 1, Dom.type = 'equispace'
case 2, Dom.type = 'chebyshev'
case 3, Dom.type = 'usergiven'
end% switch
ordInt = 4
%{
\end{matlab}
\paragraph{First configure the patch system} Establish the
microscale heterogeneity has micro-period \verb|mPeriod| on
the spatial lattice. Then \verb|configPatches2| replicates
the heterogeneity as needed to fill each patch.
\begin{matlab}
%}
mPeriod = 6
z = (0.5:mPeriod)'/mPeriod;
A = sin(2*pi*z).*sin(2*pi*z');
%{
\end{matlab}
To use a hierarchy of patches with \verb|nPatch| of~5, 9,
17, \ldots, we need up to \(N\)~patches plus one~\verb|dx|
to fit into the domain interval. Cater for up to some
full-domain simulation---can compute \(\verb|log2Nmax|=5\)
(\(\epsilon=0.06\)) within minutes:
\begin{matlab}
%}
log2Nmax = 4 % >2 up to 6 OKish
nPatchMax=2^log2Nmax+1
%{
\end{matlab}
Set the periodicity~\(\epsilon\), and other microscale
parameters.
\begin{matlab}
%}
nPeriodsPatch = 1 % any integer
nSubP = nPeriodsPatch*mPeriod+2 % for edgy int
epsilon = 2/(nPatchMax*nPeriodsPatch+1/mPeriod)
dx = epsilon/mPeriod
%{
\end{matlab}
\paragraph{For various numbers of patches} Choose five
patches to be the coarsest number of patches. Define
variables to store common results for the solutions from
differing patches. Assign \verb|Ps| to be the indices of
the common patches: for equispace set to the five common
patches, but for `chebyshev' the only common ones are the
three centre and boundary-adjacent patches.
\begin{matlab}
%}
us=[]; xs=[]; ys=[]; nPs=[];
for log2N=log2Nmax:-1:2
if log2N==log2Nmax
Ps=linspace(1,nPatchMax ...
,5-2*all(Dom.type=='chebyshev'))
else Ps=(Ps+1)/2
end
%{
\end{matlab}
Set the number of patches in \((-1,1)\):
\begin{matlab}
%}
nPatch = 2^log2N+1
%{
\end{matlab}
In the case of `usergiven', we set the standard Chebyshev
distribution of the patch-centres, which involves
overlapping of patches near the boundaries! (instead of the
coded chebyshev which has boundary layers of abutting
patches, and non-overlapping Chebyshev between the boundary
layers).
\begin{matlab}
%}
if all(Dom.type=='usergiven')
halfWidth = dx*(nSubP-1)/2;
X1 = -1+halfWidth; X2 = 1-halfWidth;
Dom.X = (X1+X2)/2-(X2-X1)/2*cos(linspace(0,pi,nPatch));
Dom.Y = Dom.X;
end
%{
\end{matlab}
Configure the patches:
\begin{matlab}
%}
configPatches2(@twoscaleDiffForce2,[-1 1],Dom,nPatch ...
,ordInt ,dx ,nSubP ,'EdgyInt',true ,'hetCoeffs',A );
%{
\end{matlab}
Compute the time-constant forcing, and store in struct
\verb|patches| for access by the microcode of
\cref{sec:twoscaleDiffForce2}.
\begin{matlab}
%}
if 1
patches.fu = 10*(patches.x+cos(pi*patches.x)+patches.y);
else patches.fu = 8+0*patches.x+0*patches.y;
end
%{
\end{matlab}
\paragraph{Solve for steady state} Set initial guess of
either zero or a subsample of the previous, next-finer,
solution. \verb|NaN| indicates patch-edge values.
Index~\verb|i| are the indices of patch-interior points, and
the number of unknowns is then its length.
\begin{matlab}
%}
if log2N==log2Nmax
u0 = zeros(nSubP,nSubP,1,1,nPatch,nPatch);
else u0 = u0(:,:,:,:,1:2:end,1:2:end);
end
u0([1 end],:,:) = nan; u0(:,[1 end],:) = nan;
patches.i = find(~isnan(u0));
nVariables = numel(patches.i)
%{
\end{matlab}
First try to solve via iterative solver \verb|bicgstab|, via
the generic patch system wrapper \verb|theRes|
(\cref{sec:theRes}).
\begin{matlab}
%}
tic;
maxIt = ceil(nVariables/10);
rhsb = theRes( zeros(size(patches.i)) );
[uSoln,flag] = bicgstab(@(u) rhsb-theRes(u),rhsb ...
,1e-9,maxIt,[],[],u0(patches.i));
bicgTime = toc
%{
\end{matlab}
However, the above often fails (and \verb|fsolve| sometimes
takes too long here), so then try a preconditioned version
of \verb|bicgstab|. The preconditioner is derived from the
Jacobian which is expensive to find (four minutes for
\(N=33\), one hour for $N=65$), but we do so as follows.
\begin{matlab}
%}
if flag>0, disp('**** bicg failed, trying ILU preconditioner')
disp(['Computing Jacobian: wait roughly ' ...
num2str(nPatch^4/4500,2) ' secs'])
tic
Jac=sparse(nVariables,nVariables);
for j=1:nVariables
Jac(:,j)=sparse( rhsb-theRes((1:nVariables)'==j) );
end
formJacTime=toc
%{
\end{matlab}
Compute an incomplete \(LU\)-factorization, and use it as
preconditioner to \verb|bicgstab|.
\begin{matlab}
%}
tic
[L,U] = ilu(Jac,struct('type','ilutp','droptol',1e-4));
LUfillFactor = (nnz(L)+nnz(U))/nnz(Jac)
[uSoln,flag] = bicgstab(@(u) rhsb-theRes(u),rhsb ...
,1e-9,maxIt,L,U,u0(patches.i));
precondSolveTime=toc
assert(flag==0,'preconditioner fails bicgstab. Lower droptol?')
end%if flag
%{
\end{matlab}
Store the solution into the patches, and give
magnitudes---Inf norm is max(abs()).
\begin{matlab}
%}
normResidual = norm(theRes(uSoln),Inf)
normSoln = norm(uSoln,Inf)
u0(patches.i) = uSoln;
u0 = patchEdgeInt2(u0);
u0( 1 ,:,:,:, 1 ,:)=0; % left edge of left patches
u0(end,:,:,:,end,:)=0; % right edge of right patches
u0(:, 1 ,:,:,:, 1 )=0; % bottom edge of bottom patches
u0(:,end,:,:,:,end)=0; % top edge of top patches
assert(normResidual<1e-5,'poor--bad solution found')
%{
\end{matlab}
Concatenate the solution on common patches into stores.
\begin{matlab}
%}
us=cat(5,us,squeeze(u0(:,:,:,:,Ps,Ps)));
xs=cat(3,xs,squeeze(patches.x(:,:,:,:,Ps,:)));
ys=cat(3,ys,squeeze(patches.y(:,:,:,:,:,Ps)));
nPs = [nPs;nPatch];
%{
\end{matlab}
End loop. Check micro-grids are aligned, then compute
errors compared to the full-domain solution (or the highest
resolution solution for the case of `usergiven').
\begin{matlab}
%}
end%for log2N
assert(max(abs(reshape(diff(xs,1,3),[],1)))<1e-12,'x-coord failure')
assert(max(abs(reshape(diff(ys,1,3),[],1)))<1e-12,'y-coord failure')
errs = us-us(:,:,:,:,1);
meanAbsErrs = mean(abs(reshape(errs,[],size(us,5))))
ratioErrs = meanAbsErrs(2:end)./meanAbsErrs(1:end-1)
%{
\end{matlab}
\paragraph{Plot solution in common patches} First reshape
arrays to suit 2D space surface plots, inserting nans to
separate patches.
\begin{matlab}
%}
x = xs(:,:,1); y = ys(:,:,1); u=us;
x(end+1,:)=nan; y(end+1,:)=nan;
u(end+1,:,:)=nan; u(:,end+1,:)=nan;
u = reshape(permute(u,[1 3 2 4 5]),numel(x),numel(y),[]);
%{
\end{matlab}
Plot the patch solution surfaces, with colour offset between
surfaces (best if \(u\)-field has a range of one): blues are
the full-domain solution, reds the coarsest patches.
\begin{matlab}
%}
figure(1), clf, colormap(jet)
for p=1:size(u,3)
mesh(x(:),y(:),u(:,:,p)',p+u(:,:,p)');
hold on;
end, hold off
view(60,55)
colorbar('Ticks',1:size(u,3) ...
,'TickLabels',[num2str(nPs) ['x';'x';'x'] num2str(nPs)]);
xlabel('space $x$'), ylabel('space $y$'), zlabel('$u(x,y)$')
ifOurCf2eps([mfilename 'us'])%optionally save
%{
\end{matlab}
\paragraph{Plot error surfaces} Plot the error surfaces,
with colour offset between surfaces (best if \(u\)-field has
a range of one): dark blue is the full-domain zero error,
reds the coarsest patches.
\begin{matlab}
%}
err=u(:,:,1)-u;
maxAbsErr=max(abs(err(:)));
figure(2), clf, colormap(jet)
for p=1:size(u,3)
mesh(x(:),y(:),err(:,:,p)',p+err(:,:,p)'/maxAbsErr);
hold on;
end, hold off
view(60,55)
colorbar('Ticks',1:size(u,3) ...
,'TickLabels',[num2str(nPs) ['x';'x';'x'] num2str(nPs)]);
xlabel('space $x$'), ylabel('space $y$')
zlabel('errors in $u(x,y)$')
ifOurCf2eps(mfilename)%optionally save
%{
\end{matlab}
\subsection{\texttt{twoscaleDiffForce2()}: microscale
discretisation inside patches of forced diffusion PDE}
\label{sec:twoscaleDiffForce2}
This function codes the lattice heterogeneous diffusion of
the \pde\ inside the patches. For 6D input arrays~\verb|u|,
\verb|x|, and~\verb|y|, computes the time derivative at each
point in the interior of a patch, output in~\verb|ut|.
\begin{matlab}
%}
function ut = twoscaleDiffForce2(t,u,patches)
dx = diff(patches.x(2:3)); % x space step
dy = diff(patches.y(2:3)); % y space step
i = 2:size(u,1)-1; % x interior points in a patch
j = 2:size(u,2)-1; % y interior points in a patch
ut = nan+u; % preallocate output array
%{
\end{matlab}
Set Dirichlet boundary value of zero around the square
domain.
\begin{matlab}
%}
u( 1 ,:,:,:, 1 ,:)=0; % left edge of left patches
u(end,:,:,:,end,:)=0; % right edge of right patches
u(:, 1 ,:,:,:, 1 )=0; % bottom edge of bottom patches
u(:,end,:,:,:,end)=0; % top edge of top patches
%{
\end{matlab}
Compute the time derivatives via stored forcing and
coefficients. Easier to code by conflating the last four
dimensions into the one~\verb|,:|.
\begin{matlab}
%}
ut(i,j,:) ...
= 2*diff(u(:,j,:),2,1)/dx^2 +2*diff(u(i,:,:),2,2)/dy^2 ...
+2*patches.cs(i,j).*( u(i+1,j+1,:) -u(i-1,j+1,:) ...
-u(i+1,j-1,:) +u(i-1,j-1,:) )/(4*dx*dy) ...
+patches.fu(i,j,:);
end%function twoscaleDiffForce2
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
odeOcts.m
|
.m
|
EquationFreeGit-master/Patch/odeOcts.m
| 624 |
utf_8
|
934d39f5a5f7fb20467f467b642ff4f4
|
% Provides Matlab-like front-end to Octave ODE solver. Uses
% non-stiff integrator as stiff ones are, surprisingly, far
% too slow. But cannot use lsode, and hence this function,
% recursively. Used by configPatches1.m, configPatches2.m,
% ensembleAverageExample.m, homogenisationExample.m,
% waterWaveExample.m, wave2D.m, and so on. AJR, 17 Aug 2020
%{
\begin{matlab}
%}
function [ts,xs] = odeOcts(dxdt,tSpan,x0)
if length(tSpan)>2, ts = tSpan;
else ts = linspace(tSpan(1),tSpan(end),21)';
end
lsode_options('integration method','non-stiff');
xs = lsode(@(x,t) dxdt(t,x),x0,ts);
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
patchEdgeInt1.m
|
.m
|
EquationFreeGit-master/Patch/patchEdgeInt1.m
| 16,444 |
utf_8
|
4d5890cdcc086ffde04fad57d210c2f8
|
% patchEdgeInt1() provides the interpolation across 1D space
% for 1D patches of simulations of a lattice system such as
% PDE discretisations. AJR & JB, Sep 2018 -- 23 Mar 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{patchEdgeInt1()}: sets patch-edge values
from interpolation over the 1D macroscale}
\label{sec:patchEdgeInt1}
Couples 1D patches across 1D space by computing their edge
values from macroscale interpolation of either the mid-patch
value \cite[]{Roberts00a, Roberts06d}, or the patch-core
average \cite[]{Bunder2013b}, or the opposite next-to-edge
values \cite[]{Bunder2020a}---this last alternative often
maintains symmetry. This function is primarily used by
\verb|patchSys1()| but is also useful for user graphics.
When using core averages (not fully implemented), assumes
the averages are sensible macroscale variables: then patch
edge values are determined by macroscale interpolation of
the core averages \citep{Bunder2013b}. \footnote{Script
\texttt{patchEdgeInt1test.m} verifies this code.}
Communicate patch-design variables via a second argument
(optional, except required for parallel computing of
\verb|spmd|), or otherwise via the global struct
\verb|patches|.
\begin{matlab}
%}
function u=patchEdgeInt1(u,patches)
if nargin<2, global patches, end
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|u| is a vector\slash array of length
$\verb|nSubP| \cdot \verb|nVars|\cdot \verb|nEnsem|\cdot
\verb|nPatch|$ where there are $\verb|nVars|\cdot
\verb|nEnsem|$ field values at each of the points in the
$\verb|nSubP| \times \verb|nPatch|$ multiscale spatial grid.
\item \verb|patches| a struct largely set by
\verb|configPatches1()|, and which includes the following.
\begin{itemize}
\item \verb|.x| is $\verb|nSubP| \times1 \times1 \times
\verb|nPatch|$ array of the spatial locations~$x_{iI}$ of
the microscale grid points in every patch. Currently it
\emph{must} be an equi-spaced lattice on the microscale
index~$i$, but may be variable spaced in macroscale
index~$I$.
\item \verb|.ordCC| is order of interpolation, integer~$\geq
-1$.
\item \verb|.periodic| indicates whether macroscale is
periodic domain, or alternatively that the macroscale has
left and right boundaries so interpolation is via divided
differences.
\item \verb|.stag| in $\{0,1\}$ is one for staggered grid
(alternating) interpolation, and zero for ordinary grid.
\item \verb|.Cwtsr| and \verb|.Cwtsl| are the coupling
coefficients for finite width interpolation---when invoking
a periodic domain.
\item \verb|.EdgyInt|, true/false, for determining
patch-edge values by interpolation:
true, from opposite-edge next-to-edge values (often
preserves symmetry);
false, from centre-patch values (original scheme).
\item \verb|.nEdge|, for each patch, the number of edge
values set by interpolation at the edge regions of each
patch (default is one).
\item \verb|.nEnsem| the number of realisations in the
ensemble.
\item \verb|.parallel| whether serial or parallel.
\item \verb|.nCore| \todo{introduced sometime but not fully
implemented yet, because prefer ensemble}
\end{itemize}
\end{itemize}
\paragraph{Output}
\begin{itemize}
\item \verb|u| is 4D array, $\verb|nSubP| \times
\verb|nVars| \times \verb|nEnsem| \times \verb|nPatch|$, of
the fields with edge values set by interpolation.
\end{itemize}
\begin{devMan}
Test for reality of the field values, and define a function
accordingly. Could be problematic if some variables are
real and some are complex, or if variables are of quite
different sizes.
\begin{matlab}
%}
if max(abs(imag(u(:))))<1e-9*max(abs(u(:)))
uclean=@(u) real(u);
else uclean=@(u) u;
end
%{
\end{matlab}
Determine the sizes of things. Any error arising in the
reshape indicates~\verb|u| has the wrong size.
\begin{matlab}
%}
[nx,~,~,Nx] = size(patches.x);
nEnsem = patches.nEnsem;
nVars = round(numel(u)/numel(patches.x)/nEnsem);
assert(numel(u) == nx*nVars*nEnsem*Nx ...
,'patchEdgeInt1: input u has wrong size for parameters')
u = reshape(u,nx,nVars,nEnsem,Nx);
%{
\end{matlab}
If the user has not defined the patch core, then we assume
it to be a single point in the middle of the patch, unless
we are interpolating from next-to-edge values.
These index vectors point to patches and their two
immediate neighbours, for periodic domain.
\begin{matlab}
%}
I = 1:Nx; Ip = mod(I,Nx)+1; Im = mod(I-2,Nx)+1;
%{
\end{matlab}
\paragraph{Implement multiple width edges by folding}
Subsample~\(x\) coordinates, noting it is only differences
that count \emph{and} the microgrid~\(x\) spacing must be
uniform.
\begin{matlab}
%}
x = patches.x;
if patches.nEdge>1
nEdge = patches.nEdge;
x = x(1:nEdge:nx,:,:,:);
nx = nx/nEdge;
u = reshape(u,nEdge,nx,nVars,nEnsem,Nx);
nVars = nVars*nEdge;
u = reshape( permute(u,[2 1 3:5]) ,nx,nVars,nEnsem,Nx);
end%if patches.nEdge
%{
\end{matlab}
Calculate centre of each patch and the surrounding core
(\verb|nx| and \verb|nCore| are both odd).
\begin{matlab}
%}
i0 = round((nx+1)/2);
c = round((patches.nCore-1)/2);
%{
\end{matlab}
\subsection{Periodic macroscale interpolation schemes}
\begin{matlab}
%}
if patches.periodic
%{
\end{matlab}
Get the size ratios of the patches, then use finite width
stencils or spectral.
\begin{matlab}
%}
r = patches.ratio(1);
if patches.ordCC>0 % then finite-width polynomial interpolation
%{
\end{matlab}
\paragraph{Lagrange interpolation gives patch-edge values}
Consequently, compute centred differences of the patch
core/edge averages/values for the macro-interpolation of all
fields. Here the domain is macro-periodic.
\begin{matlab}
%}
if patches.EdgyInt % interpolate next-to-edge values
Ux = u([2 nx-1],:,:,I);
else % interpolate mid-patch values/sums
Ux = sum( u((i0-c):(i0+c),:,:,I) ,1);
end;
%{
\end{matlab}
Just in case any last array dimension(s) are one, we force a
padding of the sizes, then adjoin the extra dimension for
the subsequent array of differences.
\begin{matlab}
%}
szUxO=size(Ux);
szUxO=[szUxO ones(1,4-length(szUxO)) patches.ordCC];
%{
\end{matlab}
Use finite difference formulas for the interpolation, so
store finite differences in these arrays. When parallel, in
order to preserve the distributed array structure we use an
index at the end for the differences.
\begin{matlab}
%}
if patches.parallel
dmu = zeros(szUxO,patches.codist); % 5D
else
dmu = zeros(szUxO); % 5D
end
%{
\end{matlab}
First compute differences, either $\mu$ and $\delta$, or
$\mu\delta$ and $\delta^2$ in space.
\begin{matlab}
%}
if patches.stag % use only odd numbered neighbours
dmu(:,:,:,I,1) = (Ux(:,:,:,Ip)+Ux(:,:,:,Im))/2; % \mu
dmu(:,:,:,I,2) = (Ux(:,:,:,Ip)-Ux(:,:,:,Im)); % \delta
Ip = Ip(Ip); Im = Im(Im); % increase shifts to \pm2
else % standard
dmu(:,:,:,I,1) = (Ux(:,:,:,Ip)-Ux(:,:,:,Im))/2; % \mu\delta
dmu(:,:,:,I,2) = (Ux(:,:,:,Ip)-2*Ux(:,:,:,I) ...
+Ux(:,:,:,Im)); % \delta^2
end%if patches.stag
%{
\end{matlab}
Recursively take $\delta^2$ of these to form successively
higher order centred differences in space.
\begin{matlab}
%}
for k = 3:patches.ordCC
dmu(:,:,:,:,k) = dmu(:,:,:,Ip,k-2) ...
-2*dmu(:,:,:,I,k-2) +dmu(:,:,:,Im,k-2);
end
%{
\end{matlab}
Interpolate macro-values to be Dirichlet edge values for
each patch \cite[]{Roberts06d, Bunder2013b}, using weights
computed in \verb|configPatches1()|. Here interpolate to
specified order.
For the case where single-point values interpolate to
patch-edge values: when we have an ensemble of
configurations, different realisations are coupled to each
other as specified by \verb|patches.le| and
\verb|patches.ri|.
\begin{matlab}
%}
if patches.nCore==1
k=1+patches.EdgyInt; % use centre/core or two edges
u(nx,:,patches.ri,I) = Ux(1,:,:,:)*(1-patches.stag) ...
+sum( shiftdim(patches.Cwtsr,-4).*dmu(1,:,:,:,:) ,5);
u(1 ,:,patches.le,I) = Ux(k,:,:,:)*(1-patches.stag) ...
+sum( shiftdim(patches.Cwtsl,-4).*dmu(k,:,:,:,:) ,5);
%{
\end{matlab}
For a non-trivial core then more needs doing: the core (one
or more) of each patch interpolates to the edge action
regions. When more than one in the core, the edge is set
depending upon near edge values so the average near the edge
is correct.
\begin{matlab}
%}
else% patches.nCore>1
error('not yet considered, july--dec 2020 ??')
u(nx,:,:,I) = Ux(:,:,I)*(1-patches.stag) ...
+ reshape(-sum(u((nx-patches.nCore+1):(nx-1),:,:,I),1) ...
+ sum( patches.Cwtsr.*dmu ),Nx,nVars);
u(1,:,:,I) = Ux(:,:,I)*(1-patches.stag) ...
+ reshape(-sum(u(2:patches.nCore,:,:,I),1) ...
+ sum( patches.Cwtsl.*dmu ),Nx,nVars);
end%if patches.nCore
%{
\end{matlab}
\paragraph{Case of spectral interpolation}
Assumes the domain is macro-periodic.
\begin{matlab}
%}
else% patches.ordCC<=0, spectral interpolation
%{
\end{matlab}
As the macroscale fields are $N$-periodic, the macroscale
Fourier transform writes the centre-patch values as $U_j =
\sum_{k}C_ke^{ik2\pi j/N}$. Then the edge-patch values
$U_{j\pm r} =\sum_{k}C_ke^{ik2\pi/N(j\pm r)}
=\sum_{k}C'_ke^{ik2\pi j/N}$ where $C'_k =
C_ke^{ikr2\pi/N}$. For \verb|Nx|~patches we resolve
`wavenumbers' $|k|<\verb|Nx|/2$, so set row vector
$\verb|ks| = k2\pi/N$ for `wavenumbers'
$\mathcode`\,="213B k = (0,1, \ldots, k_{\max}, -k_{\max},
\ldots, -1)$ for odd~$N$, and $\mathcode`\,="213B k =
(0,1, \ldots, k_{\max}, (k_{\max}+1), -k_{\max}, \ldots,
-1)$ for even~$N$.
Deal with staggered grid by doubling the number of fields
and halving the number of patches (\verb|configPatches1()|
tests that there are an even number of patches). Then the
patch-ratio is effectively halved. The patch edges are near
the middle of the gaps and swapped. \todo{Have not yet
tested whether works for Edgy Interpolation.} \todo{Have
not yet implemented multiple edge values for a staggered
grid as I am uncertain whether it makes any sense. }
\begin{matlab}
%}
if patches.stag % transform by doubling the number of fields
v = nan(size(u)); % currently to restore the shape of u
u = [u(:,:,:,1:2:Nx) u(:,:,:,2:2:Nx)];
stagShift = 0.5*[ones(1,nVars) -ones(1,nVars)];
iV = [nVars+1:2*nVars 1:nVars]; % scatter interp to alternate field
r = r/2; % ratio effectively halved
Nx = Nx/2; % halve the number of patches
nVars = nVars*2; % double the number of fields
else % the values for standard spectral
stagShift = 0;
iV = 1:nVars;
end%if patches.stag
%{
\end{matlab}
Now set wavenumbers (when \verb|Nx| is even then highest
wavenumber is~$\pi$).
\begin{matlab}
%}
kMax = floor((Nx-1)/2);
ks = shiftdim( ...
2*pi/Nx*(mod((0:Nx-1)+kMax,Nx)-kMax) ...
,-2);
%{
\end{matlab}
Compute the Fourier transform across patches of the patch
centre or next-to-edge values for all the fields. If there
are an even number of points, then if complex, treat as
positive wavenumber, but if real, treat as cosine. When
using an ensemble of configurations, different
configurations might be coupled to each other, as specified
by \verb|patches.le| and \verb|patches.ri|.
\begin{matlab}
%}
if ~patches.EdgyInt
Cleft = fft(u(i0 ,:,:,:),[],4);
Cright = Cleft;
else
Cleft = fft(u(2 ,:,:,:),[],4);
Cright= fft(u(nx-1,:,:,:),[],4);
end
%{
\end{matlab}
The inverse Fourier transform gives the edge values via a
shift a fraction~$r$ to the next macroscale grid point.
\begin{matlab}
%}
u(nx,iV,patches.ri,:) = uclean( ifft( ...
Cleft.*exp(1i*ks.*(stagShift+r)) ,[],4));
u(1 ,iV,patches.le,:) = uclean( ifft( ...
Cright.*exp(1i*ks.*(stagShift-r)) ,[],4));
%{
\end{matlab}
Restore staggered grid when appropriate. This dimensional
shifting appears to work.
Is there a better way to do this?
\begin{matlab}
%}
if patches.stag
nVars = nVars/2;
u=reshape(u,nx,nVars,2,nEnsem,Nx);
Nx = 2*Nx;
v(:,:,:,1:2:Nx) = u(:,:,1,:,:);
v(:,:,:,2:2:Nx) = u(:,:,2,:,:);
u = v;
end%if patches.stag
end%if patches.ordCC
%{
\end{matlab}
\subsection{Non-periodic macroscale interpolation}
\begin{matlab}
%}
else% patches.periodic false
assert(~patches.stag, ...
'not yet implemented staggered grids for non-periodic')
%{
\end{matlab}
Determine the order of interpolation~\verb|p|, and hence
size of the (forward) divided difference table in~\verb|F|.
\begin{matlab}
%}
if patches.ordCC<1, patches.ordCC = Nx-1; end
p = min(patches.ordCC,Nx-1);
F = nan(patches.EdgyInt+1,nVars,nEnsem,Nx,p+1);
%{
\end{matlab}
Set function values in first `column' of the table for every
variable and across ensemble. For~\verb|EdgyInt|, the
`reversal' of the next-to-edge values are because their
values are to interpolate to the opposite edge of each
patch.
\begin{matlab}
%}
if patches.EdgyInt % interpolate next-to-edge values
F(:,:,:,:,1) = u([nx-1 2],:,:,I);
X(:,:,:,:) = x([nx-1 2],:,:,I);
else % interpolate mid-patch values/sums
F(:,:,:,:,1) = sum( u((i0-c):(i0+c),:,:,I) ,1);
X(:,:,:,:) = x(i0,:,:,I);
end;
%{
\end{matlab}
Compute table of (forward) divided differences
\cite[e.g.,][]{DividedDifferences} for every variable and
across ensemble.
\begin{matlab}
%}
for q = 1:p
i = 1:Nx-q;
F(:,:,:,i,q+1) = (F(:,:,:,i+1,q)-F(:,:,:,i,q)) ...
./(X(:,:,:,i+q) -X(:,:,:,i));
end
%{
\end{matlab}
Now interpolate to the edge-values at locations~\verb|Xedge|.
\begin{matlab}
%}
Xedge = x([1 nx],:,:,:);
%{
\end{matlab}
Code Horner's evaluation of the interpolation polynomials.
Indices~\verb|i| are those of the left end of each
interpolation stencil because the table is of forward
differences.\footnote{For EdgyInt, perhaps interpret odd
order interpolation in such a way that first-order
interpolations reduces to appropriate linear interpolation
so that as patches abut the scheme is `full-domain'. May
mean left-edge and right-edge have different indices.
Explore sometime??} First alternative: the case of
order~\(p\) interpolation across the domain, asymmetric near
the boundary. Use this first alternative for now.
\begin{matlab}
%}
if true
i = max(1,min(1:Nx,Nx-ceil(p/2))-floor(p/2));
Uedge = F(:,:,:,i,p+1);
for q = p:-1:1
Uedge = F(:,:,:,i,q)+(Xedge-X(:,:,:,i+q-1)).*Uedge;
end
%{
\end{matlab}
Second alternative: lower the degree of interpolation near
the boundary to maintain the band-width of the
interpolation. Such symmetry might be essential for multi-D.
\footnote{The aim is to preserve symmetry?? Does it?? As of
Jan 2023 it only partially does---fails near boundaries, and
maybe fails with uneven spacing.}
\begin{matlab}
%}
else%if false
i = max(1,I-floor(p/2));
%{
\end{matlab}
For the tapering order of interpolation, form the interior
mask~\verb|Q| (logical) that signifies which interpolations
are to be done at order~\verb|q|. This logical mask spreads
by two as each order~\verb|q| decreases.
\begin{matlab}
%}
Q = (I-1>=floor(p/2)) & (Nx-I>=p/2);
Imid = floor(Nx/2);
%{
\end{matlab}
Initialise to highest divide difference, surrounded by zeros.
\begin{matlab}
%}
Uedge = zeros(patches.EdgyInt+1,nVars,nEnsem,Nx);
Uedge(:,:,:,Q) = F(:,:,:,i(Q),p+1);
%{
\end{matlab}
Complete Horner evaluation of the relevant polynomials.
\begin{matlab}
%}
for q = p:-1:1
Q = [Q(2:Imid) true(1,2) Q(Imid+1:end-1)]; % spread mask
Uedge(:,:,:,Q) = F(:,:,:,i(Q),q) ...
+(Xedge(:,:,:,Q)-X(:,:,:,i(Q)+q-1)).*Uedge(:,:,:,Q);
end%for q
end%if
%{
\end{matlab}
Finally, insert edge values into the array of field values,
using the required ensemble shifts.
\begin{matlab}
%}
u(1 ,:,patches.le,I) = Uedge(1,:,:,I);
u(nx,:,patches.ri,I) = Uedge(2,:,:,I);
%{
\end{matlab}
We want a user to set the extreme patch edge values
according to the microscale boundary conditions that hold at
the extremes of the domain. Consequently, unless testing,
override their computed interpolation values
with~\verb|NaN|.
\begin{matlab}
%}
if isfield(patches,'intTest')&&patches.intTest
else % usual case
u( 1,:,:, 1) = nan;
u(nx,:,:,Nx) = nan;
end%if
%{
\end{matlab}
End of the non-periodic interpolation code.
\begin{matlab}
%}
end%if patches.periodic
%{
\end{matlab}
\paragraph{Unfold multiple edges} No need to restore~\(x\).
\begin{matlab}
%}
if patches.nEdge>1
nVars = nVars/nEdge;
u = reshape( u ,nx,nEdge,nVars,nEnsem,Nx);
nx = nx*nEdge;
u = reshape( permute(u,[2 1 3:5]) ,nx,nVars,nEnsem,Nx);
end%if patches.nEdge
%{
\end{matlab}
Fin, returning the 4D array of field values.
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
BurgersExample.m
|
.m
|
EquationFreeGit-master/Patch/BurgersExample.m
| 5,063 |
utf_8
|
5e4cae9564c08774f1a2d207d37608b7
|
% Simulate a microscale space-time map of Burgers' PDE
% discretised. Simulate on spatial patches, and via
% projective integration.
% AJR, Nov 2017 -- Jul 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{BurgersExample}: simulate Burgers' PDE on patches}
\label{sec:BurgersExample}
\localtableofcontents
\cref{fig:config1Burgers} shows a previous example
simulation in time generated by the patch scheme applied to
Burgers' \pde. The code in the example of this section
similarly applies the patch scheme to a microscale
space-time map (\cref{fig:BurgersMapU}), a map derived as a
microscale space-time discretisation of Burgers'~\pde. Then
this example applies projective integration to simulate
further in time.
\begin{figure}
\centering \caption{\label{fig:BurgersMapU}a short time
simulation of the Burgers' map (\cref{sec:burgersMap}) on
patches in space. It requires many very small time-steps
only just visible in this mesh.}
\includegraphics[scale=0.9]{BurgersExampleMapU}
\end{figure}%
\subsection{Script code to simulate a microscale space-time map}
\label{sec:bescsmsts}
This first part of the script implements the following
patch\slash gap-tooth scheme (left-right arrows denote
function recursion).
\begin{enumerate}\def\itemsep{-1.5ex}
\item configPatches1
\item burgerBurst \into patchSys1 \into burgersMap
\item process results
\end{enumerate}
Establish global data struct for the microscale Burgers' map
(\cref{sec:burgersMap}) solved on \(2\pi\)-periodic domain,
with eight patches, each patch of half-size ratio~\(0.2\),
with seven points within each patch, and say fourth-order
interpolation provides edge-values that couple the patches.
\begin{matlab}
%}
global patches
nPatch = 8
ratio = 0.2
nSubP = 7
interpOrd = 4
Len = 2*pi
configPatches1(@burgersMap,[0 Len],nan,nPatch,interpOrd,ratio,nSubP);
%{
\end{matlab}
Set an initial condition, and simulate a burst of the
microscale space-time map over a time~\(0.2\) using the
function \verb|burgerBurst()| (\cref{sec:burgerBurst}).
\begin{matlab}
%}
u0 = 0.4*(1+sin(patches.x))+0.1*randn(size(patches.x));
[ts,us] = burgersBurst(0,u0,0.4);
%{
\end{matlab}
Plot the simulation. Use only the microscale values interior
to the patches by setting the edges to \verb|nan| in order
to leave gaps.
\begin{matlab}
%}
figure(1),clf
xs = patches.x; xs([1 end],:) = nan;
mesh(ts,xs(:),us')
xlabel('time $t$'), ylabel('space $x$'), zlabel('$u(x,t)$')
view(105,45)
%{
\end{matlab}
Save the plot to file to form \cref{fig:BurgersMapU}.
\begin{matlab}
%}
ifOurCf2eps([mfilename 'MapU'])
%{
\end{matlab}
\subsection{Alternatively use projective integration}
\begin{figure}
\centering \caption{\label{fig:BurgersU}macroscale
space-time field \(u(x,t)\) in a basic projective
integration of the patch scheme applied to the microscale
Burgers' map.}
\includegraphics[scale=0.9]{BurgersExampleU}
\end{figure}%
Around the microscale burst \verb|burgerBurst()|, wrap the
projective integration function \verb|PIRK2()| of
\cref{sec:PIRK2}. \cref{fig:BurgersU} shows the resultant
macroscale prediction of the patch centre values on
macroscale time-steps.
This second part of the script implements the following design.
\begin{enumerate} \def\itemsep{-1.5ex}
\item configPatches1 (done in \cref{sec:bescsmsts})
\item PIRK2 \into burgerBurst \into patchSys1 \into burgersMap
\item process results
\end{enumerate}
Mark that edge-values of patches are not to be used in the
projective extrapolation by setting initial values to \nan.
\begin{matlab}
%}
u0([1 end],:) = nan;
%{
\end{matlab}
Set the desired macroscale time-steps, and microscale
burst length over the time domain. Then projectively
integrate in time using \verb|PIRK2()| which is
second-order accurate in the macroscale time-step.
\begin{matlab}
%}
ts = linspace(0,0.5,11);
bT = 3*(ratio*Len/nPatch/(nSubP/2-1))^2
addpath('../ProjInt')
[us,tss,uss] = PIRK2(@burgersBurst,ts,u0(:),bT);
%{
\end{matlab}
Plot and save the macroscale predictions of the mid-patch
values to give the macroscale mesh-surface of
\cref{fig:BurgersU} that shows a progressing wave solution.
\begin{matlab}
%}
figure(2),clf
midP = (nSubP+1)/2;
mesh(ts,xs(midP,:),us(:,midP:nSubP:end)')
xlabel('time $t$'), ylabel('space $x$'), zlabel('$u(x,t)$')
view(120,50)
ifOurCf2eps([mfilename 'U'])
%{
\end{matlab}
Then plot and save the microscale mesh of the microscale
bursts shown in \cref{fig:BurgersMicro} (a stereo pair). The
details of the fine microscale mesh are almost invisible.
\begin{figure}
\centering \caption{\label{fig:BurgersMicro}the microscale
field \(u(x,t)\) during each of the microscale bursts used
in the projective integration. View this stereo pair
cross-eyed.}
\includegraphics[scale=0.85]{BurgersExampleMicro}
\end{figure}
\begin{matlab}
%}
figure(3),clf
for k = 1:2, subplot(2,2,k)
mesh(tss,xs(:),uss')
ylabel('space $x$'),xlabel('time $t$'),zlabel('$u(x,t)$')
axis tight, view(126-4*k,50)
end
ifOurCf2eps([mfilename 'Micro'])
%{
\end{matlab}
\input{../Patch/burgersMap.m}
\input{../Patch/burgersBurst.m}
Fin.
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
abdulleDiffEquil2.m
|
.m
|
EquationFreeGit-master/Patch/abdulleDiffEquil2.m
| 6,062 |
utf_8
|
010012d53be7d85d7b5cd2a2da563cb2
|
% Solve for steady state of multiscale heterogeneous diffusion
% in 2D on patches as an example application, varied from
% example of section 5.1 of Abdulle et al., (2020b). AJR,
% 31 Jan 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{abdulleDiffEquil2}: equilibrium of a 2D
multiscale heterogeneous diffusion via small patches}
\label{sec:abdulleDiffEquil2}
Here we find the steady state~\(u(x,y)\) to the
heterogeneous \pde\ \cite[inspired by][\S5.1]{Abdulle2020b}
\begin{equation*}
u_t=\divv[a(x,y)\grad u]+10,
\end{equation*}
on square domain \([0,1]^2\) with zero-Dirichlet BCs, for
coefficient `diffusion' matrix, varying with
period~\(\epsilon\) of (their~(45))
\begin{equation*}
a:=\frac{2+1.8\sin2\pi x/\epsilon}{2+1.8\cos2\pi y/\epsilon}
+\frac{2+\sin2\pi y/\epsilon}{2+1.8\cos2\pi x/\epsilon}.
\end{equation*}
\cref{fig:abdulleDiffEquil2} shows solutions have some
nice microscale wiggles reflecting the heterogeneity.
\begin{figure}
\centering\caption{\label{fig:abdulleDiffEquil2}%
Equilibrium of the macroscale diffusion problem of Abdulle
with boundary conditions of Dirichlet zero-value except for
\(x=0\) which is Neumann (\cref{sec:abdulleDiffEquil2}).
Here the patches have a Chebyshev-like spatial distribution.
The patch size is chosen large enough to see within.}
\includegraphics[scale=0.8]{Figs/abdulleDiffEquil2}
\end{figure}
Clear, and initiate globals.
\begin{matlab}
%}
clear all
global patches
%global OurCf2eps, OurCf2eps=true %option to save plot
%{
\end{matlab}
First establish the microscale heterogeneity has
micro-period~\verb|mPeriod| on the spatial micro-grid
lattice. Then \verb|configPatches2| replicates the
heterogeneity to fill each patch. (These diffusion
coefficients should really recognise the half-grid-point
shifts, but let's not bother.)
\begin{matlab}
%}
mPeriod = 6
x = (0.5:mPeriod)'/mPeriod; y=x';
a = (2+1.8*sin(2*pi*x))./(2+1.8*sin(2*pi*y)) ...
+(2+ sin(2*pi*y))./(2+1.8*sin(2*pi*x));
diffusivityRange = [min(a(:)) max(a(:))]
%{
\end{matlab}
Set the periodicity~\(\epsilon\), here big enough so we can
see the patches, and other microscale parameters.
\begin{matlab}
%}
epsilon = 0.04
dx = epsilon/mPeriod
nPeriodsPatch = 1 % any integer
nSubP = nPeriodsPatch*mPeriod+2 % when edgy int
%{
\end{matlab}
\paragraph{Patch configuration} Choose either Dirichlet
(default) or Neumann on the left boundary in coordination
with micro-code in \cref{sec:abdulleDiffForce2}
\begin{matlab}
%}
Dom.bcOffset = zeros(2);
if 1, Dom.bcOffset(1)=0.5; end% left Neumann
%{
\end{matlab}
Say use \(7\times7\) patches in \((0,1)^2\), fourth order
interpolation, and either `equispace' or `chebyshev':
\begin{matlab}
%}
nPatch = 7
Dom.type='chebyshev';
configPatches2(@abdulleDiffForce2,[0 1],Dom ...
,nPatch ,4 ,dx ,nSubP ,'EdgyInt',true ,'hetCoeffs',a );
%{
\end{matlab}
\paragraph{Solve for steady state} Set initial guess of
zero, with \verb|NaN| to indicate patch-edge values.
Index~\verb|i| are the indices of patch-interior points, and
the number of unknowns is then its length.
\begin{matlab}
%}
u0 = zeros(nSubP,nSubP,1,1,nPatch,nPatch);
u0([1 end],:,:) = nan; u0(:,[1 end],:) = nan;
patches.i = find(~isnan(u0));
nVariables = numel(patches.i)
%{
\end{matlab}
Solve by iteration. Use \verb|fsolve| for simplicity and
robustness (and using \verb|optimoptions| to omit trace
information), via the generic patch system wrapper
\verb|theRes| (\cref{sec:theRes}), and give magnitudes.
\begin{matlab}
%}
tic;
uSoln = fsolve(@theRes,u0(patches.i) ...
,optimoptions('fsolve','Display','off'));
solnTime = toc
normResidual = norm(theRes(uSoln))
normSoln = norm(uSoln)
%{
\end{matlab}
Store the solution vector into the patches, and interpolate,
but have not bothered to set boundary values so they stay
NaN from the interpolation.
\begin{matlab}
%}
u0(patches.i) = uSoln;
u0 = patchEdgeInt2(u0);
%{
\end{matlab}
\paragraph{Draw solution profile} Separate patches with
NaNs, then reshape arrays to suit 2D space surface plots.
\begin{matlab}
%}
figure(1), clf, colormap(0.8*hsv)
patches.x(end+1,:,:)=nan; u0(end+1,:,:)=nan;
patches.y(:,end+1,:)=nan; u0(:,end+1,:)=nan;
u = reshape(permute(squeeze(u0),[1 3 2 4]) ...
, [numel(patches.x) numel(patches.y)]);
%{
\end{matlab}
Draw the patch solution surface, with boundary-values
omitted as already~\verb|NaN| by not bothering to set them.
\begin{matlab}
%}
mesh(patches.x(:),patches.y(:),u'); view(60,55)
xlabel('space $x$'), ylabel('space $y$'), zlabel('$u(x,y)$')
ifOurCf2eps(mfilename) %optionally save plot
%{
\end{matlab}
\subsection{\texttt{abdulleDiffForce2()}: microscale
discretisation inside patches of forced diffusion PDE}
\label{sec:abdulleDiffForce2}
This function codes the lattice heterogeneous diffusion of
the \pde\ inside the patches. For 6D input arrays~\verb|u|,
\verb|x|, and~\verb|y|, computes the time derivative at each
point in the interior of a patch, output in~\verb|ut|.
\begin{matlab}
%}
function ut = abdulleDiffForce2(t,u,patches)
dx = diff(patches.x(2:3)); % x space step
dy = diff(patches.y(2:3)); % y space step
i = 2:size(u,1)-1; % x interior points in a patch
j = 2:size(u,2)-1; % y interior points in a patch
ut = nan+u; % preallocate output array
%{
\end{matlab}
Set Dirichlet boundary value of zero around the square
domain, but also cater for zero Neumann condition on the
left boundary.
\begin{matlab}
%}
u( 1 ,:,:,:, 1 ,:)=0; % left edge of left patches
u(end,:,:,:,end,:)=0; % right edge of right patches
u(:, 1 ,:,:,:, 1 )=0; % bottom edge of bottom patches
u(:,end,:,:,:,end)=0; % top edge of top patches
if 1, u(1,:,:,:,1,:)=u(2,:,:,:,1,:); end% left Neumann
%{
\end{matlab}
Compute the time derivatives via stored forcing and
coefficients. Easier to code by conflating the last four
dimensions into the one~\verb|,:|.
\begin{matlab}
%}
ut(i,j,:) = diff(patches.cs(:,j).*diff(u(:,j,:)))/dx^2 ...
+ diff(patches.cs(i,:).*diff(u(i,:,:),1,2),1,2)/dy^2 ...
+ 10;
end%function abdulleDiffForce2
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
theRes.m
|
.m
|
EquationFreeGit-master/Patch/theRes.m
| 1,064 |
utf_8
|
a07136df69b41921155bba68f9943c58
|
% This functions converts a vector of values into the
% interior values of the patches, then evaluates the time
% derivative of the system at $t=1$, and returns the vector
% of patch-interior time derivatives. AJR, 1 Feb 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{theRes()}: wrapper function to zero for equilibria}
\label{sec:theRes}
This functions converts a vector of values into the interior
values of the patches, then evaluates the time derivative of
the system at time \(t=1\), and returns the vector of
patch-interior time derivatives.
\begin{matlab}
%}
function f=theRes(u)
global patches
switch numel(size(patches.x))
case 4, pSys = @patchSys1;
v=nan(size(patches.x));
case 5, pSys = @patchSys2;
v=nan(size(patches.x+patches.y));
case 6, pSys = @patchSys3;
v=nan(size(patches.x+patches.y+patches.z));
otherwise error('number of dimensions is somehow wrong')
end%switch
v(patches.i) = u;
f = pSys(1,v(:),patches);
f = f(patches.i);
end%function theRes
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
patchSys3.m
|
.m
|
EquationFreeGit-master/Patch/patchSys3.m
| 4,266 |
utf_8
|
5fe96a5fc1710aa27ba6225006093653
|
% patchSys3() provides an interface to time integrators
% for the dynamics on patches in 3D coupled across space.
% The system must be a lattice system such as PDE
% discretisations. AJR, Aug 2020 -- 12 Apr 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{patchSys3()}: interface 3D space to time integrators}
\label{sec:patchSys3}
To simulate in time with 3D spatial patches we often need to
interface a users time derivative function with time
integration routines such as \verb|ode23| or~\verb|PIRK2|.
This function provides an interface. Communicate
patch-design variables (\cref{sec:configPatches3}) either
via the global struct~\verb|patches| or via an optional
third argument. \verb|patches| is required for the parallel
computing of \verb|spmd|, or if parameters are to be passed
though to the user microscale function.
\begin{matlab}
%}
function dudt = patchSys3(t,u,patches,varargin)
if nargin<3, global patches, end
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|u| is a vector\slash array of length
$\verb|prod(nSubP)| \cdot \verb|nVars| \cdot \verb|nEnsem|
\cdot \verb|prod(nPatch)|$ where there are $\verb|nVars|
\cdot \verb|nEnsem|$ field values at each of the points in
the $\verb|nSubP(1)| \times \verb|nSubP(2)| \times
\verb|nSubP(3)| \times \verb|nPatch(1)| \times
\verb|nPatch(2)| \times \verb|nPatch(3)|$ spatial grid.
\item \verb|t| is the current time to be passed to the
user's time derivative function.
\item \verb|patches| a struct set by \verb|configPatches3()|
with the following information used here.
\begin{itemize}
\item \verb|.fun| is the name of the user's function
\verb|fun(t,u,patches,...)| that computes the time
derivatives on the patchy lattice. The array~\verb|u| has
size $\verb|nSubP(1)| \times \verb|nSubP(2)| \times
\verb|nSubP(3)| \times \verb|nVars| \times \verb|nEsem|
\times \verb|nPatch(1)| \times \verb|nPatch(2)| \times
\verb|nPatch(3)|$. Time derivatives must be computed into
the same sized array, although herein the patch edge-values
are overwritten by zeros.
\item \verb|.x| is $\verb|nSubP(1)| \times1 \times1 \times1
\times1 \verb|nPatch(1)| \times1 \times1$ array of the
spatial locations~$x_{i}$ of the microscale
$(i,j,k)$-grid points in every patch. Currently it
\emph{must} be an equi-spaced lattice on both macro- and
microscales.
\item \verb|.y| is similarly $1 \times \verb|nSubP(2)|
\times1 \times1 \times1 \times1 \times \verb|nPatch(2)|
\times1$ array of the spatial locations~$y_{j}$ of the
microscale $(i,j,k)$-grid points in every patch.
Currently it \emph{must} be an equi-spaced lattice on both
macro- and microscales.
\item \verb|.z| is similarly $1 \times1 \times
\verb|nSubP(3)| \times1 \times1 \times1 \times1 \times
\verb|nPatch(3)|$ array of the spatial locations~$z_{k}$
of the microscale $(i,j,k)$-grid points in every patch.
Currently it \emph{must} be an equi-spaced lattice on both
macro- and microscales.
\end{itemize}
\item \verb|varargin|, optional, is arbitrary list of
parameters to be passed onto the users time-derivative
function as specified in configPatches3.
\end{itemize}
\paragraph{Output}
\begin{itemize}
\item \verb|dudt| is a vector\slash array of of time
derivatives, but with patch edge-values set to zero. It is
of total length $\verb|prod(nSubP)| \cdot \verb|nVars|
\cdot \verb|nEnsem| \cdot \verb|prod(nPatch)|$ and the same
dimensions as~\verb|u|.
\end{itemize}
\begin{devMan}
Sets the edge-face values from macroscale interpolation of
centre-patch values, and if necessary, reshapes the
fields~\verb|u| as a 8D-array. \cref{sec:patchEdgeInt3}
describes \verb|patchEdgeInt3()|.
\begin{matlab}
%}
sizeu = size(u);
u = patchEdgeInt3(u,patches);
%{
\end{matlab}
Ask the user function for the time derivatives computed in
the array, overwrite its edge\slash face values with the
dummy value of zero (as \verb|ode15s| chokes on NaNs), then
return to the user\slash integrator as same sized array as
input.
\begin{matlab}
%}
dudt = patches.fun(t,u,patches,varargin{:});
m = patches.nEdge(1);
dudt([1:m end-m+1:end],:,:,:) = 0;
m = patches.nEdge(2);
dudt(:,[1:m end-m+1:end],:,:) = 0;
m = patches.nEdge(3);
dudt(:,:,[1:m end-m+1:end],:) = 0;
dudt = reshape(dudt,sizeu);
%{
\end{matlab}
Fin.
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
wave2D.m
|
.m
|
EquationFreeGit-master/Patch/wave2D.m
| 5,167 |
utf_8
|
3f6743ba731984dba4dc6a49149c40df
|
% Simulate the linear wave PDE in 2D on patches.
% First it checks the spectrum of the system.
% AJR, Nov 2018 -- 17 Apr 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{wave2D}: example of a wave on patches in 2D}
\label{sec:wave2D}
\localtableofcontents
For \(u(x,y,t)\), test and simulate the simple wave PDE in
2D space:
\begin{equation*}
\DD tu=\delsq u\,.
\end{equation*}
This script shows one way to get started: a user's script
may have the following three steps (left-right arrows denote
function recursion).
\begin{enumerate}\def\itemsep{-1.5ex}
\item configPatches2
\item ode15s integrator \into patchSys2 \into wavePDE
\item process results
\end{enumerate}
\begin{devMan}
Establish the global data struct \verb|patches| to interface
with a function coding the wave \pde: to be solved on
\(2\pi\)-periodic domain, with \(9\times9\) patches,
spectral interpolation~(\(0\)) couples the patches, each
patch of half-size ratio~\(0.25\) (big enough for
visualisation), and with a \(5\times5\) micro-grid within
each patch.
\begin{matlab}
%}
global patches
nSubP = 5;
nPatch = 9;
configPatches2(@wavePDE,[-pi pi], nan, nPatch, 0, 0.25, nSubP);
%{
\end{matlab}
\subsection{Check on the linear stability of the wave PDE}
Construct the systems Jacobian via numerical differentiation.
Set a zero equilibrium as basis. Then find the indices of
patch-interior points as the only ones to vary in order to
construct the Jacobian.
\begin{matlab}
%}
disp('Check linear stability of the wave scheme')
uv0 = zeros(nSubP,nSubP,2,1,nPatch,nPatch);
uv0([1 end],:,:,:,:,:) = nan;
uv0(:,[1 end],:,:,:,:) = nan;
i = find(~isnan(uv0));
%{
\end{matlab}
Now construct the Jacobian. Since this is a \emph{linear}
wave \pde, use large perturbations.
\begin{matlab}
%}
small = 1;
jac = nan(length(i));
sizeJacobian = size(jac)
for j = 1:length(i)
uv = uv0(:);
uv(i(j)) = uv(i(j))+small;
tmp = patchSys2(0,uv)/small;
jac(:,j) = tmp(i);
end
%{
\end{matlab}
Now explore the eigenvalues a little: find the ten with the
biggest real-part; if these are small enough, then the
method may be good.
\begin{matlab}
%}
evals = eig(jac);
nEvals = length(evals)
[~,k] = sort(-abs(real(evals)));
evalsWithBiggestRealPart = evals(k(1:10))
if abs(real(evals(k(1))))>1e-4
warning('eigenvalue failure: real-part > 1e-4')
return, end
%{
\end{matlab}
Check that the eigenvalues are close to true waves of the
\pde\ (not yet the micro-discretised equations).
\begin{matlab}
%}
kwave = 0:(nPatch-1)/2;
freq = sort(reshape(sqrt(kwave'.^2+kwave.^2),1,[]));
freq = freq(diff([-1 freq])>1e-9);
freqerr = [freq; min(abs(imag(evals)-freq))]
%{
\end{matlab}
\subsection{Execute a simulation}
Set a Gaussian initial condition using auto-replication of
the spatial grid: here \verb|u0| and~\verb|v0| are in the
form required for computation: \(n_x\times n_y\times 1\times 1\times
N_x\times N_y\).
\begin{matlab}
%}
u0 = exp(-patches.x.^2-patches.y.^2);
v0 = zeros(size(u0));
%{
\end{matlab}
Initiate a plot of the simulation using only the microscale
values interior to the patches: set \(x\)~and \(y\)-edges to
\verb|nan| to leave the gaps. Start by showing the initial
conditions of \cref{fig:configPatches2ic} while the
simulation computes. To mesh/surf plot we need to ??
`transpose' to size \(n_x\times N_x\times n_y\times N_y\),
then reshape to size \(n_x\cdot N_x\times n_y\cdot N_y\).
\begin{matlab}
%}
x = squeeze(patches.x); y = squeeze(patches.y);
x([1 end],:) = nan; y([1 end],:) = nan;
u = reshape(permute(squeeze(u0),[1 3 2 4]), [numel(x) numel(y)]);
usurf = surf(x(:),y(:),u');
axis([-3 3 -3 3 -0.5 1]), view(60,40)
xlabel('space $x$'), ylabel('space $y$'), zlabel('$u(x,y)$')
legend('time = 0','Location','north')
colormap(hsv)
drawnow
ifOurCf2eps([mfilename 'ic'])
%{
\end{matlab}
\begin{figure}
\centering \caption{\label{fig:wave2Dic}initial
field~\(u(x,y,t)\) at time \(t=0\) of the patch scheme
applied to the simple wave~\pde: \cref{fig:wave2Dt6} plots
the computed field at time \(t=2\).}
\includegraphics[scale=0.9]{wave2Dic}
\end{figure}
Integrate in time using standard functions.
\begin{matlab}
%}
disp('Wait while we simulate u_t=v, v_t=u_xx+u_yy')
uv0 = cat(3,u0,v0);
if ~exist('OCTAVE_VERSION','builtin')
[ts,uvs] = ode23( @patchSys2,[0 6],uv0(:));
else % octave version is slower
[ts,uvs] = odeOcts(@patchSys2,linspace(0,6),uv0(:));
end
%{
\end{matlab}
Animate the computed simulation to end with
\cref{fig:wave2Dt6}. Because of the very small time-steps,
subsample to plot at most 100 times.
\begin{matlab}
%}
di = ceil(length(ts)/100);
for i = [1:di:length(ts)-1 length(ts)]
uv = patchEdgeInt2(uvs(i,:));
uv = reshape(permute(uv,[1 5 2 6 3 4]), [numel(x) numel(y) 2]);
set(usurf,'ZData', uv(:,:,1)');
legend(['time = ' num2str(ts(i),2)])
pause(0.1)
end
ifOurCf2eps([mfilename 't' num2str(ts(end))])
%{
\end{matlab}
\begin{figure}
\centering
\caption{\label{fig:wave2Dt6}field~\(u(x,y,t)\) at time
\(t=6\) of the patch scheme applied to the simple wave~\pde\
with initial condition in \cref{fig:wave2Dic}.}
\includegraphics[scale=0.9]{wave2Dt6}
\end{figure}
\input{../Patch/wavePDE.m}
\input{../Patch/odeOcts.m}
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
chanDispMicro.m
|
.m
|
EquationFreeGit-master/Patch/chanDispMicro.m
| 1,965 |
utf_8
|
c4fb6f8312af4299282260971007cb4a
|
% chanDispMicro() computes the time derivatives of
% heterogeneous advection-diffusion in 2D along a long thin
% channel on 1D array patches. AJR, Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{chanDispMicro()}: heterogeneous 2D
advection-diffusion in a long thin channel}
\label{sec:chanDispMicro}
This function codes the lattice heterogeneous diffusion
inside the patches. For 4D input arrays of
concentration~\verb|c| and spatial lattice~\verb|x| (via
edge-value interpolation of \verb|patchSys1|,
\cref{sec:patchSys1}), computes the time
derivative~\eqref{eq:ddeChanDisp} at each point in the
interior of a patch, output in~\verb|ct|. The heterogeneous
advections and diffusivities,~$u_i(y_j)$
and~$\kappa_i(y_{j+1/2})$, have previously been merged and
stored in the one array~\verb|patches.cs| (2D).
\begin{matlab}
%}
function ct = chanDispMicro(t,c,p)
[nx,ny,~,~]=size(c); % micro-grid points in patches
ix = 2:nx-1; % x interior points in a patch
dx = diff(p.x(2:3)); % x space step
dy = 2/ny; % y space step
ct = nan+c; % preallocate output array
pcs = reshape(p.cs,nx-1,[],2);
%{
\end{matlab}
Compute the cross-channel flux using `ghost' nodes at
channel boundaries, so that the flux is zero at $y=\pm1$
either because the boundary values are replicated so the
differences are zero, or because the diffusivities in
\verb|cs| are zero at the channel boundaries.
\begin{matlab}
%}
ydif = pcs(ix,1:2:end,2) ...
.*(c(ix,[1:end end],:,:)-c(ix,[1 1:end],:,:))/dy;
%{
\end{matlab}
Now evaluate advection-diffusion time
derivative~\eqref{eq:ddeChanDisp}. Could use upwind
advection and no longitudinal diffusion, or, as here,
centred advection and diffusion.
\begin{matlab}
%}
ct(ix,:,:,:) = (ydif(:,2:end,:,:)-ydif(:,1:end-1,:,:))/dy ...
+ diff(pcs(:,2:2:end,2).*diff(c))/dx^2 ...
- p.Pe*pcs(ix,2:2:end,1).*(c(ix+1,:,:,:)-c(ix-1,:,:,:))/(2*dx);
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
rotFilmSpmd.m
|
.m
|
EquationFreeGit-master/Patch/rotFilmSpmd.m
| 12,806 |
utf_8
|
5135d38aef148bbc8496b6f86a1675d5
|
% rotFilmSpmd simulates 2D fluid film flow on a rotating
% substrate with 2D patches as a Proof of Principle example
% of parallel computing with spmd. AJR, Dec 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{rotFilmSpmd}: simulation of a 2D shallow
water flow on a rotating heterogeneous substrate}
\label{sec:rotFilmSpmd}
\localtableofcontents
As an example application, consider the flow of a shallow
layer of fluid on a solid flat rotating substrate, such as
in spin coating \citep[\S II.K, e.g.]{Wilson00, Oron97} or
large-scale shallow water waves \cite[e.g.]{Dellar2005,
Hereman2009}. Let $\xv=(x,y)$ parametrise location on the
rotating substrate, and let the fluid layer have
thickness~$h(\xv , t)$ and move with depth-averaged
horizontal velocity $\vv (\xv , t)=(u,v)$. We take as given
(with its simplified physics) that the (non-dimensional)
governing set of \pde{}s is the nonlinear system
\cite[eq.~(1), e.g.]{Bunder2018a}
\begin{subequations}\label{eqs:spinddt}%
\begin{align}
\D th&=-{\nabla}\cdot (h\vv ), \label{eq:spindhdt}\\
\D t\vv&=\begin{bmatrix} -b & f\\-f &-b\end{bmatrix}\vv
-(\vv \cdot\nabla)\vv -g{\nabla}h+\divv(\nu\grad\vec{v})\,,
\label{eq:spindvdt}
\end{align}
\end{subequations}
where $b(\xv)$~represents heterogeneous `bed' drag, $f$~is
the Coriolis coefficient, $g$~is the acceleration due to
gravity, $\nu(\xv)$~is a heterogeneous `kinematic
viscosity', and we neglect surface tension.
The aim is to simulate the macroscale dynamics which (for
constant~$b$) is approximately that of the nonlinear
diffusion $\D th\approx \frac{gb}{b^2+f^2}\divv(h\grad h)$
\cite[eq.~(2)]{Bunder2018a}. But there is no known algebraic
closure for the macroscale in the case of
heterogeneous~$b(\xv)$ and~$\nu(\xv)$, nonetheless the patch
scheme automatically predicts a sensible macroscale for such
heterogeneous dynamics (\cref{fig:rotFilmSpmdtFin}).
For the microscale computation, \cref{sec:rotFilmMicro}
discretises the \pde{}s~\eqref{eqs:spinddt} in space with
$x,y$-spacing~$\delta x,\delta y$.
Choose one of four cases:
\begin{itemize}
\item \verb|theCase=1| is corresponding code without
parallelisation (in this toy problem it is much the quickest
because there is no expensive communication);
\item \verb|theCase=2| illustrates that \verb|RK2mesoPatch|
invokes \verb|spmd| computation if parallel has been
configured.
\item \verb|theCase=3| shows how users explicitly invoke
\verb|spmd|-blocks around the time integration.
\item \verb|theCase=4| invokes projective integration for
long-time simulation via short bursts of the
micro-computation, bursts done within \verb|spmd|-blocks for
parallel computing.
\end{itemize}
First, clear all to remove any existing globals, old
composites, etc---although a parallel pool persists. Then
choose the case.
\begin{matlab}
%}
clear all
theCase = 1
%{
\end{matlab}
Set micro-scale bed drag (array~1) and diffusivity
(arrays~2--3) to be a heterogeneous log-normal factor with
specified period: modify the strength of the heterogeneity
by the coefficient of~\verb|randn| from zero to perhaps one:
coefficient~$0.3$ appears a good moderate value.
\begin{matlab}
%}
mPeriod = 5
bnu = shiftdim([1 0.5 0.5],-1) ...
.*exp(0.3*randn([mPeriod mPeriod 3]));
%{
\end{matlab}
Configure the patch scheme with these choices of domain,
patches, size ratios---here each patch is square in space.
In Cases~1--2, set \verb|patches| information to be global
so the info can be used without being explicitly passed as
arguments.
\begin{matlab}
%}
if theCase<=2, global patches, end
%{
\end{matlab}
In Case~4, double the size of the domain and use more
separated patches accordingly, to maintain the spatial
microscale grid spacing to be~$0.055$. Here use fourth
order edge-based coupling between patches. Choose the
parallel option if not Case~1, which invokes
\verb|spmd|-block internally, so that field variables become
\emph{distributed} across cpus.
\begin{matlab}
%}
nSubP = 2+mPeriod
nPatch = 9
ratio = 0.2+0.2*(theCase<4)
Len = 2*pi*(1+(theCase==4))
disp('**** Setting configPatches2')
patches = configPatches2(@rotFilmMicro, [0 Len], nan ...
, nPatch, 4, ratio, nSubP, 'EdgyInt',true ...
,'hetCoeffs',bnu ,'parallel',(theCase>1) );
%{
\end{matlab}
When using parallel, any additional parameters to
\verb|patches|, such as physical parameters for the
microcode, must be set within a \verb|spmd| block (because
\verb|patches| is a co-distributed structure). Here set
frequency of substrate rotation, and strength of gravity.
\begin{matlab}
%}
f = 5, g = 1
if theCase==1, patches.f = f; patches.g = g;
else spmd, patches.f = f; patches.g = g; end
end
%{
\end{matlab}
\subsection{Simulate heterogeneous advection-diffusion}
Set initial conditions of a simulation as shown in
\cref{fig:rotFilmSpmdt0}. Here the initial condition is a
(periodic) quasi-Gaussian in~$h$ and zero velocity~\vv, with
additive random perturbations.
\begin{matlab}
%}
disp('**** Set initial condition and test dhuv0dt =')
if theCase==1
%{
\end{matlab}
When not parallel processing, invoke the usual operations.
Here add a random noise to the velocity field, but
keep~$h(x,y,0)$ smooth as shown by \cref{fig:rotFilmSpmdt0}.
The \verb|shiftdim(...,-1)| moves the given row-vector of
coefficients into the third dimension to become coefficients
of the fields~$(h,u,v)$, respectively.
\begin{matlab}
%}
huv0 = shiftdim([0.5 0 0],-1) ...
.*exp(-cos(patches.x)/2-cos(patches.y));
huv0 = huv0+0.1*shiftdim([0 1 1],-1).*rand(size(huv0));
dhuv0dt = patchSys2(0,huv0);
%{
\end{matlab}
With parallel, we must use an \verb|spmd|-block for
computations: there is no difference in Cases~2--4 here.
Also, we must sometimes explicitly tell functions how to
distribute some initial condition arrays over the cpus. Now
\verb|patchSys2| does not invoke \verb|spmd| so higher
level code must, as here. Even if \verb|patches| is global,
inside an \verb|spmd|-block we \emph{must} pass
\verb|patches| explicitly as a parameter to
\verb|patchSys2|.
\begin{matlab}
%}
else, spmd
huv0 = shiftdim([0.5 0 0],-1) ...
.*exp(-cos(patches.x)/2-cos(patches.y));
huv0 = huv0+0.1*rand(size(huv0),patches.codist);
dhuv0dt = patchSys2(0,huv0,patches)
end%spmd
end%if theCase
%{
\end{matlab}
\begin{figure}
\centering \caption{\label{fig:rotFilmSpmdt0}initial
field~$h(x,y,0)$ of the patch scheme applied to the
heterogeneous, shallow water, rotating substrate,
\pde~\eqref{eqs:spinddt}. The micro-scale sub-patch colour
displays the initial $y$-direction velocity
field~$v(x,y,0)$. \cref{fig:rotFilmSpmdtFin} plots the
roughly smooth field values at time $t=6$. In this
example the patches are relatively large, ratio~$0.4$, for
visibility.}
\includegraphics[scale=0.8]{rotFilmSpmdt0}
\end{figure}
Integrate in time, either via the automatic \verb|ode23| or
via \verb|RK2mesoPatch| which reduces communication between
patches. By default, \verb|RK2mesoPatch| does ten
micro-steps for each specified meso-step in~\verb|ts|. For
stability: with noise up to~$0.3$, need micro-steps less
than~$0.0003$; with noise~$1$, need micro-steps less
than~$0.0001$.
\begin{matlab}
%}
warning('Integrating system in time, wait a minute')
ts=0:0.003:0.3;
%{
\end{matlab}
Go to the selected case.
\begin{matlab}
%}
switch theCase
%{
\end{matlab}
\begin{enumerate}
\item For non-parallel, we could use \verb|RK2mesoPatch| as
indicated below, but instead choose to use standard
\verb|ode23| as here \verb|patchSys2| accesses patch
information via global \verb|patches|. For post-processing,
reshape each and every row of the computed solution to the
correct array size---namely that of the initial condition.
\begin{matlab}
%}
case 1
% tic,[huvs,uerrs] = RK2mesoPatch(ts,huv0);toc
[ts,huvs] = ode23(@patchSys2,[0 4],huv0(:));
huvs=reshape(huvs,[length(ts) size(huv0)]);
%{
\end{matlab}
\item In the second case, \verb|RK2mesoPatch| detects a
parallel patch code has been requested, but has only one cpu
worker, so it auto-initiates an \verb|spmd|-block for the
integration. Both this and the next case return
\emph{composite} results, so just keep one version of the
results.
\begin{matlab}
%}
case 2
huvs = RK2mesoPatch(ts,huv0);
huvs = huvs{1};
%{
\end{matlab}
\item In this third case, a user could merge this explicit
\verb|spmd|-block with the previous one that sets the
initial conditions.
\begin{matlab}
%}
case 3,spmd
huvs = RK2mesoPatch(ts,huv0,[],patches);
end%spmd
huvs = huvs{1};
%{
\end{matlab}
\item In this fourth case, use Projective Integration (PI).
Currently the PI is done serially, with parallel
\verb|spmd|-blocks only invoked inside function
\verb|aBurst()| (\cref{secRF2BfPI}) to compute each burst of
the micro-scale simulation. The macro-scale time-step needs
to be less than about~$0.1$ (which here is not much
projection). The function \verb|microBurst()| interfaces to
\verb|aBurst()| (\cref{secRF2BfPI}) in order to provide
shaped initial states, and to provide the patch information.
\begin{matlab}
%}
case 4
microBurst = @(tb0,xb0,bT) ...
aBurst(tb0 ,reshape(xb0,size(huv0)) ,patches);
ts = 0:0.1:1
huvs = PIRK2(microBurst,ts,gather(huv0(:)));
huvs = reshape(huvs,[length(ts) size(huv0)]);
%{
\end{matlab}
\end{enumerate}
End the four cases.
\begin{matlab}
%}
end%switch theCase
%{
\end{matlab}
\subsection{Plot the solution}
Optionally set to save some plots to file.
\begin{matlab}
%}
if 0, global OurCf2eps, OurCf2eps=true, end
%{
\end{matlab}
\paragraph{Animate the computed solution field over time}
\begin{matlab}
%}
figure(1), clf, colormap(0.8*jet)
%{
\end{matlab}
First get the $x$-coordinates and omit the patch-edge values
from the plot (because they are not here interpolated).
\begin{matlab}
%}
if theCase==1, x = patches.x;
y = patches.y;
else, spmd
x = gather( patches.x );
y = gather( patches.y );
end%spmd
x = x{1}; y = y{1};
end
x([1 end],:,:,:,:,:) = nan;
y(:,[1 end],:,:,:,:) = nan;
%{
\end{matlab}
Draw the field values as a patchy surface evolving over
100--200 time steps.
\begin{matlab}
%}
nTimes = length(ts)
for l = 1:ceil(nTimes/200):nTimes
%{
\end{matlab}
At each time, squeeze sub-patch data fields into three 4D
arrays, permute to get all the $x/y$-variations in the
first/last two dimensions, and and then reshape to~2D.
\begin{matlab}
%}
h = reshape( permute( squeeze( ...
huvs(l,:,:,1,1,:,:) ) ,[1 3 2 4]) ,numel(x),numel(y));
u = reshape( permute( squeeze( ...
huvs(l,:,:,2,1,:,:) ) ,[1 3 2 4]) ,numel(x),numel(y));
v = reshape( permute( squeeze( ...
huvs(l,:,:,3,1,:,:) ) ,[1 3 2 4]) ,numel(x),numel(y));
%{
\end{matlab}
Draw surface of each patch, to show both micro-scale and
macro-scale variation in space. Colour the surface according
to the velocity~$v$ in the $y$-direction.
\begin{matlab}
%}
if l==1
hp = surf(x(:),y(:),h',v');
axis([0 Len 0 Len 0 max(h(:))])
c = colorbar; c.Label.String = 'v(x,y,t)';
legend(['time = ' num2str(ts(l),'%4.2f')] ...
,'Location','north')
axis equal
xlabel('space $x$'), ylabel('space $y$'), zlabel('$h(x,y,t)$')
ifOurCf2eps([mfilename 't0'])
disp('**** pausing, press blank to begin animation')
pause
else
hp.ZData = h'; hp.CData = v';
legend(['time = ' num2str(ts(l),'%4.2f')])
pause(0.1)
end
%{
\end{matlab}
\begin{figure}
\centering \caption{\label{fig:rotFilmSpmdtFin}final
field~$h(x,y,6)$, coloured by~$v(x,y,6)$, of the patch
scheme applied to the heterogeneous, shallow water, rotating
substrate, \pde~\eqref{eqs:spinddt} with heterogeneous
factors log-normal, here distributed $\exp[\mathcal
N(0,1)]$. }
\includegraphics[scale=0.8]{rotFilmSpmdtFin}
\end{figure}
Finish the animation loop, and optionally save the final
plot to file, \cref{fig:rotFilmSpmdtFin}.
\begin{matlab}
%}
end%for over time
ifOurCf2eps([mfilename 'tFin'])
%{
\end{matlab}
\subsection{\texttt{microBurst} function for Projective Integration}
\label{secRF2BfPI}
Projective Integration stability appears to require bursts
longer than~$0.01$. Each burst is done in parallel
processing. Here use \verb|RK2mesoPatch| to take take
meso-steps, each with default ten micro-steps so the
micro-scale step is~$0.0003$. With macro-step~$0.1$, these
parameters usually give stable projective integration.
\begin{matlab}
%}
function [tbs,xbs] = aBurst(tb0,xb0,patches)
normx=max(abs(xb0(:)));
disp(['* aBurst t=' num2str(tb0) ' |x|=' num2str(normx)])
assert(normx<20,'solution exploding')
tbs = tb0+(0:0.003:0.015);
spmd
xb0 = codistributed(xb0,patches.codist);
xbs = RK2mesoPatch(tbs,xb0,[],patches);
end%spmd
xbs=reshape(xbs{1},length(tbs),[]);
end%function
%{
\end{matlab}
Fin.
\input{../Patch/rotFilmMicro.m}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
patchCwts.m
|
.m
|
EquationFreeGit-master/Patch/patchCwts.m
| 2,679 |
utf_8
|
a72c06276a2b4c05eca69bc9515e179e
|
% Compute the weightings for the polynomial
% interpolation of field values for coupling.
% AJR, 7 Aug 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{patchCwts}: weights of polynomial interpolation}
\label{sec:patchCwts}
\subsection{Introduction}
Computes the weightings for the polynomial interpolation of
field values for inter-patch coupling. Should work for any
number of dimensions as determined by the number of elements
in parameter \verb|ratio|. Used by
\verb|configPatches|\(n\) for \(n=1,2,\ldots\)\,.
\begin{matlab}
%}
function [Cwtsr,Cwtsl] = patchCwts(ratio,ordCC,stag)
%{
\end{matlab}
\paragraph{Input} \begin{itemize}
\item \verb|ratio| row vector, one element for each axis of
the spatial domain, of either the half-width or full-width
of a patch to the spacing of the patch mid-points along that
axis direction.
\item \verb|ordCC| is the order of the polynomial
interpolation for inter-patch coupling across empty space of
the macroscale patch values to the edge-values of the
patches: must be~\(2,4,\ldots\)\,.
\item \verb|stag| is true for interpolation using only odd
neighbouring patches as for staggered grids, and false for
the usual case of all neighbour coupling---as yet only
tested in 1D.
\end{itemize}
\paragraph{Output} For the \emph{global} struct \verb|patches|,
constructs the following components.
\begin{itemize}
\item \verb|Cwtsr| and \verb|Cwtsl|, when \(n\)~is the
number of elements of \verb|ratio|, are the
\(\verb|ordCC|\times n\)-array of weights for the
inter-patch interpolation onto the `right' edges and `left'
edges (respectively) with patch:macroscale ratio as
specified.
\end{itemize}
\begin{devMan}
First check, reserve storage, and define some index vectors.
\begin{matlab}
%}
assert(ordCC>0,'order of poly interp must be positive')
Cwtsr=nan(ordCC,numel(ratio));
ks = (1:2:ordCC)';
ps = (1:ordCC/2)'-1;
%{
\end{matlab}
If staggered grid, then we need something like equation~(7)
in \cite{Cao2014a}. But so far only tested for 1D??
\begin{matlab}
%}
if stag
Cwtsr(ks ,:) = [ones(size(ratio)) ...
cumprod( (ratio.^2-ks(1:end-1).^2)/4 ,1) ...
./factorial(2*ps(1:end-1)) ];
Cwtsr(ks+1,:) = [ratio/2 ...
cumprod( (ratio.^2-ks(1:end-1).^2)/4 ,1) ...
./factorial(2*ps(1:end-1)+1).*ratio/2 ];
%{
\end{matlab}
For non-staggered edge-to-edge or centre-to-edge
interpolation, use these weights.
\begin{matlab}
%}
else
Cwtsr(ks ,:) = cumprod(ratio.^2-ps.^2,1) ...
./factorial(2*ps+1)./ratio;
Cwtsr(ks+1,:) = cumprod(ratio.^2-ps.^2,1) ...
./factorial(2*ps+2);
end
Cwtsl = (-1).^((1:ordCC)'-stag).*Cwtsr;
%{
\end{matlab}
Fin.
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroDispersiveWave3.m
|
.m
|
EquationFreeGit-master/Patch/heteroDispersiveWave3.m
| 6,865 |
utf_8
|
1b4e4ff337d348654d1ae8b67f69492b
|
% Simulate in 3D on patches the heterogeneous dispersive
% waves in a fourth-order wave PDE. AJR, 16 Apr 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{heteroDispersiveWave3}: heterogeneous
Dispersive Waves from 4th order PDE}
\label{sec:heteroDispersiveWave3}
This uses small spatial patches to simulate heterogeneous
dispersive waves in 3D. The wave equation
for~\(u(x,y,z,t)\) is the fourth-order in space \pde
\begin{equation*}
u_{tt}=-\delsq(C\delsq u)
\end{equation*}
for microscale variations in scalar~\(C(x,y,z)\).
Initialise some Matlab aspects.
\begin{matlab}
%}
clear all
cMap=jet(64); cMap=0.8*cMap(7:end-7,:); % set colormap
basename = [num2str(floor(1e5*rem(now,1))) mfilename]
%global OurCf2eps, OurCf2eps=true %optional to save plots
%{
\end{matlab}
Set random heterogeneous coefficients of period two in each
of the three directions. Crudely normalise by the harmonic
mean so the macro-wave time scale is roughly one.
\begin{matlab}
%}
mPeriod = [2 2 2];
cHetr = exp(0.9*randn(mPeriod));
cHetr = cHetr*mean(1./cHetr(:))
%{
\end{matlab}
Establish global patch data struct to interface with a
function coding a fourth-order heterogeneous wave \pde: to
be solved on $[-\pi,\pi]^3$-periodic domain, with
$5^3$~patches, spectral interpolation~($0$) couples the
patches, each patch with micro-grid spacing~$0.22$
(relatively large for visualisation), and with $6^3$~points
forming each patch. (Six because two edge layers on each of
two faces, and two interior points for the \pde.)
\begin{matlab}
%}
global patches
patches = configPatches3(@heteroDispWave3,[-pi pi] ...
,'periodic', 5, 0, 0.22, mPeriod+4 ,'EdgyInt',true ...
,'hetCoeffs',cHetr ,'nEdge',2);
%{
\end{matlab}
Set a wave initial state using auto-replication of the
spatial grid, and as \cref{fig:heteroDispersiveWave3ic}
shows. This wave propagates diagonally across space.
Concatenate the two \(u,v\)-fields to be the two components
of the fourth
dimension.
\begin{matlab}
%}
u0 = 0.5+0.5*sin(patches.x+patches.y+patches.z);
v0 = -0.5*cos(patches.x+patches.y+patches.z)*3;
uv0 = cat(4,u0,v0);
%{
\end{matlab}
\begin{figure}\centering
\caption{\label{fig:heteroDispersiveWave3ic} initial
field~$u(x,y,z,t)$ at time $t=0$ of the patch scheme applied
to a heterogeneous dispersive wave~\pde:
\cref{fig:heteroDispersiveWave3fin} plots the computed field
at time $t=6$.}
\includegraphics[scale=0.9]{24168heteroDispersiveWave3ic}
\end{figure}
Integrate in time to $t=6$ using standard functions. In
Matlab \verb|ode15s| would be natural as the patch scheme is
naturally stiff, but \verb|ode23| is much quicker
\cite[Fig.~4]{Maclean2020a}.
\begin{matlab}
%}
disp('Simulate heterogeneous wave u_tt=delsq[C*delsq(u)]')
tic
[ts,us] = ode23(@patchSys3,linspace(0,6),uv0(:));
simulateTime=toc
%{
\end{matlab}
Animate the computed simulation to end with
\cref{fig:heteroDispersiveWave3fin}. Use
\verb|patchEdgeInt3| to obtain patch-face values in order to
most easily reconstruct the array data structure.
Replicate $x$, $y$, and~$z$ arrays to get individual
spatial coordinates of every data point. Then, optionally,
set faces to~\verb|nan| so the plot just shows
patch-interior data.
\begin{matlab}
%}
%%
figure(1), clf, colormap(cMap)
xs = patches.x+0*patches.y+0*patches.z;
ys = patches.y+0*patches.x+0*patches.z;
zs = patches.z+0*patches.y+0*patches.x;
if 1, xs([1:2 end-1:end],:,:,:)=nan;
xs(:,[1:2 end-1:end],:,:)=nan;
xs(:,:,[1:2 end-1:end],:)=nan;
end;%option
j=find(~isnan(xs));
%{
\end{matlab}
In the scatter plot, \verb|col()| maps the $u$-data values
to the colour of the dots.
\begin{matlab}
%}
col = @(u) sign(u).*abs(u);
%{
\end{matlab}
Loop to plot at each and every time step.
\begin{matlab}
%}
for i = 1:length(ts)
uv = patchEdgeInt3(us(i,:));
u = uv(:,:,:,1,:);
for p=1:2
subplot(1,2,p)
if (i==1)
scat(p) = scatter3(xs(j),ys(j),zs(j),'.');
axis equal, caxis(col([0 1])), view(45-4*p,42)
xlabel('$x$'), ylabel('$y$'), zlabel('$z$')
end
title(['view cross-eyed: time = ' num2str(ts(i),'%4.2f')])
set( scat(p),'CData',col(u(j)) );
end
%{
\end{matlab}
Optionally save the initial condition to graphic file for
\cref{fig:heteroDispersiveWave3ic}, and optionally save the
last plot.
\begin{matlab}
%}
if i==1,
ifOurCf2eps([basename 'ic'])
disp('Type space character to animate simulation')
pause
else pause(0.1)
end
end% i-loop over all times
ifOurCf2eps([basename 'fin'])
%{
\end{matlab}
\begin{figure}\centering
\caption{\label{fig:heteroDispersiveWave3fin}
field~$u(x,y,z,t)$ at time $t=6$ of the patch scheme applied
to the heterogeneous dispersive wave~\pde\ with initial
condition in \cref{fig:heteroDispersiveWave3ic}.}
\includegraphics[scale=0.9]{24168heteroDispersiveWave3fin}
\end{figure}
\subsection{\texttt{heteroDispWave3()}: PDE function of
4th-order heterogeneous dispersive waves}
\label{sec:heteroDispWave3}
This function codes the lattice heterogeneous waves inside
the patches. The wave \pde\ for \(u(x,y,z,t)\) and
`velocity'~\(v(x,y,z,t)\) is
\begin{equation*}
u_t=v,\quad v_t=-\delsq(C\delsq u)
\end{equation*}
for microscale variations in scalar~\(C(x,y,z)\). For 8D
input arrays~\verb|u|, \verb|x|, \verb|y|, and~\verb|z| (via
edge-value interpolation of \verb|patchSys3|,
\cref{sec:patchSys3}), computes the time derivative at each
point in the interior of a patch, output in~\verb|ut|. The
3D array of heterogeneous coefficients,~$C_{ijk}$,
$c^y_{ijk}$ and~$c^z_{ijk}$, have been stored
in~\verb|patches.cs| (3D).
Supply patch information as a third argument (required by
parallel computation), or otherwise by a global variable.
\begin{matlab}
%}
function ut = heteroDispWave3(t,u,patches)
if nargin<3, global patches, end
%{
\end{matlab}
Micro-grid space steps.
\begin{matlab}
%}
dx = diff(patches.x(2:3));
dy = diff(patches.y(2:3));
dz = diff(patches.z(2:3));
%{
\end{matlab}
First, compute \(C\delsq u\) into say~\verb|u|, using
indices for all but extreme micro-grid points. We use a
single colon to represent the last four array dimensions
because the result arrays are already dimensioned.
\begin{matlab}
%}
I = 2:size(u,1)-1; J = 2:size(u,2)-1; K = 2:size(u,3)-1;
u(I,J,K,1,:) = patches.cs(I,J,K,1,:).*( diff(u(:,J,K,1,:),2,1)/dx^2 ...
+diff(u(I,:,K,1,:),2,2)/dy^2 +diff(u(I,J,:,1,:),2,3)/dz^2 );
%{
\end{matlab}
Reserve storage, set lowercase indices to non-edge interior,
and then assign interior patch values to the heterogeneous
diffusion time derivatives.
\begin{matlab}
%}
ut = nan+u; % preallocate output array
i = I(2:end-1); j = J(2:end-1); k = K(2:end-1);
ut(i,j,k,1,:) = u(i,j,k,2,:); % du/dt=v
% dv/dt=delta^2 of above C*delta^2
ut(i,j,k,2,:) = -( diff(u(I,j,k,1,:),2,1)/dx^2 ...
+diff(u(i,J,k,1,:),2,2)/dy^2 +diff(u(i,j,K,1,:),2,3)/dz^2 );
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroWave2.m
|
.m
|
EquationFreeGit-master/Patch/heteroWave2.m
| 3,466 |
utf_8
|
e35ffe6516a19822a50855c63c3f34c9
|
% Computes the time derivatives of forced heterogeneous
% waves (slightly damped) in 2D on patches. AJR, Aug 2021
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{heteroWave2()}: heterogeneous Waves}
\label{sec:heteroWave2}
This function codes the lattice heterogeneous waves inside
the patches. The forced wave \pde\ is
\begin{equation*}
u_t=v,\quad v_t=\grad(a\divv u)+f
\end{equation*}
for scalars~\(a(t,x,y)\) and~\(f(t,x,y)\) where~\(a\) has
microscale variations. For 6D input arrays~\verb|u|,
\verb|x|, and \verb|y| (via edge-value interpolation of
\verb|patchSys2|, \cref{sec:patchSys2}), computes the
time derivative at each point in the interior of a patch,
output in~\verb|ut|. The four 2D arrays of heterogeneous
interaction coefficients,~$c_{ijk}$, have previously been
stored in~\verb|patches.cs| (3D).
Supply patch information as a third argument (required by
parallel computation), or otherwise by a global variable.
\begin{matlab}
%}
function ut = heteroWave2(t,u,patches)
if nargin<3, global patches, end
%{
\end{matlab}
Microscale space-steps, and interior point indices.
\begin{matlab}
%}
dx = diff(patches.x(2:3)); % x micro-scale step
dy = diff(patches.y(2:3)); % y micro-scale step
i = 2:size(u,1)-1; % x interior points in a patch
j = 2:size(u,2)-1; % y interior points in a patch
assert(max(abs(u(:)))<9999,"u-field exploding")
%{
\end{matlab}
Form coefficients here---odd periodic extension. To avoid
slight errors in periodicity (in full domain simulation),
first adjust any coordinates crossing \(x=\pm1\) or \(y=\pm
1\).
\begin{matlab}
%}
x=patches.x; y=patches.y;
l=find(abs(x)>1); x(l)=x(l)-sign(x(l))*2;
l=find(abs(y)>1); y(l)=y(l)-sign(y(l))*2;
%{
\end{matlab}
Then set at this time three possible forcing functions,
although only use one depending upon \verb|patches.eff|.
Forcing~\(f_1\) and~\(f_2\) are as specified by \S5.1 of
\cite{Maier2021}, whereas~\(f_3\) here is~\(f\) in
their \S5.2.
\begin{matlab}
%}
f1 = ( (abs(x)>0.4)*(20*t+230*t^2) ...
+(abs(x)<0.4)*(100*t+2300*t^2) ).*sign(x).*sign(y);
f2 = 20*t*x.*(1-abs(x)).*y.*(1-abs(y)) ...
+230*t^2*(sign(y).*x.*(1-abs(x))+sign(x).*y.*(1-abs(y)));
f3 = (5*t+50*t^2)*sin(pi*x).*sin(pi*y);
%{
\end{matlab}
Also set the heterogeneous interactions at this time.
\begin{matlab}
%}
ax = (patches.cs(:,:,1)+sin(2*pi*t)) ...
.*(patches.cs(:,:,2)+sin(2*pi*t));
ay = (patches.cs(:,:,3)+sin(2*pi*t)) ...
.*(patches.cs(:,:,4)+sin(2*pi*t));
%{
\end{matlab}
Reserve storage (using \verb|nan+u| appears quickest), and
then assign time derivatives for interior patch values due
to the heterogeneous interaction and forcing.
\begin{matlab}
%}
ut = nan+u; % preallocate output array
ut(i,j,1,:) = u(i,j,2,:);
ut(i,j,2,:) ...
= diff(ax(:,j).*diff(u(:,j,1,:),1),1)/dx^2 ...
+diff(ay(i,:).*diff(u(i,:,1,:),1,2),1,2)/dy^2 ...
+(patches.eff==1)*f1(i,j,:,:) ...
+(patches.eff==2)*f2(i,j,:,:) ...
+(patches.eff==3)*f3(i,j,:,:) ...
+ 1e-4*(diff(u(:,j,2,:),2,1)/dx^2+diff(u(i,:,2,:),2,2)/dy^2);
end% function
%{
\end{matlab}
In the last line above, the slight damping of~\(10^{-4}\)
causes microscale modes to decay at rate~\(e^{-28t}\), with
frequencies~\(2000\)--\(5000\), whereas macroscale modes
decay with rates roughly~\(0.0005\)--\(0.05\) with
frequencies~\(10\)--\(100\). This slight damping term may
correspond to the weak damping of the backward Euler scheme
adopted by \cite{Maier2021} for time integration.
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
nonDiffPDE.m
|
.m
|
EquationFreeGit-master/Patch/nonDiffPDE.m
| 876 |
utf_8
|
46e7abc9257ef65c252ffef6c7f1f550
|
% Microscale discretisation of a nonlinear diffusion PDE in
% 2D space (x,y) in 2D patches.
% AJR, 5 Apr 2019 -- Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\paragraph{Example of nonlinear diffusion PDE inside patches}
As a microscale discretisation of \(u_t=\delsq(u^3)\), code
\(\dot u_{ijkl} =\frac1{\delta x^2} (u_{i+1,j,k,l}^3
-2u_{i,j,k,l}^3 +u_{i-1,j,k,l}^3) + \frac1{\delta y^2}
(u_{i,j+1,k,l}^3 -2u_{i,j,k,l}^3 +u_{i,j-1,k,l}^3)\).
\begin{matlab}
%}
function ut = nonDiffPDE(t,u,patches)
if nargin<3, global patches, end
u = squeeze(u); % reduce to 4D
dx = diff(patches.x(1:2)); % microgrid spacing
dy = diff(patches.y(1:2));
i = 2:size(u,1)-1; j = 2:size(u,2)-1; % interior patch points
ut = nan+u; % preallocate output array
ut(i,j,:,:) = diff(u(:,j,:,:).^3,2,1)/dx^2 ...
+diff(u(i,:,:,:).^3,2,2)/dy^2;
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
homoDiffSoln2.m
|
.m
|
EquationFreeGit-master/Patch/homoDiffSoln2.m
| 12,114 |
utf_8
|
ea97f19a9c9f8e2b9ed86dec927b3395
|
% Solve for steady state of heterogeneous diffusion in 2D on
% patches as an example application. The microscale is of
% known period so we interpolate next-to-edge values to get
% opposite edge values. This version implements scenarios
% inspired by Biezemans et al. (2022) \S3, p.12. AJR, Apr
% 2022
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{homoDiffSoln2}: steady state of a 2D
heterogeneous diffusion via small patches}
\label{sec:homoDiffSoln2}
Here we find the steady state~\(u(x,y)\) to the
heterogeneous \pde
\begin{equation*}
u_t=\divv[c(x,y)\grad u]-u+f,
\quad\text{for } f=100\sin(\pi x)\sin(\pi y).
\end{equation*}
The heterogeneous diffusion~\(c\) varies over two orders of
magnitude in small space distance~\(\epsilon\). I
include~\(-u\) in the \pde\ to ensure a steady state with
periodic BCs.
\cref{sec:homoDiffSoln2Errs} gives a function that we invoke
to explore the errors in the patch scheme solution. The
spectral patch scheme is essentially exact.
\cite{Biezemans2022} discussed an example homogenisation in
2D with heterogeneity of period~\(\epsilon:=\pi/150\) in
both directions. Ensure integer multiple of heterogeneity
periods in the domain, and initially use three times
bigger~\(\epsilon\).
\begin{matlab}
%}
epsilon = 1/round(50/pi)
%{
\end{matlab}
\cite{Biezemans2022} choose microscale mesh spacing
of~$1/1024$, so the number of micro-grid points in one
period would be~\(1024\epsilon\). But \emph{initially} use
less.
\begin{matlab}
%}
mPeriod = round(128*epsilon) %round(1024*epsilon)
%{
\end{matlab}
So the migro-grid spacing is exactly
\begin{matlab}
%}
dx = epsilon/mPeriod
%{
\end{matlab}
\paragraph{Diffusivities}
Now form one period of the heterogeneity diffusivities.
\cite{Biezemans2022} used \(c=1 +100 \cos^2(\pi x/\epsilon)
\sin^2(\pi y/\epsilon) \). Need to shift phases of the
diffusivity by half-micro-grid for diffusivities in each
direction to form two diffusivity matrices on the microscale
lattice. Variables \verb|h,v| represent~\(\pi x/\epsilon\)
or~\(\pi y/\epsilon\).
\begin{matlab}
%}
cHetr=[];
v=pi*( 1:mPeriod)/mPeriod;
h=pi*(0.5:mPeriod)/mPeriod;
cHetr(:,:,1) = 1+100*cos(h').^2*sin(v).^2;
cHetr(:,:,2) = 1+100*cos(v').^2*sin(h).^2;
%{
\end{matlab}
Plot surfaces of the diffusivity.
\begin{matlab}
%}
figure(2),surf(h/pi,v/pi,cHetr(:,:,2))
hold on, surf(v/pi,h/pi,cHetr(:,:,1))
hold off, alpha 0.5, drawnow
%{
\end{matlab}
\paragraph{Patch configuration}
As is common, \cite{Biezemans2022} implemented
zero-Dirichlet BCs on $(0,1)^2$. Here these are
more-or-less encompassed by implementing periodic BCs on
$(-1,1)^2$. Initially use \(8\times8\) patches to have
\(4\times4\) patches in \((0,1)^2\), which then have patch
spacing~\(H\).
\begin{matlab}
%}
nPatch = [8 8]
H = 2./nPatch
HepsilonRatio = H/epsilon
%{
\end{matlab}
Best when each patch spans an integral number of periods
plus one grid step. The smallest such patches are
\begin{matlab}
%}
nSubP = [1 1]*mPeriod+2
%{
\end{matlab}
Consequently, the ratio of space computed on, to the space
in the domain is the product of the following ratios in each
direction, namely about~8\% here.
\begin{matlab}
%}
ratio = ((nSubP-2)*dx)./H
%{
\end{matlab}
Specify spectral interpolation. The edgy interpolation is
self-adjoint \cite[]{Bunder2020a} leading to a symmetric
matrix problem.
\begin{matlab}
%}
configPatches2(@hetDiffForce2,[-1 1 -1 1],nan,nPatch ...
,0,ratio,nSubP ,'EdgyInt',true ...
,'hetCoeffs',cHetr );
%{
\end{matlab}
\paragraph{Solve for steady state}
Set initial guess of zero, with \verb|NaN| to indicate
patch-edge values. Index~\verb|i| are the indices of
patch-interior points, and the number of unknowns is then
its length.
\begin{matlab}
%}
global patches i
u0 = zeros([nSubP,1,1,nPatch]);
u0([1 end],:,:) = nan; u0(:,[1 end],:) = nan;
i = find(~isnan(u0));
nVars = numel(i)
%{
\end{matlab}
Solve by iteration. Could use \verb|fsolve| for nonlinear
problems, but for linear it is much faster to use
Conjugate-Gradient algorithm. \verb|gmres| is competitive,
but appears to take twice as long.
% uSoln=gmres(@(u) rhsb-theRes(u),rhsb,[],1e-9,maxIt);
\begin{matlab}
%}
tic;
if 0, uSoln=fsolve(@theRes,u0(i));
%{
\end{matlab}
The above is for nonlinear \pde{}s. For linear \pde{}s,
determine the \textsc{rhs} vector, and make a function that
computes the matrix vector product.
\begin{matlab}
%}
else
maxIt = ceil(nVars/10);
rhsb = theRes(u0(i));
uSoln = pcg(@(u) rhsb-theRes(u),rhsb,1e-9,maxIt);
end
solnTime = toc
%{
\end{matlab}
Store the solution into the patches, and give magnitudes.
\begin{matlab}
%}
u0(i) = uSoln;
normSoln = norm(uSoln)
normResidual = norm(theRes(uSoln))
%{
\end{matlab}
\paragraph{Draw solution profile}
First reshape arrays to suit 2D space surface plots.
\begin{matlab}
%}
figure(1), clf, colormap(hsv)
x = squeeze(patches.x); y = squeeze(patches.y);
u = reshape(permute(squeeze(u0),[1 3 2 4]), [numel(x) numel(y)]);
%{
\end{matlab}
Draw the patch solution surface in the positive quadrant,
with edge-values omitted as already~\verb|NaN| by not
bothering to interpolate them.
\begin{matlab}
%}
surf(x(:),y(:),u'); view(60,40)
maxu = ceil(max(u(:))*10)/10;
axis([0 1 0 1 0 maxu]), caxis([0 maxu])
xlabel('$x$'), ylabel('$y$'), zlabel('$u(x,y)$')
%{
\end{matlab}
\paragraph{Assess errors in the patch scheme}
Invoke the function with desired interpolation: \(0\),~spectral; \(2,4,\ldots\),~polynomial.
\begin{matlab}
%}
errorsPatchScheme(0)
%{
\end{matlab}
\subsection{Microscale discretisation inside patches}
\paragraph{\texttt{hetDiffForce2()}: heterogeneous diffusion PDE}
This function, based upon \cref{sec:heteroDiff2}, codes the
lattice heterogeneous diffusion of the \pde\ inside the
patches. For 6D input arrays~\verb|u|, \verb|x|,
and~\verb|y|, computes the time derivativeat each point in
the interior of a patch, output in~\verb|ut|. The two 2D
array of diffusivities,~$c^x_{ij}$ and~$c^y_{ij}$, are
stored in~\verb|patches.cs| (3D).
\begin{matlab}
%}
function ut = hetDiffForce2(t,u,patches)
dx = diff(patches.x(2:3)); % x space step
dy = diff(patches.y(2:3)); % y space step
ix = 2:size(u,1)-1; % x interior points in a patch
iy = 2:size(u,2)-1; % y interior points in a patch
ut = nan+u; % preallocate output array
fu = -u+100*sin(pi*patches.x).*sin(pi*patches.y);
ut(ix,iy,:,:,:,:) ...
= diff(patches.cs(:,iy,1).*diff(u(:,iy,:,:,:,:),1),1)/dx^2 ...
+diff(patches.cs(ix,:,2).*diff(u(ix,:,:,:,:,:),1,2),1,2)/dy^2 ...
+fu(ix,iy,:,:,:,:);
end% function
%{
\end{matlab}
\paragraph{\texttt{theRes()}: function to zero}
This functions converts a vector of values into the interior
values of the patches, then evaluates the time derivative of
the system, and returns the vector of patch-interior time
derivatives.
\begin{matlab}
%}
function f=theRes(u)
global i patches
v=nan(size(patches.x+patches.y));
v(i)=u;
f=patchSys2(0,v(:),patches);
f=f(i);
end
%{
\end{matlab}
\subsection{Function to explore errors in the patch scheme}
\label{sec:homoDiffSoln2Errs}
We find the spectral interpolation patch scheme accurate to
essentially zero errors, namely errors less
than~\(10^{-10}\). Non-spectral interpolation has errors
that decrease roughly like expected power of patch spacing.
The single argument~\verb|ord| is~\(0\) for spectral
interpolation and \(2,4,\ldots\) for corresponding
polynomial interpolation schemes.
\begin{matlab}
%}
function errorsPatchScheme(ord)
warning('Assessing errors via varying number of patches')
%{
\end{matlab}
Use a hierarchy of cases with increasing number of
patches---the number increasing by~\(3^2\) from one level to
the next in the hierarchy. Then the higher resolution
patches precisely contain the lower resolution cases. The
case when index \verb|k=kMax| corresponds to the full-domain
solution. \cite{Biezemans2022} use heterogeneity of
period~\(\epsilon:=\pi/150\approx 0.021\) in both
directions, here with \verb|kMax=3| use
\(\epsilon\approx0.037\). Ensure integer multiple of
heterogeneity periods in the full domain.
\begin{matlab}
%}
kMax = 3
epsilon = 1/3^kMax
%{
\end{matlab}
\cite{Biezemans2022} choose microscale mesh spacing
of~$1/1024$, so their number of micro-grid points in one
period is~\(1024\epsilon\approx 21\). But here use less
because less is plenty enough---the issue is the accuracy of
the patch scheme to whatever micro-grid system is given,
\emph{not} the accuracy of the micro-grid system to the
\pde.
\begin{matlab}
%}
mPeriod = 9
%{
\end{matlab}
So the migro-grid spacing is
\begin{matlab}
%}
dx = epsilon/mPeriod
%{
\end{matlab}
\paragraph{Diffusivities}
Now form one period of the heterogeneity diffusivities
exactly as in above code.
\begin{matlab}
%}
cHetr=[];
v=pi*( 1:mPeriod)/mPeriod;
h=pi*(0.5:mPeriod)/mPeriod;
cHetr(:,:,1) = 1+100*cos(h').^2*sin(v).^2;
cHetr(:,:,2) = 1+100*cos(v').^2*sin(h).^2;
%{
\end{matlab}
\paragraph{Loop over different patch spacings}
\begin{matlab}
%}
for k=1:kMax
nPatch = [2 2]*3^k
%{
\end{matlab}
\paragraph{Patch configuration}
Zero-Dirichlet BCs on $(0,1)^2$ are
more-or-less encompassed by implementing periodic BCs on
$(-1,1)^2$.
\begin{matlab}
%}
H = 2./nPatch
HepsilonRatio = H/epsilon
%{
\end{matlab}
Best when each patch spans an integral number of periods
plus one grid step. The smallest such patches are
\begin{matlab}
%}
nSubP = [1 1]*mPeriod+2
%{
\end{matlab}
Consequently, the ratios of space computed on is the
following. The case \verb|k=kMax| gives a ratio of
precisely one that characterises a full-domain problem.
\begin{matlab}
%}
ratio = ((nSubP-2)*dx)./H
%{
\end{matlab}
The edgy interpolation leads to a symmetric matrix problem
\cite[]{Bunder2020a}.
\begin{matlab}
%}
configPatches2(@hetDiffForce2,[-1 1 -1 1],nan,nPatch ...
,ord,ratio,nSubP ,'EdgyInt',true ...
,'hetCoeffs',cHetr );
%{
\end{matlab}
\paragraph{Solve for steady state}
Set initial guess of zero, with \verb|NaN| to indicate
patch-edge values. Index~\verb|i| are the indices of
patch-interior points, and the number of unknowns is then
its length.
\begin{matlab}
%}
global patches i
u0 = zeros([nSubP,1,1,nPatch]);
u0([1 end],:,:) = nan; u0(:,[1 end],:) = nan;
i = find(~isnan(u0));
nVars = numel(i)
%{
\end{matlab}
For this linear problem it is fast to solve with the
Conjugate-Gradient algorithm. Determine the \textsc{rhs}
vector, and use a function that computes the matrix vector
product.
\begin{matlab}
%}
tic
rhsb = theRes(u0(i));
uSoln = pcg(@(u) rhsb-theRes(u),rhsb,1e-6,999);
solnTime = toc
%{
\end{matlab}
Store the solution into the patches, and trace magnitudes.
\begin{matlab}
%}
u0(i) = uSoln;
normSoln = norm(uSoln)
normResidual = norm(theRes(uSoln))
%{
\end{matlab}
\paragraph{End loop over different patch spacings}
Store 4D field values in cell array for post-processing.
\begin{matlab}
%}
us{k}=squeeze(u0);
end%for
%{
\end{matlab}
\paragraph{Compare errors across cases}
There are nine patches common to all grids (36~if one counts
all quadrants), indexed by the following patch indices.
\begin{matlab}
%}
disp('**** Relate errors for different patch spacing ****')
if ord==0, disp('**** Spectral interpolation between patches')
else disp(['**** Polynomial interpolation, order ' num2str(ord)])
end
i=2:nSubP-1;
I{1}=1:3;
for k=2:kMax, I{k}=3*I{k-1}-1; end
%{
\end{matlab}
Determine errors by computing difference between patch
schemes: the final patch scheme is a full-domain solution
and hence `exact'. Look at the \textsc{rms} error in each
of the patches. Find the overall error for each patch,
their ratios, and the rough order of decrease.
\begin{matlab}
%}
rmsError=[]; errorRatios=[]; orderInH=[];
for k=1:kMax-1
error{k}=us{k}(i,i,I{k},I{k})-us{kMax}(i,i,I{kMax},I{kMax});
rmsError(:,:,k)=squeeze(rms(rms(error{k})));
if (k>1)&(ord>0)
errorRatios(:,:,k-1)=rmsError(:,:,k)./rmsError(:,:,k-1);
orderInH(:,:,k-1)=-log(errorRatios(:,:,k-1))/log(3);
end
end
%{
\end{matlab}
Display the results, and end the function.
\begin{matlab}
%}
rmsError=rmsError
errorRatios=errorRatios
orderInH=orderInH
end%function
%{
\end{matlab}
Fin.
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroDiffF.m
|
.m
|
EquationFreeGit-master/Patch/heteroDiffF.m
| 1,742 |
utf_8
|
50afdf6b3a072798ee453482c350b375
|
% Computes the time derivatives of forced heterogeneous
% diffusion in 1D on patches. AJR, Apr 2019 -- 3 Jan 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{heteroDiffF()}: forced heterogeneous diffusion}
\label{sec:heteroDiffF}
This function codes the lattice heterogeneous diffusion
inside the patches with forcing and with microscale boundary
conditions on the macroscale boundaries. Computes the time
derivative at each point in the interior of a patch, output
in~\verb|ut|. The column vector of diffusivities~\(a_i\)
has been stored in struct~\verb|patches.cs|, as has the
array of forcing coefficients.
\begin{matlab}
%}
function ut = heteroDiffF(t,u,patches)
%{
\end{matlab}
Cater for the two cases: one of a non-autonomous forcing
oscillating in time when \(\verb|microTimePeriod|>0\), or
otherwise the case of an autonomous diffusion constant in
time.
\begin{matlab}
%}
global microTimePeriod
if microTimePeriod>0 % optional time fluctuations
at = cos(2*pi*t/microTimePeriod)/30;
else at=0; end
%{
\end{matlab}
Two basic parameters, and initialise result array to NaNs.
\begin{matlab}
%}
dx = diff(patches.x(2:3)); % space step
i = 2:size(u,1)-1; % interior points in a patch
ut = nan+u; % preallocate output array
%{
\end{matlab}
The macroscale Dirichlet boundary conditions are zero at the
extreme edges of the two extreme patches.
\begin{matlab}
%}
u( 1 ,:,:, 1 )=0; % left-edge of leftmost is zero
u(end,:,:,end)=0; % right-edge of rightmost is zero
%{
\end{matlab}
Code the microscale forced diffusion.
\begin{matlab}
%}
ut(i,:,:,:) = diff((patches.cs(:,1,:)+at).*diff(u))/dx^2 ...
+patches.f2(i,:,:,:)*t^2+patches.f1(i,:,:,:)*t;
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
patchSmooth3.m
|
.m
|
EquationFreeGit-master/Patch/patchSmooth3.m
| 376 |
utf_8
|
884dc8b5cd180f6a18e78fd7d0284768
|
% legacy interface patchSmooth3() auto-invokes new patchSys3()
function dudt=patchSmooth3(t,u,patches)
global smOOthCount
if isempty(smOOthCount), smOOthCount=1;
else smOOthCount=smOOthCount+1; end
l2=log2(smOOthCount);
if abs(l2-round(l2))<1e-9
warning('Use new patchSys3 instead of old patchSmooth3')
end
if nargin<3, global patches, end
dudt=patchSys3(t,u,patches);
|
github
|
uoa1184615/EquationFreeGit-master
|
configPatches3.m
|
.m
|
EquationFreeGit-master/Patch/configPatches3.m
| 33,989 |
utf_8
|
a27ad1c3bdc45bd11440e731216fa33a
|
% configPatches3() creates a data struct of the design of 3D
% patches for later use by the patch functions such as
% patchSys3(). AJR, Aug 2020 -- 12 Apr 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{configPatches3()}: configures spatial
patches in 3D}
\label{sec:configPatches3}
\localtableofcontents
Makes the struct~\verb|patches| for use by the patch\slash
gap-tooth time derivative\slash step function
\verb|patchSys3()|, and possibly other patch functions.
\cref{sec:configPatches3eg,sec:homoDiffEdgy3} list examples
of its use.
\begin{matlab}
%}
function patches = configPatches3(fun,Xlim,Dom ...
,nPatch,ordCC,dx,nSubP,varargin)
version = '2023-04-12';
%{
\end{matlab}
\paragraph{Input}
If invoked with no input arguments, then executes an example
of simulating a heterogeneous wave \pde---see
\cref{sec:configPatches3eg} for an example code.
\begin{itemize}
\item \verb|fun| is the name of the user function,
\verb|fun(t,u,patches)| or \verb|fun(t,u)| or
\verb|fun(t,u,patches,...)|, that computes time-derivatives
(or time-steps) of quantities on the 3D micro-grid within
all the 3D~patches.
\item \verb|Xlim| array/vector giving the rectangular-cuboid
macro-space domain of the computation: namely
$[\verb|Xlim(1)|, \verb|Xlim(2)|] \times [\verb|Xlim(3)|,
\verb|Xlim(4)| \times [\verb|Xlim(5)|, \verb|Xlim(6)|]$. If
\verb|Xlim| has two elements, then the domain is the cubic
domain of the same interval in all three directions.
\item \verb|Dom| sets the type of macroscale conditions for
the patches, and reflects the type of microscale boundary
conditions of the problem. If \verb|Dom| is \verb|NaN| or
\verb|[]|, then the field~\verb|u| is triply macro-periodic
in the 3D spatial domain, and resolved on equi-spaced
patches. If \verb|Dom| is a character string, then that
specifies the \verb|.type| of the following structure, with
\verb|.bcOffset| set to the default zero. Otherwise
\verb|Dom| is a structure with the following components.
\begin{itemize}
\item \verb|.type|, string, of either \verb|'periodic'| (the
default), \verb|'equispace'|, \verb|'chebyshev'|,
\verb|'usergiven'|. For all cases except \verb|'periodic'|,
users \emph{must} code into \verb|fun| the micro-grid
boundary conditions that apply at the left\slash right\slash
bottom\slash top\slash back\slash front faces of the
leftmost\slash rightmost\slash bottommost\slash
topmost\slash backmost\slash frontmost patches,
respectively.
\item \verb|.bcOffset|, optional one, three or six element
vector/array, in the cases of \verb|'equispace'| or
\verb|'chebyshev'| the patches are placed so the left\slash
right macroscale boundaries are aligned to the left\slash
right faces of the corresponding extreme patches, but offset
by \verb|bcOffset| of the sub-patch micro-grid spacing. For
example, use \verb|bcOffset=0| when the micro-code applies
Dirichlet boundary values on the extreme face micro-grid
points, whereas use \verb|bcOffset=0.5| when the microcode
applies Neumann boundary conditions halfway between the
extreme face micro-grid points. Similarly for the top,
bottom, back, and front faces.
If \verb|.bcOffset| is a scalar, then apply the same offset
to all boundaries. If three elements, then apply the first
offset to both \(x\)-boundaries, the second offset to both
\(y\)-boundaries, and the third offset to both
\(z\)-boundaries. If six elements, then apply the first two
offsets to the respective \(x\)-boundaries, the middle two
offsets to the respective \(y\)-boundaries, and the last two
offsets to the respective \(z\)-boundaries.
\item \verb|.X|, optional vector/array with \verb|nPatch(1)|
elements, in the case \verb|'usergiven'| it specifies the
\(x\)-locations of the centres of the patches---the user is
responsible the locations makes sense.
\item \verb|.Y|, optional vector/array with \verb|nPatch(2)|
elements, in the case \verb|'usergiven'| it specifies the
\(y\)-locations of the centres of the patches---the user is
responsible the locations makes sense.
\item \verb|.Z|, optional vector/array with \verb|nPatch(3)|
elements, in the case \verb|'usergiven'| it specifies the
\(z\)-locations of the centres of the patches---the user is
responsible the locations makes sense.
\end{itemize}
\item \verb|nPatch| sets the number of equi-spaced spatial
patches: if scalar, then use the same number of patches in
all three directions, otherwise \verb|nPatch(1:3)| gives the
number~($\geq1$) of patches in each direction.
\item \verb|ordCC| is the `order' of interpolation for
inter-patch coupling across empty space of the macroscale
patch values to the face-values of the patches: currently
must be~$0,2,4,\ldots$; where $0$~gives spectral
interpolation.
\item \verb|dx| (real---scalar or three elements) is usually
the sub-patch micro-grid spacing in~\(x\), \(y\) and~\(z\).
If scalar, then use the same \verb|dx| in all three
directions, otherwise \verb|dx(1:3)| gives the spacing in
each of the three directions.
However, if \verb|Dom| is~\verb|NaN| (as for pre-2023), then
\verb|dx| actually is \verb|ratio| (scalar or three elements),
namely the ratio of (depending upon \verb|EdgyInt|) either
the half-width or full-width of a patch to the equi-spacing
of the patch mid-points---adjusted a little when $\verb|nEdge|>1$. So
either $\verb|ratio|=\tfrac12$ means the patches abut and
$\verb|ratio|=1$ is overlapping patches as in holistic
discretisation, or $\verb|ratio|=1$ means the patches abut.
Small~\verb|ratio| should greatly reduce computational time.
\item \verb|nSubP| is the number of equi-spaced microscale
lattice points in each patch: if scalar, then use the same
number in all three directions, otherwise \verb|nSubP(1:3)|
gives the number in each direction. If not using
\verb|EdgyInt|, then $\verb|nSubP./nEdge|$ must be odd integer(s) so that there
is/are centre-patch lattice planes. So for the defaults
of $\verb|nEdge|=1$ and not \verb|EdgyInt|, then
\verb|nSubP| must be odd.
\item \verb|'nEdge'|, \emph{optional} (integer---scalar or three element), default=1, the width of face values set by interpolation at the
face regions of each patch. If two elements, then respectively the width in \(x,y\)-directions. The default is one (suitable
for microscale lattices with only nearest neighbour
interactions).
\item \verb|'EdgyInt'|, true/false, \emph{optional},
default=false. If true, then interpolate to left\slash
right\slash top\slash bottom\slash front\slash back
face-values from right\slash left\slash bottom\slash
top\slash back\slash front next-to-face values. If false or
omitted, then interpolate from centre-patch planes.
\item \verb|'nEnsem'|, \emph{optional-experimental},
default one, but if more, then an ensemble over this number
of realisations.
\item \verb|'hetCoeffs'|, \emph{optional}, default empty.
Supply a 3D or 4D array of microscale heterogeneous
coefficients to be used by the given microscale \verb|fun|
in each patch. Say the given array~\verb|cs| is of size
$m_x\times m_y\times m_z\times n_c$, where $n_c$~is the
number of different arrays of coefficients. For example, in
heterogeneous diffusion, $n_c=3$ for the diffusivities in
the \emph{three} different spatial directions (or $n_c=6$
for the diffusivity tensor). The coefficients are to be the
same for each and every patch. However, macroscale
variations are catered for by the $n_c$~coefficients being
$n_c$~parameters in some macroscale formula.
\begin{itemize}
\item If $\verb|nEnsem|=1$, then the array of coefficients
is just tiled across the patch size to fill up each patch,
starting from the $(1,1,1)$-point in each patch. Best accuracy
usually obtained when the periodicity of the coefficients
is a factor of \verb|nSubP-2*nEdge| for \verb|EdgyInt|, or
a factor of \verb|(nSubP-nEdge)/2| for not \verb|EdgyInt|.
\item If $\verb|nEnsem|>1$ (value immaterial), then reset
$\verb|nEnsem|:=m_x\cdot m_y\cdot m_z$ and construct an
ensemble of all $m_x\cdot m_y\cdot m_z$ phase-shifts of
the coefficients. In this scenario, the inter-patch
coupling couples different members in the ensemble. When
\verb|EdgyInt| is true, and when the coefficients are
diffusivities\slash elasticities in $x,y,z$-directions,
respectively, then this coupling cunningly preserves
symmetry.
\end{itemize}
\item \verb|'parallel'|, true/false, \emph{optional},
default=false. If false, then all patch computations are on
the user's main \textsc{cpu}---although a user may well
separately invoke, say, a \textsc{gpu} to accelerate
sub-patch computations.
If true, and it requires that you have \Matlab's Parallel
Computing Toolbox, then it will distribute the patches over
multiple \textsc{cpu}s\slash cores. In \Matlab, only one
array dimension can be split in the distribution, so it
chooses the one space dimension~$x,y,z$ corresponding to
the highest~\verb|nPatch| (if a tie, then chooses the
rightmost of~$x,y,z$). A user may correspondingly
distribute arrays with property \verb|patches.codist|, or
simply use formulas invoking the preset distributed arrays
\verb|patches.x|, \verb|patches.y|, and \verb|patches.z|. If
a user has not yet established a parallel pool, then a
`local' pool is started.
\end{itemize}
\paragraph{Output} The struct \verb|patches| is created and
set with the following components. If no output variable is
provided for \verb|patches|, then make the struct available
as a global variable.\footnote{When using \texttt{spmd}
parallel computing, it is generally best to avoid global
variables, and so instead prefer using an explicit output
variable.}
\begin{matlab}
%}
if nargout==0, global patches, end
patches.version = version;
%{
\end{matlab}
\begin{itemize}
\item \verb|.fun| is the name of the user's function
\verb|fun(t,u,patches)| or \verb|fun(t,u)| or
\verb|fun(t,u,patches,...)| that computes the time
derivatives (or steps) on the patchy lattice.
\item \verb|.ordCC| is the specified order of inter-patch
coupling.
\item \verb|.periodic|: either true, for interpolation on
the macro-periodic domain; or false, for general
interpolation by divided differences over non-periodic
domain or unevenly distributed patches.
\item \verb|.stag| is true for interpolation using only odd
neighbouring patches as for staggered grids, and false for
the usual case of all neighbour coupling---not yet
implemented.
\item \verb|.Cwtsr| and \verb|.Cwtsl| are the
$\verb|ordCC|\times 3$-array of weights for the inter-patch
interpolation onto the right\slash top\slash front and
left\slash bottom\slash back faces (respectively) with
patch:macroscale ratio as specified or as derived
from~\verb|dx|.
\item \verb|.x| (8D) is $\verb|nSubP(1)| \times1 \times1
\times1 \times1 \times \verb|nPatch(1)| \times1 \times1$
array of the regular spatial locations~$x_{iI}$ of the
microscale grid points in every patch.
\item \verb|.y| (8D) is $1 \times \verb|nSubP(2)| \times1
\times1 \times1 \times1 \times \verb|nPatch(2)| \times1$
array of the regular spatial locations~$y_{jJ}$ of the
microscale grid points in every patch.
\item \verb|.z| (8D) is $1 \times1 \times \verb|nSubP(3)|
\times1 \times1 \times1 \times1 \times \verb|nPatch(3)|$
array of the regular spatial locations~$z_{kK}$ of the
microscale grid points in every patch.
\item \verb|.ratio| $1\times 3$, only for macro-periodic
conditions, are the size ratios of every patch.
\item \verb|.nEdge| $1\times 3$, is the width of face
values set by interpolation at the face regions of each
patch, in the \(x,y,z\)-directions respectively.
\item \verb|.le|, \verb|.ri|, \verb|.bo|, \verb|.to|,
\verb|.ba|, \verb|.fr| determine inter-patch coupling of
members in an ensemble. Each a column vector of
length~\verb|nEnsem|.
\item \verb|.cs| either
\begin{itemize}
\item \verb|[]| 0D, or
\item if $\verb|nEnsem|=1$, $(\verb|nSubP(1)|-1)\times
(\verb|nSubP(2)|-1)\times (\verb|nSubP(3)|-1)\times n_c$ 4D
array of microscale heterogeneous coefficients, or
\item if $\verb|nEnsem|>1$, $(\verb|nSubP(1)|-1)\times
(\verb|nSubP(2)|-1)\times (\verb|nSubP(3)|-1)\times
n_c\times m_xm_ym_z$ 5D array of $m_xm_ym_z$~ensemble of
phase-shifts of the microscale heterogeneous coefficients.
\end{itemize}
\item \verb|.parallel|, logical: true if patches are
distributed over multiple \textsc{cpu}s\slash cores for the
Parallel Computing Toolbox, otherwise false (the default is
to activate the \emph{local} pool).
\item \verb|.codist|, \emph{optional}, describes the
particular parallel distribution of arrays over the active
parallel pool.
\end{itemize}
\subsection{If no arguments, then execute an example}
\label{sec:configPatches3eg}
\begin{matlab}
%}
if nargin==0
disp('With no arguments, simulate example of heterogeneous wave')
%{
\end{matlab}
The code here shows one way to get started: a user's script
may have the following three steps (``\into'' denotes function
recursion).
\begin{enumerate}\def\itemsep{-1.5ex}
\item configPatches3
\item ode23 integrator \into patchSys3 \into user's PDE
\item process results
\end{enumerate}
Set random heterogeneous
coefficients of period two in each of the three
directions. Crudely normalise by the harmonic mean so the
macro-wave time scale is roughly one.
\begin{matlab}
%}
mPeriod = [2 2 2];
cHetr = exp(0.9*randn([mPeriod 3]));
cHetr = cHetr*mean(1./cHetr(:))
%{
\end{matlab}
Establish global patch data struct to interface with a
function coding a nonlinear `diffusion' \pde: to be solved
on $[-\pi,\pi]^3$-periodic domain, with $5^3$~patches,
spectral interpolation~($0$) couples the patches, each patch
with micro-grid spacing~$0.22$ (relatively large for
visualisation), and with $4^3$~points forming each patch.
\begin{matlab}
%}
global patches
patches = configPatches3(@heteroWave3,[-pi pi] ...
,'periodic' , 5, 0, 0.22, mPeriod+2 ,'EdgyInt',true ...
,'hetCoeffs',cHetr);
%{
\end{matlab}
Set a wave initial state using auto-replication of the
spatial grid, and as \cref{fig:configPatches3ic} shows. This
wave propagates diagonally across space. Concatenate the two
\(u,v\)-fields to be the two components of the fourth
dimension.
\begin{matlab}
%}
u0 = 0.5+0.5*sin(patches.x+patches.y+patches.z);
v0 = -0.5*cos(patches.x+patches.y+patches.z)*sqrt(3);
uv0 = cat(4,u0,v0);
%{
\end{matlab}
\begin{figure}
\centering \caption{\label{fig:configPatches3ic}initial
field~$u(x,y,z,t)$ at time $t=0$ of the patch scheme
applied to a heterogeneous wave~\pde:
\cref{fig:configPatches3fin} plots the computed field at
time $t=6$.}
\includegraphics[scale=0.9]{configPatches3ic}
\end{figure}
Integrate in time to $t=6$ using standard functions. In
Matlab \verb|ode15s| would be natural as the patch scheme is
naturally stiff, but \verb|ode23| is much quicker
\cite[Fig.~4]{Maclean2020a}.
\begin{matlab}
%}
disp('Simulate heterogeneous wave u_tt=div[C*grad(u)]')
if ~exist('OCTAVE_VERSION','builtin')
[ts,us] = ode23(@patchSys3,linspace(0,6),uv0(:));
else %disp('octave version is very slow for me')
lsode_options('absolute tolerance',1e-4);
lsode_options('relative tolerance',1e-4);
[ts,us] = odeOcts(@patchSys3,[0 1 2],uv0(:));
end
%{
\end{matlab}
Animate the computed simulation to end with
\cref{fig:configPatches3fin}. Use \verb|patchEdgeInt3| to
obtain patch-face values in order to most easily reconstruct
the array data structure.
Replicate $x$, $y$, and~$z$ arrays to get individual
spatial coordinates of every data point. Then, optionally,
set faces to~\verb|nan| so the plot just shows
patch-interior data.
\begin{matlab}
%}
figure(1), clf, colormap(0.8*jet)
xs = patches.x+0*patches.y+0*patches.z;
ys = patches.y+0*patches.x+0*patches.z;
zs = patches.z+0*patches.y+0*patches.x;
if 1, xs([1 end],:,:,:)=nan;
xs(:,[1 end],:,:)=nan;
xs(:,:,[1 end],:)=nan;
end;%option
j=find(~isnan(xs));
%{
\end{matlab}
In the scatter plot, these functions \verb|pix()| and
\verb|col()| map the $u$-data values to the size of the
dots and to the colour of the dots, respectively.
\begin{matlab}
%}
pix = @(u) 15*abs(u)+7;
col = @(u) sign(u).*abs(u);
%{
\end{matlab}
Loop to plot at each and every time step.
\begin{matlab}
%}
for i = 1:length(ts)
uv = patchEdgeInt3(us(i,:));
u = uv(:,:,:,1,:);
for p=1:2
subplot(1,2,p)
if (i==1)| exist('OCTAVE_VERSION','builtin')
scat(p) = scatter3(xs(j),ys(j),zs(j),'filled');
axis equal, caxis(col([0 1])), view(45-5*p,25)
xlabel('$x$'), ylabel('$y$'), zlabel('$z$')
title('view stereo pair cross-eyed')
end % in matlab just update values
set(scat(p),'CData',col(u(j)) ...
,'SizeData',pix((8+xs(j)-ys(j)+zs(j))/6+0*u(j)));
legend(['time = ' num2str(ts(i),'%4.2f')],'Location','north')
end
%{
\end{matlab}
Optionally save the initial condition to graphic file for
\cref{fig:configPatches2ic}, and optionally save the last
plot.
\begin{matlab}
%}
if i==1,
ifOurCf2eps([mfilename 'ic'])
disp('Type space character to animate simulation')
pause
else pause(0.05)
end
end% i-loop over all times
ifOurCf2eps([mfilename 'fin'])
%{
\end{matlab}
\begin{figure}
\centering
\caption{\label{fig:configPatches3fin}field~$u(x,y,z,t)$
at time $t=6$ of the patch scheme applied to the
heterogeneous wave~\pde\ with initial condition in
\cref{fig:configPatches3ic}.}
\includegraphics[scale=0.9]{configPatches3fin}
\end{figure}
Upon finishing execution of the example, exit this function.
\begin{matlab}
%}
return
end%if no arguments
%{
\end{matlab}
\IfFileExists{../Patch/heteroWave3.m}{\input{../Patch/heteroWave3.m}}{}
\begin{devMan}
\subsection{Parse input arguments and defaults}
\begin{matlab}
%}
p = inputParser;
fnValidation = @(f) isa(f, 'function_handle'); %test for fn name
addRequired(p,'fun',fnValidation);
addRequired(p,'Xlim',@isnumeric);
%addRequired(p,'Dom'); % too flexible
addRequired(p,'nPatch',@isnumeric);
addRequired(p,'ordCC',@isnumeric);
addRequired(p,'dx',@isnumeric);
addRequired(p,'nSubP',@isnumeric);
addParameter(p,'nEdge',1,@isnumeric);
addParameter(p,'EdgyInt',false,@islogical);
addParameter(p,'nEnsem',1,@isnumeric);
addParameter(p,'hetCoeffs',[],@isnumeric);
addParameter(p,'parallel',false,@islogical);
%addParameter(p,'nCore',1,@isnumeric); % not yet implemented
parse(p,fun,Xlim,nPatch,ordCC,dx,nSubP,varargin{:});
%{
\end{matlab}
Set the optional parameters.
\begin{matlab}
%}
patches.nEdge = p.Results.nEdge;
if numel(patches.nEdge)==1
patches.nEdge = repmat(patches.nEdge,1,3);
end
patches.EdgyInt = p.Results.EdgyInt;
patches.nEnsem = p.Results.nEnsem;
cs = p.Results.hetCoeffs;
patches.parallel = p.Results.parallel;
%patches.nCore = p.Results.nCore;
%{
\end{matlab}
Initially duplicate parameters for three space dimensions as
needed.
\begin{matlab}
%}
if numel(Xlim)==2, Xlim = repmat(Xlim,1,3); end
if numel(nPatch)==1, nPatch = repmat(nPatch,1,3); end
if numel(dx)==1, dx = repmat(dx,1,3); end
if numel(nSubP)==1, nSubP = repmat(nSubP,1,3); end
%{
\end{matlab}
Check parameters.
\begin{matlab}
%}
assert(Xlim(1)<Xlim(2) ...
,'first pair of Xlim must be ordered increasing')
assert(Xlim(3)<Xlim(4) ...
,'second pair of Xlim must be ordered increasing')
assert(Xlim(5)<Xlim(6) ...
,'third pair of Xlim must be ordered increasing')
assert((mod(ordCC,2)==0)|all(patches.nEdge==1) ...
,'Cannot yet have nEdge>1 and staggered patch grids')
assert(all(3*patches.nEdge<=nSubP) ...
,'too many edge values requested')
assert(all(rem(nSubP,patches.nEdge)==0) ...
,'nSubP must be integer multiple of nEdge')
if ~patches.EdgyInt, assert(all(rem(nSubP./patches.nEdge,2)==1) ...
,'for non-edgyInt, nSubP./nEdge must be odd integer')
end
if (patches.nEnsem>1)&all(patches.nEdge>1)
warning('not yet tested when both nEnsem and nEdge non-one')
end
%if patches.nCore>1
% warning('nCore>1 not yet tested in this version')
% end
%{
\end{matlab}
For compatibility with pre-2023 functions, if parameter
\verb|Dom| is \verb|Nan|, then we set the \verb|ratio| to
be the value of the so-called \verb|dx| vector.
\begin{matlab}
%}
if ~isstruct(Dom), pre2023=isnan(Dom);
else pre2023=false; end
if pre2023, ratio=dx; dx=nan; end
%{
\end{matlab}
Default macroscale conditions are periodic with evenly
spaced patches.
\begin{matlab}
%}
if isempty(Dom), Dom=struct('type','periodic'); end
if (~isstruct(Dom))&isnan(Dom), Dom=struct('type','periodic'); end
%{
\end{matlab}
If \verb|Dom| is a string, then just set type to that
string, and subsequently set corresponding defaults for
others fields.
\begin{matlab}
%}
if ischar(Dom), Dom=struct('type',Dom); end
%{
\end{matlab}
We allow different macroscale domain conditions in the
different directions. But for the moment do not allow
periodic to be mixed with the others (as the interpolation
mechanism is different code)---hence why we choose
\verb|periodic| be seven characters, whereas the others are
eight characters. The different conditions are coded in
different rows of \verb|Dom.type|, so we duplicate the
string if only one row specified.
\begin{matlab}
%}
if size(Dom.type,1)==1, Dom.type=repmat(Dom.type,3,1); end
%{
\end{matlab}
Check what is and is not specified, and provide default of
Dirichlet boundaries if no \verb|bcOffset| specified when
needed. Do so for all three directions independently.
\begin{matlab}
%}
patches.periodic=false;
for p=1:3
switch Dom.type(p,:)
case 'periodic'
patches.periodic=true;
if isfield(Dom,'bcOffset')
warning('bcOffset not available for Dom.type = periodic'), end
msg=' not available for Dom.type = periodic';
if isfield(Dom,'X'), warning(['X' msg]), end
if isfield(Dom,'Y'), warning(['Y' msg]), end
if isfield(Dom,'Z'), warning(['Z' msg]), end
case {'equispace','chebyshev'}
if ~isfield(Dom,'bcOffset'), Dom.bcOffset=zeros(2,3); end
% for mixed with usergiven, following should still work
if numel(Dom.bcOffset)==1
Dom.bcOffset=repmat(Dom.bcOffset,2,3); end
if numel(Dom.bcOffset)==3
Dom.bcOffset=repmat(Dom.bcOffset(:)',2,1); end
msg=' not available for Dom.type = equispace or chebyshev';
if (p==1)& isfield(Dom,'X'), warning(['X' msg]), end
if (p==2)& isfield(Dom,'Y'), warning(['Y' msg]), end
if (p==3)& isfield(Dom,'Z'), warning(['Z' msg]), end
case 'usergiven'
% if isfield(Dom,'bcOffset')
% warning('bcOffset not available for usergiven Dom.type'), end
msg=' required for Dom.type = usergiven';
if p==1, assert(isfield(Dom,'X'),['X' msg]), end
if p==2, assert(isfield(Dom,'Y'),['Y' msg]), end
if p==3, assert(isfield(Dom,'Z'),['Z' msg]), end
otherwise
error([Dom.type ' is unknown Dom.type'])
end%switch Dom.type
end%for p
%{
\end{matlab}
\subsection{The code to make patches}
First, store the pointer to the time derivative function in
the struct.
\begin{matlab}
%}
patches.fun = fun;
%{
\end{matlab}
Second, store the order of interpolation that is to provide
the values for the inter-patch coupling conditions. Spectral
coupling is \verb|ordCC| of~$0$ or (not yet??)~$-1$.
\begin{matlab}
%}
assert((ordCC>=-1) & (floor(ordCC)==ordCC), ...
'ordCC out of allowed range integer>=-1')
%{
\end{matlab}
For odd~\verb|ordCC| do interpolation based upon odd
neighbouring patches as is useful for staggered grids.
\begin{matlab}
%}
patches.stag = mod(ordCC,2);
assert(patches.stag==0,'staggered not yet implemented??')
ordCC = ordCC+patches.stag;
patches.ordCC = ordCC;
%{
\end{matlab}
Check for staggered grid and periodic case.
\begin{matlab}
%}
if patches.stag, assert(all(mod(nPatch,2)==0), ...
'Require an even number of patches for staggered grid')
end
%{
\end{matlab}
\paragraph{Set the macro-distribution of patches}
Third, set the centre of the patches in the macroscale grid
of patches. Loop over the coordinate directions, setting
the distribution into~\verb|Q| and finally assigning to
array of corresponding direction.
\begin{matlab}
%}
for q=1:3
qq=2*q-1;
%{
\end{matlab}
Distribution depends upon \verb|Dom.type|:
\begin{matlab}
%}
switch Dom.type(q,:)
%{
\end{matlab}
%: case periodic
The periodic case is evenly spaced within the spatial
domain. Store the size ratio in \verb|patches|.
\begin{matlab}
%}
case 'periodic'
Q=linspace(Xlim(qq),Xlim(qq+1),nPatch(q)+1);
DQ=Q(2)-Q(1);
Q=Q(1:nPatch(q))+diff(Q)/2;
pEI=patches.EdgyInt;% abbreviation
pnE=patches.nEdge(q);% abbreviation
if pre2023, dx(q) = ratio(q)*DQ/(nSubP(q)-pnE*(1+pEI))*(2-pEI);
else ratio(q) = dx(q)/DQ*(nSubP(q)-pnE*(1+pEI))/(2-pEI);
end
patches.ratio=ratio;
%{
\end{matlab}
%: case equispace
The equi-spaced case is also evenly spaced but with the
extreme edges aligned with the spatial domain boundaries,
modified by the offset.
\begin{matlab}
%}
case 'equispace'
Q=linspace(Xlim(qq)+((nSubP(q)-1)/2-Dom.bcOffset(qq))*dx(q) ...
,Xlim(qq+1)-((nSubP(q)-1)/2-Dom.bcOffset(qq+1))*dx(q) ...
,nPatch(q));
DQ=diff(Q(1:2));
width=(1+patches.EdgyInt)/2*(nSubP(q)-1-patches.EdgyInt)*dx;
if DQ<width*0.999999
warning('too many equispace patches (double overlapping)')
end
%{
\end{matlab}
%: case chebyshev
The Chebyshev case is spaced according to the Chebyshev
distribution in order to reduce macro-interpolation errors,
\(Q_i \propto -\cos(i\pi/N)\), but with the extreme edges
aligned with the spatial domain boundaries, modified by the
offset, and modified by possible `boundary layers'.
\footnote{ However, maybe overlapping patches near a
boundary should be viewed as some sort of spatially analogue
of the `christmas tree' of projective integration and its
integration to a slow manifold. Here maybe the overlapping
patches allow for a `christmas tree' approach to the
boundary layers. Needs to be explored??}
\begin{matlab}
%}
case 'chebyshev'
halfWidth=dx(q)*(nSubP(q)-1)/2;
Q1 = Xlim(1)+halfWidth-Dom.bcOffset(qq)*dx(q);
Q2 = Xlim(2)-halfWidth+Dom.bcOffset(qq+1)*dx(q);
% Q = (Q1+Q2)/2-(Q2-Q1)/2*cos(linspace(0,pi,nPatch));
%{
\end{matlab}
Search for total width of `boundary layers' so that in the
interior the patches are non-overlapping Chebyshev. But
the width for assessing overlap of patches is the following
variable \verb|width|.
\begin{matlab}
%}
pEI=patches.EdgyInt; % abbreviation
pnE=patches.nEdge(q);% abbreviation
width=(1+pEI)/2*(nSubP(q)-pnE*(1+pEI))*dx(q);
for b=0:2:nPatch(q)-2
DQmin=(Q2-Q1-b*width)/2*( 1-cos(pi/(nPatch(q)-b-1)) );
if DQmin>width, break, end
end%for
if DQmin<width*0.999999
warning('too many Chebyshev patches (mid-domain overlap)')
end%if
%{
\end{matlab}
Assign the centre-patch coordinates.
\begin{matlab}
%}
Q =[ Q1+(0:b/2-1)*width ...
(Q1+Q2)/2-(Q2-Q1-b*width)/2*cos(linspace(0,pi,nPatch(q)-b)) ...
Q2+(1-b/2:0)*width ];
%{
\end{matlab}
%: case usergiven
The user-given case is entirely up to a user to specify, we
just ensure it has the correct shape of a row.
\begin{matlab}
%}
case 'usergiven'
if q==1, Q = reshape(Dom.X,1,[]); end
if q==2, Q = reshape(Dom.Y,1,[]); end
if q==3, Q = reshape(Dom.Z,1,[]); end
end%switch Dom.type
%{
\end{matlab}
Assign \(Q\)-coordinates to the correct spatial direction.
At this stage they are all rows.
\begin{matlab}
%}
if q==1, X=Q; end
if q==2, Y=Q; end
if q==3, Z=Q; end
end%for q
%{
\end{matlab}
\paragraph{Construct the micro-grids}
Fourth, construct the microscale grid in each patch, centred
about the given mid-points~\verb|X,Y,Z|. Reshape the grid to be
8D to suit dimensions (micro,Vars,Ens,macro).
\begin{matlab}
%}
xs = dx(1)*( (1:nSubP(1))-mean(1:nSubP(1)) );
patches.x = reshape( xs'+X ...
,nSubP(1),1,1,1,1,nPatch(1),1,1);
ys = dx(2)*( (1:nSubP(2))-mean(1:nSubP(2)) );
patches.y = reshape( ys'+Y ...
,1,nSubP(2),1,1,1,1,nPatch(2),1);
zs = dx(3)*( (1:nSubP(3))-mean(1:nSubP(3)) );
patches.z = reshape( zs'+Z ...
,1,1,nSubP(3),1,1,1,1,nPatch(3));
%{
\end{matlab}
\paragraph{Pre-compute weights for macro-periodic} In the
case of macro-periodicity, precompute the weightings to
interpolate field values for coupling. \todo{Might sometime
extend to coupling via derivative values.}
\begin{matlab}
%}
if patches.periodic
ratio = reshape(ratio,1,3); % force to be row vector
patches.ratio = ratio;
if ordCC>0
[Cwtsr,Cwtsl] = patchCwts(ratio,ordCC,patches.stag);
patches.Cwtsr = Cwtsr; patches.Cwtsl = Cwtsl;
end%if
end%if patches.periodic
%{
\end{matlab}
\subsection{Set ensemble inter-patch communication}
For \verb|EdgyInt| or centre interpolation respectively,
\begin{itemize}
\item the right-face\slash centre realisations
\verb|1:nEnsem| are to interpolate to left-face~\verb|le|,
and
\item the left-face\slash centre realisations
\verb|1:nEnsem| are to interpolate to~\verb|re|.
\end{itemize}
\verb|re| and \verb|li| are `transposes' of each other as
\verb|re(li)=le(ri)| are both \verb|1:nEnsem|. Similarly for
bottom-face\slash centre interpolation to top-face
via~\verb|to|, top-face\slash centre interpolation to
bottom-face via~\verb|bo|, back-face\slash centre
interpolation to front-face via~\verb|fr|, and
front-face\slash centre interpolation to back-face
via~\verb|ba|.
The default is nothing shifty. This setting reduces the
number of if-statements in function \verb|patchEdgeInt3()|.
\begin{matlab}
%}
nE = patches.nEnsem;
patches.le = 1:nE; patches.ri = 1:nE;
patches.bo = 1:nE; patches.to = 1:nE;
patches.ba = 1:nE; patches.fr = 1:nE;
%{
\end{matlab}
However, if heterogeneous coefficients are supplied via
\verb|hetCoeffs|, then do some non-trivial replications.
First, get microscale periods, patch size, and replicate
many times in order to subsequently sub-sample: \verb|nSubP|
times should be enough. If \verb|cs| is more then 4D, then
the higher-dimensions are reshaped into the 4th dimension.
\begin{matlab}
%}
if ~isempty(cs)
[mx,my,mz,nc] = size(cs);
nx = nSubP(1); ny = nSubP(2); nz = nSubP(3);
cs = repmat(cs,nSubP);
%{
\end{matlab}
If only one member of the ensemble is required, then
sub-sample to patch size, and store coefficients in
\verb|patches| as is.
\begin{matlab}
%}
if nE==1, patches.cs = cs(1:nx-1,1:ny-1,1:nz-1,:); else
%{
\end{matlab}
But for $\verb|nEnsem|>1$ an ensemble of
$m_xm_ym_z$~phase-shifts of the coefficients is
constructed from the over-supply. Here code phase-shifts
over the periods---the phase shifts are like
Hankel-matrices.
\begin{matlab}
%}
patches.nEnsem = mx*my*mz;
patches.cs = nan(nx-1,ny-1,nz-1,nc,mx,my,mz);
for k = 1:mz
ks = (k:k+nz-2);
for j = 1:my
js = (j:j+ny-2);
for i = 1:mx
is = (i:i+nx-2);
patches.cs(:,:,:,:,i,j,k) = cs(is,js,ks,:);
end
end
end
patches.cs = reshape(patches.cs,nx-1,ny-1,nz-1,nc,[]);
%{
\end{matlab}
Further, set a cunning left\slash right\slash bottom\slash
top\slash front\slash back realisation of inter-patch
coupling. The aim is to preserve symmetry in the system
when also invoking \verb|EdgyInt|. What this coupling does
without \verb|EdgyInt| is unknown. Use auto-replication.
\begin{matlab}
%}
mmx=(0:mx-1)'; mmy=0:my-1; mmz=shiftdim(0:mz-1,-1);
le = mod(mmx+mod(nx-2,mx),mx)+1;
patches.le = reshape( le+mx*(mmy+my*mmz) ,[],1);
ri = mod(mmx-mod(nx-2,mx),mx)+1;
patches.ri = reshape( ri+mx*(mmy+my*mmz) ,[],1);
bo = mod(mmy+mod(ny-2,my),my)+1;
patches.bo = reshape( 1+mmx+mx*(bo-1+my*mmz) ,[],1);
to = mod(mmy-mod(ny-2,my),my)+1;
patches.to = reshape( 1+mmx+mx*(to-1+my*mmz) ,[],1);
ba = mod(mmz+mod(nz-2,mz),mz)+1;
patches.ba = reshape( 1+mmx+mx*(mmy+my*(ba-1)) ,[],1);
fr = mod(mmz-mod(nz-2,mz),mz)+1;
patches.fr = reshape( 1+mmx+mx*(mmy+my*(fr-1)) ,[],1);
%{
\end{matlab}
Issue warning if the ensemble is likely to be affected by
lack of scale separation. \todo{Need to justify this and
the arbitrary threshold more carefully??}
\begin{matlab}
%}
if prod(ratio)*patches.nEnsem>0.9, warning( ...
'Probably poor scale separation in ensemble of coupled phase-shifts')
scaleSeparationParameter = ratio*patches.nEnsem
end
%{
\end{matlab}
End the two if-statements.
\begin{matlab}
%}
end%if-else nEnsem>1
end%if not-empty(cs)
%{
\end{matlab}
\paragraph{If parallel code} then first assume this is not
within an \verb|spmd|-environment, and so we invoke
\verb|spmd...end| (which starts a parallel pool if not
already started). At this point, the global \verb|patches|
is copied for each worker processor and so it becomes
\emph{composite} when we distribute any one of the fields.
Hereafter, {\em all fields in the global variable
\verb|patches| must only be referenced within an
\verb|spmd|-environment.}%
\footnote{If subsequently outside spmd, then one must use
functions like \texttt{getfield(patches\{1\},'a')}.}
\begin{matlab}
%}
if patches.parallel
spmd
%{
\end{matlab}
Second, decide which dimension is to be sliced among
parallel workers (for the moment, do not consider slicing
the ensemble). Choose the direction of most patches, biased
towards the last.
\begin{matlab}
%}
[~,pari]=max(nPatch+0.01*(1:3));
patches.codist=codistributor1d(5+pari);
%{
\end{matlab}
\verb|patches.codist.Dimension| is the index that is split
among workers. Then distribute the appropriate coordinate
direction among the workers: the function must be invoked
inside an \verb|spmd|-group in order for this to work---so
we do not need \verb|parallel| in argument list.
\begin{matlab}
%}
switch pari
case 1, patches.x=codistributed(patches.x,patches.codist);
case 2, patches.y=codistributed(patches.y,patches.codist);
case 3, patches.z=codistributed(patches.z,patches.codist);
otherwise
error('should never have bad index for parallel distribution')
end%switch
end%spmd
%{
\end{matlab}
If not parallel, then clean out \verb|patches.codist| if it exists.
May not need, but safer.
\begin{matlab}
%}
else% not parallel
if isfield(patches,'codist'), rmfield(patches,'codist'); end
end%if-parallel
%{
\end{matlab}
\paragraph{Fin}
\begin{matlab}
%}
end% function
%{
\end{matlab}
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
quasiLogAxes.m
|
.m
|
EquationFreeGit-master/Patch/quasiLogAxes.m
| 7,787 |
utf_8
|
2e66a0625f73ccf8ead15ef75a5bde19
|
% quasiLogAxes() transforms selected axes of the given plot
% to a quasi-log axes (via asinh), including possibly
% transforming color axis. AJR, 25 Sep 2021 -- 18 Apr 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{quasiLogAxes()}: transforms some axes of a
plot to quasi-log}
\label{sec:quasiLogAxes}
This function rescales some coordinates and labels the axes
of the given 2D or 3D~plot. The original aim was to
effectively show the complex spectrum of multiscale systems
such as the patch scheme. The eigenvalues are over a wide
range of magnitudes, but are signed. So we use a nonlinear
asinh transformation of the axes, and then label the axes
with reasonable ticks. The nonlinear rescaling is useful in
other scenarios also.
\begin{matlab}
%}
function quasiLogAxes(handle,xScale,yScale,zScale,cScale)
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|handle|: handle to your plot to transform, for
example, obtained by \verb|handle=plot(...)|
\item \verb|xScale| (optional, default~inf): if inf, then no
transformation is done in the `x'-coordinate. Otherwise, when
\verb|xScale| is not inf, transforms the plot \(x\)-coordinates
with the \(\text{asinh}()\) function so that
\begin{itemize}
\item for \(|x|\lesssim x_{\text{scale}}\) the x-axis scaling
is approximately linear, whereas
\item for \(|x|\gtrsim x_{\text{scale}}\) the x-axis scaling
is approximately signed-logarithmic.
\end{itemize}
\item \verb|yScale| (optional, default~inf): corresponds to
\verb|xScale| for the second axis scaling.
\item \verb|zScale| (optional, default~inf): corresponds to
\verb|xScale| for a third axis scaling if it exists.
\item \verb|cScale| (optional, default~inf): corresponds to
\verb|xScale| but for a colormap, and colorbar scaling if
one exists.
\end{itemize}
\paragraph{Output} None, just the transformed plot.
\paragraph{Example}
If invoked with no arguments, then execute an example.
\begin{matlab}
%}
if nargin==0
% generate some data
n=99; fast=(rand(n,1)<0.8);
z = -rand(n,1).*(1+1e3*fast)+1i*randn(n,1).*(5+1e2*fast);
% plot data and transform axes
handle = plot(real(z),imag(z),'.');
xlabel('real-part'), ylabel('imag-part')
quasiLogAxes(handle,1,10);
return
end% example
%{
\end{matlab}
Default values for scaling, \verb|inf| denotes no
transformation of that axis.
\begin{matlab}
%}
if nargin<5, cScale=inf; end
if nargin<4, zScale=inf; end
if nargin<3, yScale=inf; end
if nargin<2, xScale=inf; end
%{
\end{matlab}
\begin{devMan}
Get current limits of the plot to use if the user has set
them already. And also get the pointer to the axes and to
the figure of the plot.
\begin{matlab}
%}
xlim0=xlim; ylim0=ylim; zlim0=zlim; clim0=caxis;
theAxes = get(handle(1),'parent');
theFig = get(theAxes,'parent');
%{
\end{matlab}
Find overall factors so the data is nonlinearly mapped to
order oneish---so that then pgfplots et al.\ do not think
there is an overall scaling factor on the axes.
\begin{matlab}
%}
xFac=1e-99; yFac=xFac; zFac=xFac; cFac=xFac;
for k=1:length(handle)
if ~isinf(xScale)
temp = asinh(handle(k).XData/xScale);
xFac = max(xFac, max(abs(temp(:)),[],'omitnan') );
end
if ~isinf(yScale)
temp = asinh(handle(k).YData/yScale);
yFac = max(yFac, max(abs(temp(:)),[],'omitnan') );
end
if ~isinf(zScale)
temp = asinh(handle(k).ZData/zScale);
zFac = max(zFac, max(abs(temp(:)),[],'omitnan') );
end
if ~isinf(cScale)
temp = asinh(handle(k).CData/cScale);
cFac = max(cFac, max(abs(temp(:)),[],'omitnan') );
end
end%for
xFac=9/xFac; yFac=9/yFac; zFac=9/zFac; cFac=9/cFac;
%{
\end{matlab}
Scale the plot data in the plot \verb|handle|. Give an
error if it appears that the plot-data has already been
transformed. Color data has to be transformed first
because usually there is automatic flow from z-data to c-data.
\begin{matlab}
%}
for k=1:length(handle)
assert(~strcmp(handle(k).UserData,'quasiLogAxes'), ...
'Replot graph---it appears plot data is already transformed')
if ~isinf(cScale)
handle(k).CData = cFac*asinh(handle(k).CData/cScale);
clim1=[min(handle(k).CData(:)) max(handle(k).CData(:))];
end
if ~isinf(xScale)
handle(k).XData = xFac*asinh(handle(k).XData/xScale);
xlim1=[min(handle(k).XData(:)) max(handle(k).XData(:))];
end
if ~isinf(yScale)
handle(k).YData = yFac*asinh(handle(k).YData/yScale);
ylim1=[min(handle(k).YData(:)) max(handle(k).YData(:))];
end
if ~isinf(zScale)
handle(k).ZData = zFac*asinh(handle(k).ZData/zScale);
zlim1=[min(handle(k).ZData(:)) max(handle(k).ZData(:))];
end
handle(k).UserData = 'quasiLogAxes';
end%for
%{
\end{matlab}
Set 4\%~padding around all margins of transformed
data---crude but serviceable. Unless the axis had already
been manually set, in which case use the transformed set
limits.
\begin{matlab}
%}
if ~isinf(xScale),
if xlim('mode')=="manual"
xlim1=xFac*asinh(xlim0/xScale);
else xlim1=xlim1+0.04*diff(xlim1)*[-1 1];
end, end
if ~isinf(yScale),
if ylim('mode')=="manual"
ylim1=yFac*asinh(ylim0/yScale);
else ylim1=ylim1+0.04*diff(ylim1)*[-1 1];
end, end
if ~isinf(zScale),
if zlim('mode')=="manual"
zlim1=zFac*asinh(zlim0/zScale);
else zlim1=zlim1+0.04*diff(zlim1)*[-1 1];
end, end
if ~isinf(cScale),
if theAxes.CLimMode=="manual"
clim1=cFac*asinh(clim0/cScale);
else clim1=clim1+ 0*diff(clim1)*[-1 1];
end, end
%{
\end{matlab}
\paragraph{Scale axes, and tick marks on axes}
\begin{matlab}
%}
if ~isinf(xScale)
xlim(xlim1);
tickingQuasiLogAxes(theAxes,'X',xlim1,xScale,xFac)
end%if
if ~isinf(yScale)
ylim(ylim1);
tickingQuasiLogAxes(theAxes,'Y',ylim1,yScale,yFac)
end%if
if ~isinf(zScale)
zlim(zlim1);
tickingQuasiLogAxes(theAxes,'Z',zlim1,zScale,zFac)
end%if
%{
\end{matlab}
But for color, only tick when we find a colorbar.
\begin{matlab}
%}
if ~isinf(cScale)
caxis(clim1);
for p=1:numel(theFig.Children)
ca = theFig.Children(p);
if class(ca) == "matlab.graphics.illustration.ColorBar"
tickingQuasiLogAxes(ca,'C',clim1,cScale,cFac)
break
end
end
end%if
%{
\end{matlab}
Turn the grid on by default.
\begin{matlab}
%}
grid on
end%function
%{
\end{matlab}
\subsection{\texttt{tickingQuasiLogAxes()}: typeset ticks
and labels on an axis}
\begin{matlab}
%}
function tickingQuasiLogAxes(ca,Q,qlim1,qScale,qFac)
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|ca|: pointer to axes/colorbar dataset.
\item \verb|Q|: character, either \texttt{X,Y,Z,C}.
\item \verb|qlim1|: the scaled limits of the axis.
\item \verb|qScale|: the scaling parameter for the axis.
\item \verb|qFac|: the scaling factor for the axis.
\end{itemize}
\paragraph{Output} None, just the ticked and labelled axes.
Get the order of magnitude of the horizontal data.
\begin{matlab}
%}
qmax=max(abs(qlim1));
qmag=floor(log10(qScale*sinh(qmax/qFac)));
%{
\end{matlab}
Form a range of ticks, geometrically spaced, trim off the
small values that would be too dense near zero (omit those
within 6\% of \verb|qmax|).
\begin{matlab}
%}
ticks=10.^(qmag+(-7:0));
j=find(ticks>qScale*sinh(0.06*qmax/qFac));
nj=length(j);
if nj<3, ticks=[1;2;5]*ticks(j);
elseif nj<5, ticks=[1;3]*ticks(j);
else ticks=ticks(j);
end
ticks=sort([0;ticks(:);-ticks(:)]);
%{
\end{matlab}
Set the ticks in place according to the transformation.
\begin{matlab}
%}
if Q=='C', p='s'; Q=''; else p=''; end
set(ca,[Q 'Tick' p],qFac*asinh(ticks/qScale) ...
,[Q 'TickLabel' p],cellstr(num2str(ticks,4)))
if Q=='X', set(ca,[Q 'TickLabelRotation'],40), end
end%function qScaling
%{
\end{matlab}
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
idealWavePDE.m
|
.m
|
EquationFreeGit-master/Patch/idealWavePDE.m
| 2,006 |
utf_8
|
0a2c31be68f6887c7e2742e1896492fe
|
% Codes the ideal wave PDE on a staggered 1D grid inside
% patches in space. Used by waterWaveExample.m
% AJR, 4 Apr 2019 -- Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{idealWavePDE()}: ideal wave PDE}
\label{sec:idealWavePDE}
This function codes the staggered lattice equation inside
the patches for the ideal wave \pde\ system \(h_t=-u_x\) and
\(u_t=-h_x\). Here code for a staggered micro-grid,
index~\(i\), of staggered macroscale patches, index~\(j\):
the array
\begin{equation*}
U_{ij}=\begin{cases} u_{ij}&i+j\text{ even},\\
h_{ij}& i+j\text{ odd}.
\end{cases}
\end{equation*}
The output~\verb|Ut| contains the merged time derivatives of
the two staggered fields. So set the micro-grid spacing and
reserve space for time derivatives.
\begin{matlab}
%}
function Ut = idealWavePDE(t,U,patches)
dx = diff(patches.x(2:3));
U = squeeze(U);
Ut = nan(size(U)); ht = Ut;
%{
\end{matlab}
Compute the \pde\ derivatives only at interior micro-grid
points of the patches.
\begin{matlab}
%}
i = 2:size(U,1)-1;
%{
\end{matlab}
Here `wastefully' compute time derivatives for both \pde{}s
at all grid points---for simplicity---and then merge the
staggered results. Since \(\dot h_{ij} \approx -(u_{i+1,j}
-u_{i-1,j}) /(2\cdot dx) =-(U_{i+1,j} -U_{i-1,j}) /(2\cdot
dx)\) as adding\slash subtracting one from the index of a
\(h\)-value is the location of the neighbouring \(u\)-value
on the staggered micro-grid.
\begin{matlab}
%}
ht(i,:) = -(U(i+1,:)-U(i-1,:))/(2*dx);
%{
\end{matlab}
Since \(\dot u_{ij} \approx -(h_{i+1,j} -h_{i-1,j}) /(2\cdot
dx) =-(U_{i+1,j} -U_{i-1,j}) /(2\cdot dx)\) as adding\slash
subtracting one from the index of a \(u\)-value is the
location of the neighbouring \(h\)-value on the staggered
micro-grid.
\begin{matlab}
%}
Ut(i,:) = -(U(i+1,:)-U(i-1,:))/(2*dx);
%{
\end{matlab}
Then overwrite the unwanted~\(\dot u_{ij}\) with the
corresponding wanted~\(\dot h_{ij}\).
\begin{matlab}
%}
Ut(patches.hPts) = ht(patches.hPts);
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroWave.m
|
.m
|
EquationFreeGit-master/Patch/heteroWave.m
| 1,364 |
utf_8
|
b07ccdac5060fc6af73096ee80efc543
|
% Computes the time derivatives of heterogeneous wave
% in 1D on patches. Used by homoWaveEdgy1.m,
% AJR, 26 Nov 2019 -- Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{heteroWave()}: wave in heterogeneous
media with weak viscous damping}
\label{sec:heteroWave}
This function codes the lattice heterogeneous wave equation,
with weak viscosity, inside the patches. For 3D input
array~\verb|u| (\(u_{ij} = \verb|u(i,1,j)|\) and \(v_{ij} =
\verb|u(i,2,j)|\)) and 2D array~\verb|x| (obtained in full
via edge-value interpolation of \verb|patchSys1|,
\cref{sec:patchSys1}), computes the time derivatives at
each point in the interior of a patch, output in~\verb|ut|:
\begin{equation*}
\D t{u_{ij}}=v_{ij}\,,\quad
\D t{v_{ij}}= \frac1{dx^2}\delta[c_{i-1/2}\delta u_{ij}]
+\frac{0.02}{dx^2}\delta^2 v_{ij}\,.
\end{equation*}
The column vector (or possibly array) of diffusion
coefficients~\(c_i\) have previously been stored in
struct~\verb|patches|.
\begin{matlab}
%}
function ut = heteroWave(t,u,patches)
u = squeeze(u);
dx = diff(patches.x(2:3)); % space step
i = 2:size(u,1)-1; % interior points in a patch
ut = nan(size(u)); % preallocate output array
ut(i,1,:) = u(i,2,:); % du/dt=v then dvdt=
ut(i,2,:) = diff(patches.cs.*diff(u(:,1,:)))/dx^2 ...
+0.02*diff(u(:,2,:),2)/dx^2;
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroLanLif1D.m
|
.m
|
EquationFreeGit-master/Patch/heteroLanLif1D.m
| 2,278 |
utf_8
|
d1c006eb26861126db87a18042695e33
|
% Computes the time derivatives of heterogeneous
% Landau--Lifshitz PDE on 1D lattice within spatial patches.
% From Leitenmaier & Runborg, arxiv.org/abs/2108.09463 and
% used by homoLanLif1D.m AJR, Sep 2021
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{heteroLanLif1D()}: heterogeneous Landau--Lifshitz PDE}
\label{sec:heteroLanLif1D}
This function codes the lattice heterogeneous
Landau--Lifshitz PDE \cite[(1.1)]{Leitenmaier2021} inside
patches in 1D space. For 4D input array~\verb|M| storing
the three components of~\Mv\ (via edge-value interpolation
of \verb|patchSys1|, \cref{sec:patchSys1}), computes
the time derivative at each point in the interior of a
patch, output in~\verb|Mt|. The column vector of
coefficients \(c_i=1+\tfrac12\sin(2\pi x_i/\epsilon)\) have
previously been stored in struct~\verb|patches.cs|.
\begin{itemize}
\item With \verb|ex5p1=0| computes the example \textsc{ex1}
\cite[p.6]{Leitenmaier2021}.
\item With \verb|ex5p1=1| computes the first 'locally
periodic' example \cite[p.27]{Leitenmaier2021}.
\end{itemize}
\begin{matlab}
%}
function Mt = heteroLanLif1D(t,M,patches)
global alpha ex5p1
dx = diff(patches.x(2:3)); % space step
%{
\end{matlab}
Compute the heterogeneous \(\Hv:=\divv(a\grad\Mv)\)
\begin{matlab}
%}
a = patches.cs ...
+ex5p1*(0.1+0.25*sin(2*pi*(patches.x(2:end,:,:,:)-dx/2)+1.1));
H = diff(a.*diff(M))/dx^2;
%{
\end{matlab}
At each microscale grid point, compute the cross-products
\(\Mv\times \Hv\) and \(\Mv\times(\Mv\times \Hv)\) to then
give the time derivative \(\Mv_t=-\Mv\times \Hv -\alpha \Mv\times (\Mv\times \Hv)\) \cite[(1.1)]{Leitenmaier2021}:
\begin{matlab}
%}
i = 2:size(M,1)-1; % interior points in a patch
MH=nan+H; % preallocate for MxH
MH(:,3,:,:) = M(i,1,:,:).*H(:,2,:,:)-M(i,2,:,:).*H(:,1,:,:);
MH(:,2,:,:) = M(i,3,:,:).*H(:,1,:,:)-M(i,1,:,:).*H(:,3,:,:);
MH(:,1,:,:) = M(i,2,:,:).*H(:,3,:,:)-M(i,3,:,:).*H(:,2,:,:);
MMH=nan+H; % preallocate for MxMxH
MMH(:,3,:,:)= M(i,1,:,:).*MH(:,2,:,:)-M(i,2,:,:).*MH(:,1,:,:);
MMH(:,2,:,:)= M(i,3,:,:).*MH(:,1,:,:)-M(i,1,:,:).*MH(:,3,:,:);
MMH(:,1,:,:)= M(i,2,:,:).*MH(:,3,:,:)-M(i,3,:,:).*MH(:,2,:,:);
Mt = nan+M; % preallocate output array
Mt(i,:,:,:) = -MH-alpha*MMH;
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroDiff3.m
|
.m
|
EquationFreeGit-master/Patch/heteroDiff3.m
| 1,917 |
utf_8
|
1cd7c41c67dc5bb4941e58b0db310269
|
% heteroDiff3() computes the time derivatives of
% heterogeneous diffusion in 3D on patches. Adapted from 2D
% heterogeneous diffusion. JEB & AJR, May--Sep 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{heteroDiff3()}: heterogeneous diffusion}
\label{sec:heteroDiff3}
This function codes the lattice heterogeneous diffusion
inside the patches. For 8D input array~\verb|u| (via
edge-value interpolation of \verb|patchEdgeInt3|, such as by
\verb|patchSys3|, \cref{sec:patchSys3}), computes the
time derivative~\cref{eq:HomogenisationExample} at each
point in the interior of a patch, output in~\verb|ut|. The
three 3D array of diffusivities,~$c^x_{ijk}$, $c^y_{ijk}$
and~$c^z_{ijk}$, have previously been stored
in~\verb|patches.cs| (4+D).
Supply patch information as a third argument (required by
parallel computation), or otherwise by a global variable.
\begin{matlab}
%}
function ut = heteroDiff3(t,u,patches)
if nargin<3, global patches, end
%{
\end{matlab}
Microscale space-steps.
Q: is using \verb|i,j,k| slower than \verb|2:end-1|??
\begin{matlab}
%}
dx = diff(patches.x(2:3)); % x micro-scale step
dy = diff(patches.y(2:3)); % y micro-scale step
dz = diff(patches.z(2:3)); % z micro-scale step
i = 2:size(u,1)-1; % x interior points in a patch
j = 2:size(u,2)-1; % y interior points in a patch
k = 2:size(u,3)-1; % y interior points in a patch
%{
\end{matlab}
Reserve storage and then assign interior patch values to the
heterogeneous diffusion time derivatives. Using \verb|nan+u|
appears quicker than \verb|nan(size(u),patches.codist)|
\begin{matlab}
%}
ut = nan+u; % reserve storage
ut(i,j,k,:,:,:,:,:) ...
= diff(patches.cs(:,j,k,1,:).*diff(u(:,j,k,:,:,:,:,:),1),1)/dx^2 ...
+diff(patches.cs(i,:,k,2,:).*diff(u(i,:,k,:,:,:,:,:),1,2),1,2)/dy^2 ...
+diff(patches.cs(i,j,:,3,:).*diff(u(i,j,:,:,:,:,:,:),1,3),1,3)/dz^2;
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
monoscaleDiffEquil2.m
|
.m
|
EquationFreeGit-master/Patch/monoscaleDiffEquil2.m
| 6,326 |
utf_8
|
e4091ed6937be2ebd3555e0a9985035e
|
% Solve for steady state of monoscale heterogeneous
% diffusion in 2D on patches as an example application, from
% section 5.2 of Freese, 2211.13731. AJR, 31 Jan 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{monoscaleDiffEquil2}: equilibrium of a 2D
monoscale heterogeneous diffusion via small patches}
\label{sec:monoscaleDiffEquil2}
Here we find the steady state~\(u(x,y)\), see
\cref{fig:monoscaleDiffEquil2}, to the heterogeneous \pde
(inspired by Freese et al.\footnote{ \protect
\url{http://arxiv.org/abs/2211.13731}} \S5.2)
\begin{equation*}
u_t=A(x,y)\grad\grad u-f,
\end{equation*}
on domain \([-1,1]^2\) with Dirichlet BCs, for coefficient
pseudo-diffusion matrix
\begin{equation*}
A:=\begin{bmatrix} 2& a\\a & 2 \end{bmatrix}
\quad \text{with } a:=\sign(xy)
\text{ or }a:=\sin(\pi x)\sin(\pi y),
\end{equation*}
and for forcing~\(f(x,y)\) such that the exact equilibrium
is \(u = x\big(1-e^{1-|x|}\big) y \big(1-e^{1-|y|}\big)\).
But for simplicity, let's obtain \(u = x(1-x^2) y(1-y^2)\)
for which we code~\(f\) later---as determined by this Reduce
algebra code.
\begin{figure}
\centering\begin{tabular}{@{}c@{\ }c@{}}
\parbox[t]{10em}{\caption{\label{fig:monoscaleDiffEquil2}%
Equilibrium of the macroscale diffusion problem of Freese
with Dirichlet zero-value boundary conditions
(\cref{sec:monoscaleDiffEquil2}). The patch size is not
small so we can see the patches.}} &
\def\extraAxisOptions{label shift={-1.5ex}}
\raisebox{-\height}{\input{../Patch/Figs/monoscaleDiffEquil2}}
\end{tabular}
\end{figure}
%let { df(sign(~x),~x)=>0
% , df(abs(~x),~x)=>sign(x)
% , abs(~x)^2=>abs(x), sign(~x)^2=>1 };
%u:=x*(1-exp(1-abs(x)))*y*(1-exp(1-abs(y)));
\begin{verbatim}
on gcd; factor sin;
u:=x*(1-x^2)*y*(1-y^2);
a:=sin(pi*x)*sin(pi*y);
f:=2*df(u,x,x)+2*a*df(u,x,y)+2*df(u,y,y);
\end{verbatim}
Clear, and initiate globals.
\begin{matlab}
%}
clear all
global patches
%global OurCf2eps, OurCf2eps=true %option to save plot
%{
\end{matlab}
\paragraph{Patch configuration} Initially use \(7\times7\)
patches in the square \((-1,1)^2\). For continuous forcing
we may have small patches of any reasonable microgrid
spacing---here the microgrid error dominates.
\begin{matlab}
%}
nPatch = 7
nSubP = 5
dx = 0.03
%{
\end{matlab}
Specify some order of interpolation.
\begin{matlab}
%}
configPatches2(@monoscaleDiffForce2,[-1 1 -1 1],'equispace' ...
,nPatch ,4 ,dx ,nSubP ,'EdgyInt',true );
%{
\end{matlab}
Compute the time-constant coefficient and time-constant
forcing, and store them in struct \verb|patches| for access
by the microcode of \cref{sec:monoscaleDiffForce2}.
\begin{matlab}
%}
x=patches.x; y=patches.y;
patches.A = sin(pi*x).*sin(pi*y);
patches.fu = ...
+2*patches.A.*(9*x.^2.*y.^2-3*x.^2-3*y.^2+1) ...
+12*x.*y.*(x.^2+y.^2-2);
%{
\end{matlab}
By construction, the \pde\ has analytic solution
\begin{matlab}
%}
uAnal = x.*(1-x.^2).*y.*(1-y.^2);
%{
\end{matlab}
\paragraph{Solve for steady state} Set initial guess of
zero, with \verb|NaN| to indicate patch-edge values.
Index~\verb|i| are the indices of patch-interior points, and
the number of unknowns is then its length.
\begin{matlab}
%}
u0 = zeros(nSubP,nSubP,1,1,nPatch,nPatch);
u0([1 end],:,:) = nan; u0(:,[1 end],:) = nan;
patches.i = find(~isnan(u0));
nVariables = numel(patches.i)
%{
\end{matlab}
Solve by iteration. Use \verb|fsolve| for simplicity and
robustness (using \verb|optimoptions| to omit its trace
information), and give magnitudes.
\begin{matlab}
%}
tic;
uSoln = fsolve(@theRes,u0(patches.i) ...
,optimoptions('fsolve','Display','off'));
solnTime = toc
normResidual = norm(theRes(uSoln))
normSoln = norm(uSoln)
normError = norm(uSoln-uAnal(patches.i))
%{
\end{matlab}
Store the solution vector into the patches, and interpolate,
but have not bothered to set boundary values so they stay
NaN from the interpolation.
\begin{matlab}
%}
u0(patches.i) = uSoln;
u0 = patchEdgeInt2(u0);
%{
\end{matlab}
\paragraph{Draw solution profile} Separate patches with
NaNs, then reshape arrays to suit 2D space surface plots.
\begin{matlab}
%}
figure(1), clf, colormap(0.8*hsv)
x(end+1,:,:)=nan; u0(end+1,:,:)=nan;
y(:,end+1,:)=nan; u0(:,end+1,:)=nan;
u = reshape(permute(squeeze(u0),[1 3 2 4]), [numel(x) numel(y)]);
%{
\end{matlab}
Draw the patch solution surface, with boundary-values omitted as
already~\verb|NaN| by not bothering to set them.
\begin{matlab}
%}
mesh(x(:),y(:),u'); view(60,55)
xlabel('space $x$'), ylabel('space $y$'), zlabel('$u(x,y)$')
ifOurCf2tex(mfilename)%optionally save
%{
\end{matlab}
\subsection{\texttt{monoscaleDiffForce2()}: microscale
discretisation inside patches of forced diffusion PDE}
\label{sec:monoscaleDiffForce2}
This function codes the lattice heterogeneous diffusion of
the \pde\ inside the patches. For 6D input arrays~\verb|u|,
\verb|x|, and~\verb|y|, computes the time derivative at each
point in the interior of a patch, output in~\verb|ut|.
\begin{matlab}
%}
function ut = monoscaleDiffForce2(t,u,patches)
dx = diff(patches.x(2:3)); % x space step
dy = diff(patches.y(2:3)); % y space step
i = 2:size(u,1)-1; % x interior points in a patch
j = 2:size(u,2)-1; % y interior points in a patch
ut = nan+u; % preallocate output array
%{
\end{matlab}
Set Dirichlet boundary value of zero around the square
domain.
\begin{matlab}
%}
u( 1 ,:,:,:, 1 ,:)=0; % left edge of left patches
u(end,:,:,:,end,:)=0; % right edge of right patches
u(:, 1 ,:,:,:, 1 )=0; % bottom edge of bottom patches
u(:,end,:,:,:,end)=0; % top edge of top patches
%{
\end{matlab}
Or code some function variation around the boundary, such as
a function of~\(y\) on the left boundary, and a (constant)
function of~\(x\) at the top boundary.
\begin{matlab}
%}
if 0
u(1,:,:,:,1,:)=(1+patches.y)/2; % left edge of left patches
u(:,end,:,:,:,end)=1; % top edge of top patches
end%if
%{
\end{matlab}
Compute the time derivatives via stored forcing and
coefficients. Easier to code by conflating the last four
dimensions into the one~\verb|,:|.
\begin{matlab}
%}
ut(i,j,:) ...
= 2*diff(u(:,j,:),2,1)/dx^2 +2*diff(u(i,:,:),2,2)/dy^2 ...
+2*patches.A(i,j,:).*( u(i+1,j+1,:) -u(i-1,j+1,:) ...
-u(i+1,j-1,:) +u(i-1,j-1,:) )/(4*dx*dy) ...
-patches.fu(i,j,:);
end%function monoscaleDiffForce2
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroDiff2.m
|
.m
|
EquationFreeGit-master/Patch/heteroDiff2.m
| 1,225 |
utf_8
|
967c37585d32078f06055a9df468bf78
|
% Computes the time derivatives of heterogeneous diffusion
% in 2D on patches. Adapted from 1D heterogeneous diffusion.
% JEB & AJR, May 2020 -- Nov 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{heteroDiff2()}: heterogeneous diffusion}
\label{sec:heteroDiff2}
This function codes the lattice heterogeneous diffusion
inside the patches. For 6D input arrays~\verb|u|, \verb|x|,
and~\verb|y| (via edge-value interpolation of
\verb|patchSys2|, \cref{sec:patchSys2}), computes the
time derivative~\cref{eq:HomogenisationExample} at each
point in the interior of a patch, output in~\verb|ut|. The
two 2D array of diffusivities,~$c^x_{ij}$ and~$c^y_{ij}$,
have previously been stored in~\verb|patches.cs| (3D).
\begin{matlab}
%}
function ut = heteroDiff2(t,u,patches)
dx = diff(patches.x(2:3)); % x space step
dy = diff(patches.y(2:3)); % y space step
ix = 2:size(u,1)-1; % x interior points in a patch
iy = 2:size(u,2)-1; % y interior points in a patch
ut = nan+u; % preallocate output array
ut(ix,iy,:,:,:,:) ...
= diff(patches.cs(:,iy,1,:).*diff(u(:,iy,:,:,:,:),1),1)/dx^2 ...
+diff(patches.cs(ix,:,2,:).*diff(u(ix,:,:,:,:,:),1,2),1,2)/dy^2;
end% function
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
randAdvecDiffEquil2.m
|
.m
|
EquationFreeGit-master/Patch/randAdvecDiffEquil2.m
| 6,879 |
utf_8
|
c3f92a6065c8df6d796e18a6ea619c96
|
% Solve for steady state of two-scale heterogeneous
% diffusion in 2D on patches as an example application
% involving Neumann boundary conditions, from section 6.2 of
% Bonizzoni, 2211.15221. AJR, 1 Feb 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{randAdvecDiffEquil2}: equilibrium of a 2D
random heterogeneous advection-diffusion via small patches}
\label{sec:randAdvecDiffEquil2}
Here we find the steady state~\(u(x,y)\) of the
heterogeneous \pde\ (inspired by Bonizzoni et al.\footnote{
\protect \url{http://arxiv.org/abs/2211.15221}} \S6.2)
\begin{equation*}
u_t=\mu_1\delsq u -(\cos\mu_2,\sin\mu_2)\cdot\grad u -u +f\,,
\end{equation*}
on domain \([0,1]^2\) with Neumann boundary conditions, for
microscale random pseudo-diffusion and pseudo-advection
coefficients, \(\mu_1(x,y)\in[0.01,0.1]\)\footnote{More
interesting microscale structure arises here for~\(\mu_1\) a
factor of three smaller.} and \(\mu_2(x,y)\in[0,2\pi)\), and
for forcing
\begin{equation*}
f(x,y):=\exp\left[-\frac{(x-\mu_3)^2+(y-\mu_4)^2}{\mu_5^2}\right],
\end{equation*}
smoothly varying in space for fixed \(\mu_3, \mu_4 \in
[0.25,0.75]\) and \(\mu_5 \in [0.1,0.25]\). The above
system is dominantly diffusive for lengths scales
\(\ell<0.01 = \min\mu_1\). Due to the randomness, we get
different solutions each execution of this code.
\cref{fig:randAdvecDiffEquil2} plots one example. A
physical interpretation of the solution field is confounded
because the problem is pseudo-advection-diffusion due to the
varying coefficients being outside the \(\grad\)~operator.
\begin{figure}
\centering\caption{\label{fig:randAdvecDiffEquil2}%
Equilibrium of the macroscale diffusion problem of Bonizzoni
et al.\ with Neumann boundary conditions of zero
(\cref{sec:randAdvecDiffEquil2}). Here the patches have a
equispaced spatial distribution. The microscale periodicity,
and hence the patch size, is chosen large enough to see
within.}
\includegraphics[scale=0.8]{Figs/randAdvecDiffEquil2}
\end{figure}
Clear, and initiate globals.
\begin{matlab}
%}
clear all
global patches
%global OurCf2eps, OurCf2eps=true %option to save plot
%{
\end{matlab}
First establish the microscale heterogeneity has
micro-period \verb|mPeriod| on the spatial lattice. Then
\verb|configPatches2| replicates the heterogeneity to fill
each patch.
\begin{matlab}
%}
mPeriod = 4
mu1 = 0.01*10.^rand(mPeriod)
mu2 = 2*pi*rand(mPeriod)
cs = cat(3,mu1,cos(mu2),sin(mu2));
meanDiffAdvec=squeeze(mean(mean(cs)))
%{
\end{matlab}
Set the periodicity~\(\epsilon\), here big enough so we can
see the patches, and other microscale parameters.
\begin{matlab}
%}
epsilon = 0.04
dx = epsilon/mPeriod
nPeriodsPatch = 1 % any integer
nSubP = nPeriodsPatch*mPeriod+2 % for edgy int
%{
\end{matlab}
\paragraph{Patch configuration}
Say use \(7\times7\) patches in \((0,1)^2\), fourth order
interpolation, either `equispace' or `chebyshev', and the
offset for Neumann boundary conditions:
\begin{matlab}
%}
nPatch = 7
Dom.type= 'equispace';
Dom.bcOffset = 0.5;
configPatches2(@randAdvecDiffForce2,[0 1],Dom ...
,nPatch ,4 ,dx ,nSubP ,'EdgyInt',true ,'hetCoeffs',cs );
%{
\end{matlab}
Compute the time-constant forcing, and store in struct
\verb|patches| for access by the microcode of
\cref{sec:randAdvecDiffForce2}.
\begin{matlab}
%}
mu = [ 0.25+0.5*rand(1,2) 0.1+0.15*rand ]
patches.fu = exp(-((patches.x-mu(1)).^2 ...
+(patches.y-mu(2)).^2)/mu(3)^2);
%{
\end{matlab}
\paragraph{Solve for steady state}
Set initial guess of zero, with \verb|NaN| to indicate
patch-edge values. Index~\verb|i| are the indices of
patch-interior points, store in global patches for access by
\verb|theRes|, and the number of unknowns is then its
number of elements.
\begin{matlab}
%}
u0 = zeros(nSubP,nSubP,1,1,nPatch,nPatch);
u0([1 end],:,:) = nan; u0(:,[1 end],:) = nan;
patches.i = find(~isnan(u0));
nVariables = numel(patches.i)
%{
\end{matlab}
Solve by iteration. Use \verb|fsolve| for simplicity and
robustness (and using \verb|optimoptions| to omit trace
information), via the generic patch system wrapper
\verb|theRes| (\cref{sec:theRes}).
\begin{matlab}
%}
tic;
uSoln = fsolve(@theRes,u0(patches.i) ...
,optimoptions('fsolve','Display','off'));
solnTime = toc
normResidual = norm(theRes(uSoln))
normSoln = norm(uSoln)
%{
\end{matlab}
Store the solution vector into the patches, and interpolate,
but have not bothered to set boundary values so they stay
NaN from the interpolation.
\begin{matlab}
%}
u0(patches.i) = uSoln;
u0 = patchEdgeInt2(u0);
%{
\end{matlab}
\paragraph{Draw solution profile} Separate patches with
NaNs, then reshape arrays to suit 2D space surface plots.
\begin{matlab}
%}
figure(1), clf, colormap(0.8*hsv)
patches.x(end+1,:,:)=nan; u0(end+1,:,:)=nan;
patches.y(:,end+1,:)=nan; u0(:,end+1,:)=nan;
u = reshape(permute(squeeze(u0),[1 3 2 4]) ...
, [numel(patches.x) numel(patches.y)]);
%{
\end{matlab}
Draw the patch solution surface, with boundary-values
omitted as already~\verb|NaN| by not bothering to set them.
\begin{matlab}
%}
mesh(patches.x(:),patches.y(:),u'); view(60,55)
xlabel('space $x$'), ylabel('space $y$'), zlabel('$u(x,y)$')
ifOurCf2eps(mfilename) %optionally save plot
%{
\end{matlab}
\subsection{\texttt{randAdvecDiffForce2()}: microscale
discretisation inside patches of forced diffusion PDE}
\label{sec:randAdvecDiffForce2}
This function codes the lattice heterogeneous diffusion of
the \pde\ inside the patches. For 6D input arrays~\verb|u|,
\verb|x|, and~\verb|y|, computes the time derivative at each
point in the interior of a patch, output in~\verb|ut|.
\begin{matlab}
%}
function ut = randAdvecDiffForce2(t,u,patches)
dx = diff(patches.x(2:3)); % x space step
dy = diff(patches.y(2:3)); % y space step
i = 2:size(u,1)-1; % x interior points in a patch
j = 2:size(u,2)-1; % y interior points in a patch
ut = nan+u; % preallocate output array
%{
\end{matlab}
Set Neumann boundary condition of zero derivative around the
square domain: that is, the edge value equals the
next-to-edge value.
\begin{matlab}
%}
u( 1 ,:,:,:, 1 ,:)=u( 2 ,:,:,:, 1 ,:); % left edge of left patches
u(end,:,:,:,end,:)=u(end-1,:,:,:,end,:); % right edge of right patches
u(:, 1 ,:,:,:, 1 )=u(:, 2 ,:,:,:, 1 ); % bottom edge of bottom patches
u(:,end,:,:,:,end)=u(:,end-1,:,:,:,end); % top edge of top patches
%{
\end{matlab}
Compute the time derivatives via stored forcing and
coefficients. Easier to code by conflating the last four
dimensions into the one~\verb|,:|.
\begin{matlab}
%}
ut(i,j,:) ...
= patches.cs(i,j,1).*(diff(u(:,j,:),2,1)/dx^2 ...
+diff(u(i,:,:),2,2)/dy^2)...
-patches.cs(i,j,2).*(u(i+1,j,:)-u(i-1,j,:))/(2*dx) ...
-patches.cs(i,j,3).*(u(i,j+1,:)-u(i,j-1,:))/(2*dy) ...
-u(i,j,:) +patches.fu(i,j,:);
end%function randAdvecDiffForce2
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
SwiftHohenberg2dPattern.m
|
.m
|
EquationFreeGit-master/Patch/SwiftHohenberg2dPattern.m
| 7,101 |
utf_8
|
6c2e0a887d3bc61c45a03c76692f2e90
|
% Simulate Swift--Hohenberg PDE in 2D on patches as an
% example application of patches in 2D space with pairs of
% edge points needing to be interpolated between patches.
% AJR, 13 Apr 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{SwiftHohenberg2dPattern}: patterns of the
Swift--Hohenberg PDE in 2D on patches}
\label{sec:SwiftHohenberg2dPattern}
\localtableofcontents
\cref{fig:SwiftHohenberg2dPattern2,fig:SwiftHohenberg2dPattern3%
,fig:SwiftHohenberg2dPattern4,fig:SwiftHohenberg2dPattern5%
,fig:SwiftHohenberg2dPattern6,fig:SwiftHohenberg2dPattern7}
show an example simulation in time generated by the patch
scheme applied to the patterns arising from the 2D
Swift--Hohenberg \pde.
Consider a lattice of values~\(u_i(t)\), with lattice
spacing~\(dx\), and governed by a microscale centred
discretisation of the Swift--Hohenberg \pde
\begin{equation}
\partial_tu = -(1+\delsq/k_0^2)^2u+\Ra u-u^3,
\label{eq:SwiftHohenberg2dPattern}
\end{equation}
with various boundary conditions at \(x,y=0,L\). For \Ra\
just above critical, say \(\Ra=0.1\), the system rapidly
evolves to spatial quasi-periodic solutions with period\({}
\approx 0.24\) when wavenumber parameter \(k_0 = 26\).
These spatial oscillations are here resolved on a micro-grid
of spacing~\(0.042\). On medium times these spatial
oscillations grow to near equilibrium amplitude
of~\(\sqrt{\Ra}\), and over very long times the phases of
the oscillations evolve in space to adapt to the boundaries.
Set the desired microscale periodicity, and correspondingly
choose random microscale diffusion coefficients (with
subscripts shifted by a half).
\begin{matlab}
%}
clear all
cMap=jet(64); cMap=0.8*cMap(7:end-7,:); % set colormap
basename = ['r' num2str(floor(1e5*rem(now,1))) mfilename]
%global OurCf2eps, OurCf2eps=true %optional to save plots
Ra = 0.2 % Ra>0 leads to patterns
nGapFac = 2
waveLength = 0.5/nGapFac
nPtsPeriod = 6
dx = waveLength/nPtsPeriod
k0 = 2.1*pi/waveLength
%{
\end{matlab}
The above factor~\(2.1\) is close to \(3/\sqrt2=2.1213\) for
which \((\pm1,\pm2)\) modes have same linear growth-rate
as~\((\pm2,0)\) modes.
Establish global data struct~\verb|patches| for the
Swift--Hohenberg \pde\ on some square domain. For
simplicity, use five patches in each direction. Quartic
(fourth-order) interpolation \(\verb|ordCC|=4\) provides
values for the inter-patch coupling conditions. Set
\verb|bcOffset| for different boundary conditions around the
square domain.
\begin{matlab}
%}
nPatch = 5
nSubP = 2*nPtsPeriod+4
Len = nPatch;
ordCC = 4;
dom.type='equispace';
dom.bcOffset=[0.5 0.5;1.0 1.5]
patches = configPatches2(@SwiftHohenbergPDE,[0 Len],dom ...
,nPatch,ordCC,dx,nSubP,'EdgyInt',true,'nEdge',2);
xs=squeeze(patches.x);
ys=squeeze(patches.y);
%{
\end{matlab}
\subsubsection{Simulate in time}
Set an initial condition, and here integrate forward in time
using a standard method for stiff systems. Integrate the
interface \verb|patchSys2| (\cref{sec:patchSys2}) to the
microscale differential equations (despite the extreme
stiffness, \verb|ode23| is ten times quicker than
\verb|ode15s|). Because pattern evolution is eventually
phase-diffusion, here sample the pattern at quadratically
varying times.
\begin{matlab}
%}
fprintf('\n**** Simulate in time\n')
u0 = 0.3*( -1+2*rand(size(patches.x+patches.y)) );
Ts=400*linspace(0,1,97).^2;
tic
[ts,us] = ode23(@patchSys2, Ts, u0(:),[],patches,k0,Ra);
simulateTime = toc
us = reshape(us',nSubP,nSubP,nPatch,nPatch,[]);
%{
\end{matlab}
\foreach \p in {2,...,7}{%
\begin{SCfigure}\centering
\caption{\label{fig:SwiftHohenberg2dPattern\p} pattern field
\(u(x,y,t)\) in the patch scheme applied to a microscale
discretisation of the 2D Swift--Hohenberg \pde. \ifcase\p\or
\or%2
At this early time much of the random sub-patch microstrucre
has decayed leaving some random marginal modes starting to
grow.
\or%3
By now the local sub-patch patterns have reached a
quasi-equilibrium amplitude.
\or%4
Patterns within the patches are evolving to the preferred
rolls, but with weak coupling to other patches.
\or%5
Can see different effects arising at different types of
boundaries.
\else \ldots
\fi}
\includegraphics[scale=0.9]{r26336SwiftHohenberg2dPattern\p}
\end{SCfigure}
}%end foreach
Plot the simulation such as that shown in
\cref{fig:SwiftHohenberg2dPattern2,fig:SwiftHohenberg2dPattern3%
,fig:SwiftHohenberg2dPattern4,fig:SwiftHohenberg2dPattern5%
,fig:SwiftHohenberg2dPattern6,fig:SwiftHohenberg2dPattern7}
First, reshape the data, omitting edge values.
\begin{matlab}
%}
xs([1:2 end-1:end],:) = nan;
ys([1:2 end-1:end],:) = nan;
us = reshape( permute(us,[1 3 2 4 5]) ...
,nSubP*nPatch,nSubP*nPatch,[]);
uRange=[min(us(:)) max(us(:))];
%{
\end{matlab}
Second, plot six examples of the evolving pattern,
equi-spaced in time-index.
\begin{matlab}
%}
plots = round( 1+linspace(0,1,7)*(numel(ts)-1) )
for p=2:numel(plots)
figure(p),clf
mesh(xs(:),ys(:),us(:,:,plots(p))')
axis equal, view(0,90)
caxis(uRange), colormap(cMap), colorbar
xlabel('space $x$'), ylabel('space $y$'), zlabel('$u(x,y,t)$')
title(['time = ' num2str(ts(plots(p)),3)])
ifOurCf2eps([basename num2str(p)],[12 11])
end%for p
%{
\end{matlab}
Third, plot animation in time: starts after a key press.
\begin{matlab}
%}
%%
figure(1),clf
cf=mesh(xs(:),ys(:),us(:,:,1)');
axis equal, view(0,90)
caxis(uRange), colormap(cMap), colorbar
xlabel('space $x$'), ylabel('space $y$'), zlabel('$u(x,y,t)$')
title(['time = ' num2str(ts(1),3)])
ca=gca;
disp('Press any key to start animation'),pause
for p=2:numel(ts)
cf.ZData=us(:,:,p)';
cf.CData=us(:,:,p)';
ca.Title.String=['time = ' num2str(ts(p),3)];
pause(0.1)
end
%{
\end{matlab}
Fin.
\subsection{The Swift--Hohenberg PDE and BCs inside patches}
As a microscale discretisation of Swift--Hohenberg \pde\
\(u_t= -(1+\delsq/k_0^2)^2u +\Ra u -u^3\), here code
straightforward centred discretisation in space.
\begin{matlab}
%}
function ut=SwiftHohenbergPDE(t,u,patches,k0,Ra)
dx=diff(patches.x(1:2)); % microscale spacing
dy=diff(patches.y(1:2)); % microscale spacing
i=3:size(u,1)-2; % interior points in patches
j=3:size(u,2)-2; % interior points in patches
%{
\end{matlab}
Code various boundary conditions. For slightly simpler
coding, squeeze out the two singleton dimensions.
\begin{matlab}
%}
u = squeeze(u);
u(1:2,:,1,:)=0; % u=u_x=0 at x=0
u(:,1:2,:,1)=0; % u=u_y=0 at y=0
u(end-1,:,end,:)=0; % u=0 at x=L
u(end ,:,end,:)=-u(end-2,:,end,:); % u_x=0 at x=L
u(:,end-1,:,end)=-u(:,end-2,:,end); % u_y=0 at y=L
u(:,end ,:,end)=-u(:,end-3,:,end); % u_yyy=0 at y=L
%{
\end{matlab}
Here code straightforward centred discretisation in space.
\begin{matlab}
%}
ut=nan+u; % preallocate output array
v = u(2:end-1,2:end-1,:,:) ...
+( diff(u(:,2:end-1,:,:),2,1)/dx^2 ...
+diff(u(2:end-1,:,:,:),2,2)/dy^2 )/k0^2;
ut(i,j,:,:) = -( v(2:end-1,2:end-1,:,:) ...
+( diff(v(:,2:end-1,:,:),2,1)/dx^2 ...
+diff(v(2:end-1,:,:,:),2,2)/dy^2 )/k0^2 ) ...
+Ra*u(i,j,:,:) -u(i,j,:,:).^3;
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
burgersBurst.m
|
.m
|
EquationFreeGit-master/Patch/burgersBurst.m
| 1,246 |
utf_8
|
f0f01615492ea5012e74cba12816fd17
|
% Simulates a burst in time of a microscale map that is
% applied on patches in space. Used by BurgersExample.m
% AJR, 4 Apr 2019
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{burgerBurst()}: code a burst of the patch map}
\label{sec:burgerBurst}
\begin{matlab}
%}
function [ts, us] = burgersBurst(ti, ui, bT)
%{
\end{matlab}
First find and set the number of microscale time-steps.
\begin{matlab}
%}
global patches
dt = diff(patches.x(2:3))^2/2;
ndt = ceil(bT/dt -0.2);
ts = ti+(0:ndt)'*dt;
%{
\end{matlab}
Use \verb|patchSys1()| (\cref{sec:patchSys1}) to apply
the microscale map over all time-steps in the burst. The
\verb|patchSys1()| interface provides the interpolated
edge-values of each patch. Store the results in rows to be
consistent with \ode\ and projective integrators.
\begin{matlab}
%}
us = nan(ndt+1,numel(ui));
us(1,:) = reshape(ui,1,[]);
for j = 1:ndt
ui = patchSys1(ts(j),ui);
us(j+1,:) = reshape(ui,1,[]);
end
%{
\end{matlab}
Linearly interpolate (extrapolate) to get the field values
at the precise final time of the burst. Then return.
\begin{matlab}
%}
ts(ndt+1) = ti+bT;
us(ndt+1,:) = us(ndt,:) ...
+ diff(ts(ndt:ndt+1))/dt*diff(us(ndt:ndt+1,:));
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
RK2mesoPatch.m
|
.m
|
EquationFreeGit-master/Patch/RK2mesoPatch.m
| 8,304 |
utf_8
|
66bc19ff089e5222de4554896a9d96e6
|
% RK2mesoPatch() is a simple example of Runge--Kutta, 2nd
% order, integration of a given deterministic system on
% patches. AJR, Sept 2020 -- Dec 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{RK2mesoPatch()}}
\label{sec:RK2mesoPatch}
This is a Runge--Kutta, 2nd order, integration of a given
deterministic system of \ode{}s on patches. It invokes
meso-time updates of the patch-edge values in order to
reduce interpolation costs, and uses a linear variation in
edge-values over the meso-time-step \cite[case
\(Q=2\)]{Bunder2015a}. This function is aimed primarily for
large problems executed on a computer cluster to markedly
reduce expensive communication between computers.
If using within projective integration, it appears quite
tricky to get all the time-steps chosen appropriately. One
has to choose times for: the micro-scale time-step, the
meso-time interval between communications, the longer
meso-time burst length, and the macro-scale integration
time-step.
\begin{matlab}
%}
function [xs,errs] = RK2mesoPatch(ts,x0,nMicro,patches)
if nargin<4, global patches, end
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|patches.fun()| is a function such as
\verb|dxdt=fun(t,x,patches)| that computes the right-hand
side of the \ode\ \(d\xv/dt=\fv(t,\xv)\) where \xv~is a
vector\slash array, \(t\)~is a scalar, and the result~\fv\
is a correspondingly sized vector\slash array.
\item \verb|x0| is an vector\slash array of initial values at
the time \verb|ts(1)|.
\item \verb|ts| is a vector of meso-scale times to compute
the approximate solution, say in~\(\RR^\ell\) for
\(\ell\geq2\)\,.
\item \verb|nMicro|, optional, default~\(10\), is the number
of micro-time-steps taken for each meso-scale time-step.
\item \verb|patches| struct set by \verb|configPatches|\(n\) and
provided as either as parameter, or as a global variable.
\end{itemize}
\paragraph{Output}
\begin{itemize}
\item \verb|xs|, 5/7/9D (depending upon~\verb|nD|) array
of length~\(\ell \times\cdots\) of approximate solution
vector\slash array at the specified times. But, if using
parallel computing via \verb|spmd|, then \verb|xs| is a
\emph{composite} 5/7/9D~array, so outside of an
\verb|spmd|-block access a single copy of the array
via~\verb|xs{1}|. Similarly for~\verb|errs|.
\item \verb|errs|, column vector in \(\RR^{\ell}\) of local
error estimate for the step from~\(t_{k-1}\) to~\(t_k\).
\end{itemize}
\begin{devMan}
\paragraph{Code of RK2 integration}
Set default number of micro-scale time-steps in each
requested meso-scale step of~\verb|ts|. Cannot use
\verb|nargin| inside explicit \verb|spmd|, but can use it if
the \verb|spmd| is already active from the code that invokes
this function.
\begin{matlab}
%}
if nargin<3|isempty(nMicro), nMicro=10; end
%{
\end{matlab}
If patches are set to be in parallel (there must be a
parallel pool), but only one worker available (i.e., not
already inside \verb|spmd|), then invoke function
recursively inside \verb|spmd|. Q:~is \verb|numlabs| defined
without the parallel computing toolbox??
\begin{matlab}
%}
if isequal(class(patches),'Composite') && numlabs==1
spmd,
[xs,errs] = RK2mesoPatch(ts,x0,nMicro,patches);
end% spmd
assert(isequal(class(xs) ,'Composite'),' xs not composite')
assert(isequal(class(errs),'Composite'),'errs not composite')
return
end
%{
\end{matlab}
Set the number of space dimensions from the number stored
patch-size ratios.
\begin{matlab}
%}
nD = length(patches.ratio);
%{
\end{matlab}
Set the micro-time-steps and create storage for outputs.
\begin{matlab}
%}
dt = diff(ts)/nMicro;
xs = nan([numel(ts) size(x0)]);
errs = nan(numel(ts),1);
%{
\end{matlab}
Initialise first result to the given initial condition, and
evaluate the initial time derivative into~\verb|f1|. Use
inter-patch interpolation to ensure edge values of the
initial condition are defined and are reasonable.
\footnote{These \texttt{gather()} functions cause all-to-all
interprocessor communication once every meso-step. Maybe
better to use distributed array instead, (although need to
then need to put time index last instead of first??), but we
need to do some inter-cpu communication in order to estimate
errors.}
\begin{matlab}
%}
switch nD
case 1, x0 = patchEdgeInt1(x0,patches);
xs(1,:,:,:,:) = gather(x0);
case 2, x0 = patchEdgeInt2(x0,patches);
xs(1,:,:,:,:,:,:) = gather(x0);
case 3, x0 = patchEdgeInt3(x0,patches);
xs(1,:,:,:,:,:,:,:,:) = gather(x0);
end;%switch nD
errs(1) = 0;
f1 = patches.fun(ts(1),x0,patches);
%{
\end{matlab}
Compute the meso-time-steps from~\(t_k\) to~\(t_{k+1}\),
copying the derivative~\verb|f1| at the end of the last
micro-time-step to be the derivative at the start of this
one.
\begin{matlab}
%}
for k = 1:numel(dt)
%{
\end{matlab}
Perform meso-time burst with the new interpolation for edge
values, and an interpolation of the time derivatives to get
derivative estimates of the edge-values.
\begin{matlab}
%}
switch nD
case 1, dx0 = patchEdgeInt1(f1,patches);
case 2, dx0 = patchEdgeInt2(f1,patches);
case 3, dx0 = patchEdgeInt3(f1,patches);
end;%switch nD
%{
\end{matlab}
Perform the micro-time steps.
\begin{matlab}
%}
for m=1:nMicro
f0 = f1;
% assert(iscodistributed(f0),'f0 not codist')
%{
\end{matlab}
For all micro-time derivative evaluations, include that the
edge values are varying according to the estimate made at
the start of the meso-time-step.
\begin{matlab}
%}
switch nD
case 1, f0([1 end],:,:,:)=dx0([1 end],:,:,:);
case 2, f0([1 end],:,:,:,:,:)=dx0([1 end],:,:,:,:,:);
f0(:,[1 end],:,:,:,:)=dx0(:,[1 end],:,:,:,:);
case 3
f0([1 end],:,:,:,:,:,:,:)=dx0([1 end],:,:,:,:,:,:,:);
f0(:,[1 end],:,:,:,:,:,:)=dx0(:,[1 end],:,:,:,:,:,:);
f0(:,:,[1 end],:,:,:,:,:)=dx0(:,:,[1 end],:,:,:,:,:);
end;%switch nD
% assert(iscodistributed(f0),'f0 not codist two')
%{
\end{matlab}
Simple second-order accurate Runge--Kutta micro-scale
time-step.
\begin{matlab}
%}
xh = x0+f0*dt(k)/2;
% assert(iscodistributed(xh),'xh not codist')
fh = patches.fun(ts(k)+dt(k)*(m-0.5),xh,patches);
% assert(iscodistributed(fh),'fh not codist one')
switch nD
case 1, fh([1 end],:,:,:)=dx0([1 end],:,:,:);
case 2, fh([1 end],:,:,:,:,:)=dx0([1 end],:,:,:,:,:);
fh(:,[1 end],:,:,:,:)=dx0(:,[1 end],:,:,:,:);
case 3
fh([1 end],:,:,:,:,:,:,:)=dx0([1 end],:,:,:,:,:,:,:);
fh(:,[1 end],:,:,:,:,:,:)=dx0(:,[1 end],:,:,:,:,:,:);
fh(:,:,[1 end],:,:,:,:,:)=dx0(:,:,[1 end],:,:,:,:,:);
end;%switch nD
% assert(iscodistributed(fh),'fh not codist two')
x0 = x0+fh*dt(k);
% assert(iscodistributed(x0),'x0 not codist two')
%{
\end{matlab}
End the burst of micro-time-steps.
\begin{matlab}
%}
end
%{
\end{matlab}
At the end of each meso-step burst, refresh the interpolate
of the edge values, evaluate time-derivative, and
temporarily fill-in edges of derivatives (to ensure error
estimate is reasonable).
\begin{matlab}
%}
switch nD
case 1, x0 = patchEdgeInt1(x0,patches);
xs(k+1,:,:,:,:) = gather(x0);
case 2, x0 = patchEdgeInt2(x0,patches);
xs(k+1,:,:,:,:,:,:) = gather(x0);
case 3, x0 = patchEdgeInt3(x0,patches);
xs(k+1,:,:,:,:,:,:,:,:) = gather(x0);
end;%switch nD
% assert(iscodistributed(x0),'x0 not codist three')
f1 = patches.fun(ts(k+1),x0,patches);
switch nD
case 1, f1([1 end],:,:,:)=dx0([1 end],:,:,:);
case 2, f1([1 end],:,:,:,:,:)=dx0([1 end],:,:,:,:,:);
f1(:,[1 end],:,:,:,:)=dx0(:,[1 end],:,:,:,:);
case 3
f1([1 end],:,:,:,:,:,:,:)=dx0([1 end],:,:,:,:,:,:,:);
f1(:,[1 end],:,:,:,:,:,:)=dx0(:,[1 end],:,:,:,:,:,:);
f1(:,:,[1 end],:,:,:,:,:)=dx0(:,:,[1 end],:,:,:,:,:);
end;%switch nD
%{
\end{matlab}
Use the time derivative at~\(t_{k+1}\) to estimate an error
by storing the difference with what Simpson's rule would
estimate over the last micro-time step performed.
\begin{matlab}
%}
f0=f0-2*fh+f1;
% assert(iscodistributed(f0),'f2ndDeriv not codist')
errs(k+1) = sqrt(gather(mean(f0(:).^2,'omitnan')))*dt(k)/6;
end%for-loop
end%function
%{
\end{matlab}
End of the function with results returned in~\verb|xs|
and~\verb|errs|.
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
BurgersPDE.m
|
.m
|
EquationFreeGit-master/Patch/BurgersPDE.m
| 894 |
utf_8
|
cdf32fe35b336290773eee01ed34c16b
|
% A microscale discretisation of Burgers' PDE on a lattice x.
% AJR 5 Apr 2019 -- Jun 2020
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\paragraph{Example of Burgers PDE inside patches}
As a microscale discretisation of Burgers' \pde\
\(u_t=u_{xx}-30uu_x\), here code \(\dot u_{ij}
=\frac1{\delta x^2} (u_{i+1,j}-2u_{i,j}+u_{i-1,j})
-30u_{ij} \frac1{2\delta x}(u_{i+1,j}-u_{i-1,j})\).
Here there is only one field variable, and one in the
ensemble, so for simpler coding of the PDE we squeeze them
out (with no need to reshape when via patchSys1()).
\begin{matlab}
%}
function ut=BurgersPDE(t,u,patches)
u=squeeze(u); % omit singleton dimensions
dx=diff(patches.x(1:2)); % microscale spacing
i=2:size(u,1)-1; % interior points in patches
ut=nan+u; % preallocate output array
ut(i,:)=diff(u,2)/dx^2 ...
-30*u(i,:).*(u(i+1,:)-u(i-1,:))/(2*dx);
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
twoscaleDiffEquil2.m
|
.m
|
EquationFreeGit-master/Patch/twoscaleDiffEquil2.m
| 5,874 |
utf_8
|
72360a5657f3775100239977c8460614
|
% Solve for steady state of twoscale heterogeneous diffusion
% in 2D on patches as an example application, from section
% 5.3.1 of Freese, 2211.13731. AJR, 31 Jan 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{twoscaleDiffEquil2}: equilibrium of a 2D
twoscale heterogeneous diffusion via small patches}
\label{sec:twoscaleDiffEquil2}
Here we find the steady state~\(u(x,y)\) to the
heterogeneous \pde\ (inspired by Freese et al.\footnote{
\protect \url{http://arxiv.org/abs/2211.13731}} \S5.3.1)
\begin{equation*}
u_t=A(x,y)\grad\grad u-f,
\end{equation*}
on domain \([-1,1]^2\) with Dirichlet BCs, for coefficient
`diffusion' matrix, varying with period~\(2\epsilon\) on the
microscale \(\epsilon=2^{-7}\), of
\begin{equation*}
A:=\begin{bmatrix} 2& a\\a & 2 \end{bmatrix}
\quad \text{with } a:=\sin(\pi x/\epsilon)\sin(\pi y/\epsilon),
\end{equation*}
and for forcing \(f:=(x+\cos3\pi x)y^3\).
\begin{figure}
\centering\begin{tabular}{@{}c@{\ }c@{}}
\parbox[t]{10em}{\caption{\label{fig:twoscaleDiffEquil2}%
Equilibrium of the multiscale diffusion problem of Freese
with Dirichlet zero-value boundary conditions
(\cref{sec:twoscaleDiffEquil2}). The patch size is not
small so we can see the patches and the sub-patch grid. The
solution~\(u(x,y)\) is boringly smooth.}} &
\def\extraAxisOptions{label shift={-1.5ex}}
\raisebox{-\height}{\input{../Patch/Figs/twoscaleDiffEquil2}}
\end{tabular}
\end{figure}
Clear, and initiate globals.
\begin{matlab}
%}
clear all
global patches
%global OurCf2eps, OurCf2eps=true %option to save plot
%{
\end{matlab}
First establish the microscale heterogeneity has
micro-period \verb|mPeriod| on the spatial lattice. Set the
phase of the heterogeneity so that each patch centre is a
point of symmetry of the diffusivity. Then
\verb|configPatches2| replicates the heterogeneity to fill
each patch.
\begin{matlab}
%}
mPeriod = 6
z = (0.5:mPeriod)'/mPeriod;
A = sin(2*pi*z).*sin(2*pi*z');
%{
\end{matlab}
Set the periodicity, via~\(\epsilon\), and other microscale
parameters.
\begin{matlab}
%}
nPeriodsPatch = 1 % any integer
epsilon = 2^(-6) % 4 or 5 to see the patches
dx = (2*epsilon)/mPeriod
nSubP = nPeriodsPatch*mPeriod+2 % for edgy int
%{
\end{matlab}
\paragraph{Patch configuration}
Say use \(7\times7\) patches in \((-1,1)^2\), fourth order
interpolation, and either `equispace' or `chebyshev':
\begin{matlab}
%}
nPatch = 7
configPatches2(@twoscaleDiffForce2,[-1 1],'equispace' ...
,nPatch ,4 ,dx ,nSubP ,'EdgyInt',true ,'hetCoeffs',A );
%{
\end{matlab}
Compute the time-constant forcing, and store in struct
\verb|patches| for access by the microcode of
\cref{sec:twoscaleDiffForce2}.
\begin{matlab}
%}
x = patches.x; y = patches.y;
patches.fu = 100*(x+cos(3*pi*x)).*y.^3;
%{
\end{matlab}
\paragraph{Solve for steady state}
Set initial guess of zero, with \verb|NaN| to indicate
patch-edge values. Index~\verb|i| are the indices of
patch-interior points, and the number of unknowns is then
its length.
\begin{matlab}
%}
u0 = zeros(nSubP,nSubP,1,1,nPatch,nPatch);
u0([1 end],:,:) = nan; u0(:,[1 end],:) = nan;
patches.i = find(~isnan(u0));
nVariables = numel(patches.i)
%{
\end{matlab}
Solve by iteration. Use \verb|fsolve| for simplicity and
robustness (and using \verb|optimoptions| to omit trace
information), via the generic patch system wrapper
\verb|theRes| (\cref{sec:theRes}), and give magnitudes.
\begin{matlab}
%}
tic;
uSoln = fsolve(@theRes,u0(patches.i) ...
,optimoptions('fsolve','Display','off'));
solveTime = toc
normResidual = norm(theRes(uSoln))
normSoln = norm(uSoln)
%{
\end{matlab}
Store the solution vector into the patches, and interpolate,
but have not bothered to set boundary values so they stay
NaN from the interpolation.
\begin{matlab}
%}
u0(patches.i) = uSoln;
u0 = patchEdgeInt2(u0);
%{
\end{matlab}
\paragraph{Draw solution profile} Separate patches with
NaNs, then reshape arrays to suit 2D space surface plots.
\begin{matlab}
%}
figure(1), clf, colormap(0.8*hsv)
x(end+1,:,:)=nan; u0(end+1,:,:)=nan;
y(:,end+1,:)=nan; u0(:,end+1,:)=nan;
u = reshape(permute(squeeze(u0),[1 3 2 4]), [numel(x) numel(y)]);
%{
\end{matlab}
Draw the patch solution surface, with boundary-values omitted as
already~\verb|NaN| by not bothering to set them.
\begin{matlab}
%}
mesh(x(:),y(:),u'); view(60,55)
xlabel('space $x$'), ylabel('space $y$'), zlabel('$u(x,y)$')
ifOurCf2tex(mfilename)%optionally save
%{
\end{matlab}
\subsection{\texttt{twoscaleDiffForce2()}: microscale
discretisation inside patches of forced diffusion PDE}
\label{sec:twoscaleDiffForce2}
This function codes the lattice heterogeneous diffusion of
the \pde\ inside the patches. For 6D input arrays~\verb|u|,
\verb|x|, and~\verb|y|, computes the time derivative at each
point in the interior of a patch, output in~\verb|ut|.
\begin{matlab}
%}
function ut = twoscaleDiffForce2(t,u,patches)
dx = diff(patches.x(2:3)); % x space step
dy = diff(patches.y(2:3)); % y space step
i = 2:size(u,1)-1; % x interior points in a patch
j = 2:size(u,2)-1; % y interior points in a patch
ut = nan+u; % preallocate output array
%{
\end{matlab}
Set Dirichlet boundary value of zero around the square
domain.
\begin{matlab}
%}
u( 1 ,:,:,:, 1 ,:)=0; % left edge of left patches
u(end,:,:,:,end,:)=0; % right edge of right patches
u(:, 1 ,:,:,:, 1 )=0; % bottom edge of bottom patches
u(:,end,:,:,:,end)=0; % top edge of top patches
%{
\end{matlab}
Compute the time derivatives via stored forcing and
coefficients. Easier to code by conflating the last four
dimensions into the one~\verb|,:|.
\begin{matlab}
%}
ut(i,j,:) ...
= 2*diff(u(:,j,:),2,1)/dx^2 +2*diff(u(i,:,:),2,2)/dy^2 ...
+2*patches.cs(i,j).*( u(i+1,j+1,:) -u(i-1,j+1,:) ...
-u(i+1,j-1,:) +u(i-1,j-1,:) )/(4*dx*dy) ...
-patches.fu(i,j,:);
end%function twoscaleDiffForce2
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
patchSys1.m
|
.m
|
EquationFreeGit-master/Patch/patchSys1.m
| 3,219 |
utf_8
|
a1fb92a0c8cf097956420a725b7f4cef
|
% patchSys1() provides an interface to time integrators for
% the dynamics on patches coupled across space. The system
% must be a lattice system such as PDE discretisations.
% AJR, Nov 2017 -- 31 Mar 2023
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{patchSys1()}: interface 1D space to time integrators}
\label{sec:patchSys1}
To simulate in time with 1D spatial patches we often need to
interface a user's time derivative function with time
integration routines such as \verb|ode23| or~\verb|PIRK2|.
This function provides an interface. Communicate
patch-design variables (\cref{sec:configPatches1}) either
via the global struct~\verb|patches| or via an optional
third argument. \verb|patches| is required for the parallel
computing of \verb|spmd|, or if parameters are to be passed
though to the user microscale function.
\begin{matlab}
%}
function dudt=patchSys1(t,u,patches,varargin)
if nargin<3, global patches, end
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|u| is a vector\slash array of length
$\verb|nSubP| \cdot \verb|nVars| \cdot \verb|nEnsem| \cdot
\verb|nPatch|$ where there are $\verb|nVars| \cdot
\verb|nEnsem|$ field values at each of the points in the
$\verb|nSubP|\times \verb|nPatch|$ grid.
\item \verb|t| is the current time to be passed to the
user's time derivative function.
\item \verb|patches| a struct set by \verb|configPatches1()|
with the following information used here.
\begin{itemize}
\item \verb|.fun| is the name of the user's function
\verb|fun(t,u,patches,...)| that computes the time
derivatives on the patchy lattice. The array~\verb|u| has
size $\verb|nSubP| \times \verb|nVars| \times \verb|nEnsem|
\times \verb|nPatch|$. Time derivatives should be computed
into the same sized array, then herein the patch edge values
are overwritten by zeros.
\item \verb|.x| is $\verb|nSubP| \times1 \times1 \times
\verb|nPatch|$ array of the spatial locations~$x_{i}$ of
the microscale grid points in every patch. Currently it
\emph{must} be an equi-spaced lattice on the microscale.
\end{itemize}
\item \verb|varargin|, optional, is arbitrary number of
parameters to be passed onto the users time-derivative
function as specified in configPatches1.
\end{itemize}
\paragraph{Output}
\begin{itemize}
\item \verb|dudt| is a vector\slash array of of time
derivatives, but with patch edge-values set to zero. It is
of total length $\verb|nSubP| \cdot \verb|nVars| \cdot
\verb|nEnsem| \cdot \verb|nPatch|$ and the same dimensions
as~\verb|u|.
\end{itemize}
\begin{devMan}
Reshape the fields~\verb|u| as a 4D-array, and sets the edge
values from macroscale interpolation of centre-patch values.
\cref{sec:patchEdgeInt1} describes \verb|patchEdgeInt1()|.
\begin{matlab}
%}
sizeu = size(u);
u = patchEdgeInt1(u,patches);
%{
\end{matlab}
Ask the user function for the time derivatives computed in
the array, overwrite its edge values with the dummy value of
zero (as \verb|ode15s| chokes on NaNs), then return to the
user\slash integrator as same sized array as input.
\begin{matlab}
%}
dudt=patches.fun(t,u,patches,varargin{:});
n=patches.nEdge;
dudt([1:n end-n+1:end],:,:,:) = 0;
dudt=reshape(dudt,sizeu);
%{
\end{matlab}
Fin.
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
heteroBurst.m
|
.m
|
EquationFreeGit-master/Patch/heteroBurst.m
| 790 |
utf_8
|
44b59399b9035cafb49f59bce3a33b3a
|
% Simulates a burst of the system linked to by the
% configuration of patches. Used by homogenisationExample.m,
% homoDiffEdgy1.m, and maybe homoLanLif1D.m
% AJR, 4 Apr 2019 -- Sep 2021
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{heteroBurst()}: a burst of heterogeneous diffusion}
\label{sec:heteroBurst}
This code integrates in time the derivatives computed by
\verb|heteroDiff| from within the patch coupling of
\verb|patchSys1|. Try~\verb|ode23| or \verb|rk2Int|,
although \verb|ode45| may give smoother results.
\begin{matlab}
%}
function [ts, ucts] = heteroBurst(ti, ui, bT)
if ~exist('OCTAVE_VERSION','builtin')
[ts,ucts] = ode23( @patchSys1,[ti ti+bT],ui(:));
else % octave version
[ts,ucts] = rk2Int(@patchSys1,[ti ti+bT],ui(:));
end
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
MMburstAcc.m
|
.m
|
EquationFreeGit-master/ProjInt/MMburstAcc.m
| 1,007 |
utf_8
|
212e6fa96d948014cf0e9a656327a56c
|
% Short explanation for users typing "help fun"
% Author, date
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\paragraph{Code an accurate burst of Michaelis--Menten enzyme kinetics}
Integrate a burst of length~\verb|bT| of the \ode{}s for the
Michaelis--Menten enzyme kinetics at parameter~\(\epsilon\)
inherited from above. Code \textsc{ode}s in
function~\verb|dMMdt| with variables \(x=\verb|x(1)|\) and
\(y=\verb|x(2)|\). Starting at time~\verb|ti|, and
state~\verb|xi| (row), we here use \verb|ode45| for accurate
integrate in time.
\begin{matlab}
%}
function [ts, xs] = MMburstAcc(ti, xi, bT)
global MMepsilon
dMMdt = @(t,x) [ -x(1)+(x(1)+0.5)*x(2)
1/MMepsilon*( x(1)-(x(1)+1)*x(2) ) ];
if ~exist('OCTAVE_VERSION','builtin')
odeopts = odeset('RelTol',1e-8,'AbsTol',1e-8);
[ts, xs] = ode45(dMMdt, [ti ti+bT], xi, odeopts);
else % octave version, by default errors = 1e-8
ts = linspace(ti,ti+bT,11);
xs = lsode(@(x,t) dMMdt(t,x),xi,ts);
end
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
PIRK2.m
|
.m
|
EquationFreeGit-master/ProjInt/PIRK2.m
| 14,636 |
utf_8
|
c60cf0db519e9b6bec8fb3ed5d3a4783
|
% PIRK2() implements second-order Projective Integration
% with a user-specified microsolver. The macrosolver adapts
% the explicit second-order Runge--Kutta Improved Euler
% scheme. JM and AJR, Oct 2018 -- Oct 2020. Execute with no
% arguments to see an example.
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{PIRK2()}: projective integration of second-order accuracy}
\label{sec:PIRK2}
\localtableofcontents
\subsection{Introduction}
This Projective Integration scheme implements a macroscale
scheme that is analogous to the second-order Runge--Kutta
Improved Euler integration.
\begin{matlab}
%}
function [x, tms, xms, rm, svf] = PIRK2(microBurst, tSpan, x0, bT)
%{
\end{matlab}
\paragraph{Input}
If there are no input arguments, then this function applies
itself to the Michaelis--Menton example: see the code in
\cref{sec:pirk2eg} as a basic template of how to use.
\begin{itemize}
\item \verb|microBurst()|, a user-coded function that
computes a short-time burst of the microscale simulation.
\begin{verbatim}
[tOut, xOut] = microBurst(tStart, xStart, bT)
\end{verbatim}
\begin{itemize}
\item Inputs:
\verb|tStart|,~the start time of a burst of simulation;
\(\verb|xStart|\),~the row \(n\)-vector of the starting
state; \verb|bT|, \emph{optional}, the total time to
simulate in the burst---if your \verb|microBurst()|
determines the burst time, then replace~\verb|bT| in the
argument list by~\verb|varargin|.
\item Outputs:
\verb|tOut|,~the column vector of solution times; and
\verb|xOut|,~an array in which each \emph{row} contains
the system state at corresponding times.
\end{itemize}
Be wary that for very large scale separations (such as
\verb|MMepsilon<1e-5| in the Michaelis--Menten example),
microscale integration by error-controlled variable-step
routines (such as \verb|ode23/45|) often generate microscale
variations that ruin the projective extrapolation of
\verb|PIRK2()|. In such cases, a fixed time-step microscale
integrator is much better (such as \verb|rk2Int()|).
\item \verb|tSpan| is an \(\ell\)-vector of times at which
the user requests output, of which the first element is
always the initial time. \verb|PIRK2()| does not use
adaptive time-stepping; the macroscale time-steps are
(nearly) the steps between elements of \verb|tSpan|.
\item \verb|x0| is an \(n\)-vector of initial values at the
initial time \verb|tSpan(1)|. Elements of~\verb|x0| may be
\verb|NaN|: such \verb|Nan|s are carried in the simulation
through to the output, and often represent boundaries\slash
edges in spatial fields.
\item \verb|bT|, \emph{optional}, either missing, or
empty~(\verb|[]|), or a scalar: if a given scalar, then it
is the length of the micro-burst simulations---the minimum
amount of time needed for the microscale simulation to relax
to the slow manifold; else if missing or~\verb|[]|, then
\verb|microBurst()| must itself determine the length of a
burst.
\begin{matlab}
%}
if nargin<4, bT=[]; end
%{
\end{matlab}
\end{itemize}
\paragraph{Choose a long enough burst length}
Suppose: firstly, you have some desired relative
accuracy~\(\varepsilon\) that you wish to achieve (e.g.,
\(\varepsilon\approx0.01\) for two digit accuracy);
secondly, the slow dynamics of your system occurs at
rate\slash frequency of magnitude about~\(\alpha\); and
thirdly, the rate of \emph{decay} of your fast modes are
faster than the lower bound~\(\beta\) (e.g., if three fast
modes decay roughly like \(e^{-12t}, e^{-34t}, e^{-56t}\)
then \(\beta\approx 12\)).
\begin{figure}
\centering
\def\aD{\alpha\Delta}\def\bD{\beta\Delta}\def\dD{\delta/\Delta}
\caption{\label{fig:bTlength}Need macroscale step~\(\Delta\)
such that $|\aD|\lesssim\sqrt{6\varepsilon}$ for given
relative error~\(\varepsilon\) and slow rate~\(\alpha\), and
then $\dD\gtrsim\frac1{\bD}\log|\bD|$ determines the minimum
required burst length~\(\delta\) for every given fast
rate~\(\beta\).}
\tikzsetnextfilename{ProjInt/bTlength}
\begin{tikzpicture}
\begin{loglogaxis}[xlabel={$\bD$}
,ylabel={$(\dD)_{\min}$}
,domain=2.7:1000 ,grid=both ]
\addplot+[no marks]{ln(x)/x};
\end{loglogaxis}
\end{tikzpicture}
\end{figure}
Then set
\begin{enumerate}
\item a macroscale time-step, \(\Delta=\verb|diff(tSpan)|\),
such that \(\alpha\Delta\approx\sqrt{6\varepsilon}\), and
\item a microscale burst length, \(\delta=\verb|bT| \gtrsim
\frac1\beta\log|\beta\Delta|\), see \cref{fig:bTlength}.
\end{enumerate}
\paragraph{Output}
If there are no output arguments specified, then a plot is
drawn of the computed solution~\verb|x| versus \verb|tSpan|.
\begin{itemize}
\item \verb|x|, an \(\ell \times n\) array of the
approximate solution vector. Each row is an estimated state
at the corresponding time in \verb|tSpan|. The simplest
usage is then \verb|x = PIRK2(microBurst,tSpan,x0,bT)|.
However, microscale details of the underlying Projective
Integration computations may be helpful. \verb|PIRK2()|
provides up to four optional outputs of the microscale
bursts.
\item \verb|tms|, optional, is an \(L\) dimensional column
vector containing the microscale times within the burst
simulations, each burst separated by~\verb|NaN|;
\item \verb|xms|, optional, is an \(L\times n\) array of the
corresponding microscale states---each rows is an accurate
estimate of the state at the corresponding time~\verb|tms|
and helps visualise details of the solution.
\item \verb|rm|, optional, a struct containing the
`remaining' applications of the microBurst required by the
Projective Integration method during the calculation of the
macrostep: \begin{itemize}
\item \verb|rm.t|~is a column vector of microscale times; and
\item \verb|rm.x|~is the array of corresponding burst states.
\end{itemize}
The states \verb|rm.x| do not have the same physical
interpretation as those in \verb|xms|; the \verb|rm.x| are
required in order to estimate the slow vector field during
the calculation of the Runge--Kutta increments, and do
\emph{not} accurately approximate the macroscale dynamics.
\item \verb|svf|, optional, a struct containing the
Projective Integration estimates of the slow vector field.
\begin{itemize}
\item \verb|svf.t| is a \(2\ell\) dimensional column vector
containing all times at which the Projective Integration
scheme is extrapolated along microBurst data to form a
macrostep.
\item \verb|svf.dx| is a \(2\ell\times n\) array containing
the estimated slow vector field.
\end{itemize}
\end{itemize}
\subsection{If no arguments, then execute an example}
\label{sec:pirk2eg}
\begin{matlab}
%}
if nargin==0
%{
\end{matlab}
\paragraph{Example code for Michaelis--Menton dynamics} The
Michaelis--Menten enzyme kinetics is expressed as a
singularly perturbed system of differential equations for
\(x(t)\) and~\(y(t)\):
\begin{equation*}
\frac{dx}{dt}=-x+(x+\tfrac12)y \quad\text{and}\quad
\frac{dy}{dt}=\frac1\epsilon\big[x-(x+1)y\big]
\end{equation*}
(encoded in function \verb|MMburst()| in the next
paragraph). With initial conditions \(x(0)=1\) and
\(y(0)=0\), the following code computes and plots a solution
over time \(0\leq t\leq6\) for parameter
\(\epsilon=0.05\)\,. Since the rate of decay is
\(\beta\approx 1/\epsilon\) we choose a burst length
\(\epsilon\log(\Delta/\epsilon)\) as here the macroscale
time-step \(\Delta=1\).
\begin{matlab}
%}
global MMepsilon
MMepsilon = 0.05
ts = 0:6
bT = MMepsilon*log( (ts(2)-ts(1))/MMepsilon )
[x,tms,xms] = PIRK2(@MMburst, ts, [1;0], bT);
figure, plot(ts,x,'o:',tms,xms)
title('Projective integration of Michaelis--Menten enzyme kinetics')
xlabel('time t'), legend('x(t)','y(t)')
%{
\end{matlab}
Upon finishing execution of the example, exit this function.
\begin{matlab}
%}
return
end%if no arguments
%{
\end{matlab}
\input{../ProjInt/MMburst.m}
\input{../ProjInt/odeOct.m}
\begin{devMan}
\subsection{The projective integration code}
Determine the number of time-steps and preallocate storage
for macroscale estimates.
\begin{matlab}
%}
nT=length(tSpan);
x=nan(nT,length(x0));
%{
\end{matlab}
Get the number of expected outputs and set logical indices
to flag what data should be saved.
\begin{matlab}
%}
nArgOut=nargout();
saveMicro = (nArgOut>1);
saveFullMicro = (nArgOut>3);
saveSvf = (nArgOut>4);
%{
\end{matlab}
Run a preliminary application of the microBurst on the given
initial state to help relax to the slow manifold. This is
done in addition to the microBurst in the main loop, because
the initial state is often far from the attracting slow
manifold. Require the user to input and output rows of the
system state.
\begin{matlab}
%}
x0 = reshape(x0,1,[]);
[relax_t,relax_x0] = microBurst(tSpan(1),x0,bT);
%{
\end{matlab}
Use the end point of this preliminary microBurst as the
initial state for the loop of macro-steps.
\begin{matlab}
%}
tSpan(1) = relax_t(end);
x(1,:)=relax_x0(end,:);
%{
\end{matlab}
If saving information, then record the first application of
the microBurst. Allocate cell arrays for times and states
for outputs requested by the user, as concatenating cells is
much faster than iteratively extending arrays.
\begin{matlab}
%}
if saveMicro
tms = cell(nT,1);
xms = cell(nT,1);
tms{1} = reshape(relax_t,[],1);
xms{1} = relax_x0;
if saveFullMicro
rm.t = cell(nT,1);
rm.x = cell(nT,1);
if saveSvf
svf.t = nan(2*nT-2,1);
svf.dx = nan(2*nT-2,length(x0));
end
end
end
%{
\end{matlab}
\paragraph{Loop over the macroscale time-steps}
Also set an initial rounding tolerance for checking.
\begin{matlab}
%}
roundingTol = 1e-8;
for jT = 2:nT
T = tSpan(jT-1);
%{
\end{matlab}
If two applications of the microBurst would cover one entire
macroscale time-step, then do so (setting some internal
states to \verb|NaN|); else proceed to projective step.
\begin{matlab}
%}
if ~isempty(bT) && 2*abs(bT)>=abs(tSpan(jT)-T) && bT*(tSpan(jT)-T)>0
[t1,xm1] = microBurst(T, x(jT-1,:), tSpan(jT)-T);
x(jT,:) = xm1(end,:);
t2 = nan; xm2 = nan(1,size(xm1,2));
dx1 = xm2; dx2 = xm2;
else
%{
\end{matlab}
Run the first application of the microBurst; since this
application directly follows from the initial conditions, or
from the latest macrostep, this microscale information is
physically meaningful as a simulation of the system. Extract
the size of the final time-step.
\begin{matlab}
%}
[t1,xm1] = microBurst(T, x(jT-1,:), bT);
%{
\end{matlab}
To estimate the derivative by numerical differentiation, we
balance approximation error~\(\|\ddot x\|/dt\) with
round-off error~\(\|x\|\epsilon/dt\) by the optimal
time-step \(dt\approx\sqrt(\|x\|\epsilon/\|\ddot x\|)\).
Omit~\(\|\ddot x\|\) as we do not know it. Also,
limit~\(dt\) to at most the last tenth of the burst, and at
least one step.
\begin{matlab}
%}
nt = length(t1);
optdt = min(0.1*(t1(nt)-t1(1)),sqrt(max(rms(xm1))*1e-15));
[~,kt] = min(abs(t1(nt)-optdt-t1(1:nt-1)));
ktnt = [kt nt];
del = t1(nt)-t1(kt);
%{
\end{matlab}
Check for round-off error, and decrease tolerance so that
warnings are not repeated unless things get worse.
\begin{matlab}
%}
xt = [reshape(t1(ktnt),[],1) xm1(ktnt,:)];
if norm(diff(xt))/norm(xt,'fro') < roundingTol
warning(['significant round-off error in 1st projection at T=' num2str(T)])
roundingTol = roundingTol/10;
end
%{
\end{matlab}
Find the needed time-step to reach time \verb|tSpan(n+1)|
and form a first estimate \verb|dx1| of the slow vector
field.
\begin{matlab}
%}
Dt = tSpan(jT)-t1(end);
dx1 = (xm1(nt,:)-xm1(kt,:))/del;
%{
\end{matlab}
Project along \verb|dx1| to form an intermediate
approximation of~\verb|x|; run another application of the
microBurst and form a second estimate of the slow vector
field (assuming the burst length is the same, or nearly so).
\begin{matlab}
%}
xint = xm1(end,:) + (Dt-(t1(end)-t1(1)))*dx1;
[t2,xm2] = microBurst(T+Dt, xint, bT);
%{
\end{matlab}
As before, choose~\(dt\) as best we can to estimate
derivative.
\begin{matlab}
%}
nt = length(t2);
optdt = min(0.1*(t2(nt)-t2(1)),sqrt(max(rms(xm2))*1e-15));
[~,kt] = min(abs(t2(nt)-optdt-t2(1:nt-1)));
ktnt = [kt nt];
del = t2(nt)-t2(kt);
dx2 = (xm2(nt,:)-xm2(kt,:))/del;
%{
\end{matlab}
Check for round-off error, and decrease tolerance so that
warnings are not repeated unless things get worse.
\begin{matlab}
%}
xt = [reshape(t2(ktnt),[],1) xm2(ktnt,:)];
if norm(diff(xt))/norm(xt,'fro') < roundingTol
warning(['significant round-off error in 2nd projection at T=' num2str(T)])
roundingTol = roundingTol/10;
end
%{
\end{matlab}
Use the weighted average of the estimates of the slow vector
field to take a macro-step.
\begin{matlab}
%}
x(jT,:) = xm1(end,:) + Dt*(dx1+dx2)/2;
%{
\end{matlab}
Now end the if-statement that tests whether a projective
step saves simulation time.
\begin{matlab}
%}
end
%{
\end{matlab}
If saving trusted microscale data, then populate the cell
arrays for the current loop iterate with the time-steps and
output of the first application of the microBurst. Separate
bursts by~\verb|NaN|s.
\begin{matlab}
%}
if saveMicro
tms{jT} = [reshape(t1,[],1); nan];
xms{jT} = [xm1; nan(1,size(xm1,2))];
%{
\end{matlab}
If saving all microscale data, then repeat for the remaining
applications of the microBurst.
\begin{matlab}
%}
if saveFullMicro
rm.t{jT} = [reshape(t2,[],1); nan];
rm.x{jT} = [xm2; nan(1,size(xm2,2))];
%{
\end{matlab}
If saving Projective Integration estimates of the slow
vector field, then populate the corresponding cells with
times and estimates.
\begin{matlab}
%}
if saveSvf
svf.t(2*jT-3:2*jT-2) = [t1(end); t2(end)];
svf.dx(2*jT-3:2*jT-2,:) = [dx1; dx2];
end
end
end
%{
\end{matlab}
End the main loop over all the macro-steps.
\begin{matlab}
%}
end
%{
\end{matlab}
Overwrite \verb|x(1,:)| with the specified initial condition
\verb|tSpan(1)|.
\begin{matlab}
%}
x(1,:) = reshape(x0,1,[]);
%{
\end{matlab}
For additional requested output, concatenate all the cells
of time and state data into two arrays.
\begin{matlab}
%}
if saveMicro
tms = cell2mat(tms);
xms = cell2mat(xms);
if saveFullMicro
rm.t = cell2mat(rm.t);
rm.x = cell2mat(rm.x);
end
end
%{
\end{matlab}
\subsection{If no output specified, then plot the simulation}
\begin{matlab}
%}
if nArgOut==0
figure, plot(tSpan,x,'o:')
title('Projective Simulation with PIRK2')
end
%{
\end{matlab}
This concludes \verb|PIRK2()|.
\begin{matlab}
%}
end
%{
\end{matlab}
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
PIG.m
|
.m
|
EquationFreeGit-master/ProjInt/PIG.m
| 14,213 |
utf_8
|
3c6184d1532de0ab1665b48a3cdd7008
|
% PIG implements Projective Integration scheme with any
% inbuilt integrator or user-specified integrator for the
% slow-time macroscale, and with any inbuilt/user-specified
% microsolver. JM & AJR, Sept 2018 -- Apr 2019.
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{PIG()}: Projective Integration via a General macroscale integrator}
\label{sec:PIG}
\localtableofcontents
\subsection{Introduction}
This is a Projective Integration scheme when the macroscale
integrator is any specified coded method. The advantage is
that one may use \script's inbuilt integration functions,
with all their sophisticated error control and adaptive
time-stepping, to do the macroscale integration\slash
simulation.
By default, for the microscale simulations \verb|PIG()| uses
`constraint-defined manifold computing', \verb|cdmc()|
(\cref{sec:cdmc}). This algorithm, initiated by
\cite{Gear05}, uses a backward projection so that the
simulation time is unchanged after running the microscale
simulator.
\begin{matlab}
%}
function [T,X,tms,xms,svf] = PIG(macroInt,microBurst,Tspan,x0 ...
,restrict,lift,cdmcFlag)
%{
\end{matlab}
\paragraph{Inputs:}
\begin{itemize}
\item \verb|macroInt()|, the numerical method that the user
wants to apply on a slow-time macroscale. Either specify a
standard \script\ integration function (such as
\verb|'ode23'| or~\verb|'ode45|'), or code your own
integration function using standard arguments. That is, if
you code your own, then it must be
\begin{equation*}
\verb|[Ts,Xs] = macroInt(F,Tspan,X0)|
\end{equation*}
where \begin{itemize}
\item function \verb|F(T,X)| notionally evaluates the time
derivatives \(d\Xv/dt\) at any time;
\item \verb|Tspan| is either the macro-time interval, or the
vector of macroscale times at which macroscale values are to
be returned; and
\item \verb|X0| are the initial values of~\(\Xv\) at time
\verb|Tspan(1)|.
\end{itemize}
Then the \(i\)th~\emph{row} of~\verb|Xs|, \verb|Xs(i,:)|, is
to be the vector~\(\Xv(t)\) at time \(t=\verb|Ts(i)|\).
Remember that in \verb|PIG()| the function \verb|F(T,X)| is
to be estimated by Projective Integration.
\item \verb|microBurst()| is a function that produces output
from the user-specified code for a burst of microscale
simulation. The function must internally specify\slash
decide how long a burst it is to use. Usage
\begin{equation*}
\verb|[tbs,xbs] = microBurst(tb0,xb0)|
\end{equation*}
\emph{Inputs:} \verb|tb0| is the start time of a burst;
\verb|xb0|~is the \(n\)-vector microscale state at the start
of a burst.
\emph{Outputs:} \verb|tbs|, the vector of solution times;
and \verb|xbs|, the corresponding microscale states.
\item \verb|Tspan|, a vector of macroscale times at which
the user requests output. The first element is always the
initial time. If \verb|macroInt| reports adaptively
selected time steps (e.g., \verb|ode45|), then \verb|Tspan|
consists of an initial and final time only.
\item \verb|x0|, the \(n\)-vector of initial microscale
values at the initial time~\verb|Tspan(1)|.
\end{itemize}
\paragraph{Optional Inputs:}
\verb|PIG()| allows for none, two or three additional inputs
after~\verb|x0|. If you distinguish distinct microscale and
macroscale states and your aim is to do Projective
Integration on the macroscale only, then lifting and
restriction functions must be provided to convert between
them. Usage \verb|PIG(...,restrict,lift)|:
\begin{itemize}
\item \verb|restrict(x)|, a function that takes an input
high-dimensional, \(n\)-D, microscale state~\xv\ and
computes the corresponding low-dimensional, \(N\)-D,
macroscale state~\Xv;
\item \verb|lift(X,xApprox)|, a function that converts an
input low-dimensional, \(N\)-D, macroscale state~\Xv\ to a
corresponding high-dimensional, \(n\)-D, microscale
state~\xv, given that \verb|xApprox| is a recently computed
microscale state on the slow manifold.
\end{itemize}
Either both \verb|restrict()| and \verb|lift()| are to be
defined, or neither. If neither are defined, then they are
assumed to be identity functions, so that \verb|N=n| in the
following.
If desired, the default constraint-defined manifold
computing microsolver may be disabled, via
\verb|PIG(...,restrict,lift,cdmcFlag)|
\begin{itemize}
\item \verb|cdmcFlag|, \emph{any} seventh input to
\verb|PIG()|, will disable \verb|cdmc()|, e.g., the string
\verb|'cdmc off'|.
\end{itemize}
If the \verb|cdmcFlag| is to be set without using a
\verb|restrict()| or \verb|lift()| function, then use empty
matrices~\verb|[]| for the restrict and lift functions.
\paragraph{Output}
Between zero and five outputs may be requested. If there are
no output arguments specified, then a plot is drawn of the
computed solution~\verb|X| versus~\verb|T|. Most often you
would store the first two output results of \verb|PIG()|,
via say \verb|[T,X] = PIG(...)|.
\begin{itemize}
\item \verb|T|, an \(L\)-vector of times at which
\verb|macroInt| produced results.
\item \verb|X|, an \(L \times N\) array of the computed
solution: the \(i\)th~\emph{row} of~\verb|X|, \verb|X(i,:)|,
is to be the macro-state vector~\(\Xv(t)\) at time
\(t=\verb|T(i)|\).
\end{itemize}
However, microscale details of the underlying Projective
Integration computations may be helpful, and so \verb|PIG()|
provides some optional outputs of the microscale bursts, via
\verb|[T,X,tms,xms] = PIG(...)|
\begin{itemize}
\item \verb|tms|, optional, is an \(\ell\)-dimensional column
vector containing microscale times with bursts,
each burst separated by~\verb|NaN|;
\item \verb|xms|, optional, is an \(\ell\times n\) array of
the corresponding microscale states.
\end{itemize}
In some contexts it may be helpful to see directly how
Projective Integration approximates a reduced slow vector
field, via \verb|[T,X,tms,xms,svf] = PIG(...)| in which
\begin{itemize}
\item \verb|svf|, optional, a struct containing the
Projective Integration estimates of the slow vector field.
\begin{itemize}
\item \verb|svf.T| is a \(\hat L\)-dimensional column
vector containing all times at which the microscale
simulation data is extrapolated to form an estimate of
\(d\xv/dt\) in \verb|macroInt()|.
\item \verb|svf.dX| is a \(\hat L\times N\) array containing
the estimated slow vector field.
\end{itemize}
\end{itemize}
If \verb|macroInt()| is, for example, the forward Euler
method (or the Runge--Kutta method), then \(\hat L = L\) (or
\(\hat L = 4L \)).
\subsection{If no arguments, then execute an example}
\label{sec:pigeg}
\begin{matlab}
%}
if nargin==0
%{
\end{matlab}
\begin{figure}
\centering
\caption{\label{fig:PIGsing}Projective Integration by
\texttt{PIG} of the example system~\eqref{eq:PIGsing} with
\(\epsilon=10^{-3}\) (\cref{sec:pigeg}). The macroscale
solution~\(X(t)\) is represented by just the blue circles.
The microscale bursts are the microscale states
\((x_1(t),x_2(t)) = (\text{red},\text{yellow})\) dots.}
\includegraphics[scale=0.9]{PIGsing}
\end{figure}%
As a basic example, consider a microscale system of the
singularly perturbed system of differential equations
\begin{equation}
\frac{dx_1}{dt}=\cos(x_1)\sin(x_2)\cos(t) \quad\text{and}\quad
\frac{dx_2}{dt}=\frac1\epsilon\big[\cos(x_1)-x_2\big].
\label{eq:PIGsing}
\end{equation}
The macroscale variable is \(X(t)=x_1(t)\), and the
evolution \(dX/dt\) is unclear. With initial condition
\(X(0)=1\), the following code computes and plots a solution
of the system~\eqref{eq:PIGsing} over time \(0\leq t\leq6\)
for parameter \(\epsilon=10^{-3}\)(\cref{fig:PIGsing}).
Whenever needed by \verb|microBurst()|, the microscale
system~\eqref{eq:PIGsing} is initialised (`\verb|lift|ed')
using \(x_2(t) = x_2^{\text{approx}}\) (yellow dots in
\cref{fig:PIGsing}).
First we code the right-hand side function of the microscale
system~\eqref{eq:PIGsing} of \ode{}s.
\begin{matlab}
%}
epsilon = 1e-3;
dxdt=@(t,x) [ cos(x(1))*sin(x(2))*cos(t)
( cos(x(1))-x(2) )/epsilon ];
%{
\end{matlab}
Second, we code microscale bursts, here using the standard
\verb|ode45()|. We choose a burst length \(2 \epsilon
\log(1/\epsilon)\) as the rate of decay is \(\beta\approx
1/\epsilon\) but we do not know the macroscale time-step
invoked by \verb|macroInt()|, so blithely assume
\(\Delta\le1\) and then double the usual formula for safety.
\begin{matlab}
%}
bT = 2*epsilon*log(1/epsilon)
if ~exist('OCTAVE_VERSION','builtin')
micB='ode45'; else micB='rk2Int'; end
microBurst = @(tb0, xb0) feval(micB,dxdt,[tb0 tb0+bT],xb0);
%{
\end{matlab}
Third, code functions to convert between macroscale and
microscale states.
\begin{matlab}
%}
restrict = @(x) x(1);
lift = @(X,xApprox) [X; xApprox(2)];
%{
\end{matlab}
Fourth, invoke \verb|PIG| to use \script's
\verb|ode23|\slash\verb|lsode|, say, on the macroscale slow
evolution. Integrate the micro-bursts over \(0\leq t\leq6\)
from initial condition \(\xv(0)=(1,0)\). You could set
\verb|Tspan=[0 -6]| to integrate backward in macroscale time
with forward microscale bursts \cite[]{Gear03b, Frewen2009}.
\begin{matlab}
%}
Tspan = [0 6];
x0 = [1;0];
if ~exist('OCTAVE_VERSION','builtin')
macInt='ode23'; else macInt='odeOct'; end
[Ts,Xs,tms,xms] = PIG(macInt,microBurst,Tspan,x0,restrict,lift);
%{
\end{matlab}
Plot output of this projective integration.
\begin{matlab}
%}
figure, plot(Ts,Xs,'o:',tms,xms,'.')
title('Projective integration of singularly perturbed ODE')
xlabel('time t')
legend('X(t) = x_1(t)','x_1(t) micro bursts','x_2(t) micro bursts')
%{
\end{matlab}
Upon finishing execution of the example, exit this function.
\begin{matlab}
%}
return
end%if no arguments
%{
\end{matlab}
\begin{devMan}
\subsection{The projective integration code}
If no lifting/restriction functions are provided, then
assign them to be the identity functions.
\begin{matlab}
%}
if nargin < 5 || isempty(restrict)
lift=@(X,xApprox) X;
restrict=@(x) x;
end
%{
\end{matlab}
Get the number of expected outputs and set logical indices
to flag what data should be saved.
\begin{matlab}
%}
nArgOut = nargout();
saveMicro = (nArgOut>2);
saveSvf = (nArgOut>4);
%{
\end{matlab}
Find the number of time-steps at which output is expected,
and the number of variables.
\begin{matlab}
%}
nT = length(Tspan)-1;
nx = length(x0);
nX = length(restrict(x0));
%{
\end{matlab}
Reformulate the microsolver to use \verb|cdmc()|, unless
flagged otherwise. The result is that the solution from
microBurst will terminate at the given initial time.
% Should be OK in Octave.
\begin{matlab}
%}
if nargin<7
microBurst = @(t,x) cdmc(microBurst,t,x);
else
warning(['A ' class(cdmcFlag) ' seventh input to PIG().'...
' PIG will not use constraint-defined manifold computing.'])
end
%{
\end{matlab}
Execute a preliminary application of the microBurst on the
initial state. This is done in addition to the microBurst in
the main loop, because the initial state is often far from
the attracting slow manifold.
\begin{matlab}
%}
[relaxT,x0MicroRelax] = microBurst(Tspan(1),x0);
xMicroLast = x0MicroRelax(end,:).';
X0Relax = restrict(xMicroLast);
%{
\end{matlab}
Update the initial time.
\begin{matlab}
%}
Tspan(1) = relaxT(end);
%{
\end{matlab}
Allocate cell arrays for times and states for any of the
outputs requested by the user. If saving information, then
record the first application of the microBurst. It is
unknown a priori how many applications of microBurst will be
required; this code may be run more efficiently if the
correct number is used in place of \verb|nT+1| as the
dimension of the cell arrays.
\begin{matlab}
%}
if saveMicro
tms=cell(nT+1,1); xms=cell(nT+1,1);
n=1;
tms{n} = reshape(relaxT,[],1);
xms{n} = x0MicroRelax;
if saveSvf
svf.T = cell(nT+1,1);
svf.dX = cell(nT+1,1);
else
svf = [];
end
else
tms = []; xms = []; svf = [];
end
%{
\end{matlab}
\paragraph{Define a function of macro simulation}
The idea of \verb|PIG()| is to use the output from the
\verb|microBurst()| to approximate an unknown function
\verb|F(t,X)| that computes \(d\Xv/dt\). This approximation
is then used in the system\slash user-defined `coarse
solver' \verb|macroInt()|. The approximation is computed in
the function
\begin{matlab}
%}
function [dXdt]=PIFun(t,X)
%{
\end{matlab}
Run a microBurst from the given macroscale initial values.
\begin{matlab}
%}
x = lift(X,xMicroLast);
[tTmp,xMicroTmp] = microBurst(t,reshape(x,[],1));
xMicroLast = xMicroTmp(end,:).';
%{
\end{matlab}
Compute the standard Projective Integration approximation of
the slow vector field.
\begin{matlab}
%}
X2 = restrict(xMicroTmp(end,:));
X1 = restrict(xMicroTmp(end-1,:));
dt = tTmp(end)-tTmp(end-1);
dXdt = (X2 - X1).'/dt;
%{
\end{matlab}
Save the microscale data, and the Projective Integration
slow vector field, if requested.
\begin{matlab}
%}
if saveMicro
n=n+1;
tms{n} = [reshape(tTmp,[],1); nan];
xms{n} = [xMicroTmp; nan(1,nx)];
if saveSvf
svf.T{n-1} = t;
svf.dX{n-1} = dXdt;
end
end
end% PIFun function
%{
\end{matlab}
\paragraph{Invoke the macroscale integration}
Integrate \verb|PIF()| with the user-specified simulator
\verb|macroInt()|. For some reason, in \script\ we need to
use a one-line function, \verb|PIF|, that invokes the above
macroscale function, \verb|PIFun|. We also need to use
\verb|feval| because \verb|macroInt()| has multiple outputs.
\begin{matlab}
%}
PIF = @(t,x) PIFun(t,x);
[T,X] = feval(macroInt,PIF,Tspan,X0Relax.');
%{
\end{matlab}
Overwrite \verb|X(1,:)| and \verb|T(1)|, which a user
expects to be \verb|X0| and \verb|Tspan(1)| respectively,
with the given initial conditions.
\begin{matlab}
%}
X(1,:) = restrict(x0);
T(1) = Tspan(1);
%{
\end{matlab}
Concatenate all the additional requested outputs into arrays.
\begin{matlab}
%}
if saveMicro
tms = cell2mat(tms);
xms = cell2mat(xms);
if saveSvf
svf.T = cell2mat(svf.T);
svf.dX = cell2mat(svf.dX);
end
end
%{
\end{matlab}
\subsection{If no output specified, then plot the simulation}
\begin{matlab}
%}
if nArgOut==0
figure, plot(T,X,'o:')
title('Projective Simulation via PIG')
end
%{
\end{matlab}
This concludes \verb|PIG()|.
\begin{matlab}
%}
end
%{
\end{matlab}
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
linearBurst.m
|
.m
|
EquationFreeGit-master/ProjInt/linearBurst.m
| 839 |
utf_8
|
ebbd668d39488aea22480543f39ccaa4
|
% Used by PIRKexample.m
% AJR, Apr 2019
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\paragraph{A micro-burst simulation}
Used by \verb|PIRKexample.m|. Code the micro-burst function
using simple Euler steps. As a rule of thumb, the time-steps
\verb|dt| should satisfy $\verb|dt| \le
1/|\verb|fastband|(1)|$ and the time to simulate with each
application of the microsolver, \verb|bT|, should be larger
than or equal to $1/|\verb|fastband|(2)|$. We set the
integration scheme to be used in the microsolver. Since the
time-steps are so small, we just use the forward Euler
scheme
\begin{matlab}
%}
function [ts, xs] = linearBurst(ti, xi, varargin)
global dxdt
dt = 0.001;
ts = ti+(0:dt:0.05)';
nts = length(ts);
xs = NaN(nts,length(xi));
xs(1,:)=xi;
for k=2:nts
xi = xi + dt*dxdt(ts(k),xi.').';
xs(k,:)=xi;
end
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
PIRK4.m
|
.m
|
EquationFreeGit-master/ProjInt/PIRK4.m
| 12,579 |
utf_8
|
9c94ee399df7625a5a3c86670fa4309f
|
% PIRK4 implements fourth-order Projective Integration with
% a user-specified microsolver. The macrosolver adapts the
% explicit fourth-order Runge--Kutta scheme.
% JM, Oct 2018--Apr 2019.
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{PIRK4()}: projective integration of fourth-order accuracy}
\label{sec:PIRK4}
\localtableofcontents
\subsection{Introduction}
This Projective Integration scheme implements a macrosolver
analogous to the fourth-order Runge--Kutta method.
\begin{matlab}
%}
function [x, tms, xms, rm, svf] = PIRK4(microBurst, tSpan, x0, bT)
%{
\end{matlab}
See \cref{sec:PIRK2} as the inputs and outputs are the
same as \verb|PIRK2()|.
\paragraph{If no arguments, then execute an example}
\begin{matlab}
%}
if nargin==0
%{
\end{matlab}
\subparagraph{Example of Michaelis--Menton backwards in time}
The Michaelis--Menten enzyme kinetics is expressed as a
singularly perturbed system of differential equations for
\(x(t)\) and~\(y(t)\) (encoded in function \verb|MMburst|):
\begin{equation*}
\frac{dx}{dt}=-x+(x+\tfrac12)y \quad\text{and}\quad
\frac{dy}{dt}=\frac1\epsilon\big[x-(x+1)y\big].
\end{equation*}
With initial conditions \(x(0)=y(0)=0.2\), the following
code uses forward time bursts in order to integrate
backwards in time to \(t=-5\) \cite[e.g.]{Frewen2009}. It
plots the computed solution over time \(-5\leq t\leq0\) for
parameter \(\epsilon=0.1\)\,. Since the rate of decay is
\(\beta\approx 1/\epsilon\) we choose a burst length
\(\epsilon\log(|\Delta|/\epsilon)\) as here the macroscale
time-step \(\Delta=-1\).
\begin{matlab}
%}
global MMepsilon
MMepsilon = 0.1
ts = 0:-1:-5
bT = MMepsilon*log(abs(ts(2)-ts(1))/MMepsilon)
[x,tms,xms,rm,svf] = PIRK4(@MMburst, ts, 0.2*[1;1], bT);
figure, plot(ts,x,'o:',tms,xms)
xlabel('time t'), legend('x(t)','y(t)')
title('Backwards-time projective integration of Michaelis--Menten')
%{
\end{matlab}
Upon finishing execution of the example, exit this function.
\begin{matlab}
%}
return
end%if no arguments
%{
\end{matlab}
\input{../ProjInt/MMburst.m}
\input{../ProjInt/odeOct.m}
\begin{devMan}
\paragraph{Input}
If there are no input arguments, then this function applies
itself to the Michaelis--Menton example: see the code in
\cref{sec:pirk2eg} as a basic template of how to use.
\begin{itemize}
\item \verb|microBurst()|, a user-coded function that
computes a short-time burst of the microscale simulation.
\begin{verbatim}
[tOut, xOut] = microBurst(tStart, xStart, bT)
\end{verbatim}
\begin{itemize}
\item Inputs:
\verb|tStart|,~the start time of a burst of simulation;
\(\verb|xStart|\),~the row \(n\)-vector of the starting
state; \verb|bT|, \emph{optional}, the total time to
simulate in the burst---if your \verb|microBurst()|
determines the burst time, then replace~\verb|bT| in the
argument list by~\verb|varargin|.
\item Outputs:
\verb|tOut|,~the column vector of solution times; and
\verb|xOut|,~an array in which each \emph{row} contains
the system state at corresponding times.
\end{itemize}
\item \verb|tSpan| is an \(\ell\)-vector of times at which
the user requests output, of which the first element is
always the initial time. \verb|PIRK4()| does not use
adaptive time-stepping; the macroscale time-steps are
(nearly) the steps between elements of \verb|tSpan|.
\item \verb|x0| is an \(n\)-vector of initial values at the
initial time \verb|tSpan(1)|. Elements of~\verb|x0| may be
\verb|NaN|: such \verb|Nan|s are carried in the simulation
through to the output, and often represent boundaries\slash
edges in spatial fields.
\item \verb|bT|, \emph{optional}, either missing, or
empty~(\verb|[]|), or a scalar: if a given scalar, then it
is the length of the micro-burst simulations---the minimum
amount of time needed for the microscale simulation to relax
to the slow manifold; else if missing or~\verb|[]|, then
\verb|microBurst()| must itself determine the length of a
burst.
\begin{matlab}
%}
if nargin<4, bT=[]; end
%{
\end{matlab}
\end{itemize}
\paragraph{Output}
If there are no output arguments specified, then a plot is
drawn of the computed solution~\verb|x| versus \verb|tSpan|.
\begin{itemize}
\item \verb|x|, an \(\ell \times n\) array of the
approximate solution vector. Each row is an estimated state
at the corresponding time in \verb|tSpan|. The simplest
usage is then \verb|x = PIRK4(microBurst,tSpan,x0,bT)|.
However, microscale details of the underlying Projective
Integration computations may be helpful. \verb|PIRK4()|
provides up to four optional outputs of the microscale
bursts.
\item \verb|tms|, optional, is an \(L\) dimensional column
vector containing the microscale times within the burst
simulations, each burst separated by~\verb|NaN|;
\item \verb|xms|, optional, is an \(L\times n\) array of the
corresponding microscale states---each rows is an accurate
estimate of the state at the corresponding time~\verb|tms|
and helps visualise details of the solution.
\item \verb|rm|, optional, a struct containing the
`remaining' applications of the microBurst required by the
Projective Integration method during the calculation of the
macrostep: \begin{itemize}
\item \verb|rm.t|~is a column vector of microscale times; and
\item \verb|rm.x|~is the array of corresponding burst states.
\end{itemize}
The states \verb|rm.x| do not have the same physical
interpretation as those in \verb|xms|; the \verb|rm.x| are
required in order to estimate the slow vector field during
the calculation of the Runge--Kutta increments, and do
\emph{not} accurately approximate the macroscale dynamics.
\item \verb|svf|, optional, a struct containing the
Projective Integration estimates of the slow vector field.
\begin{itemize}
\item \verb|svf.t| is a \(4\ell\) dimensional column vector
containing all times at which the Projective Integration
scheme is extrapolated along microBurst data to form a
macrostep.
\item \verb|svf.dx| is a \(4\ell\times n\) array containing
the estimated slow vector field.
\end{itemize}
\end{itemize}
\subsection{The projective integration code}
Determine the number of time-steps and preallocate storage
for macroscale estimates.
\begin{matlab}
%}
nT = length(tSpan);
x = nan(nT,length(x0));
%{
\end{matlab}
Get the number of expected outputs and set logical indices
to flag what data should be saved.
\begin{matlab}
%}
nArgOut = nargout();
saveMicro = (nArgOut>1);
saveFullMicro = (nArgOut>3);
saveSvf = (nArgOut>4);
%{
\end{matlab}
Run a preliminary application of the micro-burst on the
initial state to help relax to the slow manifold. This is
done in addition to the micro-burst in the main loop,
because the initial state is often far from the attracting
slow manifold. Require the user to input and output rows of
the system state.
\begin{matlab}
%}
x0 = reshape(x0,1,[]);
[relax_t,relax_x0] = microBurst(tSpan(1),x0,bT);
%{
\end{matlab}
Use the end point of the micro-burst as the initial state
for the macroscale time-steps.
\begin{matlab}
%}
tSpan(1) = relax_t(end);
x(1,:) = relax_x0(end,:);
%{
\end{matlab}
If saving information, then record the first application of
the micro-burst. Allocate cell arrays for times and states
for outputs requested by the user, as concatenating cells is
much faster than iteratively extending arrays.
\begin{matlab}
%}
if saveMicro
tms = cell(nT,1);
xms = cell(nT,1);
tms{1} = reshape(relax_t,[],1);
xms{1} = relax_x0;
if saveFullMicro
rm.t = cell(nT,1);
rm.x = cell(nT,1);
if saveSvf
svf.t = nan(4*nT-4,1);
svf.dx = nan(4*nT-4,length(x0));
end
end
end
%{
\end{matlab}
\paragraph{Loop over the macroscale time-steps}
\begin{matlab}
%}
for jT = 2:nT
T = tSpan(jT-1);
%{
\end{matlab}
If four applications of the micro-burst would cover the
entire macroscale time-step, then do so (setting some
internal states to \verb|NaN|); else proceed to projective
step.
\begin{matlab}
%}
if ~isempty(bT) && 4*abs(bT)>=abs(tSpan(jT)-T) && bT*(tSpan(jT)-T)>0
[t1,xm1] = microBurst(T, x(jT-1,:), tSpan(jT)-T);
x(jT,:) = xm1(end,:);
t2=nan; xm2=nan(1,size(xm1,2));
t3=nan; t4=nan; xm3=xm2; xm4 = xm2; dx1=xm2; dx2=xm2;
else
%{
\end{matlab}
Run the first application of the micro-burst; since this
application directly follows from the initial conditions, or
from the latest macrostep, this microscale information is
physically meaningful as a simulation of the system. Extract
the size of the final time-step.
\begin{matlab}
%}
[t1,xm1] = microBurst(T, x(jT-1,:), bT);
del = t1(end)-t1(end-1);
%{
\end{matlab}
Check for round-off error.
\begin{matlab}
%}
xt = [reshape(t1(end-1:end),[],1) xm1(end-1:end,:)];
roundingTol = 1e-8;
if norm(diff(xt))/norm(xt,'fro') < roundingTol
warning(['significant round-off error in 1st projection at T=' num2str(T)])
end
%{
\end{matlab}
Find the needed time-step to reach time \verb|tSpan(n+1)|
and form a first estimate \verb|dx1| of the slow vector
field.
\begin{matlab}
%}
Dt = tSpan(jT)-t1(end);
dx1 = (xm1(end,:)-xm1(end-1,:))/del;
%{
\end{matlab}
\emph{Assume} burst times are the same length for this
macro-step, or effectively so (recall that \verb|bT| may be
empty as it may be only coded and known in
\verb|microBurst()|).
\begin{matlab}
%}
abT = t1(end)-t1(1);
%{
\end{matlab}
Project along \verb|dx1| to form an intermediate
approximation of \verb|x|; run another application of the
micro-burst and form a second estimate of the slow vector
field.
\begin{matlab}
%}
xint = xm1(end,:) + (Dt/2-abT)*dx1;
[t2,xm2] = microBurst(T+Dt/2, xint, bT);
del = t2(end)-t2(end-1);
dx2 = (xm2(end,:)-xm2(end-1,:))/del;
xint = xm1(end,:) + (Dt/2-abT)*dx2;
[t3,xm3] = microBurst(T+Dt/2, xint, bT);
del = t3(end)-t3(end-1);
dx3 = (xm3(end,:)-xm3(end-1,:))/del;
xint = xm1(end,:) + (Dt-abT)*dx3;
[t4,xm4] = microBurst(T+Dt, xint, bT);
del = t4(end)-t4(end-1);
dx4 = (xm4(end,:)-xm4(end-1,:))/del;
%{
\end{matlab}
Check for round-off error.
\begin{matlab}
%}
xt = [reshape(t2(end-1:end),[],1) xm2(end-1:end,:)];
if norm(diff(xt))/norm(xt,'fro') < roundingTol
warning(['significant round-off error in 2nd projection at T=' num2str(T)])
end
%{
\end{matlab}
Use the weighted average of the estimates of the slow vector
field to take a macrostep.
\begin{matlab}
%}
x(jT,:) = xm1(end,:) + Dt*(dx1 + 2*dx2 + 2*dx3 + dx4)/6;
%{
\end{matlab}
Now end the if-statement that tests whether a projective
step saves simulation time.
\begin{matlab}
%}
end
%{
\end{matlab}
If saving trusted microscale data, then populate the cell
arrays for the current loop iterate with the time-steps and
output of the first application of the micro-burst. Separate
bursts by~\verb|NaN|s.
\begin{matlab}
%}
if saveMicro
tms{jT} = [reshape(t1,[],1); nan];
xms{jT} = [xm1; nan(1,size(xm1,2))];
%{
\end{matlab}
If saving all microscale data, then repeat for the remaining
applications of the micro-burst.
\begin{matlab}
%}
if saveFullMicro
rm.t{jT} = [reshape(t2,[],1); nan;...
reshape(t3,[],1); nan;...
reshape(t4,[],1); nan];
rm.x{jT} = [xm2; nan(1,size(xm2,2));...
xm3; nan(1,size(xm2,2));...
xm4; nan(1,size(xm2,2))];
%{
\end{matlab}
If saving Projective Integration estimates of the slow
vector field, then populate the corresponding cells with
times and estimates.
\begin{matlab}
%}
if saveSvf
svf.t(4*jT-7:4*jT-4) = [t1(end); t2(end); t3(end); t4(end)];
svf.dx(4*jT-7:4*jT-4,:) = [dx1; dx2; dx3; dx4];
end
end
end
%{
\end{matlab}
End of the main loop of all macro-steps.
\begin{matlab}
%}
end
%{
\end{matlab}
Overwrite \verb|x(1,:)| with the specified initial state
\verb|tSpan(1)|.
\begin{matlab}
%}
x(1,:) = reshape(x0,1,[]);
%{
\end{matlab}
For additional requested output, concatenate all the cells
of time and state data into two arrays.
\begin{matlab}
%}
if saveMicro
tms = cell2mat(tms);
xms = cell2mat(xms);
if saveFullMicro
rm.t = cell2mat(rm.t);
rm.x = cell2mat(rm.x);
end
end
%{
\end{matlab}
\subsection{If no output specified, then plot the simulation}
\begin{matlab}
%}
if nArgOut==0
figure, plot(tSpan,x,'o:')
title('Projective Simulation with PIRK4')
end
%{
\end{matlab}
This concludes \verb|PIRK4()|.
\begin{matlab}
%}
end
%{
\end{matlab}
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
rk2Int.m
|
.m
|
EquationFreeGit-master/ProjInt/rk2Int.m
| 2,458 |
utf_8
|
73bec46b0040fb09024a567dff167dee
|
% rk2Int() is a simple example of Runge--Kutta, 2nd order,
% integration of a given deterministic ODE. Used by
% PIG.m, PIGExample.m, PIGExplore.m, homogenisationExample.m
% AJR, 4 Apr 2019
%{
This is a simple example of Runge--Kutta, 2nd order,
integration of a given deterministic \ode.
\begin{matlab}
%}
function [ts,xs,errs] = rk2Int(dxdt,ts,x0)
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|dxdt()| is a function such as
\verb|dxdt=dxdt(t,x)| that computes the right-hand side of
the \ode\ \(d\xv/dt=\fv(\xv,t)\) where \xv~is a column
vector, say in \(\RR^n\) for \(n\geq1\)\,, \(t\)~is a
scalar, and the result~\fv\ is a column vector in~\(\RR^n\).
\item \verb|x0| is an \(\RR^n\) vector of initial values at
the time \verb|ts(1)|.
\item \verb|ts| is the begin and end times of the integration,
or vector of proposed micro-times.
\end{itemize}
\paragraph{Output}
\begin{itemize}
\item \verb|ts|, vector of~$\ell$ times (guess $\ell=11$).
\item \verb|xs|, array in \(\RR^{\ell\times n}\) of
approximate solution row vector at the specified times.
\end{itemize}
Compute the time-steps and create storage for outputs. Guess
that ten time-steps is often adequate, but need at least
sixty for homogenisationExample.
\begin{matlab}
%}
ndt=max(10,numel(ts)-1);
maxtry=6;
for itry=1:maxtry
ts = linspace(ts(1),ts(end),ndt+1).';
dt = diff(ts);
xs = nan(numel(x0),numel(ts));
errs = nan(numel(ts),1);
%{
\end{matlab}
Initialise first result to the given initial condition, and
evaluate the initial time derivative into~\verb|f1|.
\begin{matlab}
%}
xs(:,1) = x0(:);
errs(1) = 0;
f1 = dxdt(ts(1),xs(:,1));
%{
\end{matlab}
Compute the time-steps from~\(t_k\) to~\(t_{k+1}\), copying
the derivative~\verb|f1| at the end of the last time-step to
be the derivative at the start of this one.
\begin{matlab}
%}
for k = 1:ndt
f0 = f1;
%{
\end{matlab}
Simple second-order accurate time-step.
\begin{matlab}
%}
xh = xs(:,k)+f0*dt(k)/2;
fh = dxdt(ts(k)+dt(k)/2,xh);
xs(:,k+1) = xs(:,k)+fh*dt(k);
f1 = dxdt(ts(k+1),xs(:,k+1));
%{
\end{matlab}
Use the time derivative at~\(t_{k+1}\) to estimate an error
by storing the difference with what Simpson's rule would
estimate.
\begin{matlab}
%}
errs(k+1) = norm(f0-2*fh+f1)*dt(k)/6;
end
if norm(errs)<0.01*norm(xs(~isnan(xs)),1),break,end
ndt = ndt*3; % triple the number of time steps
if itry==maxtry, error('too many step reductions in rk2Int'), end
end% try-loop
xs = xs.';
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
MMburst.m
|
.m
|
EquationFreeGit-master/ProjInt/MMburst.m
| 914 |
utf_8
|
cc5651080a0e3428d0668d1514648bbb
|
% Short explanation for users typing "help fun"
% Author, date
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\paragraph{Code a burst of Michaelis--Menten enzyme kinetics}
Integrate a burst of length~\verb|bT| of the \ode{}s for the
Michaelis--Menten enzyme kinetics at parameter~\(\epsilon\)
inherited from above. Code \textsc{ode}s in
function~\verb|dMMdt| with variables \(x=\verb|x(1)|\) and
\(y=\verb|x(2)|\). Starting at time~\verb|ti|, and
state~\verb|xi| (row), we here simply use \script's
\verb|ode23|\slash\verb|lsode| to integrate a burst in time.
\begin{matlab}
%}
function [ts, xs] = MMburst(ti, xi, bT)
global MMepsilon
dMMdt = @(t,x) [ -x(1)+(x(1)+0.5)*x(2)
1/MMepsilon*( x(1)-(x(1)+1)*x(2) ) ];
if ~exist('OCTAVE_VERSION','builtin')
[ts, xs] = ode23(dMMdt, [ti ti+bT], xi);
else % octave version
[ts, xs] = odeOct(dMMdt, [ti ti+bT], xi);
end
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
odeOct.m
|
.m
|
EquationFreeGit-master/ProjInt/odeOct.m
| 508 |
utf_8
|
8be6e35ca6931e20fbe997ac46b7015f
|
% Provides Matlab-like front-end to Octave ODE solver. But
% cannot use lsode, and hence this, recursively. Used by
% MMburst.m, PIG.m, PIGExample.m, PIGExplore.m
% AJR, 4 Apr 2019
%{
\begin{matlab}
%}
function [ts,xs] = odeOct(dxdt,tSpan,x0)
if length(tSpan)>2, ts = tSpan;
else ts = linspace(tSpan(1),tSpan(end),21);
end
% mimic ode45 and ode23, but much slower for non-PI
lsode_options('integration method','non-stiff');
xs = lsode(@(x,t) dxdt(t,x),x0,ts);
end
%{
\end{matlab}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
bbgen.m
|
.m
|
EquationFreeGit-master/ProjInt/bbgen.m
| 1,062 |
utf_8
|
98e5f46df9870706e518440596bec800
|
%Generate a `black-box' microsolver suitable for PI from a standard
%numerical method, an ordinary differential equation, and a given upper
%bound on the time-step.
%JM,Sept 2018.
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\subsection{\texttt{bbgen()}}
\label{sec:bbgen}
\verb|bbgen()| is a simple function that takes a standard numerical method and produces a black-box solver of the type required by the PI schemes.
\begin{matlab}
%}
function bb = bbgen(solver,f,dt)
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|solver|, a standard numerical solver for ordinary differential equations
\item \verb|f|, a function f(t,x) taking time and state inputs
\item \verb|dt|, a time-step suitable for simulation with \verb|f|
\end{itemize}
\paragraph{Output}
\verb|bb = bb|\((t_{in},x_{in},T)\) a `black-box' microsolver that initialises at \( (t_{in},x_{in}) \) and simulates forward a duration \(T\).
\begin{devMan}
\begin{matlab}
%}
bb = @(t_in,x_in,T) feval(solver,f,...
linspace(t_in,t_in+T,1+ceil(T/dt)),x_in);
end
%{
\end{matlab}
\end{devMan}
%}
|
github
|
uoa1184615/EquationFreeGit-master
|
cdmc.m
|
.m
|
EquationFreeGit-master/ProjInt/cdmc.m
| 2,446 |
utf_8
|
d12a17f15f6948ac9b557d3c5befd9a3
|
% Relax a given initial condition to one onto the slow
% manifold by two steps of the 'xmas-tree' algorithm.
% JM & AJR, July 2018 -- Apr 2019
%!TEX root = ../Doc/eqnFreeDevMan.tex
%{
\section{\texttt{cdmc()}: constraint defined manifold computing}
\label{sec:cdmc}
The function \verb|cdmc()| iteratively applies the given
micro-burst and then projects backward to the initial time.
The cumulative effect is to relax the variables to the
attracting slow manifold, while keeping the `final' time for
the output the same as the input time.
\begin{matlab}
%}
function [ts, xs] = cdmc(microBurst, t0, x0)
%{
\end{matlab}
\paragraph{Input}
\begin{itemize}
\item \verb|microBurst()|, a black-box micro-burst function
suitable for Projective Integration. See any of
\verb|PIRK2()|, \verb|PIRK4()|, or \verb|PIG()| for a
description of \verb|microBurst()|.
\item \verb|t0|, an initial time.
\item \verb|x0|, an initial state vector.
\end{itemize}
\paragraph{Output}
\begin{itemize}
\item \verb|ts|, a vector of times.
\item \verb|xs|, an array of state estimates produced by
\verb|microBurst()|.
\end{itemize}
This function is a wrapper for the micro-burst. For instance
if the problem of interest is a dynamical system that is not
too stiff, and which is simulated by the micro-burst function
\verb|sol(t,x)|, one would invoke \verb|cdmc()| by defining
\begin{verbatim}
cdmcSol = @(t,x) cdmc(sol,t,x)|
\end{verbatim}
and thereafter use \verb|cdmcSol()| in place of \verb|sol()|
as the microBurst in any Projective Integration scheme. The
original microBurst \verb|sol()| could create large errors
if used in the \verb|PIG()| scheme, but the output via
\verb|cdmc()| should not.
\begin{devMan}
Begin with a standard application of the micro-burst. Need
\verb|feval| as \verb|microBurst| has multiple outputs.
\begin{matlab}
%}
[t1,x1] = feval(microBurst,t0,x0);
bT = t1(end)-t1(1);
%{
\end{matlab}
Project backwards to before the initial time, then simulate
just one burst forward to obtain a simulation burst that
ends at the original~\verb|t0|.
\begin{matlab}
%}
dxdt = (x1(end,:) - x1(end-1,:))/(t1(end) - t1(end-1));
x0 = x1(end,:)-2*bT*dxdt;
t0 = t1(1)-bT;
[t2,x2] = feval(microBurst,t0,x0.');
%{
\end{matlab}
Return both sets of output(?), although only \verb|(t2,x2)|
should be used in Projective Integration---maybe safer to
return only \verb|(t2,x2)|.
\begin{matlab}
%}
ts = [t1(:); t2(:)];
xs = [x1; x2];
%{
\end{matlab}
\end{devMan}
%}
|
github
|
PamirGhimire/visualServoing_ROSProject-master
|
checkratio.m
|
.m
|
visualServoing_ROSProject-master/IBVS_QrPointsBased/MATLAB/qrIdentifierDetection/checkratio.m
| 442 |
utf_8
|
60435c493dc630593aabf762f3736443
|
%% Function for checking bwbwb ratio for qr identifier detection
% correct ratio = 1:1:3:1:1
% Pamir Ghimire, December 10, 2017
function ratiopositive = checkratio(b1, w1, b2, w2, b3)
ratiopositive = false;
input = [b1, w1, b2, w2, b3];
desired = [1, 1, 3, 1, 1];
input = input / min(input);
tolerance = 0.7;
if (norm(input - desired, 2) < tolerance)
ratiopositive = true;
else
ratiopositive = false;
end
end
|
github
|
SergioMarreroMarrero/OldWorks-master
|
TaluDMet.m
|
.m
|
OldWorks-master/3)slopeSoftwareOptimized/Software/3.Interfaz guide/TaluDMet.m
| 41,106 |
utf_8
|
9f552c51f9fcf44098c45a2dd8fe1693
|
function varargout = TaluDMet(varargin)
% TALUDMET MATLAB code for TaluDMet.fig
% TALUDMET, by itself, creates coordenadax new TALUDMET or raises the existing
% singleton*.
%
% Altura = TALUDMET returns the handle to coordenadax new TALUDMET or the handle to
% the existing singleton*.
%
% TALUDMET('CALLBACK',hObject,eventData,handles,...) calls the local
% function named CALLBACK in TALUDMET.M with the given input arguments.
%
% TALUDMET('Property','Value',...) creates coordenadax new TALUDMET or raises the
% existing singleton*. Starting from the left, property value pairs are
% applied to the GUI before TaluDMet_OpeningFcn gets called. An
% unrecognized property name or invalid value makes property application
% stop. All inputs are passed to TaluDMet_OpeningFcn via varargin.
%
% *See GUI Options on GUIDE's Tools menu. Choose "GUI allows only one
% instance to run (singleton)".
%
% See also: GUIDE, GUIDATA, GUIHANDLES
% Edit the above text to modify the response to help TaluDMet
% Last Modified by GUIDE v2.5 21-Jul-2015 12:58:14
% Begin initialization code - DO NOT EDIT
gui_Singleton = 1;
gui_State = struct('gui_Name', mfilename, ...
'gui_Singleton', gui_Singleton, ...
'gui_OpeningFcn', @TaluDMet_OpeningFcn, ...
'gui_OutputFcn', @TaluDMet_OutputFcn, ...
'gui_LayoutFcn', [] , ...
'gui_Callback', []);
if nargin && ischar(varargin{1})
gui_State.gui_Callback = str2func(varargin{1});
end
if nargout
[varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
gui_mainfcn(gui_State, varargin{:});
end
% End initialization code - DO NOT EDIT
% --- Executes just before TaluDMet is made visible.
function TaluDMet_OpeningFcn(hObject, eventdata, handles, varargin)
% This function has no output args, see OutputFcn.
% hObject handle to figure
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% varargin command line arguments to TaluDMet (see VARARGIN)
inicio
% Choose default command line output for TaluDMet
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% UIWAIT makes TaluDMet wait for user response (see UIRESUME)
% uiwait(handles.figure1);
% --- Outputs from this function are returned to the command line.
function varargout = TaluDMet_OutputFcn(hObject, eventdata, handles)
% varargout cell array for returning output args (see VARARGOUT);
% hObject handle to figure
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Get default command line output from handles structure
varargout{1} = handles.output;
% --- Executes on button press in dibujar.
function dibujar_Callback(hObject, eventdata, handles)
% hObject handle to dibujar (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
%% Cambio de valores
% Borrados
try
delete(handles.SuperficieTalud)
end
borradosparaploteos;
borrarelmarco
H=handles.H;
B=handles.B;
try
axes(handles.axes1);
% TALUD
[x_talud,y_talud]=dibujotalud(H,B);
handles.SuperficieTalud=plot(x_talud,y_talud,'k','LineWidth',3);
axis equal
hold on
catch
errordlg('Faltan datos por introducir','ERROR');
end
a=handles.a;
b=handles.b;
R=handles.R;
rebanadas=handles.rebanadas;
try
% DESlIZAMIENTO
dibujodeslizamiento
%% Plot Circunferencia
handles.Circunferencia=plot(x_circunfe,y_circunfe,'y','LineWidth',1);
%% Plot centro de circunferencia
handles.CentroCirc=plot(a,b,'*r','LineWidth',2);
%% Plot arco de circunferencia
handles.ArcoCircunf=plot(arco_talud_x,arco_talud_y,'r','LineWidth',2);
axis equal
end
vp=[handles.vpx handles.vpy];
vf=[handles.vfx handles.vfy];
vh=[handles.vpx handles.vfy];
vb=[handles.vfx handles.vpy];
try
% Lineas
handles.lynea1=line([vp(1) vh(1)],[vp(2) vh(2)]);
handles.lynea2=line([vh(1) vf(1)],[vh(2) vf(2)]);
handles.lynea3=line([vf(1) vb(1)],[vf(2) vb(2)]);
handles.lynea4=line([vb(1) vp(1)],[vb(2) vp(2)]);
% Puntos
handles.punto1=plot(vp(1),vp(2),'b*');
handles.punto2=plot(vh(1),vh(2),'b*');
handles.punto3=plot(vb(1),vb(2),'b*');
handles.punto4=plot(vf(1),vf(2),'b*');
end
handles.output = hObject;
guidata(hObject, handles);
% --- Executes on button press in limpiar.
function limpiar_Callback(hObject, eventdata, handles)
% hObject handle to limpiar (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
cla;
% set(gca,'YTickLabel',[],'XTickLabel',[])
% legend('off')
set(handles.rgeometricos,'String','')
set(handles.rtalud,'String','')
handles.output = hObject;
guidata(hObject, handles);
% --- Executes on button press in RESET.
function RESET_Callback(hObject, eventdata, handles)
% hObject handle to RESET (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
reseteo
handles.output = hObject;
guidata(hObject, handles);
function coordenadax_Callback(hObject, eventdata, handles)
% hObject handle to coordenadax (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of coordenadax as text
% str2double(get(hObject,'String')) returns contents of coordenadax as coordenadax double
handles.a=str2double(get(hObject,'String'));
% Choose default command line output for TaluDMet
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function coordenadax_CreateFcn(hObject, eventdata, handles)
% hObject handle to coordenadax (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have coordenadax white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function coordenaday_Callback(hObject, eventdata, handles)
% hObject handle to coordenaday (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of coordenaday as text
% str2double(get(hObject,'String')) returns contents of coordenaday as coordenadax double
handles.b=str2double(get(hObject,'String'));
% Choose default command line output for TaluDMet
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function coordenaday_CreateFcn(hObject, eventdata, handles)
% hObject handle to coordenaday (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have coordenadax white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function Radio_Callback(hObject, eventdata, handles)
% hObject handle to Radio (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of Radio as text
% str2double(get(hObject,'String')) returns contents of Radio as coordenadax double
handles.R=str2double(get(hObject,'String'));
% Choose default command line output for TaluDMet
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function Radio_CreateFcn(hObject, eventdata, handles)
% hObject handle to Radio (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have coordenadax white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function Base_Callback(hObject, eventdata, handles)
% hObject handle to COORDENADAY (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of COORDENADAY as text
% str2double(get(hObject,'String')) returns contents of COORDENADAY as coordenadax double
handles.B=str2double(get(hObject,'String'));
% Choose default command line output for TaluDMet
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function Base_CreateFcn(hObject, eventdata, handles)
% hObject handle to COORDENADAY (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have coordenadax white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function Altura_Callback(hObject, eventdata, handles)
% hObject handle to Altura (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of Altura as text
% str2double(get(hObject,'String')) returns contents of Altura as coordenadax double
handles.H=str2double(get(hObject,'String'));
% Choose default command line output for TaluDMet
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function Altura_CreateFcn(hObject, eventdata, handles)
% hObject handle to Altura (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have coordenadax white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function angulorozamiento_Callback(hObject, eventdata, handles)
% hObject handle to angulorozamiento (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of angulorozamiento as text
% str2double(get(hObject,'String')) returns contents of angulorozamiento as coordenadax double
handles.fi=str2double(get(hObject,'String'));
% Choose default command line output for TaluDMet
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function angulorozamiento_CreateFcn(hObject, eventdata, handles)
% hObject handle to angulorozamiento (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have coordenadax white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function Especifico_Callback(hObject, eventdata, handles)
% hObject handle to Especifico (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of Especifico as text
% str2double(get(hObject,'String')) returns contents of Especifico as coordenadax double
handles.gd=str2double(get(hObject,'String'));
% Choose default command line output for TaluDMet
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function Especifico_CreateFcn(hObject, eventdata, handles)
% hObject handle to Especifico (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have coordenadax white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function Cohesion_Callback(hObject, eventdata, handles)
% hObject handle to Cohesion (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of Cohesion as text
% str2double(get(hObject,'String')) returns contents of Cohesion as coordenadax double
handles.C=str2double(get(hObject,'String'));
% Choose default command line output for TaluDMet
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function Cohesion_CreateFcn(hObject, eventdata, handles)
% hObject handle to Cohesion (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have coordenadax white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
% --- Executes on slider movement.
function slider1_Callback(hObject, eventdata, handles)
% hObject handle to slider1 (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'Value') returns position of slider
% get(hObject,'Min') and get(hObject,'Max') to determine range of slider
slideRadio;
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function slider1_CreateFcn(hObject, eventdata, handles)
% hObject handle to slider1 (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: slider controls usually have coordenadax light gray background.
if isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor',[.9 .9 .9]);
end
% --- Executes on button press in centromanual.
function centromanual_Callback(hObject, eventdata, handles)
% hObject handle to centromanual (see GCBO)
% eventdata reserved - to be defined in coordenadax future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
centroginput
%% Plot centro de circunferencia
handles.output = hObject;
guidata(hObject, handles);
% --- Executes on button press in geometria.
function geometria_Callback(hObject, eventdata, handles)
% hObject handle to geometria (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
geometricos;
% --- Executes on button press in Talud.
function Talud_Callback(hObject, eventdata, handles)
% hObject handle to Talud (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
propiedadestalud
% --- Executes on button press in Factor.
function Factor_Callback(hObject, eventdata, handles)
% hObject handle to Factor (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% --- Executes on button press in rejas.
function rejas_Callback(hObject, eventdata, handles)
% hObject handle to rejas (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hint: get(hObject,'Value') returns toggle state of rejas
if (get(hObject,'Value') == get(hObject,'Max'))
axes(handles.axes1);
grid on
else
axes(handles.axes1);
grid off
end
handles.output = hObject;
guidata(hObject, handles);
% --- Executes on button press in analizar.
function analizar_Callback(hObject, eventdata, handles)
% hObject handle to analizar (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
flujo
handles.output = hObject;
guidata(hObject, handles);
% --- Executes on selection change in tipodeanalisis.
function tipodeanalisis_Callback(hObject, eventdata, handles)
% hObject handle to tipodeanalisis (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: contents = cellstr(get(hObject,'String')) returns tipodeanalisis contents as cell array
% contents{get(hObject,'Value')} returns selected item from tipodeanalisis
analisistipo
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function tipodeanalisis_CreateFcn(hObject, eventdata, handles)
% hObject handle to tipodeanalisis (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: popupmenu controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function puntospormetro_Callback(hObject, eventdata, handles)
% hObject handle to puntospormetro (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of puntospormetro as text
% str2double(get(hObject,'String')) returns contents of puntospormetro as a double
precision=str2double(get(hObject,'String'));
handles.ppm1=round(1/precision);
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function puntospormetro_CreateFcn(hObject, eventdata, handles)
% hObject handle to puntospormetro (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
% --- Executes on button press in marco.
function marco_Callback(hObject, eventdata, handles)
% hObject handle to marco (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
dibujarelmarco
handles.output = hObject;
guidata(hObject, handles);
% --- Executes on selection change in metodousado.
function metodousado_Callback(hObject, eventdata, handles)
% hObject handle to metodousado (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: contents = cellstr(get(hObject,'String')) returns metodousado contents as cell array
% contents{get(hObject,'Value')} returns selected item from metodousado
metodousar
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function metodousado_CreateFcn(hObject, eventdata, handles)
% hObject handle to metodousado (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: popupmenu controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function reb_Callback(hObject, eventdata, handles)
% hObject handle to reb (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of reb as text
% str2double(get(hObject,'String')) returns contents of reb as a double
handles.rebanadas=str2double(get(hObject,'String'));
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function reb_CreateFcn(hObject, eventdata, handles)
% hObject handle to reb (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
% --------------------------------------------------------------------
function uipushtool1_ClickedCallback(hObject, eventdata, handles)
% hObject handle to uipushtool1 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
cargardatos;
handles.output = hObject;
guidata(hObject, handles);
% --- Executes on button press in LIMPIARPANTALLA.
function LIMPIARPANTALLA_Callback(hObject, eventdata, handles)
% hObject handle to LIMPIARPANTALLA (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
cla;
set(gca,'YTickLabel',[],'XTickLabel',[])
legend('off')
function limiteFS_Callback(hObject, eventdata, handles)
% hObject handle to limiteFS (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of limiteFS as text
% str2double(get(hObject,'String')) returns contents of limiteFS as a double
handles.limFS=str2double(get(hObject,'String'));
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function limiteFS_CreateFcn(hObject, eventdata, handles)
% hObject handle to limiteFS (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function PuntosR_Callback(hObject, eventdata, handles)
% hObject handle to PuntosR (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of PuntosR as text
% str2double(get(hObject,'String')) returns contents of PuntosR as a double
handles.PTR=str2double(get(hObject,'String'));
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function PuntosR_CreateFcn(hObject, eventdata, handles)
% hObject handle to PuntosR (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function pbasex_Callback(hObject, eventdata, handles)
% hObject handle to pbasex (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of pbasex as text
% str2double(get(hObject,'String')) returns contents of pbasex as a double
handles.vbx=str2double(get(hObject,'String'));
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function pbasex_CreateFcn(hObject, eventdata, handles)
% hObject handle to pbasex (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function palturax_Callback(hObject, eventdata, handles)
% hObject handle to palturax (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of palturax as text
% str2double(get(hObject,'String')) returns contents of palturax as a double
handles.vhx=str2double(get(hObject,'String'));
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function palturax_CreateFcn(hObject, eventdata, handles)
% hObject handle to palturax (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function pinicialx_Callback(hObject, eventdata, handles)
% hObject handle to pinicialx (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of pinicialx as text
% str2double(get(hObject,'String')) returns contents of pinicialx as a double
handles.vpx=str2double(get(hObject,'String'));
% handles.vhx=handles.vpx;
% set(handles.palturax,'String',num2str(handles.vpx))
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function pinicialx_CreateFcn(hObject, eventdata, handles)
% hObject handle to pinicialx (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function pfinalx_Callback(hObject, eventdata, handles)
% hObject handle to pfinalx (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of pfinalx as text
% str2double(get(hObject,'String')) returns contents of pfinalx as a double
handles.vfx=str2double(get(hObject,'String'));
% handles.vbx=handles.vfx;
% set(handles.pbasex,'String',num2str(handles.vbx))
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function pfinalx_CreateFcn(hObject, eventdata, handles)
% hObject handle to pfinalx (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
% --- Executes on button press in borramarco.
function borramarco_Callback(hObject, eventdata, handles)
% hObject handle to borramarco (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
limpio='';
set(handles.pinicialx,'String',limpio)
set(handles.pinicialy,'String',limpio)
% set(handles.palturax,'String',limpio)
% set(handles.palturay,'String',limpio)
% set(handles.pbasex,'String',limpio)
% set(handles.pbasey,'String',limpio)
set(handles.pfinalx,'String',limpio)
set(handles.pfinaly,'String',limpio)
handles.vpx=nan;
handles.vpy=nan;
handles.vhx=nan;
handles.vhy=nan;
handles.vbx=nan;
handles.vby=nan;
handles.vfx=nan;
handles.vfy=nan;
borrarelmarco
handles.output = hObject;
guidata(hObject, handles);
function pinicialy_Callback(hObject, eventdata, handles)
% hObject handle to pinicialy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of pinicialy as text
% str2double(get(hObject,'String')) returns contents of pinicialy as a double
vpy=str2double(get(hObject,'String'));
handles.vpy=vpy;
% handles.vby=vpy;
% set(handles.pbasey,'String',num2str(vpy));
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function pinicialy_CreateFcn(hObject, eventdata, handles)
% hObject handle to pinicialy (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function palturay_Callback(hObject, eventdata, handles)
% hObject handle to palturay (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of palturay as text
% str2double(get(hObject,'String')) returns contents of palturay as a double
handles.vhy=str2double(get(hObject,'String'));
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function palturay_CreateFcn(hObject, eventdata, handles)
% hObject handle to palturay (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function pbasey_Callback(hObject, eventdata, handles)
% hObject handle to pbasey (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of pbasey as text
% str2double(get(hObject,'String')) returns contents of pbasey as a double
handles.vby=str2double(get(hObject,'String'));
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function pbasey_CreateFcn(hObject, eventdata, handles)
% hObject handle to pbasey (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function pfinaly_Callback(hObject, eventdata, handles)
% hObject handle to pfinaly (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of pfinaly as text
% str2double(get(hObject,'String')) returns contents of pfinaly as a double
vfy=str2double(get(hObject,'String'));
handles.vfy=vfy;
% handles.vhy=handles.vfy;
% set(handles.palturay,'String',num2str(vfy))
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function pfinaly_CreateFcn(hObject, eventdata, handles)
% hObject handle to pfinaly (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function puntospormetro2_Callback(hObject, eventdata, handles)
% hObject handle to puntospormetro2 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of puntospormetro2 as text
% str2double(get(hObject,'String')) returns contents of puntospormetro2 as a double
handles.ppm2=str2double(get(hObject,'String'));
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function puntospormetro2_CreateFcn(hObject, eventdata, handles)
% hObject handle to puntospormetro2 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function entorno_Callback(hObject, eventdata, handles)
% hObject handle to entorno (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of entorno as text
% str2double(get(hObject,'String')) returns contents of entorno as a double
handles.ampl=str2double(get(hObject,'String'));
handles.output = hObject;
guidata(hObject, handles);
% --- Executes during object creation, after setting all properties.
function entorno_CreateFcn(hObject, eventdata, handles)
% hObject handle to entorno (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function edit26_Callback(hObject, eventdata, handles)
% hObject handle to edit26 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit26 as text
% str2double(get(hObject,'String')) returns contents of edit26 as a double
% --- Executes during object creation, after setting all properties.
function edit26_CreateFcn(hObject, eventdata, handles)
% hObject handle to edit26 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function edit27_Callback(hObject, eventdata, handles)
% hObject handle to edit27 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit27 as text
% str2double(get(hObject,'String')) returns contents of edit27 as a double
% --- Executes during object creation, after setting all properties.
function edit27_CreateFcn(hObject, eventdata, handles)
% hObject handle to edit27 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function edit28_Callback(hObject, eventdata, handles)
% hObject handle to edit28 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit28 as text
% str2double(get(hObject,'String')) returns contents of edit28 as a double
% --- Executes during object creation, after setting all properties.
function edit28_CreateFcn(hObject, eventdata, handles)
% hObject handle to edit28 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles empty - handles not created until after all CreateFcns called
% Hint: edit controls usually have a white background on Windows.
% See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
% --------------------------------------------------------------------
function uipushtool2_ClickedCallback(hObject, eventdata, handles)
% hObject handle to uipushtool2 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
guardardatos
% --- Executes on button press in marcopt.
function marcopt_Callback(hObject, eventdata, handles)
% hObject handle to marcopt (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
[ca,cb] = ginput(1);
handles.ca=ca;
handles.cb=cb;
l1=handles.ampl;
% Punto inicial
vp=[ca cb]-l1/2;
vp=round(vp*1000)/1000;
% Punto final
vf=[ca cb]+l1/2;
vf=round(vf*1000)/1000;
% Vertice de altura. Altura
vh=[vp(1) vf(2)];
% Vertice de longitud. Base
vb=[vf(1) vp(2)];
axes(handles.axes1)
verticesx=sort([vp(1) vf(1)]);
verticesy=sort([vp(2) vf(2)]);
a1=verticesx(1);a2=verticesx(2);
b1=verticesy(1);b2=verticesy(2);
handles.rectangulo31 =plot([a1,a2,a2,a1,a1],[b1,b1,b2,b2,b1]);
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% --- Executes on button press in marcoborro.
function marcoborro_Callback(hObject, eventdata, handles)
% hObject handle to marcoborro (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
try
delete(handles.rectangulo31);
end
|
github
|
SergioMarreroMarrero/OldWorks-master
|
DSSStartup.m
|
.m
|
OldWorks-master/6)Montecarlo-LowVoltage/Software/apoyo/DSSStartup.m
| 425 |
utf_8
|
5a54c00cbb8bef30ce34a677898c1229
|
%--------------------------------------------------------------------------
function [Start,Obj,Text] = DSSStartup
% Function for starting up the DSS
%instantiate the DSS Object
Obj = actxserver('OpenDSSEngine.DSS');
%
%Start the DSS. Only needs to be executed the first time w/in a
%Matlab session
Start = Obj.Start(0);
% Define the text interface
Text = Obj.Text;
|
github
|
SergioMarreroMarrero/OldWorks-master
|
startAndGoal.m
|
.m
|
OldWorks-master/2)AutonomoDrive/Planification/Matlab/startAndGoal.m
| 1,716 |
utf_8
|
ebf0bf51e9e758eb6bfe21b6154955c9
|
function [start,goal] =startAndGoal(map,option)
switch lower(option)
case 'r'
freeVal=find(map==0);
% Start
indexStart=round(rand*length(freeVal));
start=freeVal(indexStart);
freeVal(indexStart)=[];
%Goal
indexGoal=round(rand*length(freeVal));
goal=freeVal(indexGoal);
case 'm'
outsideMapStart=1;outsideMapGoal=1; % para entrar en el while
plotMap(map);
%start
while outsideMapStart
[x,y] = ginput(1);
posStartRow=round(y);
posStartColumn=round(x);
outsideMapStart=map(posStartRow,posStartColumn)~=0;
end
%Goal
while outsideMapGoal
[x,y] = ginput(1);
posGoalRow=round(y);
posGoalColumn=round(x);
outsideMapGoal=map(posGoalRow,posGoalColumn)~=0;
end
start=sub2ind(size(map),posStartRow,posStartColumn);
goal=sub2ind(size(map),posGoalRow,posGoalColumn);
case 'ml'
outsideMapStart=1; % para entrar en el while
plotMap(map);
%start
while outsideMapStart
[x,y] = ginput(1);
posStartRow=round(y);
posStartColumn=round(x);
outsideMapStart=map(posStartRow,posStartColumn)~=0;
end
start=sub2ind(size(map),posStartRow,posStartColumn);
goal=1;
end
end
function [mapcode]=plotMap(map)
close all
mapcode=imagesc(map);
grid on
[nrow,ncolumn]=size(map);
set(gca,'xtick',1.5:1:nrow-0.5);
set(gca,'ytick',1.5:1:ncolumn-0.5);
end
|
github
|
SergioMarreroMarrero/OldWorks-master
|
localizationFunction.m
|
.m
|
OldWorks-master/2)AutonomoDrive/Planification/Matlab/localizacion/localizationFunction.m
| 5,378 |
utf_8
|
c71bf82b62c4a08f04b780adfa1d0f14
|
function [ whereTheRobotIs] = localizationFunction( map,posStart )
% 1) Definicion de algunas variables iniciales
blockCode=-5;positionRobotcode=-10;posWhereRobotCouldBecode=-15;
map(map==1)=blockCode;
originalMap=map;mapPosition=map;
pathThatRobotHaveDone=[]; % Store the path walked
%2)
pathThatRobotHaveDone=[pathThatRobotHaveDone posStart]; % Iniciamos el camino hecho con la posicion de partida
%3)
%%%%%%%%%%%%%%%PLOT%%%%%%%%%%%%%%%%%
mapPosition(pathThatRobotHaveDone)=positionRobotcode; % Actualizamos el camino hecho, incorporando en el mapa la la variable pathThatRobotHaveDone
plotMap(mapPosition);
pause;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%4)
%%%%%%%%%%%%%%%%WHERE IS THE ROBOT%%%%%%%%%%%%%%%%%%%%%%%
[currentEnvMatrix]=whatRobotSee(map,posStart);
[numMapRow,numMapColumn]=size(map);
%5)
%%%%Con este algoritmo creamos inicialmente los puntos en los que el robot
%%%%podria estar, es decir, todos los puntos del mapa (en los que haya un cero)
l=0;
for i=2:numMapRow-1 % i,j=1 el borde del mapa es un muro
for j= 2:numMapColumn-1
l=l+1;
pointsWhereRobotCouldBe(l)=sub2ind(size(map),i,j);
end
end
%6)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[pointsWhereRobotCouldBe]=matchingMapEnviromentMatrix(map,currentEnvMatrix,pointsWhereRobotCouldBe);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%7)
%%%%%%%%%%%%%%%PLOT%%%%%%%%%%%%%%%%%
mapPosition(pathThatRobotHaveDone)=positionRobotcode;
plotMap(mapPosition);
putPoints(pointsWhereRobotCouldBe,map,'r*');
pause;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%8)
while length(pointsWhereRobotCouldBe)>1
%8)
%%%%%%%%%%%%%%%%MOVE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
direction=selectDirection2Move(map,pathThatRobotHaveDone,blockCode);
[pointsWhereRobotCouldBeAfterMove]=robotMove(direction,pointsWhereRobotCouldBe,map); % Averiguamos donde esta robot despues de moverse
pathThatRobotHaveDone=[pathThatRobotHaveDone pointsWhereRobotCouldBeAfterMove(pointsWhereRobotCouldBe==pathThatRobotHaveDone(end))]; % Actualizamos la nueva posicion del robot
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%9)
%%%%%%%%%%%%%%%PLOT%%%%%%%%%%%%%%%%%
mapPosition(pathThatRobotHaveDone)=positionRobotcode;
plotMap(mapPosition);
putPoints(pointsWhereRobotCouldBe,map,'r*');
putPoints(pointsWhereRobotCouldBeAfterMove,map,'y*');
pause;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%10)
%%%%%%%%%%%%%%%%WHERE IS THE ROBOT%%%%%%%%%%%%%%%%%%%%%%%
[currentEnvMatrix]=whatRobotSee(map,pathThatRobotHaveDone(end));
[pointsWhereRobotCouldBe]=matchingMapEnviromentMatrix(map,currentEnvMatrix,pointsWhereRobotCouldBeAfterMove);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%PLOT%%%%%%%%%%%%%%%%%
mapPosition(pathThatRobotHaveDone)=positionRobotcode;
plotMap(mapPosition);
putPoints(pointsWhereRobotCouldBe,map,'r*');
pause;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
end
whereTheRobotIs=pointsWhereRobotCouldBe;
end
function enviromentMatrix=whatRobotSee(map,pos)
[posRow,posColumn]=ind2sub(size(map),pos);
enviromentMatrix=nan(3); % iniciamos la variable
j=0;
for i=[-1 0 1] %1)
j=j+1;
enviromentMatrix(j,:)=map(posRow+i,posColumn-1:posColumn+1); %2)
end
end
function pointsWhereRobotCouldBe=matchingMapEnviromentMatrix(map,enviromentMatrix,candidates)
j=0;
for matchPos=candidates %1)Para cada punto posible en que el robot se encuentre
enviromentMatrixMatching=whatRobotSee(map,matchPos);%2) Sacamos la matriz del entorno de cada punto del mapa
prob2behere=sum(sum(enviromentMatrixMatching==enviromentMatrix))==length(enviromentMatrix)^2; %2)Sumamos el numero de aciertos. Se suma dos veces porque sum() suma solo por columnas
if prob2behere %3) Almacenamos el resultado en la lista pointsWhereRobotCouldBe
j=j+1;
pointsWhereRobotCouldBe(j)=matchPos;
end
end
end
function direction=selectDirection2Move(map,pathThatRobotHaveDone,blockCode)
map(pathThatRobotHaveDone)=blockCode;
[enviromentMatrix]=whatRobotSee(map,pathThatRobotHaveDone(end));
if enviromentMatrix(4)==0
direction=4;
elseif enviromentMatrix(6)==0
direction=6;
elseif enviromentMatrix(8)==0
direction=8;
elseif enviromentMatrix(2)==0
direction=2;
else
disp('Something wrong')
end
end
function pointsWhereRobotCouldBeAfterMove=robotMove(direction,pointsWhereRobotCouldBe,map)
[currentRow,currentColumn]=ind2sub(size(map),pointsWhereRobotCouldBe);
if direction==4 % Up resto 1 a fila
refreshRow=currentRow-1;
refreshtColumn=currentColumn;
elseif direction==6 % Down sumo 1 a fila
refreshRow=currentRow+1;
refreshtColumn=currentColumn;
elseif direction==2 % Left resto 1 a columna
refreshRow=currentRow;
refreshtColumn=currentColumn-1;
elseif direction==8 % Right sumo 1 a columna
refreshRow=currentRow;
refreshtColumn=currentColumn+1;
else
disp('Something wrong inside robotMove')
end
pointsWhereRobotCouldBeAfterMove=sub2ind(size(map),refreshRow,refreshtColumn);
end
function putPoints(pointsWhereRobotCouldBe,map,codePlot)
[pointColumn,pointRow]=ind2sub(size(map),pointsWhereRobotCouldBe);
hold on
plot(pointRow,pointColumn,codePlot);
end
|
github
|
idnavid/misc-master
|
demo_gif.m
|
.m
|
misc-master/demo_gif.m
| 409 |
utf_8
|
43aefadbc9e033d9ceb47347da232fff
|
function demo_gif()
filename = './clock.gif';
x = 0:0.1:100;
for i = 1:100
plot(x,i*x/1000);
ylim([0 10])
save_gif(filename,i)
end
end
function save_gif(filename,n)
frame = getframe;
im = frame2im(frame);
[imind,cm] = rgb2ind(im,256);
% Write to the GIF File
if n == 1
imwrite(imind,cm,filename,'gif', 'Loopcount',inf);
else
imwrite(imind,cm,filename,'gif','WriteMode','append');
end
end
|
github
|
JuXinCheng/rtklib_2.4.2-master
|
plotlexion.m
|
.m
|
rtklib_2.4.2-master/util/testlex/plotlexion.m
| 1,593 |
utf_8
|
1333e9ae36ebce0dfcb9da8ab7972734
|
function plotlexion(file,index)
%
% plot lex ionosphere correction error
%
% 2010/12/09 0.1 new
%
if nargin<1, file='LEXION_20101204'; end
if nargin<2, index=2; end
eval(file);
td=caltomjd(epoch);
time=time(index);
ep=mjdtocal(td+(time+0.5)/86400);
ts=sprintf('%04.0f/%02.0f/%02.0f %02.0f:%02.0f',ep(1:5));
% plot lex ion
figure('color','w');
plotmap(tec(:,:,index),lons,lats,['LEX Vertical Ionosphere Delay (L1) (m): ',ts]);
% plot igs ion
for i=1:length(lons)
for j=1:length(lats)
ion(j,i)=ion_tec(td,time,[0 pi/2],[lats(j),lons(i),0],'../lexerrdata','igr');
end
end
figure('color','w');
plotmap(ion,lons,lats,['IGR Vertical Ionosphere Delay (L1) (m): ',ts]);
% plot vion map ----------------------------------------------------------------
function plotmap(ion,lons,lats,ti)
fn='Times New Roman';
pos=[0.01 0.01 0.91 0.92];
cent=[137 35];
scale=8;
gray=[.5 .5 .5];
range=0:0.01:10;
gmt('mmap','proj','eq','cent',cent,'base',cent,'scale',6,'pos',pos,'fontname',fn);
[lon,lat]=meshgrid(lons,lats);
[xs,ys,zs]=gmt('lltoxy',lon,lat);
[c,h]=contourf(xs,ys,ion,range);
set(h,'edgecolor','none');
caxis(range([1,end]))
gmt('mcoast','lcolor','none','scolor','none','ccolor',gray);
gmt('mgrid','gint',5,'lint',5,'color',gray);
lonr=[141.0 129.0 126.7 146.0 146.0 141.0]; % lex tec coverage
latr=[ 45.5 34.7 26.0 26.0 45.5 45.5];
lonp=[130.0 118.0 115.7 157.0 157.0 130.0];
latp=[ 56.5 45.7 15.0 15.0 56.5 56.5];
gmt('mplot',lonr,latr,'k');
%gmt('mplot',lonp,latp,gray);
title(ti);
ggt('colorbarv',[0.94,0.015,0.015,0.92],range([1,end]),'',...
'fontname',fn);
|
github
|
JuXinCheng/rtklib_2.4.2-master
|
testionex.m
|
.m
|
rtklib_2.4.2-master/test/utest/testionex.m
| 326,974 |
utf_8
|
e8fe871f5e28acb961451acf230d028f
|
function testionex
[tec,rms]=testdata1;
range=0:0.01:10;
figure
[c,h]=contourf(0:2:360,90:-2:-90,tec,range);
set(h,'edgecolor','none');
caxis(range([1,end]));
title('vertical iono delay');
figure
[c,h]=contourf(0:2:360,90:-2:-90,sqrt(rms),range);
set(h,'edgecolor','none');
caxis(range([1,end]));
title('vertical iono delay std');
tec=testdata2;
figure, hold on, box on, grid on
gray=[.5 .5 .5];
time=0:30:86400*2-30;
plot(tec(:,1)/3600,tec(:,2),'.');
plot(tec(:,1)/3600,tec(:,2)-tec(:,3),'-','color',gray);
plot(tec(:,1)/3600,tec(:,2)+tec(:,3),'-','color',gray);
xlim(tec([1,end],1)/3600);
ylim([0,10]);
xlabel('time (hr)');
ylabel('iono delay (m)');
% test data 1 for tec
function [tec,rms]=testdata1
tec=[
nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
0.68 0.68 0.68 0.68 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.71 0.71 0.71 0.72 0.72 0.72 0.72 0.72 0.72 0.72 0.73 0.73 0.73 0.73 0.73 0.73 0.74 0.74 0.74 0.74 0.74 0.74 0.74 0.74 0.75 0.75 0.75 0.75 0.74 0.74 0.74 0.74 0.75 0.75 0.75 0.75 0.75 0.76 0.76 0.76 0.76 0.76 0.77 0.77 0.77 0.77 0.78 0.78 0.79 0.79 0.79 0.79 0.80 0.80 0.81 0.81 0.81 0.81 0.81 0.82 0.82 0.82 0.83 0.83 0.84 0.84 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.86 0.86 0.87 0.87 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.85 0.84 0.84 0.84 0.84 0.84 0.83 0.83 0.83 0.82 0.82 0.82 0.82 0.82 0.81 0.81 0.80 0.79 0.79 0.79 0.78 0.78 0.77 0.77 0.77 0.76 0.76 0.76 0.75 0.75 0.75 0.74 0.74 0.74 0.73 0.73 0.73 0.72 0.71 0.71 0.70 0.70 0.69 0.69 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68
0.62 0.62 0.62 0.63 0.63 0.63 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.65 0.65 0.66 0.66 0.66 0.66 0.66 0.66 0.66 0.66 0.66 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.68 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.71 0.71 0.71 0.71 0.71 0.71 0.72 0.72 0.72 0.73 0.73 0.73 0.74 0.74 0.74 0.75 0.75 0.76 0.76 0.77 0.77 0.78 0.78 0.79 0.80 0.81 0.81 0.81 0.82 0.83 0.84 0.84 0.85 0.86 0.86 0.87 0.87 0.87 0.87 0.87 0.88 0.88 0.88 0.89 0.89 0.89 0.89 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.87 0.87 0.87 0.87 0.86 0.85 0.85 0.84 0.83 0.83 0.83 0.82 0.81 0.81 0.80 0.79 0.78 0.78 0.77 0.76 0.75 0.74 0.73 0.73 0.72 0.72 0.72 0.71 0.70 0.70 0.69 0.69 0.69 0.68 0.68 0.67 0.66 0.65 0.64 0.64 0.64 0.63 0.62 0.62 0.62 0.62 0.62 0.61 0.61 0.61 0.61 0.61 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.62 0.62 0.62 0.62 0.62 0.62
0.56 0.57 0.57 0.57 0.57 0.57 0.58 0.59 0.59 0.59 0.59 0.60 0.60 0.60 0.61 0.61 0.61 0.61 0.61 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.63 0.63 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.65 0.66 0.66 0.67 0.67 0.67 0.67 0.67 0.67 0.68 0.68 0.69 0.69 0.69 0.69 0.70 0.71 0.71 0.72 0.73 0.73 0.74 0.75 0.75 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.85 0.86 0.87 0.87 0.88 0.89 0.89 0.89 0.89 0.89 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.91 0.90 0.90 0.89 0.89 0.89 0.88 0.87 0.86 0.86 0.85 0.84 0.83 0.82 0.80 0.79 0.78 0.77 0.76 0.75 0.74 0.72 0.71 0.70 0.69 0.69 0.68 0.67 0.66 0.66 0.65 0.64 0.63 0.62 0.62 0.62 0.61 0.60 0.59 0.58 0.57 0.57 0.57 0.57 0.56 0.55 0.55 0.55 0.55 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.55 0.55 0.56 0.56 0.56 0.56 0.56 0.56
0.54 0.53 0.52 0.52 0.53 0.54 0.54 0.55 0.55 0.55 0.55 0.55 0.55 0.56 0.56 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.59 0.59 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.61 0.62 0.62 0.62 0.62 0.63 0.63 0.63 0.63 0.64 0.64 0.65 0.66 0.66 0.67 0.68 0.68 0.69 0.69 0.70 0.71 0.71 0.73 0.74 0.75 0.77 0.78 0.79 0.79 0.80 0.82 0.83 0.83 0.84 0.85 0.86 0.88 0.88 0.88 0.88 0.90 0.91 0.91 0.91 0.92 0.93 0.94 0.94 0.94 0.95 0.95 0.96 0.96 0.96 0.95 0.94 0.93 0.91 0.90 0.89 0.87 0.86 0.84 0.82 0.80 0.78 0.76 0.75 0.74 0.72 0.71 0.70 0.69 0.67 0.66 0.65 0.63 0.62 0.61 0.60 0.59 0.58 0.58 0.57 0.57 0.56 0.55 0.55 0.54 0.53 0.53 0.52 0.51 0.51 0.50 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.50 0.50 0.50 0.50 0.51 0.52 0.52 0.52 0.52 0.52 0.52 0.52 0.53 0.54
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0.52 0.53 0.55 0.56 0.56 0.57 0.56 0.56 0.55 0.55 0.55 0.55 0.55 0.56 0.57 0.58 0.60 0.62 0.65 0.67 0.70 0.72 0.74 0.76 0.78 0.80 0.80 0.81 0.81 0.81 0.81 0.81 0.80 0.79 0.78 0.76 0.75 0.74 0.72 0.71 0.70 0.69 0.69 0.69 0.69 0.70 0.72 0.74 0.75 0.77 0.78 0.81 0.83 0.85 0.87 0.89 0.93 0.96 0.99 1.01 1.04 1.07 1.10 1.14 1.17 1.20 1.24 1.28 1.32 1.35 1.38 1.41 1.45 1.47 1.50 1.53 1.55 1.58 1.60 1.63 1.66 1.68 1.71 1.73 1.76 1.79 1.81 1.84 1.86 1.87 1.88 1.90 1.92 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.92 1.91 1.89 1.87 1.85 1.82 1.79 1.75 1.71 1.67 1.64 1.61 1.57 1.52 1.48 1.44 1.40 1.35 1.30 1.25 1.20 1.16 1.11 1.07 1.02 0.98 0.95 0.91 0.89 0.86 0.83 0.80 0.76 0.73 0.70 0.68 0.66 0.64 0.62 0.60 0.59 0.57 0.57 0.57 0.57 0.55 0.53 0.52 0.52 0.52 0.52 0.52 0.52 0.53 0.54 0.53 0.52 0.51 0.50 0.49 0.47 0.46 0.45 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.45 0.46 0.47 0.47 0.47 0.48 0.48 0.49 0.51 0.52
0.61 0.62 0.64 0.65 0.66 0.67 0.68 0.68 0.68 0.67 0.67 0.67 0.68 0.69 0.71 0.73 0.74 0.76 0.77 0.79 0.82 0.84 0.86 0.89 0.91 0.93 0.92 0.92 0.92 0.92 0.92 0.90 0.89 0.88 0.87 0.85 0.83 0.81 0.79 0.78 0.76 0.76 0.75 0.75 0.76 0.76 0.78 0.79 0.81 0.82 0.83 0.86 0.88 0.91 0.93 0.96 0.99 1.02 1.05 1.09 1.12 1.15 1.19 1.23 1.26 1.29 1.33 1.37 1.41 1.45 1.48 1.52 1.55 1.58 1.61 1.64 1.67 1.70 1.72 1.75 1.77 1.79 1.81 1.84 1.86 1.89 1.91 1.92 1.94 1.95 1.96 1.97 1.98 1.98 1.98 1.97 1.97 1.96 1.96 1.96 1.96 1.94 1.92 1.90 1.88 1.86 1.84 1.81 1.78 1.75 1.72 1.69 1.65 1.61 1.57 1.53 1.49 1.45 1.41 1.36 1.32 1.27 1.22 1.18 1.14 1.10 1.06 1.02 0.99 0.95 0.91 0.88 0.85 0.81 0.77 0.74 0.71 0.68 0.65 0.63 0.61 0.59 0.57 0.56 0.56 0.56 0.54 0.53 0.52 0.52 0.52 0.52 0.53 0.54 0.54 0.55 0.55 0.55 0.54 0.53 0.53 0.52 0.51 0.50 0.49 0.48 0.48 0.48 0.48 0.49 0.49 0.50 0.51 0.51 0.52 0.52 0.53 0.55 0.56 0.59 0.61
0.71 0.73 0.75 0.77 0.79 0.81 0.82 0.83 0.83 0.83 0.83 0.84 0.84 0.85 0.86 0.88 0.89 0.91 0.92 0.94 0.96 0.98 1.01 1.02 1.03 1.05 1.04 1.02 1.01 1.00 0.99 0.98 0.96 0.95 0.93 0.92 0.89 0.87 0.85 0.84 0.82 0.82 0.81 0.82 0.82 0.83 0.84 0.85 0.86 0.86 0.87 0.90 0.93 0.95 0.98 1.00 1.04 1.07 1.10 1.14 1.18 1.21 1.25 1.29 1.33 1.36 1.41 1.45 1.49 1.53 1.57 1.60 1.64 1.68 1.71 1.74 1.77 1.79 1.82 1.84 1.87 1.89 1.90 1.92 1.94 1.96 1.98 2.00 2.01 2.02 2.03 2.03 2.03 2.02 2.02 2.01 2.00 2.00 1.99 1.98 1.96 1.95 1.93 1.91 1.89 1.87 1.85 1.82 1.80 1.77 1.75 1.71 1.68 1.64 1.60 1.56 1.53 1.49 1.45 1.41 1.37 1.33 1.29 1.25 1.21 1.17 1.13 1.09 1.05 1.01 0.96 0.93 0.89 0.85 0.82 0.78 0.74 0.70 0.67 0.65 0.63 0.60 0.58 0.57 0.56 0.55 0.55 0.54 0.54 0.54 0.55 0.56 0.57 0.58 0.58 0.59 0.59 0.59 0.59 0.58 0.57 0.57 0.57 0.56 0.55 0.54 0.53 0.53 0.53 0.54 0.54 0.55 0.56 0.57 0.58 0.59 0.62 0.64 0.66 0.69 0.71
0.82 0.85 0.88 0.90 0.93 0.95 0.96 0.98 0.99 1.00 1.00 1.01 1.01 1.02 1.03 1.04 1.05 1.06 1.08 1.09 1.11 1.13 1.14 1.15 1.15 1.16 1.14 1.12 1.10 1.08 1.06 1.04 1.02 1.00 0.98 0.96 0.94 0.93 0.91 0.89 0.87 0.87 0.87 0.87 0.88 0.89 0.89 0.90 0.91 0.91 0.92 0.94 0.97 0.99 1.02 1.04 1.07 1.11 1.14 1.18 1.22 1.26 1.30 1.34 1.38 1.42 1.47 1.51 1.55 1.59 1.63 1.68 1.72 1.75 1.79 1.82 1.84 1.87 1.90 1.92 1.95 1.96 1.98 2.00 2.02 2.04 2.05 2.06 2.08 2.08 2.09 2.08 2.08 2.07 2.06 2.06 2.05 2.04 2.02 2.00 1.97 1.95 1.93 1.91 1.89 1.87 1.85 1.83 1.81 1.78 1.76 1.73 1.70 1.66 1.63 1.59 1.56 1.52 1.49 1.45 1.42 1.38 1.34 1.30 1.26 1.22 1.18 1.14 1.10 1.06 1.02 0.98 0.94 0.90 0.86 0.82 0.78 0.74 0.70 0.68 0.65 0.63 0.61 0.59 0.58 0.57 0.57 0.57 0.58 0.59 0.59 0.61 0.63 0.64 0.65 0.66 0.66 0.66 0.66 0.65 0.65 0.65 0.65 0.64 0.63 0.61 0.61 0.60 0.60 0.60 0.61 0.62 0.63 0.65 0.67 0.69 0.71 0.74 0.76 0.79 0.82
0.94 0.97 1.00 1.03 1.06 1.09 1.11 1.13 1.14 1.15 1.16 1.16 1.17 1.17 1.18 1.19 1.20 1.21 1.22 1.24 1.25 1.26 1.27 1.28 1.27 1.26 1.24 1.22 1.19 1.16 1.12 1.09 1.07 1.04 1.02 1.00 0.99 0.97 0.96 0.94 0.92 0.92 0.92 0.92 0.93 0.94 0.94 0.95 0.95 0.96 0.96 0.98 1.00 1.03 1.05 1.08 1.11 1.15 1.18 1.22 1.26 1.31 1.35 1.39 1.43 1.47 1.51 1.55 1.59 1.64 1.69 1.73 1.77 1.81 1.85 1.88 1.91 1.93 1.96 1.99 2.01 2.03 2.05 2.07 2.09 2.11 2.12 2.13 2.13 2.14 2.15 2.14 2.13 2.13 2.12 2.11 2.10 2.09 2.06 2.03 2.00 1.97 1.94 1.92 1.91 1.89 1.86 1.84 1.81 1.79 1.77 1.74 1.71 1.68 1.65 1.63 1.59 1.55 1.52 1.48 1.45 1.41 1.37 1.34 1.30 1.26 1.22 1.18 1.15 1.11 1.07 1.03 0.99 0.95 0.90 0.86 0.82 0.78 0.75 0.72 0.69 0.67 0.64 0.62 0.61 0.60 0.61 0.62 0.63 0.64 0.66 0.68 0.69 0.71 0.73 0.74 0.75 0.75 0.75 0.75 0.74 0.74 0.74 0.73 0.72 0.71 0.70 0.70 0.70 0.70 0.70 0.72 0.74 0.76 0.77 0.79 0.82 0.85 0.88 0.91 0.94
1.07 1.10 1.12 1.15 1.19 1.22 1.24 1.26 1.27 1.29 1.30 1.31 1.31 1.32 1.32 1.32 1.33 1.34 1.35 1.37 1.38 1.39 1.39 1.39 1.38 1.36 1.34 1.31 1.28 1.23 1.19 1.15 1.12 1.09 1.07 1.04 1.02 1.00 0.98 0.97 0.96 0.95 0.95 0.95 0.96 0.97 0.97 0.97 0.98 0.98 0.99 1.01 1.03 1.06 1.09 1.12 1.15 1.19 1.22 1.27 1.32 1.35 1.39 1.43 1.47 1.51 1.56 1.60 1.65 1.69 1.74 1.78 1.82 1.85 1.89 1.93 1.96 2.00 2.02 2.04 2.06 2.08 2.10 2.12 2.15 2.18 2.19 2.20 2.21 2.22 2.22 2.22 2.21 2.21 2.20 2.19 2.17 2.15 2.13 2.09 2.06 2.03 2.00 1.97 1.94 1.92 1.88 1.85 1.82 1.80 1.77 1.75 1.73 1.70 1.67 1.64 1.61 1.58 1.54 1.51 1.48 1.44 1.40 1.36 1.33 1.30 1.27 1.23 1.20 1.16 1.12 1.08 1.04 1.00 0.95 0.91 0.87 0.83 0.80 0.77 0.75 0.72 0.69 0.68 0.68 0.68 0.69 0.69 0.71 0.73 0.75 0.77 0.79 0.81 0.82 0.84 0.85 0.86 0.86 0.85 0.84 0.84 0.83 0.82 0.82 0.81 0.81 0.81 0.82 0.82 0.83 0.85 0.87 0.88 0.90 0.91 0.94 0.97 1.01 1.04 1.07
1.16 1.19 1.22 1.26 1.29 1.32 1.34 1.36 1.38 1.39 1.40 1.41 1.42 1.42 1.43 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 1.49 1.48 1.47 1.44 1.41 1.37 1.32 1.28 1.23 1.19 1.15 1.11 1.08 1.06 1.04 1.02 1.00 0.98 0.98 0.98 0.98 0.98 0.99 0.99 0.99 0.99 1.00 1.00 1.03 1.05 1.08 1.12 1.16 1.20 1.23 1.28 1.32 1.37 1.41 1.45 1.49 1.53 1.58 1.62 1.66 1.70 1.75 1.80 1.84 1.88 1.92 1.95 1.98 2.01 2.04 2.06 2.08 2.10 2.13 2.15 2.18 2.20 2.23 2.25 2.27 2.29 2.30 2.30 2.30 2.30 2.30 2.29 2.28 2.26 2.23 2.21 2.17 2.14 2.10 2.06 2.03 1.99 1.95 1.92 1.88 1.85 1.81 1.78 1.76 1.74 1.72 1.70 1.68 1.65 1.61 1.58 1.55 1.52 1.48 1.45 1.42 1.38 1.35 1.32 1.29 1.25 1.21 1.17 1.13 1.09 1.05 1.01 0.97 0.94 0.90 0.86 0.84 0.81 0.80 0.78 0.78 0.78 0.79 0.80 0.81 0.83 0.85 0.88 0.90 0.92 0.93 0.95 0.96 0.97 0.97 0.97 0.97 0.96 0.95 0.95 0.94 0.93 0.92 0.92 0.93 0.93 0.94 0.95 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.11 1.14 1.16
1.23 1.27 1.30 1.33 1.36 1.40 1.42 1.44 1.45 1.46 1.48 1.48 1.48 1.49 1.49 1.50 1.51 1.52 1.53 1.55 1.56 1.57 1.57 1.57 1.57 1.56 1.53 1.49 1.45 1.40 1.35 1.30 1.25 1.20 1.17 1.13 1.11 1.09 1.07 1.05 1.03 1.03 1.02 1.02 1.01 1.01 1.01 1.01 1.02 1.03 1.05 1.08 1.11 1.14 1.18 1.22 1.26 1.30 1.35 1.39 1.43 1.47 1.51 1.55 1.59 1.63 1.68 1.72 1.77 1.82 1.87 1.91 1.95 1.99 2.02 2.05 2.07 2.09 2.11 2.13 2.15 2.17 2.19 2.22 2.26 2.29 2.32 2.34 2.36 2.38 2.39 2.39 2.40 2.40 2.39 2.38 2.36 2.33 2.30 2.26 2.22 2.18 2.13 2.09 2.05 2.00 1.96 1.92 1.88 1.84 1.81 1.79 1.77 1.75 1.73 1.72 1.69 1.65 1.62 1.59 1.56 1.53 1.50 1.47 1.44 1.41 1.38 1.35 1.31 1.27 1.22 1.19 1.15 1.12 1.07 1.03 0.99 0.96 0.93 0.92 0.91 0.90 0.89 0.89 0.90 0.91 0.93 0.95 0.97 1.00 1.03 1.05 1.07 1.08 1.09 1.10 1.10 1.10 1.10 1.09 1.09 1.08 1.07 1.05 1.04 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.10 1.11 1.12 1.13 1.15 1.17 1.19 1.21 1.23
1.29 1.32 1.35 1.38 1.41 1.44 1.45 1.47 1.49 1.50 1.52 1.52 1.52 1.52 1.53 1.53 1.55 1.56 1.58 1.60 1.61 1.62 1.62 1.63 1.63 1.63 1.59 1.55 1.51 1.46 1.41 1.35 1.30 1.26 1.21 1.17 1.15 1.13 1.12 1.10 1.08 1.07 1.06 1.06 1.05 1.04 1.05 1.05 1.06 1.08 1.10 1.14 1.17 1.21 1.25 1.30 1.34 1.39 1.43 1.46 1.50 1.54 1.58 1.62 1.66 1.70 1.74 1.79 1.84 1.89 1.93 1.98 2.02 2.05 2.08 2.11 2.13 2.15 2.17 2.18 2.20 2.22 2.24 2.27 2.31 2.35 2.38 2.41 2.44 2.46 2.48 2.49 2.50 2.50 2.49 2.48 2.46 2.43 2.39 2.34 2.30 2.25 2.21 2.16 2.11 2.06 2.01 1.97 1.92 1.88 1.84 1.82 1.80 1.78 1.77 1.76 1.73 1.70 1.67 1.64 1.61 1.58 1.56 1.53 1.50 1.48 1.45 1.41 1.37 1.33 1.28 1.25 1.21 1.17 1.13 1.09 1.06 1.03 1.02 1.01 1.01 1.01 1.01 1.01 1.03 1.05 1.07 1.10 1.13 1.16 1.18 1.20 1.22 1.23 1.23 1.23 1.23 1.22 1.22 1.20 1.19 1.18 1.17 1.15 1.14 1.12 1.13 1.14 1.16 1.17 1.18 1.19 1.20 1.20 1.20 1.21 1.22 1.24 1.26 1.28 1.29
1.34 1.37 1.39 1.41 1.43 1.45 1.46 1.48 1.50 1.51 1.52 1.52 1.52 1.53 1.53 1.54 1.56 1.58 1.60 1.62 1.63 1.64 1.65 1.66 1.66 1.66 1.63 1.60 1.56 1.50 1.45 1.40 1.35 1.30 1.26 1.21 1.19 1.17 1.15 1.14 1.13 1.11 1.10 1.09 1.08 1.08 1.09 1.09 1.11 1.14 1.17 1.21 1.25 1.29 1.34 1.39 1.43 1.47 1.51 1.55 1.58 1.62 1.66 1.69 1.73 1.77 1.81 1.86 1.90 1.95 1.99 2.03 2.07 2.11 2.14 2.17 2.19 2.21 2.23 2.23 2.24 2.27 2.29 2.33 2.36 2.40 2.44 2.49 2.52 2.54 2.57 2.58 2.59 2.59 2.58 2.57 2.54 2.51 2.47 2.42 2.37 2.33 2.28 2.23 2.18 2.13 2.08 2.02 1.97 1.93 1.88 1.86 1.84 1.83 1.81 1.80 1.77 1.75 1.73 1.70 1.68 1.65 1.62 1.60 1.57 1.55 1.51 1.48 1.44 1.40 1.35 1.31 1.27 1.23 1.19 1.16 1.14 1.12 1.11 1.11 1.11 1.12 1.13 1.14 1.17 1.19 1.22 1.25 1.28 1.31 1.34 1.35 1.37 1.38 1.37 1.36 1.34 1.33 1.31 1.29 1.27 1.25 1.24 1.23 1.23 1.22 1.23 1.23 1.24 1.25 1.27 1.27 1.28 1.28 1.28 1.28 1.29 1.31 1.32 1.33 1.34
1.38 1.40 1.42 1.43 1.44 1.45 1.46 1.48 1.49 1.49 1.49 1.50 1.51 1.51 1.52 1.53 1.55 1.57 1.58 1.59 1.61 1.63 1.65 1.66 1.67 1.67 1.65 1.62 1.58 1.53 1.48 1.43 1.37 1.33 1.29 1.25 1.22 1.20 1.18 1.17 1.15 1.14 1.13 1.12 1.11 1.10 1.12 1.13 1.15 1.19 1.22 1.27 1.32 1.37 1.43 1.48 1.52 1.56 1.59 1.63 1.67 1.70 1.74 1.77 1.80 1.83 1.87 1.91 1.95 2.00 2.05 2.08 2.12 2.16 2.19 2.22 2.24 2.26 2.28 2.30 2.32 2.34 2.36 2.39 2.43 2.47 2.52 2.57 2.61 2.63 2.65 2.66 2.67 2.67 2.66 2.65 2.61 2.57 2.53 2.49 2.45 2.40 2.35 2.30 2.24 2.19 2.14 2.09 2.04 1.99 1.95 1.92 1.90 1.88 1.86 1.85 1.83 1.81 1.79 1.77 1.75 1.73 1.70 1.68 1.65 1.62 1.59 1.56 1.52 1.47 1.43 1.38 1.34 1.30 1.27 1.23 1.21 1.20 1.19 1.20 1.20 1.22 1.24 1.27 1.30 1.33 1.36 1.38 1.42 1.45 1.49 1.50 1.51 1.50 1.48 1.46 1.44 1.41 1.38 1.36 1.33 1.32 1.31 1.30 1.30 1.30 1.30 1.30 1.31 1.32 1.33 1.34 1.34 1.35 1.35 1.35 1.35 1.36 1.37 1.37 1.38
1.42 1.43 1.44 1.44 1.44 1.45 1.45 1.46 1.47 1.46 1.45 1.46 1.46 1.46 1.48 1.49 1.51 1.53 1.55 1.56 1.58 1.61 1.63 1.64 1.65 1.65 1.63 1.60 1.57 1.52 1.48 1.43 1.37 1.33 1.29 1.25 1.23 1.21 1.19 1.17 1.15 1.13 1.12 1.11 1.11 1.10 1.12 1.14 1.17 1.21 1.26 1.32 1.38 1.44 1.50 1.56 1.60 1.64 1.68 1.72 1.75 1.78 1.80 1.83 1.86 1.89 1.93 1.96 2.01 2.05 2.10 2.14 2.18 2.21 2.25 2.29 2.31 2.34 2.36 2.39 2.41 2.43 2.45 2.48 2.53 2.57 2.61 2.66 2.69 2.71 2.72 2.73 2.73 2.72 2.70 2.69 2.65 2.61 2.57 2.53 2.49 2.44 2.39 2.33 2.28 2.23 2.18 2.14 2.09 2.05 2.01 1.99 1.96 1.94 1.93 1.92 1.90 1.88 1.86 1.84 1.82 1.80 1.79 1.76 1.73 1.70 1.67 1.64 1.60 1.55 1.51 1.46 1.42 1.37 1.34 1.30 1.29 1.28 1.28 1.29 1.31 1.33 1.35 1.38 1.42 1.45 1.48 1.51 1.54 1.57 1.60 1.60 1.60 1.59 1.56 1.54 1.50 1.46 1.42 1.39 1.36 1.34 1.33 1.32 1.32 1.32 1.32 1.32 1.33 1.35 1.37 1.38 1.39 1.40 1.41 1.41 1.41 1.42 1.42 1.42 1.42
1.45 1.45 1.45 1.45 1.44 1.44 1.44 1.44 1.44 1.43 1.42 1.41 1.41 1.41 1.43 1.44 1.45 1.47 1.49 1.52 1.55 1.57 1.60 1.61 1.62 1.62 1.60 1.58 1.55 1.50 1.46 1.41 1.37 1.33 1.29 1.25 1.23 1.20 1.18 1.16 1.13 1.12 1.10 1.10 1.10 1.10 1.13 1.16 1.20 1.26 1.32 1.38 1.45 1.51 1.57 1.63 1.68 1.73 1.77 1.79 1.82 1.84 1.86 1.88 1.91 1.93 1.97 2.01 2.05 2.09 2.14 2.19 2.23 2.28 2.33 2.37 2.40 2.43 2.46 2.49 2.52 2.55 2.58 2.61 2.65 2.70 2.73 2.77 2.79 2.81 2.82 2.81 2.81 2.79 2.76 2.73 2.69 2.64 2.60 2.55 2.51 2.46 2.41 2.37 2.32 2.27 2.23 2.19 2.15 2.11 2.07 2.04 2.02 2.00 1.98 1.96 1.95 1.94 1.93 1.91 1.90 1.89 1.88 1.86 1.83 1.80 1.77 1.73 1.70 1.65 1.60 1.55 1.51 1.46 1.42 1.38 1.37 1.36 1.37 1.38 1.40 1.43 1.45 1.48 1.51 1.55 1.57 1.60 1.63 1.65 1.67 1.67 1.67 1.66 1.62 1.59 1.54 1.50 1.45 1.41 1.37 1.36 1.35 1.34 1.34 1.33 1.34 1.34 1.35 1.38 1.40 1.42 1.44 1.45 1.46 1.47 1.47 1.47 1.46 1.46 1.45
1.49 1.48 1.48 1.47 1.45 1.43 1.42 1.42 1.41 1.40 1.39 1.38 1.37 1.37 1.38 1.39 1.41 1.42 1.44 1.47 1.51 1.53 1.56 1.57 1.58 1.58 1.56 1.54 1.51 1.47 1.43 1.39 1.35 1.31 1.27 1.24 1.21 1.18 1.15 1.13 1.11 1.10 1.09 1.09 1.10 1.11 1.15 1.19 1.24 1.31 1.38 1.45 1.52 1.58 1.65 1.71 1.76 1.81 1.85 1.87 1.89 1.91 1.92 1.94 1.96 1.98 2.02 2.06 2.10 2.15 2.19 2.25 2.31 2.36 2.42 2.47 2.51 2.55 2.58 2.62 2.66 2.70 2.74 2.77 2.82 2.86 2.89 2.92 2.94 2.95 2.95 2.93 2.92 2.89 2.84 2.80 2.74 2.68 2.63 2.58 2.54 2.49 2.45 2.40 2.35 2.31 2.27 2.23 2.19 2.16 2.12 2.09 2.07 2.05 2.03 2.01 2.00 2.00 2.00 1.99 1.99 1.97 1.96 1.94 1.92 1.90 1.87 1.84 1.80 1.75 1.71 1.66 1.60 1.56 1.52 1.47 1.46 1.45 1.45 1.46 1.48 1.50 1.52 1.55 1.58 1.61 1.63 1.66 1.68 1.70 1.72 1.72 1.72 1.70 1.66 1.62 1.58 1.53 1.48 1.44 1.39 1.38 1.36 1.35 1.34 1.34 1.35 1.36 1.37 1.40 1.43 1.46 1.48 1.50 1.51 1.53 1.52 1.52 1.51 1.50 1.49
1.54 1.52 1.50 1.48 1.46 1.43 1.41 1.40 1.39 1.38 1.38 1.37 1.35 1.35 1.35 1.36 1.37 1.38 1.41 1.44 1.47 1.49 1.52 1.53 1.53 1.53 1.51 1.49 1.46 1.43 1.40 1.36 1.31 1.27 1.24 1.20 1.17 1.14 1.12 1.10 1.08 1.07 1.07 1.08 1.10 1.12 1.17 1.22 1.28 1.36 1.44 1.51 1.59 1.66 1.72 1.79 1.83 1.88 1.91 1.94 1.96 1.98 1.99 2.01 2.03 2.05 2.09 2.13 2.17 2.22 2.27 2.33 2.40 2.46 2.52 2.58 2.63 2.68 2.73 2.78 2.83 2.88 2.93 2.98 3.02 3.07 3.10 3.14 3.15 3.14 3.14 3.11 3.08 3.03 2.96 2.90 2.83 2.76 2.70 2.65 2.59 2.54 2.50 2.45 2.40 2.35 2.31 2.27 2.23 2.20 2.17 2.14 2.12 2.10 2.08 2.06 2.05 2.05 2.05 2.05 2.06 2.05 2.03 2.02 2.00 1.98 1.96 1.93 1.90 1.86 1.81 1.76 1.71 1.66 1.61 1.56 1.55 1.53 1.53 1.53 1.54 1.55 1.57 1.59 1.62 1.64 1.66 1.68 1.69 1.71 1.73 1.73 1.73 1.71 1.68 1.64 1.60 1.55 1.51 1.47 1.43 1.41 1.39 1.37 1.37 1.36 1.37 1.39 1.41 1.44 1.47 1.50 1.53 1.55 1.57 1.59 1.58 1.58 1.56 1.55 1.54
1.58 1.55 1.52 1.49 1.46 1.43 1.41 1.39 1.38 1.37 1.36 1.36 1.35 1.34 1.34 1.33 1.34 1.36 1.38 1.40 1.43 1.45 1.47 1.48 1.48 1.48 1.46 1.44 1.41 1.38 1.35 1.31 1.27 1.23 1.19 1.15 1.13 1.10 1.08 1.06 1.04 1.05 1.05 1.07 1.09 1.12 1.19 1.25 1.32 1.40 1.48 1.56 1.65 1.72 1.79 1.85 1.90 1.94 1.98 2.00 2.03 2.04 2.06 2.07 2.09 2.11 2.16 2.20 2.25 2.31 2.37 2.44 2.51 2.58 2.66 2.73 2.79 2.84 2.91 2.98 3.05 3.11 3.17 3.22 3.28 3.33 3.37 3.41 3.42 3.42 3.41 3.36 3.32 3.25 3.16 3.07 2.98 2.90 2.83 2.76 2.70 2.64 2.59 2.54 2.48 2.42 2.36 2.30 2.26 2.22 2.19 2.19 2.18 2.16 2.13 2.09 2.09 2.08 2.08 2.09 2.09 2.09 2.09 2.09 2.08 2.06 2.04 2.02 1.99 1.95 1.92 1.86 1.80 1.74 1.69 1.64 1.61 1.59 1.58 1.58 1.58 1.59 1.60 1.61 1.63 1.64 1.65 1.67 1.68 1.69 1.70 1.70 1.70 1.70 1.68 1.66 1.62 1.58 1.54 1.51 1.48 1.46 1.44 1.43 1.42 1.41 1.43 1.44 1.46 1.48 1.51 1.54 1.58 1.60 1.62 1.64 1.63 1.63 1.61 1.59 1.58
1.63 1.59 1.54 1.51 1.47 1.44 1.42 1.40 1.39 1.38 1.38 1.37 1.35 1.34 1.34 1.33 1.34 1.35 1.36 1.38 1.40 1.42 1.43 1.44 1.43 1.43 1.41 1.39 1.36 1.32 1.28 1.24 1.20 1.17 1.13 1.10 1.07 1.05 1.03 1.02 1.01 1.02 1.03 1.05 1.08 1.12 1.19 1.26 1.34 1.43 1.52 1.60 1.69 1.76 1.83 1.90 1.95 2.00 2.04 2.07 2.09 2.11 2.12 2.14 2.16 2.18 2.23 2.29 2.35 2.43 2.50 2.58 2.66 2.75 2.83 2.91 2.99 3.07 3.15 3.24 3.33 3.40 3.47 3.54 3.61 3.68 3.72 3.77 3.79 3.78 3.77 3.71 3.64 3.56 3.46 3.35 3.26 3.16 3.08 3.00 2.93 2.86 2.79 2.72 2.65 2.58 2.50 2.42 2.35 2.29 2.23 2.22 2.22 2.21 2.21 2.20 2.18 2.15 2.14 2.14 2.13 2.13 2.13 2.13 2.13 2.13 2.11 2.09 2.06 2.02 1.98 1.92 1.86 1.81 1.76 1.70 1.68 1.65 1.63 1.62 1.61 1.61 1.61 1.61 1.62 1.63 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.69 1.68 1.67 1.65 1.62 1.60 1.58 1.56 1.54 1.53 1.52 1.52 1.52 1.52 1.53 1.55 1.57 1.59 1.62 1.64 1.67 1.69 1.70 1.70 1.69 1.68 1.65 1.63
1.69 1.64 1.58 1.54 1.50 1.46 1.45 1.43 1.41 1.40 1.39 1.38 1.36 1.35 1.34 1.33 1.33 1.34 1.35 1.36 1.38 1.39 1.40 1.40 1.39 1.38 1.36 1.34 1.31 1.27 1.23 1.18 1.14 1.10 1.07 1.04 1.02 1.00 0.98 0.97 0.97 0.98 1.00 1.02 1.07 1.11 1.18 1.25 1.34 1.44 1.54 1.63 1.72 1.80 1.87 1.95 2.00 2.06 2.10 2.14 2.17 2.19 2.20 2.23 2.25 2.28 2.34 2.40 2.48 2.56 2.64 2.73 2.82 2.92 3.01 3.11 3.21 3.31 3.41 3.52 3.63 3.72 3.81 3.90 3.99 4.08 4.14 4.19 4.22 4.21 4.21 4.14 4.08 3.99 3.87 3.76 3.66 3.56 3.47 3.37 3.27 3.18 3.09 2.99 2.89 2.79 2.69 2.60 2.51 2.44 2.37 2.35 2.33 2.32 2.31 2.30 2.27 2.24 2.22 2.20 2.18 2.17 2.17 2.16 2.16 2.16 2.14 2.12 2.10 2.06 2.03 1.97 1.91 1.86 1.81 1.76 1.73 1.70 1.68 1.66 1.64 1.63 1.62 1.62 1.62 1.62 1.63 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.69 1.69 1.69 1.68 1.68 1.66 1.65 1.65 1.64 1.64 1.64 1.64 1.64 1.64 1.64 1.65 1.66 1.68 1.70 1.72 1.75 1.77 1.76 1.75 1.73 1.71 1.69
1.74 1.68 1.63 1.58 1.53 1.49 1.47 1.45 1.43 1.42 1.41 1.40 1.38 1.37 1.35 1.34 1.34 1.33 1.34 1.34 1.35 1.36 1.36 1.36 1.35 1.34 1.32 1.29 1.25 1.21 1.17 1.13 1.08 1.05 1.01 0.98 0.96 0.94 0.93 0.93 0.92 0.94 0.97 1.00 1.05 1.10 1.17 1.25 1.34 1.45 1.56 1.65 1.75 1.84 1.91 1.98 2.05 2.11 2.17 2.21 2.25 2.28 2.30 2.33 2.37 2.40 2.47 2.54 2.62 2.70 2.79 2.89 2.99 3.10 3.21 3.33 3.45 3.57 3.69 3.81 3.94 4.06 4.18 4.30 4.41 4.52 4.59 4.67 4.70 4.70 4.71 4.66 4.61 4.54 4.43 4.32 4.21 4.10 3.98 3.86 3.74 3.62 3.49 3.36 3.23 3.09 2.97 2.85 2.74 2.65 2.56 2.52 2.48 2.45 2.43 2.41 2.37 2.33 2.30 2.27 2.23 2.22 2.20 2.18 2.18 2.17 2.15 2.14 2.11 2.07 2.04 1.99 1.93 1.88 1.84 1.80 1.77 1.74 1.71 1.69 1.66 1.65 1.64 1.63 1.63 1.62 1.63 1.63 1.64 1.65 1.67 1.69 1.71 1.72 1.74 1.75 1.76 1.77 1.77 1.77 1.76 1.77 1.77 1.77 1.77 1.76 1.75 1.74 1.73 1.73 1.73 1.75 1.76 1.78 1.80 1.82 1.81 1.80 1.78 1.76 1.74
1.78 1.73 1.68 1.62 1.57 1.52 1.49 1.47 1.45 1.44 1.43 1.42 1.40 1.39 1.37 1.35 1.34 1.33 1.33 1.33 1.32 1.33 1.34 1.33 1.32 1.31 1.27 1.24 1.20 1.16 1.11 1.07 1.03 1.00 0.96 0.93 0.91 0.90 0.89 0.89 0.89 0.91 0.94 0.98 1.03 1.09 1.17 1.26 1.36 1.47 1.58 1.68 1.78 1.87 1.95 2.03 2.10 2.17 2.24 2.29 2.34 2.38 2.42 2.46 2.50 2.54 2.61 2.68 2.76 2.86 2.95 3.06 3.16 3.28 3.41 3.54 3.68 3.82 3.96 4.11 4.25 4.40 4.56 4.71 4.84 4.98 5.06 5.15 5.21 5.23 5.25 5.23 5.21 5.17 5.09 5.02 4.88 4.75 4.62 4.47 4.33 4.18 4.02 3.85 3.67 3.49 3.34 3.18 3.04 2.91 2.78 2.72 2.65 2.60 2.56 2.53 2.48 2.42 2.37 2.33 2.29 2.26 2.23 2.20 2.18 2.16 2.15 2.13 2.10 2.07 2.03 1.99 1.94 1.90 1.86 1.83 1.80 1.77 1.74 1.71 1.68 1.66 1.65 1.64 1.64 1.63 1.64 1.65 1.66 1.69 1.71 1.74 1.77 1.79 1.82 1.84 1.86 1.87 1.88 1.88 1.88 1.89 1.90 1.90 1.89 1.88 1.86 1.84 1.83 1.83 1.82 1.82 1.82 1.83 1.84 1.86 1.85 1.85 1.83 1.80 1.78
1.82 1.77 1.71 1.66 1.61 1.56 1.53 1.51 1.48 1.46 1.45 1.43 1.41 1.39 1.37 1.35 1.33 1.32 1.31 1.31 1.30 1.30 1.30 1.29 1.28 1.27 1.23 1.20 1.16 1.12 1.07 1.03 0.99 0.96 0.93 0.89 0.88 0.87 0.86 0.87 0.88 0.90 0.92 0.96 1.02 1.09 1.19 1.28 1.39 1.50 1.61 1.71 1.82 1.91 1.99 2.08 2.16 2.23 2.31 2.37 2.44 2.49 2.54 2.59 2.63 2.68 2.76 2.84 2.92 3.02 3.12 3.23 3.35 3.48 3.61 3.75 3.91 4.06 4.22 4.38 4.55 4.73 4.92 5.09 5.24 5.39 5.49 5.60 5.68 5.73 5.78 5.79 5.81 5.80 5.76 5.73 5.61 5.50 5.36 5.20 5.03 4.85 4.67 4.47 4.24 4.01 3.81 3.61 3.42 3.24 3.07 2.96 2.86 2.78 2.71 2.65 2.58 2.52 2.46 2.40 2.34 2.30 2.26 2.22 2.19 2.16 2.14 2.12 2.09 2.06 2.03 1.99 1.95 1.92 1.88 1.85 1.82 1.79 1.75 1.72 1.69 1.68 1.66 1.66 1.66 1.66 1.68 1.70 1.72 1.75 1.79 1.83 1.86 1.89 1.91 1.93 1.95 1.97 1.98 1.99 2.00 2.01 2.02 2.02 2.01 2.00 1.98 1.96 1.94 1.92 1.90 1.89 1.89 1.89 1.89 1.90 1.89 1.87 1.86 1.84 1.82
1.84 1.80 1.75 1.70 1.66 1.61 1.58 1.55 1.52 1.49 1.46 1.43 1.41 1.38 1.36 1.33 1.32 1.31 1.30 1.28 1.27 1.27 1.26 1.26 1.25 1.24 1.20 1.17 1.13 1.09 1.05 1.00 0.96 0.92 0.90 0.88 0.87 0.85 0.85 0.86 0.86 0.89 0.92 0.97 1.04 1.10 1.21 1.32 1.43 1.54 1.65 1.76 1.86 1.96 2.05 2.14 2.23 2.31 2.39 2.47 2.55 2.61 2.68 2.74 2.79 2.85 2.93 3.00 3.09 3.19 3.29 3.42 3.55 3.69 3.82 3.96 4.11 4.26 4.42 4.59 4.75 4.96 5.16 5.35 5.52 5.69 5.80 5.92 6.01 6.08 6.16 6.21 6.26 6.29 6.29 6.30 6.24 6.18 6.08 5.93 5.77 5.58 5.38 5.16 4.91 4.66 4.42 4.17 3.94 3.71 3.48 3.34 3.19 3.07 2.96 2.85 2.76 2.66 2.58 2.51 2.44 2.38 2.31 2.26 2.22 2.19 2.16 2.14 2.11 2.08 2.04 2.00 1.96 1.93 1.90 1.86 1.84 1.81 1.78 1.75 1.71 1.70 1.69 1.69 1.70 1.71 1.74 1.77 1.80 1.85 1.89 1.92 1.96 1.99 2.01 2.04 2.05 2.06 2.08 2.09 2.10 2.11 2.12 2.11 2.10 2.09 2.06 2.04 2.02 2.00 1.98 1.97 1.95 1.94 1.93 1.93 1.91 1.90 1.88 1.86 1.84
1.85 1.81 1.77 1.73 1.69 1.65 1.62 1.58 1.54 1.51 1.47 1.44 1.41 1.38 1.35 1.32 1.30 1.29 1.27 1.26 1.25 1.24 1.24 1.24 1.22 1.21 1.18 1.14 1.11 1.06 1.02 0.98 0.94 0.91 0.89 0.87 0.86 0.86 0.86 0.87 0.88 0.91 0.94 0.99 1.05 1.12 1.24 1.35 1.47 1.59 1.70 1.82 1.93 2.03 2.13 2.23 2.32 2.41 2.50 2.60 2.69 2.76 2.83 2.90 2.96 3.02 3.09 3.16 3.25 3.35 3.46 3.60 3.74 3.87 4.01 4.15 4.29 4.43 4.58 4.74 4.89 5.09 5.29 5.48 5.66 5.84 5.96 6.08 6.19 6.28 6.38 6.45 6.52 6.58 6.64 6.69 6.68 6.68 6.62 6.50 6.39 6.19 5.99 5.78 5.56 5.33 5.03 4.74 4.46 4.19 3.93 3.74 3.56 3.40 3.26 3.13 3.00 2.87 2.76 2.66 2.57 2.49 2.41 2.34 2.30 2.25 2.22 2.18 2.15 2.11 2.08 2.04 2.00 1.97 1.93 1.90 1.87 1.84 1.82 1.78 1.75 1.74 1.73 1.74 1.76 1.78 1.82 1.86 1.90 1.95 1.99 2.02 2.05 2.07 2.09 2.11 2.12 2.13 2.14 2.15 2.17 2.17 2.17 2.17 2.16 2.15 2.13 2.11 2.08 2.05 2.03 2.01 1.98 1.96 1.94 1.92 1.91 1.90 1.88 1.87 1.85
1.84 1.81 1.78 1.75 1.72 1.69 1.65 1.61 1.57 1.53 1.49 1.45 1.42 1.38 1.35 1.32 1.29 1.27 1.25 1.24 1.23 1.22 1.22 1.21 1.20 1.19 1.16 1.13 1.09 1.05 1.01 0.97 0.93 0.90 0.88 0.87 0.87 0.87 0.87 0.89 0.90 0.93 0.96 1.01 1.08 1.16 1.27 1.39 1.51 1.64 1.77 1.88 2.00 2.11 2.21 2.32 2.42 2.52 2.63 2.73 2.83 2.91 2.99 3.06 3.12 3.17 3.24 3.32 3.41 3.52 3.62 3.77 3.91 4.05 4.19 4.32 4.45 4.57 4.70 4.84 4.98 5.16 5.34 5.51 5.68 5.86 5.98 6.10 6.21 6.31 6.41 6.49 6.58 6.67 6.76 6.86 6.90 6.95 6.94 6.88 6.82 6.65 6.48 6.29 6.07 5.85 5.54 5.23 4.93 4.66 4.39 4.17 3.96 3.77 3.60 3.44 3.28 3.13 2.99 2.86 2.73 2.63 2.54 2.46 2.40 2.33 2.29 2.25 2.20 2.16 2.12 2.09 2.06 2.02 1.98 1.95 1.92 1.89 1.86 1.83 1.80 1.79 1.79 1.80 1.83 1.86 1.91 1.96 2.01 2.05 2.10 2.12 2.15 2.17 2.18 2.19 2.18 2.18 2.18 2.18 2.19 2.19 2.19 2.18 2.18 2.18 2.15 2.13 2.10 2.07 2.04 2.01 1.98 1.95 1.92 1.90 1.88 1.87 1.86 1.85 1.84
1.83 1.81 1.80 1.77 1.74 1.71 1.67 1.63 1.59 1.55 1.51 1.47 1.42 1.38 1.35 1.32 1.29 1.26 1.24 1.23 1.21 1.21 1.20 1.20 1.18 1.17 1.14 1.12 1.08 1.05 1.01 0.97 0.93 0.91 0.89 0.88 0.88 0.88 0.89 0.90 0.91 0.95 0.98 1.03 1.11 1.19 1.31 1.43 1.56 1.69 1.83 1.95 2.07 2.19 2.30 2.41 2.52 2.63 2.74 2.86 2.97 3.05 3.13 3.20 3.26 3.31 3.38 3.45 3.55 3.66 3.78 3.92 4.07 4.20 4.34 4.47 4.57 4.68 4.79 4.90 5.02 5.17 5.31 5.46 5.61 5.75 5.86 5.97 6.07 6.17 6.26 6.35 6.44 6.54 6.67 6.80 6.89 6.98 7.03 7.03 7.03 6.91 6.78 6.61 6.39 6.17 5.89 5.60 5.33 5.08 4.83 4.60 4.37 4.16 3.95 3.75 3.58 3.42 3.25 3.09 2.93 2.81 2.70 2.60 2.51 2.43 2.38 2.32 2.27 2.22 2.18 2.14 2.11 2.08 2.05 2.02 1.98 1.94 1.91 1.88 1.86 1.86 1.86 1.88 1.92 1.96 2.02 2.07 2.12 2.17 2.22 2.24 2.27 2.28 2.28 2.28 2.25 2.23 2.21 2.19 2.17 2.17 2.17 2.17 2.17 2.17 2.14 2.11 2.09 2.05 2.02 1.98 1.95 1.91 1.89 1.86 1.85 1.84 1.83 1.83 1.83
1.82 1.81 1.81 1.79 1.76 1.74 1.70 1.66 1.62 1.57 1.53 1.48 1.44 1.39 1.35 1.32 1.29 1.26 1.24 1.22 1.20 1.20 1.20 1.20 1.18 1.17 1.14 1.12 1.08 1.05 1.01 0.97 0.94 0.92 0.91 0.89 0.89 0.89 0.90 0.91 0.93 0.96 0.99 1.05 1.12 1.20 1.33 1.46 1.60 1.74 1.88 2.01 2.14 2.27 2.38 2.50 2.61 2.72 2.84 2.96 3.09 3.16 3.24 3.31 3.37 3.43 3.50 3.57 3.66 3.78 3.90 4.04 4.18 4.31 4.43 4.55 4.65 4.75 4.85 4.94 5.03 5.14 5.25 5.36 5.47 5.57 5.65 5.74 5.82 5.89 5.96 6.04 6.13 6.24 6.37 6.51 6.63 6.76 6.85 6.91 6.97 6.89 6.81 6.69 6.52 6.35 6.11 5.87 5.63 5.40 5.16 4.94 4.72 4.50 4.29 4.08 3.89 3.70 3.52 3.34 3.17 3.02 2.87 2.74 2.65 2.55 2.48 2.41 2.34 2.29 2.24 2.21 2.19 2.16 2.13 2.09 2.06 2.03 2.00 1.97 1.95 1.95 1.96 1.99 2.04 2.09 2.16 2.22 2.28 2.33 2.39 2.41 2.43 2.42 2.40 2.37 2.32 2.27 2.22 2.19 2.16 2.15 2.15 2.14 2.13 2.13 2.10 2.08 2.05 2.02 2.00 1.96 1.92 1.89 1.87 1.85 1.83 1.81 1.81 1.81 1.82
1.82 1.82 1.82 1.80 1.78 1.76 1.72 1.69 1.64 1.59 1.54 1.49 1.45 1.41 1.37 1.34 1.31 1.28 1.26 1.24 1.23 1.22 1.22 1.21 1.20 1.18 1.16 1.13 1.10 1.06 1.02 0.99 0.95 0.93 0.91 0.89 0.90 0.90 0.91 0.93 0.94 0.97 1.00 1.06 1.14 1.21 1.35 1.48 1.63 1.77 1.92 2.06 2.20 2.33 2.46 2.58 2.69 2.81 2.93 3.04 3.16 3.25 3.33 3.40 3.45 3.50 3.58 3.65 3.74 3.85 3.96 4.10 4.24 4.37 4.50 4.62 4.72 4.81 4.90 4.97 5.05 5.12 5.19 5.27 5.33 5.40 5.43 5.46 5.49 5.52 5.56 5.63 5.70 5.81 5.94 6.07 6.21 6.35 6.46 6.54 6.63 6.61 6.60 6.53 6.42 6.31 6.13 5.95 5.77 5.57 5.37 5.17 4.97 4.77 4.56 4.35 4.15 3.95 3.76 3.57 3.39 3.23 3.07 2.93 2.81 2.69 2.61 2.53 2.46 2.40 2.34 2.32 2.29 2.27 2.25 2.22 2.20 2.17 2.15 2.13 2.10 2.12 2.14 2.18 2.23 2.29 2.36 2.44 2.50 2.54 2.58 2.59 2.59 2.57 2.51 2.46 2.38 2.30 2.23 2.18 2.13 2.12 2.11 2.10 2.10 2.10 2.08 2.06 2.03 2.00 1.97 1.94 1.91 1.89 1.86 1.84 1.82 1.81 1.81 1.81 1.82
1.83 1.82 1.81 1.80 1.78 1.76 1.73 1.69 1.65 1.60 1.55 1.51 1.47 1.43 1.40 1.37 1.34 1.32 1.30 1.29 1.28 1.27 1.26 1.25 1.24 1.22 1.19 1.16 1.13 1.09 1.04 1.01 0.98 0.95 0.93 0.91 0.92 0.92 0.93 0.94 0.95 0.98 1.02 1.08 1.16 1.25 1.38 1.52 1.66 1.82 1.97 2.12 2.26 2.40 2.52 2.64 2.75 2.87 2.99 3.10 3.21 3.30 3.38 3.45 3.51 3.56 3.63 3.70 3.78 3.89 3.99 4.12 4.26 4.40 4.53 4.66 4.76 4.86 4.95 5.02 5.09 5.14 5.19 5.22 5.25 5.27 5.25 5.23 5.22 5.21 5.20 5.25 5.31 5.39 5.51 5.63 5.76 5.90 6.02 6.13 6.23 6.26 6.29 6.28 6.23 6.17 6.06 5.94 5.81 5.65 5.49 5.31 5.13 4.95 4.75 4.55 4.36 4.16 3.97 3.78 3.59 3.43 3.26 3.12 2.99 2.86 2.78 2.69 2.61 2.56 2.50 2.47 2.45 2.42 2.40 2.38 2.37 2.35 2.33 2.32 2.31 2.33 2.36 2.40 2.46 2.52 2.59 2.66 2.70 2.74 2.77 2.76 2.74 2.70 2.62 2.54 2.44 2.35 2.27 2.22 2.17 2.15 2.12 2.12 2.12 2.12 2.11 2.09 2.07 2.04 2.01 1.98 1.95 1.92 1.89 1.86 1.85 1.83 1.82 1.83 1.83
1.84 1.83 1.81 1.79 1.77 1.75 1.72 1.69 1.65 1.61 1.57 1.53 1.49 1.46 1.43 1.40 1.39 1.37 1.36 1.36 1.35 1.34 1.33 1.31 1.30 1.28 1.25 1.22 1.18 1.13 1.08 1.05 1.02 0.99 0.97 0.95 0.95 0.95 0.95 0.96 0.97 1.00 1.04 1.10 1.19 1.29 1.42 1.56 1.71 1.86 2.02 2.17 2.32 2.45 2.57 2.68 2.80 2.91 3.02 3.13 3.23 3.31 3.38 3.45 3.51 3.57 3.63 3.70 3.78 3.89 3.99 4.12 4.26 4.40 4.54 4.68 4.79 4.91 5.00 5.08 5.16 5.20 5.23 5.25 5.24 5.23 5.17 5.12 5.06 5.01 4.96 4.98 5.00 5.05 5.15 5.24 5.37 5.49 5.61 5.73 5.85 5.91 5.98 6.01 6.01 6.01 5.95 5.88 5.79 5.67 5.55 5.39 5.23 5.06 4.88 4.70 4.51 4.32 4.13 3.94 3.76 3.60 3.45 3.31 3.18 3.05 2.97 2.88 2.81 2.75 2.69 2.66 2.63 2.61 2.60 2.58 2.57 2.56 2.55 2.55 2.55 2.58 2.61 2.65 2.71 2.77 2.82 2.88 2.91 2.93 2.96 2.93 2.90 2.83 2.73 2.63 2.53 2.43 2.35 2.30 2.25 2.22 2.20 2.19 2.20 2.20 2.19 2.18 2.16 2.13 2.10 2.07 2.03 1.99 1.95 1.92 1.89 1.87 1.85 1.85 1.84
1.86 1.84 1.82 1.80 1.78 1.76 1.72 1.69 1.66 1.62 1.58 1.55 1.52 1.49 1.47 1.45 1.44 1.44 1.43 1.43 1.43 1.42 1.41 1.39 1.37 1.35 1.32 1.28 1.24 1.19 1.14 1.10 1.07 1.04 1.01 0.99 0.98 0.98 0.98 0.99 1.00 1.03 1.06 1.13 1.22 1.32 1.46 1.59 1.74 1.90 2.07 2.21 2.35 2.48 2.59 2.71 2.82 2.93 3.03 3.13 3.22 3.28 3.34 3.40 3.45 3.51 3.58 3.65 3.74 3.85 3.97 4.10 4.24 4.39 4.54 4.69 4.82 4.95 5.07 5.16 5.26 5.30 5.33 5.34 5.31 5.29 5.21 5.13 5.04 4.96 4.88 4.86 4.84 4.86 4.92 4.98 5.09 5.20 5.31 5.42 5.53 5.62 5.70 5.77 5.82 5.87 5.85 5.82 5.77 5.67 5.58 5.44 5.29 5.13 4.97 4.81 4.63 4.44 4.26 4.08 3.90 3.76 3.62 3.49 3.37 3.25 3.17 3.09 3.02 2.96 2.90 2.87 2.85 2.83 2.83 2.82 2.81 2.80 2.80 2.80 2.81 2.85 2.88 2.92 2.97 3.02 3.06 3.11 3.14 3.14 3.15 3.10 3.05 2.97 2.85 2.73 2.64 2.55 2.47 2.42 2.37 2.34 2.32 2.31 2.31 2.32 2.31 2.30 2.28 2.25 2.22 2.17 2.13 2.08 2.03 1.99 1.95 1.92 1.89 1.88 1.86
1.90 1.87 1.83 1.81 1.79 1.77 1.74 1.70 1.67 1.64 1.61 1.58 1.56 1.54 1.53 1.53 1.53 1.53 1.52 1.52 1.51 1.50 1.48 1.46 1.44 1.41 1.38 1.35 1.31 1.26 1.22 1.17 1.13 1.09 1.07 1.04 1.03 1.01 1.01 1.02 1.02 1.06 1.09 1.15 1.25 1.35 1.48 1.62 1.77 1.92 2.08 2.22 2.36 2.49 2.59 2.70 2.80 2.90 3.00 3.09 3.18 3.22 3.26 3.30 3.35 3.39 3.48 3.58 3.68 3.80 3.91 4.06 4.21 4.37 4.53 4.69 4.85 5.00 5.14 5.26 5.37 5.43 5.49 5.51 5.48 5.46 5.36 5.27 5.18 5.08 4.98 4.93 4.87 4.85 4.88 4.90 4.98 5.06 5.14 5.23 5.31 5.41 5.50 5.60 5.68 5.76 5.78 5.80 5.78 5.71 5.63 5.50 5.36 5.21 5.05 4.89 4.72 4.55 4.38 4.22 4.06 3.93 3.80 3.68 3.57 3.46 3.38 3.30 3.24 3.19 3.13 3.11 3.08 3.07 3.06 3.05 3.05 3.05 3.06 3.07 3.09 3.12 3.16 3.20 3.24 3.28 3.32 3.36 3.37 3.37 3.36 3.28 3.21 3.11 3.00 2.89 2.79 2.68 2.60 2.55 2.50 2.47 2.45 2.44 2.44 2.44 2.43 2.42 2.40 2.37 2.34 2.29 2.23 2.18 2.13 2.08 2.03 1.99 1.95 1.93 1.90
1.94 1.91 1.87 1.85 1.82 1.80 1.76 1.72 1.69 1.65 1.62 1.61 1.60 1.59 1.59 1.59 1.60 1.61 1.61 1.61 1.60 1.58 1.55 1.53 1.50 1.48 1.44 1.40 1.36 1.32 1.28 1.23 1.18 1.14 1.11 1.08 1.06 1.04 1.04 1.04 1.05 1.09 1.12 1.19 1.27 1.36 1.49 1.62 1.76 1.91 2.07 2.20 2.33 2.45 2.55 2.66 2.76 2.85 2.95 3.03 3.12 3.15 3.17 3.21 3.25 3.29 3.40 3.50 3.62 3.74 3.86 4.02 4.17 4.33 4.51 4.68 4.86 5.03 5.20 5.34 5.49 5.58 5.67 5.72 5.72 5.73 5.66 5.59 5.50 5.41 5.31 5.24 5.16 5.12 5.12 5.11 5.15 5.19 5.24 5.30 5.36 5.45 5.54 5.63 5.72 5.82 5.85 5.88 5.86 5.80 5.74 5.61 5.48 5.33 5.17 5.00 4.84 4.68 4.52 4.37 4.23 4.10 3.98 3.87 3.77 3.67 3.58 3.50 3.43 3.38 3.33 3.32 3.31 3.31 3.32 3.33 3.34 3.35 3.36 3.36 3.37 3.40 3.44 3.47 3.51 3.54 3.56 3.59 3.58 3.54 3.50 3.42 3.34 3.24 3.14 3.03 2.92 2.81 2.73 2.67 2.62 2.60 2.58 2.56 2.54 2.53 2.53 2.52 2.51 2.49 2.47 2.41 2.35 2.29 2.23 2.17 2.11 2.06 2.01 1.98 1.94
1.99 1.95 1.92 1.89 1.86 1.83 1.79 1.75 1.72 1.69 1.66 1.66 1.66 1.66 1.66 1.67 1.68 1.69 1.70 1.69 1.68 1.66 1.63 1.60 1.58 1.55 1.51 1.47 1.43 1.39 1.35 1.30 1.24 1.20 1.16 1.12 1.10 1.08 1.07 1.08 1.08 1.12 1.15 1.21 1.29 1.37 1.50 1.62 1.75 1.89 2.03 2.16 2.28 2.39 2.49 2.59 2.68 2.78 2.86 2.94 3.02 3.06 3.10 3.14 3.19 3.24 3.35 3.45 3.56 3.69 3.81 3.96 4.11 4.28 4.46 4.64 4.83 5.03 5.22 5.40 5.58 5.71 5.84 5.93 5.98 6.02 5.99 5.95 5.90 5.83 5.76 5.68 5.59 5.53 5.50 5.47 5.48 5.50 5.53 5.57 5.61 5.68 5.75 5.82 5.90 5.98 6.00 6.02 6.01 5.94 5.88 5.75 5.62 5.48 5.32 5.16 5.00 4.84 4.70 4.56 4.43 4.31 4.19 4.08 3.98 3.87 3.79 3.71 3.64 3.59 3.53 3.53 3.53 3.54 3.55 3.57 3.59 3.61 3.62 3.63 3.64 3.65 3.67 3.69 3.72 3.74 3.74 3.73 3.70 3.63 3.56 3.47 3.39 3.30 3.21 3.12 3.02 2.93 2.85 2.79 2.72 2.70 2.68 2.66 2.64 2.63 2.63 2.63 2.63 2.61 2.60 2.54 2.48 2.42 2.34 2.27 2.20 2.14 2.08 2.03 1.99
2.06 2.02 1.98 1.95 1.91 1.88 1.84 1.79 1.76 1.73 1.71 1.71 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.76 1.75 1.72 1.70 1.67 1.65 1.62 1.58 1.54 1.50 1.45 1.41 1.35 1.30 1.25 1.21 1.16 1.14 1.11 1.10 1.11 1.12 1.15 1.18 1.24 1.31 1.39 1.50 1.62 1.74 1.87 2.01 2.12 2.23 2.33 2.42 2.51 2.60 2.68 2.76 2.84 2.92 2.97 3.03 3.09 3.15 3.22 3.31 3.41 3.51 3.63 3.75 3.89 4.04 4.20 4.38 4.57 4.78 4.99 5.21 5.42 5.63 5.81 5.98 6.12 6.21 6.31 6.32 6.34 6.34 6.30 6.27 6.18 6.10 6.03 5.98 5.93 5.93 5.93 5.94 5.97 5.99 6.03 6.08 6.12 6.16 6.20 6.21 6.23 6.20 6.13 6.06 5.94 5.81 5.67 5.51 5.36 5.20 5.05 4.91 4.78 4.65 4.54 4.42 4.31 4.21 4.10 4.01 3.93 3.86 3.81 3.75 3.75 3.74 3.75 3.77 3.78 3.81 3.84 3.85 3.85 3.86 3.86 3.87 3.87 3.88 3.89 3.85 3.82 3.75 3.65 3.55 3.46 3.37 3.29 3.22 3.15 3.08 3.01 2.95 2.88 2.82 2.79 2.76 2.74 2.73 2.72 2.73 2.73 2.73 2.71 2.70 2.65 2.60 2.54 2.46 2.38 2.30 2.23 2.16 2.11 2.06
2.15 2.10 2.05 2.01 1.97 1.93 1.88 1.83 1.80 1.77 1.74 1.75 1.76 1.77 1.79 1.80 1.81 1.82 1.82 1.81 1.80 1.78 1.75 1.73 1.70 1.66 1.63 1.59 1.55 1.50 1.45 1.40 1.35 1.30 1.25 1.20 1.17 1.14 1.13 1.13 1.14 1.17 1.21 1.26 1.33 1.40 1.51 1.62 1.74 1.87 2.00 2.10 2.20 2.29 2.36 2.44 2.51 2.59 2.67 2.74 2.82 2.89 2.95 3.02 3.11 3.19 3.28 3.37 3.46 3.57 3.68 3.82 3.95 4.11 4.29 4.47 4.69 4.91 5.15 5.39 5.64 5.85 6.07 6.25 6.40 6.55 6.63 6.71 6.75 6.75 6.75 6.69 6.63 6.57 6.51 6.46 6.44 6.43 6.42 6.43 6.43 6.44 6.45 6.46 6.46 6.46 6.45 6.45 6.41 6.34 6.27 6.15 6.02 5.89 5.73 5.58 5.44 5.30 5.16 5.03 4.90 4.79 4.67 4.56 4.46 4.35 4.26 4.17 4.09 4.04 3.99 3.97 3.95 3.95 3.96 3.97 3.99 4.01 4.02 4.02 4.02 4.02 4.02 4.02 4.00 3.99 3.92 3.85 3.75 3.62 3.49 3.40 3.30 3.23 3.19 3.15 3.11 3.07 3.02 2.97 2.92 2.88 2.84 2.82 2.82 2.81 2.81 2.80 2.79 2.78 2.77 2.74 2.70 2.64 2.57 2.49 2.41 2.33 2.26 2.20 2.15
2.24 2.18 2.12 2.07 2.02 1.96 1.92 1.87 1.83 1.80 1.77 1.78 1.80 1.82 1.84 1.87 1.87 1.87 1.86 1.85 1.83 1.81 1.78 1.75 1.72 1.69 1.66 1.62 1.58 1.54 1.49 1.44 1.39 1.33 1.28 1.22 1.19 1.15 1.14 1.15 1.15 1.20 1.24 1.30 1.36 1.43 1.53 1.64 1.75 1.88 2.01 2.10 2.20 2.27 2.33 2.39 2.45 2.52 2.58 2.66 2.73 2.80 2.87 2.95 3.04 3.13 3.22 3.30 3.39 3.49 3.59 3.72 3.86 4.01 4.18 4.35 4.58 4.81 5.05 5.32 5.59 5.83 6.08 6.31 6.51 6.72 6.85 6.98 7.07 7.11 7.14 7.11 7.08 7.04 7.00 6.97 6.94 6.91 6.89 6.86 6.84 6.81 6.78 6.76 6.73 6.71 6.68 6.65 6.61 6.53 6.46 6.35 6.23 6.10 5.97 5.83 5.69 5.56 5.42 5.29 5.16 5.05 4.94 4.83 4.72 4.61 4.51 4.42 4.34 4.28 4.22 4.19 4.16 4.14 4.13 4.12 4.12 4.12 4.13 4.13 4.14 4.14 4.14 4.12 4.08 4.04 3.95 3.85 3.73 3.59 3.44 3.35 3.26 3.20 3.18 3.15 3.12 3.10 3.07 3.05 3.02 2.99 2.97 2.94 2.91 2.87 2.85 2.84 2.83 2.83 2.83 2.80 2.77 2.73 2.66 2.60 2.51 2.43 2.36 2.30 2.24
2.32 2.26 2.20 2.14 2.07 2.00 1.95 1.89 1.85 1.82 1.78 1.80 1.82 1.85 1.89 1.93 1.92 1.91 1.89 1.86 1.83 1.81 1.78 1.75 1.72 1.69 1.66 1.63 1.60 1.57 1.53 1.48 1.42 1.36 1.29 1.23 1.19 1.16 1.14 1.15 1.17 1.22 1.27 1.33 1.41 1.48 1.58 1.69 1.80 1.91 2.03 2.11 2.19 2.25 2.30 2.35 2.40 2.46 2.51 2.58 2.64 2.71 2.79 2.87 2.96 3.06 3.14 3.21 3.30 3.39 3.48 3.61 3.74 3.88 4.05 4.21 4.43 4.64 4.89 5.16 5.43 5.71 6.00 6.25 6.49 6.72 6.90 7.08 7.21 7.30 7.39 7.39 7.39 7.38 7.35 7.33 7.29 7.25 7.20 7.14 7.08 7.04 6.99 6.94 6.90 6.85 6.80 6.74 6.69 6.62 6.55 6.45 6.34 6.23 6.12 6.01 5.88 5.75 5.63 5.52 5.41 5.30 5.19 5.08 4.98 4.87 4.77 4.67 4.58 4.51 4.44 4.40 4.37 4.34 4.32 4.29 4.28 4.26 4.25 4.25 4.24 4.24 4.23 4.20 4.13 4.07 3.96 3.85 3.74 3.62 3.49 3.41 3.32 3.26 3.23 3.19 3.18 3.17 3.16 3.15 3.14 3.11 3.07 3.03 2.98 2.93 2.90 2.88 2.86 2.86 2.85 2.84 2.82 2.78 2.72 2.66 2.58 2.50 2.44 2.38 2.32
2.38 2.31 2.25 2.18 2.10 2.02 1.96 1.90 1.85 1.82 1.79 1.81 1.83 1.86 1.91 1.97 1.95 1.93 1.90 1.87 1.83 1.80 1.78 1.74 1.71 1.68 1.65 1.63 1.60 1.57 1.54 1.48 1.43 1.36 1.30 1.23 1.20 1.16 1.15 1.17 1.18 1.24 1.30 1.37 1.46 1.54 1.64 1.73 1.84 1.94 2.05 2.12 2.19 2.25 2.29 2.33 2.37 2.42 2.47 2.52 2.58 2.65 2.72 2.79 2.88 2.97 3.05 3.12 3.21 3.29 3.38 3.50 3.61 3.74 3.90 4.05 4.25 4.46 4.70 4.96 5.23 5.52 5.82 6.09 6.35 6.61 6.81 7.01 7.18 7.31 7.44 7.47 7.50 7.50 7.48 7.47 7.42 7.37 7.31 7.23 7.15 7.08 7.01 6.95 6.89 6.84 6.78 6.73 6.67 6.61 6.55 6.46 6.37 6.28 6.19 6.10 5.99 5.88 5.78 5.68 5.58 5.48 5.38 5.28 5.18 5.07 4.97 4.87 4.78 4.71 4.63 4.59 4.54 4.50 4.48 4.45 4.42 4.40 4.37 4.36 4.34 4.31 4.29 4.23 4.15 4.08 3.97 3.87 3.77 3.67 3.57 3.50 3.43 3.37 3.34 3.30 3.29 3.28 3.26 3.25 3.24 3.20 3.16 3.10 3.04 2.98 2.94 2.91 2.89 2.88 2.87 2.85 2.83 2.80 2.74 2.69 2.62 2.56 2.49 2.44 2.38
2.40 2.33 2.26 2.19 2.11 2.03 1.97 1.90 1.86 1.83 1.79 1.81 1.82 1.85 1.91 1.96 1.94 1.92 1.89 1.86 1.83 1.79 1.76 1.72 1.69 1.66 1.64 1.61 1.59 1.56 1.54 1.48 1.42 1.36 1.29 1.23 1.20 1.17 1.16 1.18 1.20 1.27 1.34 1.42 1.51 1.61 1.70 1.79 1.88 1.98 2.08 2.14 2.21 2.26 2.30 2.33 2.37 2.40 2.44 2.49 2.54 2.61 2.67 2.74 2.82 2.90 2.97 3.04 3.12 3.20 3.28 3.39 3.49 3.61 3.75 3.88 4.08 4.27 4.49 4.75 5.00 5.29 5.57 5.85 6.12 6.38 6.61 6.83 7.01 7.16 7.31 7.35 7.39 7.41 7.40 7.39 7.34 7.28 7.21 7.12 7.02 6.93 6.84 6.77 6.71 6.65 6.61 6.57 6.52 6.47 6.42 6.36 6.29 6.23 6.15 6.08 6.00 5.93 5.84 5.76 5.67 5.59 5.50 5.41 5.31 5.22 5.12 5.03 4.94 4.86 4.78 4.72 4.67 4.62 4.59 4.56 4.52 4.48 4.45 4.41 4.38 4.34 4.29 4.23 4.15 4.08 3.99 3.90 3.81 3.73 3.65 3.59 3.54 3.49 3.46 3.42 3.40 3.39 3.37 3.35 3.32 3.26 3.20 3.14 3.07 3.00 2.96 2.92 2.89 2.87 2.85 2.83 2.81 2.77 2.73 2.68 2.62 2.56 2.50 2.45 2.40
2.37 2.30 2.23 2.16 2.09 2.02 1.96 1.91 1.86 1.83 1.80 1.81 1.81 1.84 1.88 1.92 1.91 1.90 1.87 1.84 1.81 1.77 1.73 1.70 1.66 1.63 1.61 1.59 1.57 1.54 1.52 1.46 1.40 1.34 1.28 1.22 1.19 1.17 1.18 1.21 1.24 1.32 1.39 1.48 1.58 1.67 1.76 1.85 1.94 2.02 2.11 2.17 2.23 2.28 2.32 2.36 2.38 2.41 2.45 2.49 2.54 2.59 2.65 2.71 2.78 2.85 2.92 2.98 3.05 3.12 3.20 3.29 3.38 3.48 3.61 3.73 3.91 4.08 4.29 4.53 4.76 5.03 5.29 5.55 5.82 6.08 6.31 6.54 6.73 6.87 7.02 7.06 7.10 7.12 7.12 7.12 7.06 7.00 6.92 6.82 6.71 6.61 6.50 6.42 6.37 6.32 6.29 6.26 6.23 6.20 6.17 6.13 6.10 6.05 6.01 5.97 5.91 5.86 5.80 5.74 5.68 5.61 5.54 5.46 5.38 5.29 5.20 5.11 5.03 4.95 4.86 4.80 4.75 4.69 4.64 4.60 4.55 4.50 4.45 4.40 4.35 4.31 4.26 4.21 4.14 4.07 4.00 3.92 3.86 3.79 3.73 3.68 3.63 3.59 3.56 3.53 3.50 3.48 3.45 3.41 3.36 3.29 3.22 3.14 3.07 2.99 2.95 2.90 2.86 2.83 2.81 2.78 2.75 2.72 2.67 2.63 2.57 2.51 2.46 2.42 2.37
2.31 2.24 2.18 2.11 2.06 2.00 1.94 1.88 1.84 1.81 1.79 1.80 1.81 1.83 1.85 1.87 1.86 1.85 1.83 1.79 1.75 1.72 1.69 1.66 1.62 1.59 1.57 1.55 1.53 1.50 1.48 1.43 1.37 1.32 1.27 1.22 1.20 1.19 1.20 1.24 1.28 1.37 1.45 1.54 1.64 1.74 1.83 1.92 2.00 2.08 2.16 2.21 2.26 2.31 2.34 2.37 2.40 2.42 2.46 2.50 2.55 2.60 2.65 2.71 2.78 2.84 2.89 2.95 3.00 3.07 3.13 3.21 3.29 3.38 3.48 3.59 3.74 3.90 4.09 4.30 4.51 4.75 4.98 5.22 5.47 5.72 5.93 6.14 6.32 6.45 6.58 6.62 6.67 6.69 6.68 6.67 6.62 6.57 6.49 6.39 6.28 6.17 6.06 5.98 5.93 5.88 5.85 5.83 5.81 5.80 5.80 5.78 5.77 5.76 5.74 5.73 5.71 5.68 5.65 5.62 5.59 5.53 5.47 5.41 5.34 5.28 5.20 5.12 5.05 4.98 4.90 4.84 4.77 4.71 4.65 4.60 4.53 4.47 4.41 4.35 4.30 4.26 4.21 4.16 4.11 4.06 4.00 3.94 3.89 3.84 3.78 3.74 3.71 3.67 3.64 3.60 3.57 3.54 3.49 3.44 3.38 3.29 3.21 3.12 3.04 2.96 2.90 2.84 2.79 2.76 2.73 2.70 2.66 2.63 2.59 2.55 2.50 2.45 2.40 2.35 2.31
2.23 2.16 2.10 2.04 1.99 1.95 1.89 1.84 1.80 1.77 1.75 1.76 1.77 1.78 1.79 1.80 1.80 1.79 1.77 1.73 1.69 1.65 1.61 1.58 1.55 1.53 1.51 1.49 1.46 1.44 1.41 1.37 1.33 1.29 1.25 1.20 1.21 1.21 1.24 1.29 1.33 1.42 1.51 1.61 1.70 1.80 1.89 1.97 2.05 2.12 2.19 2.24 2.30 2.34 2.36 2.38 2.40 2.43 2.46 2.51 2.56 2.60 2.65 2.70 2.76 2.82 2.87 2.92 2.97 3.02 3.07 3.14 3.21 3.30 3.39 3.48 3.62 3.75 3.91 4.10 4.28 4.49 4.70 4.91 5.12 5.34 5.52 5.69 5.84 5.95 6.06 6.10 6.15 6.16 6.13 6.10 6.05 6.00 5.93 5.85 5.76 5.68 5.61 5.53 5.46 5.38 5.35 5.32 5.31 5.31 5.32 5.32 5.33 5.34 5.36 5.37 5.37 5.37 5.36 5.34 5.33 5.29 5.26 5.22 5.18 5.13 5.08 5.02 4.96 4.91 4.85 4.79 4.73 4.67 4.61 4.56 4.50 4.44 4.38 4.32 4.26 4.22 4.17 4.13 4.08 4.03 3.98 3.93 3.88 3.85 3.81 3.77 3.73 3.70 3.66 3.63 3.59 3.54 3.48 3.40 3.31 3.22 3.12 3.03 2.93 2.84 2.78 2.72 2.67 2.64 2.60 2.57 2.53 2.50 2.47 2.45 2.40 2.35 2.31 2.27 2.23
2.13 2.07 2.00 1.95 1.90 1.85 1.81 1.77 1.74 1.73 1.71 1.72 1.73 1.74 1.75 1.76 1.74 1.73 1.70 1.66 1.61 1.57 1.53 1.50 1.47 1.45 1.44 1.42 1.40 1.38 1.36 1.33 1.30 1.27 1.24 1.21 1.23 1.24 1.27 1.33 1.39 1.48 1.57 1.66 1.75 1.85 1.93 2.01 2.08 2.15 2.21 2.26 2.31 2.35 2.38 2.41 2.42 2.44 2.47 2.52 2.57 2.60 2.64 2.68 2.72 2.77 2.82 2.87 2.91 2.96 3.00 3.08 3.15 3.23 3.31 3.40 3.51 3.62 3.76 3.92 4.08 4.26 4.44 4.62 4.79 4.96 5.10 5.25 5.36 5.44 5.53 5.56 5.59 5.58 5.55 5.51 5.46 5.41 5.36 5.29 5.23 5.17 5.12 5.06 4.98 4.90 4.87 4.84 4.82 4.82 4.83 4.84 4.86 4.88 4.91 4.94 4.95 4.97 4.98 4.99 4.99 4.98 4.98 4.96 4.94 4.91 4.88 4.85 4.81 4.77 4.73 4.69 4.65 4.60 4.55 4.50 4.44 4.38 4.32 4.27 4.22 4.17 4.12 4.08 4.03 3.99 3.94 3.90 3.86 3.83 3.80 3.76 3.73 3.69 3.64 3.60 3.54 3.48 3.39 3.29 3.19 3.09 2.99 2.89 2.78 2.68 2.63 2.57 2.53 2.49 2.46 2.43 2.39 2.37 2.34 2.32 2.28 2.24 2.20 2.17 2.13
2.01 1.95 1.89 1.84 1.79 1.75 1.72 1.70 1.68 1.67 1.66 1.67 1.69 1.70 1.71 1.72 1.70 1.68 1.65 1.60 1.55 1.50 1.46 1.42 1.40 1.38 1.37 1.35 1.34 1.33 1.31 1.29 1.28 1.26 1.25 1.24 1.26 1.28 1.32 1.38 1.45 1.53 1.62 1.71 1.79 1.88 1.95 2.03 2.09 2.15 2.21 2.26 2.30 2.34 2.38 2.42 2.44 2.46 2.48 2.52 2.56 2.59 2.62 2.65 2.68 2.71 2.75 2.80 2.84 2.89 2.93 3.00 3.08 3.15 3.24 3.33 3.42 3.51 3.63 3.77 3.91 4.06 4.22 4.36 4.49 4.62 4.72 4.82 4.90 4.96 5.01 5.02 5.03 5.01 4.98 4.95 4.90 4.85 4.81 4.75 4.70 4.67 4.64 4.60 4.53 4.47 4.44 4.40 4.38 4.38 4.38 4.39 4.41 4.43 4.46 4.49 4.52 4.55 4.57 4.59 4.60 4.62 4.63 4.64 4.64 4.63 4.63 4.62 4.60 4.58 4.56 4.54 4.52 4.49 4.45 4.41 4.37 4.32 4.27 4.22 4.18 4.14 4.10 4.05 4.01 3.96 3.92 3.87 3.83 3.80 3.76 3.72 3.68 3.63 3.57 3.52 3.44 3.36 3.26 3.15 3.03 2.92 2.81 2.71 2.62 2.52 2.47 2.42 2.38 2.35 2.32 2.28 2.25 2.22 2.20 2.18 2.14 2.11 2.07 2.04 2.01
1.86 1.82 1.77 1.73 1.69 1.66 1.64 1.62 1.61 1.61 1.62 1.64 1.66 1.67 1.68 1.68 1.66 1.64 1.60 1.55 1.50 1.45 1.41 1.37 1.34 1.32 1.31 1.30 1.29 1.28 1.28 1.28 1.27 1.27 1.28 1.28 1.31 1.34 1.38 1.45 1.51 1.59 1.67 1.75 1.83 1.90 1.96 2.02 2.08 2.13 2.18 2.22 2.27 2.32 2.36 2.41 2.44 2.47 2.49 2.52 2.55 2.57 2.60 2.62 2.63 2.65 2.69 2.72 2.76 2.81 2.85 2.93 3.00 3.08 3.17 3.26 3.35 3.43 3.54 3.66 3.78 3.91 4.03 4.14 4.23 4.32 4.38 4.45 4.49 4.52 4.54 4.53 4.51 4.49 4.46 4.44 4.39 4.35 4.31 4.27 4.22 4.21 4.20 4.18 4.14 4.11 4.07 4.04 4.02 4.01 4.00 4.01 4.02 4.04 4.06 4.08 4.11 4.14 4.17 4.19 4.22 4.24 4.27 4.29 4.30 4.32 4.33 4.34 4.35 4.35 4.36 4.36 4.36 4.34 4.32 4.30 4.28 4.25 4.22 4.18 4.15 4.12 4.08 4.05 4.00 3.96 3.90 3.85 3.80 3.76 3.72 3.66 3.60 3.53 3.47 3.41 3.31 3.21 3.10 2.99 2.87 2.75 2.64 2.54 2.45 2.37 2.32 2.27 2.23 2.20 2.17 2.14 2.11 2.08 2.05 2.03 1.99 1.96 1.93 1.90 1.86
1.72 1.69 1.66 1.63 1.61 1.59 1.58 1.57 1.57 1.58 1.59 1.61 1.63 1.64 1.65 1.66 1.63 1.60 1.57 1.51 1.46 1.41 1.36 1.32 1.29 1.27 1.26 1.25 1.25 1.26 1.27 1.28 1.29 1.31 1.33 1.35 1.38 1.41 1.46 1.52 1.58 1.65 1.72 1.78 1.84 1.90 1.95 1.99 2.04 2.08 2.13 2.17 2.22 2.27 2.32 2.37 2.41 2.45 2.48 2.51 2.53 2.55 2.57 2.59 2.60 2.61 2.63 2.65 2.69 2.73 2.78 2.85 2.92 3.00 3.09 3.18 3.28 3.38 3.48 3.58 3.69 3.78 3.88 3.96 4.01 4.06 4.10 4.14 4.15 4.15 4.14 4.11 4.09 4.06 4.03 3.99 3.96 3.93 3.90 3.87 3.85 3.84 3.84 3.82 3.80 3.78 3.76 3.74 3.73 3.72 3.70 3.70 3.70 3.71 3.71 3.72 3.74 3.77 3.80 3.83 3.86 3.89 3.92 3.94 3.97 3.99 4.03 4.06 4.09 4.10 4.12 4.14 4.16 4.17 4.17 4.17 4.17 4.16 4.15 4.13 4.11 4.08 4.06 4.02 3.97 3.93 3.88 3.83 3.77 3.70 3.64 3.57 3.51 3.44 3.36 3.28 3.18 3.07 2.96 2.84 2.71 2.60 2.49 2.39 2.31 2.22 2.18 2.13 2.09 2.06 2.03 2.00 1.98 1.95 1.92 1.88 1.85 1.82 1.79 1.75 1.72
1.62 1.60 1.58 1.57 1.55 1.54 1.54 1.53 1.54 1.56 1.58 1.60 1.62 1.63 1.62 1.62 1.59 1.57 1.53 1.47 1.42 1.38 1.33 1.30 1.27 1.25 1.25 1.25 1.26 1.28 1.29 1.32 1.34 1.37 1.40 1.44 1.48 1.51 1.56 1.61 1.67 1.71 1.76 1.80 1.84 1.87 1.91 1.94 1.98 2.01 2.05 2.09 2.14 2.19 2.24 2.29 2.34 2.38 2.42 2.45 2.48 2.51 2.53 2.55 2.56 2.58 2.59 2.60 2.63 2.66 2.70 2.76 2.83 2.91 3.00 3.09 3.19 3.29 3.38 3.48 3.58 3.65 3.71 3.77 3.81 3.85 3.87 3.89 3.89 3.86 3.84 3.82 3.79 3.76 3.73 3.70 3.67 3.64 3.62 3.61 3.60 3.58 3.57 3.56 3.56 3.56 3.55 3.53 3.52 3.50 3.48 3.47 3.46 3.46 3.46 3.46 3.48 3.51 3.53 3.56 3.58 3.60 3.63 3.66 3.69 3.72 3.76 3.80 3.83 3.87 3.90 3.93 3.96 3.99 4.00 4.02 4.03 4.04 4.04 4.03 4.03 4.00 3.98 3.95 3.90 3.86 3.81 3.75 3.68 3.61 3.53 3.46 3.38 3.30 3.22 3.14 3.03 2.93 2.82 2.70 2.58 2.48 2.37 2.28 2.20 2.12 2.07 2.02 1.98 1.95 1.93 1.90 1.87 1.84 1.81 1.78 1.74 1.70 1.67 1.64 1.62
1.55 1.55 1.54 1.53 1.52 1.51 1.51 1.51 1.53 1.55 1.57 1.58 1.59 1.59 1.58 1.58 1.55 1.52 1.48 1.44 1.39 1.35 1.31 1.28 1.27 1.25 1.26 1.27 1.28 1.31 1.34 1.37 1.41 1.44 1.49 1.53 1.57 1.62 1.66 1.70 1.74 1.77 1.80 1.83 1.85 1.87 1.89 1.91 1.93 1.96 1.99 2.03 2.07 2.12 2.17 2.21 2.26 2.31 2.35 2.39 2.42 2.45 2.47 2.49 2.50 2.52 2.53 2.54 2.56 2.59 2.61 2.68 2.74 2.82 2.90 2.99 3.08 3.18 3.27 3.36 3.45 3.50 3.55 3.59 3.62 3.65 3.65 3.66 3.65 3.63 3.61 3.58 3.55 3.52 3.49 3.46 3.44 3.42 3.41 3.41 3.40 3.40 3.40 3.40 3.40 3.41 3.40 3.39 3.38 3.36 3.34 3.32 3.30 3.29 3.29 3.29 3.30 3.31 3.33 3.35 3.37 3.39 3.42 3.44 3.47 3.50 3.54 3.58 3.61 3.65 3.69 3.73 3.77 3.81 3.83 3.86 3.88 3.89 3.91 3.91 3.91 3.89 3.88 3.85 3.81 3.77 3.71 3.65 3.59 3.51 3.43 3.34 3.26 3.17 3.08 2.99 2.89 2.79 2.69 2.58 2.47 2.38 2.29 2.20 2.12 2.05 2.00 1.95 1.91 1.89 1.86 1.83 1.80 1.77 1.74 1.70 1.67 1.63 1.60 1.58 1.55
1.53 1.53 1.52 1.52 1.51 1.50 1.50 1.51 1.52 1.54 1.56 1.56 1.57 1.56 1.55 1.55 1.51 1.48 1.45 1.41 1.37 1.33 1.30 1.28 1.27 1.26 1.28 1.29 1.32 1.35 1.39 1.43 1.48 1.52 1.57 1.61 1.66 1.70 1.74 1.77 1.80 1.82 1.84 1.85 1.86 1.87 1.88 1.88 1.90 1.93 1.95 1.99 2.03 2.07 2.12 2.16 2.21 2.26 2.30 2.33 2.37 2.39 2.41 2.43 2.44 2.46 2.47 2.48 2.50 2.52 2.54 2.60 2.66 2.73 2.81 2.89 2.98 3.06 3.14 3.22 3.29 3.34 3.38 3.41 3.44 3.46 3.45 3.45 3.44 3.43 3.42 3.39 3.36 3.33 3.30 3.27 3.27 3.26 3.26 3.26 3.26 3.27 3.28 3.29 3.30 3.31 3.30 3.29 3.28 3.26 3.24 3.22 3.21 3.19 3.19 3.18 3.18 3.19 3.20 3.21 3.23 3.24 3.26 3.28 3.30 3.32 3.36 3.39 3.43 3.47 3.50 3.55 3.59 3.63 3.67 3.70 3.72 3.75 3.76 3.77 3.78 3.77 3.76 3.73 3.70 3.66 3.60 3.55 3.48 3.40 3.33 3.24 3.15 3.06 2.96 2.87 2.77 2.67 2.57 2.48 2.38 2.30 2.22 2.14 2.07 2.00 1.96 1.91 1.88 1.85 1.83 1.80 1.77 1.74 1.70 1.67 1.64 1.61 1.58 1.56 1.53
1.55 1.54 1.53 1.52 1.52 1.52 1.52 1.52 1.53 1.55 1.56 1.56 1.56 1.55 1.54 1.52 1.49 1.46 1.42 1.39 1.35 1.32 1.30 1.29 1.29 1.29 1.31 1.33 1.36 1.40 1.44 1.49 1.55 1.59 1.64 1.69 1.72 1.76 1.80 1.83 1.86 1.86 1.87 1.87 1.87 1.87 1.87 1.87 1.88 1.90 1.93 1.97 2.01 2.05 2.09 2.14 2.18 2.22 2.26 2.29 2.33 2.34 2.36 2.37 2.39 2.41 2.41 2.42 2.44 2.46 2.49 2.54 2.58 2.64 2.71 2.79 2.87 2.94 3.01 3.07 3.12 3.16 3.20 3.22 3.24 3.26 3.26 3.27 3.26 3.25 3.24 3.22 3.21 3.19 3.16 3.14 3.14 3.14 3.15 3.16 3.16 3.18 3.19 3.21 3.22 3.23 3.22 3.22 3.21 3.19 3.17 3.16 3.15 3.13 3.12 3.11 3.12 3.12 3.13 3.13 3.14 3.14 3.15 3.16 3.18 3.19 3.22 3.24 3.27 3.31 3.34 3.39 3.43 3.47 3.51 3.54 3.57 3.60 3.62 3.63 3.64 3.63 3.63 3.61 3.58 3.55 3.50 3.45 3.39 3.31 3.23 3.15 3.06 2.97 2.88 2.78 2.68 2.58 2.48 2.40 2.32 2.24 2.17 2.10 2.04 1.99 1.95 1.91 1.88 1.85 1.83 1.80 1.78 1.74 1.71 1.67 1.64 1.61 1.59 1.57 1.55
1.59 1.58 1.57 1.56 1.56 1.56 1.56 1.56 1.56 1.57 1.58 1.57 1.56 1.55 1.53 1.51 1.48 1.45 1.41 1.38 1.35 1.33 1.31 1.30 1.31 1.32 1.35 1.38 1.41 1.45 1.48 1.54 1.59 1.65 1.69 1.74 1.78 1.82 1.84 1.86 1.88 1.88 1.88 1.88 1.87 1.87 1.87 1.87 1.88 1.90 1.92 1.95 1.99 2.04 2.08 2.13 2.17 2.21 2.24 2.26 2.29 2.31 2.33 2.34 2.35 2.35 2.36 2.37 2.38 2.41 2.44 2.47 2.50 2.55 2.62 2.70 2.75 2.81 2.86 2.91 2.96 2.98 3.01 3.03 3.05 3.07 3.08 3.08 3.09 3.09 3.09 3.08 3.07 3.06 3.04 3.02 3.03 3.05 3.06 3.07 3.09 3.10 3.12 3.14 3.15 3.17 3.16 3.15 3.15 3.14 3.13 3.12 3.11 3.10 3.09 3.09 3.09 3.09 3.09 3.09 3.09 3.09 3.09 3.09 3.09 3.10 3.11 3.13 3.15 3.18 3.21 3.25 3.29 3.33 3.37 3.41 3.44 3.47 3.49 3.50 3.51 3.51 3.51 3.50 3.48 3.46 3.41 3.37 3.31 3.25 3.18 3.09 3.00 2.91 2.82 2.73 2.64 2.55 2.46 2.38 2.31 2.24 2.18 2.12 2.07 2.01 1.98 1.95 1.92 1.89 1.87 1.84 1.82 1.79 1.75 1.72 1.69 1.66 1.63 1.61 1.59
1.66 1.64 1.62 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.60 1.58 1.56 1.53 1.50 1.47 1.44 1.42 1.40 1.37 1.36 1.35 1.34 1.35 1.35 1.38 1.41 1.44 1.49 1.53 1.58 1.64 1.69 1.74 1.79 1.82 1.86 1.88 1.90 1.91 1.91 1.91 1.91 1.90 1.89 1.89 1.89 1.91 1.93 1.95 1.99 2.02 2.06 2.10 2.14 2.17 2.21 2.23 2.26 2.28 2.29 2.31 2.32 2.32 2.33 2.33 2.33 2.34 2.36 2.38 2.42 2.45 2.49 2.54 2.59 2.63 2.68 2.71 2.74 2.77 2.80 2.83 2.85 2.87 2.89 2.89 2.90 2.91 2.91 2.92 2.91 2.91 2.91 2.91 2.92 2.92 2.93 2.95 2.97 2.99 3.01 3.03 3.05 3.06 3.08 3.08 3.08 3.08 3.08 3.08 3.08 3.08 3.07 3.07 3.06 3.06 3.07 3.07 3.06 3.06 3.06 3.06 3.06 3.05 3.05 3.06 3.08 3.09 3.11 3.14 3.17 3.19 3.23 3.27 3.31 3.33 3.36 3.38 3.39 3.40 3.41 3.41 3.41 3.39 3.38 3.35 3.31 3.27 3.21 3.16 3.08 3.01 2.93 2.84 2.75 2.67 2.59 2.52 2.44 2.37 2.32 2.26 2.21 2.16 2.12 2.08 2.05 2.02 1.98 1.95 1.91 1.88 1.85 1.81 1.77 1.74 1.71 1.68 1.67 1.66
1.74 1.72 1.69 1.68 1.68 1.67 1.67 1.67 1.67 1.67 1.66 1.64 1.62 1.59 1.56 1.52 1.49 1.46 1.44 1.42 1.40 1.39 1.38 1.38 1.39 1.39 1.42 1.45 1.48 1.53 1.57 1.62 1.68 1.73 1.77 1.82 1.85 1.89 1.91 1.93 1.95 1.95 1.95 1.94 1.94 1.94 1.94 1.94 1.95 1.98 2.00 2.04 2.07 2.10 2.13 2.16 2.19 2.21 2.23 2.25 2.26 2.27 2.28 2.29 2.30 2.30 2.30 2.30 2.31 2.33 2.34 2.37 2.39 2.42 2.45 2.49 2.52 2.55 2.58 2.60 2.62 2.64 2.66 2.68 2.70 2.71 2.72 2.73 2.73 2.74 2.75 2.75 2.76 2.76 2.77 2.78 2.79 2.80 2.82 2.85 2.87 2.89 2.91 2.93 2.95 2.96 2.98 2.99 2.99 3.00 3.01 3.01 3.02 3.02 3.02 3.01 3.02 3.03 3.04 3.04 3.03 3.03 3.03 3.02 3.02 3.02 3.02 3.03 3.04 3.06 3.08 3.10 3.13 3.16 3.19 3.22 3.25 3.27 3.29 3.31 3.33 3.34 3.35 3.35 3.34 3.33 3.31 3.28 3.25 3.20 3.16 3.10 3.04 2.98 2.91 2.84 2.77 2.70 2.63 2.57 2.50 2.45 2.39 2.34 2.29 2.24 2.20 2.17 2.13 2.09 2.04 2.00 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.75 1.74
1.83 1.80 1.77 1.76 1.75 1.74 1.74 1.74 1.74 1.73 1.71 1.69 1.67 1.64 1.60 1.57 1.54 1.51 1.48 1.46 1.45 1.44 1.43 1.43 1.43 1.44 1.46 1.49 1.52 1.56 1.60 1.66 1.71 1.76 1.81 1.85 1.89 1.92 1.95 1.96 1.98 1.99 1.99 1.99 1.99 2.00 2.00 2.00 2.02 2.04 2.06 2.09 2.12 2.14 2.17 2.19 2.20 2.22 2.23 2.24 2.25 2.26 2.26 2.27 2.27 2.27 2.27 2.28 2.28 2.30 2.31 2.33 2.34 2.36 2.38 2.40 2.42 2.44 2.46 2.48 2.50 2.51 2.53 2.54 2.55 2.56 2.57 2.57 2.58 2.58 2.58 2.59 2.59 2.60 2.61 2.63 2.64 2.65 2.67 2.70 2.72 2.74 2.76 2.78 2.81 2.83 2.84 2.86 2.88 2.89 2.91 2.92 2.93 2.94 2.94 2.95 2.96 2.97 2.98 2.99 2.99 2.98 2.98 2.98 2.98 2.98 2.98 2.98 2.99 3.01 3.03 3.05 3.08 3.11 3.13 3.16 3.19 3.21 3.24 3.26 3.28 3.29 3.31 3.31 3.31 3.31 3.29 3.27 3.25 3.22 3.19 3.14 3.10 3.05 3.00 2.95 2.89 2.83 2.78 2.72 2.67 2.61 2.55 2.49 2.44 2.38 2.34 2.30 2.25 2.20 2.15 2.10 2.06 2.01 1.97 1.92 1.89 1.86 1.84 1.83 1.83
1.90 1.88 1.85 1.83 1.83 1.82 1.82 1.82 1.81 1.79 1.77 1.75 1.73 1.70 1.67 1.63 1.60 1.58 1.55 1.53 1.51 1.50 1.49 1.48 1.49 1.49 1.51 1.53 1.56 1.60 1.64 1.70 1.75 1.80 1.85 1.89 1.93 1.96 1.99 2.01 2.03 2.03 2.04 2.05 2.06 2.06 2.07 2.08 2.09 2.11 2.13 2.14 2.16 2.18 2.19 2.21 2.21 2.22 2.23 2.23 2.23 2.24 2.24 2.24 2.24 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.31 2.32 2.33 2.34 2.36 2.37 2.38 2.39 2.40 2.41 2.43 2.43 2.43 2.43 2.43 2.43 2.43 2.43 2.43 2.43 2.43 2.43 2.44 2.46 2.47 2.48 2.50 2.53 2.55 2.57 2.59 2.61 2.64 2.67 2.69 2.71 2.73 2.75 2.77 2.79 2.81 2.83 2.85 2.86 2.88 2.89 2.90 2.91 2.92 2.92 2.92 2.92 2.92 2.92 2.93 2.94 2.95 2.96 2.98 3.00 3.03 3.06 3.09 3.12 3.15 3.18 3.20 3.22 3.24 3.26 3.28 3.29 3.29 3.30 3.29 3.28 3.26 3.24 3.22 3.19 3.16 3.13 3.10 3.06 3.02 2.97 2.93 2.87 2.82 2.76 2.71 2.65 2.59 2.53 2.48 2.43 2.37 2.31 2.25 2.20 2.15 2.10 2.05 2.00 1.97 1.93 1.91 1.91 1.90
1.96 1.95 1.93 1.91 1.91 1.90 1.89 1.87 1.86 1.85 1.83 1.82 1.80 1.77 1.75 1.72 1.69 1.66 1.63 1.60 1.58 1.57 1.56 1.56 1.56 1.56 1.58 1.60 1.63 1.67 1.70 1.75 1.80 1.84 1.89 1.93 1.97 2.01 2.04 2.06 2.08 2.09 2.10 2.11 2.12 2.13 2.13 2.14 2.15 2.17 2.19 2.20 2.21 2.21 2.21 2.21 2.21 2.22 2.22 2.21 2.21 2.21 2.21 2.21 2.21 2.21 2.21 2.22 2.23 2.23 2.24 2.25 2.27 2.28 2.29 2.31 2.31 2.32 2.33 2.33 2.34 2.34 2.35 2.35 2.34 2.34 2.33 2.31 2.30 2.30 2.29 2.28 2.28 2.27 2.27 2.27 2.29 2.31 2.33 2.34 2.35 2.38 2.41 2.43 2.46 2.48 2.51 2.54 2.56 2.59 2.61 2.64 2.67 2.69 2.72 2.74 2.76 2.78 2.80 2.81 2.83 2.83 2.84 2.84 2.85 2.86 2.87 2.88 2.90 2.92 2.94 2.96 2.99 3.02 3.05 3.09 3.12 3.16 3.19 3.20 3.21 3.23 3.25 3.27 3.28 3.30 3.30 3.30 3.29 3.28 3.26 3.24 3.22 3.20 3.18 3.15 3.12 3.09 3.04 2.99 2.94 2.89 2.84 2.78 2.71 2.65 2.59 2.54 2.48 2.42 2.35 2.30 2.24 2.18 2.13 2.08 2.04 2.00 1.98 1.97 1.96
2.04 2.02 1.99 1.98 1.97 1.96 1.95 1.93 1.91 1.90 1.89 1.87 1.86 1.84 1.82 1.80 1.78 1.75 1.73 1.70 1.68 1.66 1.65 1.63 1.63 1.62 1.64 1.66 1.69 1.73 1.77 1.82 1.86 1.90 1.94 1.98 2.02 2.06 2.09 2.12 2.14 2.16 2.17 2.18 2.19 2.19 2.20 2.21 2.21 2.22 2.23 2.23 2.22 2.22 2.22 2.22 2.21 2.20 2.20 2.19 2.18 2.18 2.17 2.17 2.18 2.18 2.18 2.18 2.19 2.21 2.23 2.24 2.25 2.27 2.28 2.29 2.30 2.31 2.31 2.31 2.31 2.31 2.31 2.31 2.30 2.29 2.27 2.26 2.25 2.23 2.21 2.19 2.18 2.17 2.16 2.16 2.16 2.16 2.17 2.18 2.19 2.21 2.23 2.25 2.28 2.30 2.33 2.35 2.38 2.41 2.43 2.46 2.49 2.52 2.55 2.58 2.60 2.62 2.65 2.66 2.68 2.70 2.72 2.73 2.75 2.77 2.78 2.79 2.81 2.84 2.86 2.90 2.93 2.97 3.00 3.03 3.07 3.10 3.13 3.15 3.18 3.20 3.23 3.24 3.26 3.27 3.28 3.29 3.29 3.29 3.29 3.28 3.26 3.25 3.23 3.21 3.18 3.15 3.11 3.07 3.03 2.98 2.93 2.87 2.82 2.76 2.71 2.65 2.59 2.52 2.46 2.39 2.33 2.27 2.22 2.17 2.13 2.09 2.07 2.05 2.04
2.11 2.08 2.06 2.04 2.03 2.02 2.00 1.99 1.97 1.96 1.95 1.94 1.92 1.91 1.90 1.89 1.87 1.85 1.83 1.80 1.78 1.76 1.74 1.72 1.71 1.70 1.71 1.73 1.76 1.80 1.83 1.88 1.92 1.96 2.00 2.05 2.08 2.12 2.15 2.17 2.20 2.21 2.22 2.23 2.23 2.24 2.24 2.24 2.25 2.25 2.25 2.24 2.23 2.22 2.21 2.21 2.19 2.18 2.17 2.16 2.15 2.14 2.13 2.13 2.14 2.15 2.15 2.16 2.18 2.20 2.21 2.23 2.25 2.27 2.28 2.30 2.30 2.30 2.31 2.31 2.31 2.30 2.29 2.28 2.27 2.26 2.25 2.24 2.22 2.20 2.17 2.15 2.12 2.10 2.09 2.08 2.07 2.06 2.06 2.07 2.08 2.09 2.10 2.11 2.13 2.15 2.17 2.19 2.22 2.24 2.27 2.30 2.33 2.36 2.39 2.42 2.44 2.47 2.49 2.52 2.54 2.56 2.58 2.60 2.62 2.65 2.67 2.69 2.71 2.74 2.77 2.81 2.85 2.89 2.93 2.96 3.00 3.03 3.07 3.10 3.13 3.15 3.18 3.20 3.22 3.24 3.26 3.27 3.28 3.28 3.28 3.27 3.27 3.26 3.24 3.23 3.21 3.18 3.15 3.12 3.09 3.05 3.01 2.96 2.91 2.86 2.81 2.75 2.69 2.63 2.56 2.50 2.43 2.37 2.32 2.26 2.22 2.17 2.14 2.13 2.11
2.20 2.17 2.14 2.12 2.10 2.08 2.07 2.05 2.04 2.02 2.01 2.00 1.99 1.98 1.98 1.97 1.95 1.94 1.92 1.90 1.89 1.86 1.83 1.82 1.80 1.79 1.80 1.82 1.84 1.88 1.91 1.95 1.99 2.02 2.06 2.10 2.13 2.16 2.19 2.22 2.24 2.25 2.26 2.27 2.27 2.27 2.27 2.26 2.26 2.26 2.25 2.24 2.22 2.21 2.19 2.18 2.16 2.15 2.13 2.12 2.11 2.10 2.09 2.10 2.10 2.11 2.13 2.15 2.16 2.19 2.21 2.23 2.25 2.27 2.29 2.31 2.32 2.32 2.32 2.32 2.32 2.31 2.29 2.28 2.28 2.27 2.25 2.24 2.22 2.20 2.17 2.14 2.11 2.09 2.07 2.05 2.04 2.03 2.02 2.01 2.01 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.09 2.12 2.14 2.17 2.19 2.22 2.25 2.27 2.30 2.32 2.34 2.37 2.40 2.42 2.44 2.46 2.49 2.52 2.55 2.57 2.60 2.64 2.67 2.71 2.75 2.79 2.83 2.87 2.91 2.95 2.99 3.03 3.06 3.09 3.11 3.14 3.17 3.19 3.21 3.22 3.23 3.23 3.24 3.24 3.24 3.23 3.22 3.21 3.20 3.18 3.17 3.15 3.13 3.10 3.07 3.03 2.98 2.93 2.88 2.83 2.77 2.71 2.65 2.58 2.52 2.46 2.41 2.35 2.31 2.27 2.23 2.21 2.20
2.29 2.26 2.23 2.20 2.18 2.16 2.14 2.12 2.10 2.09 2.07 2.06 2.05 2.05 2.05 2.05 2.03 2.02 2.01 2.00 1.98 1.96 1.93 1.92 1.90 1.89 1.90 1.91 1.94 1.97 2.00 2.03 2.06 2.09 2.12 2.15 2.18 2.20 2.23 2.25 2.28 2.28 2.29 2.29 2.29 2.29 2.28 2.27 2.26 2.25 2.24 2.22 2.21 2.19 2.17 2.15 2.13 2.11 2.10 2.08 2.07 2.07 2.06 2.07 2.08 2.09 2.11 2.13 2.16 2.18 2.21 2.24 2.27 2.29 2.31 2.33 2.34 2.36 2.36 2.36 2.35 2.34 2.33 2.32 2.31 2.31 2.29 2.27 2.25 2.23 2.21 2.19 2.16 2.14 2.11 2.08 2.07 2.05 2.04 2.02 2.00 1.99 1.99 1.99 2.00 2.00 2.01 2.01 2.02 2.04 2.05 2.07 2.09 2.11 2.13 2.15 2.17 2.19 2.21 2.24 2.26 2.28 2.30 2.33 2.36 2.39 2.42 2.46 2.49 2.52 2.55 2.60 2.64 2.69 2.73 2.77 2.81 2.86 2.90 2.94 2.97 3.00 3.03 3.05 3.08 3.10 3.12 3.13 3.15 3.16 3.17 3.17 3.18 3.17 3.17 3.16 3.16 3.16 3.15 3.15 3.14 3.12 3.09 3.06 3.01 2.97 2.92 2.87 2.82 2.76 2.71 2.65 2.59 2.54 2.49 2.44 2.40 2.36 2.33 2.31 2.29
2.35 2.33 2.32 2.29 2.27 2.24 2.22 2.20 2.19 2.17 2.16 2.15 2.13 2.12 2.12 2.11 2.10 2.10 2.09 2.08 2.06 2.04 2.02 2.01 1.99 1.98 1.99 1.99 2.01 2.05 2.08 2.10 2.12 2.14 2.17 2.19 2.21 2.23 2.25 2.27 2.29 2.29 2.29 2.29 2.29 2.29 2.28 2.26 2.25 2.24 2.22 2.21 2.19 2.17 2.15 2.13 2.11 2.09 2.07 2.06 2.05 2.05 2.05 2.05 2.07 2.08 2.10 2.13 2.16 2.19 2.22 2.26 2.29 2.32 2.34 2.37 2.38 2.40 2.40 2.40 2.40 2.40 2.39 2.38 2.38 2.37 2.36 2.34 2.33 2.32 2.31 2.28 2.25 2.23 2.20 2.18 2.15 2.12 2.10 2.08 2.06 2.05 2.04 2.03 2.02 2.01 2.01 2.01 2.01 2.01 2.01 2.03 2.04 2.05 2.06 2.06 2.08 2.10 2.12 2.13 2.14 2.17 2.20 2.22 2.25 2.27 2.31 2.34 2.37 2.40 2.44 2.48 2.53 2.57 2.62 2.66 2.70 2.74 2.78 2.82 2.86 2.88 2.91 2.94 2.96 2.99 3.01 3.03 3.04 3.06 3.07 3.08 3.08 3.09 3.09 3.09 3.09 3.10 3.10 3.10 3.10 3.08 3.06 3.04 3.00 2.97 2.93 2.89 2.85 2.80 2.74 2.69 2.64 2.59 2.56 2.52 2.48 2.44 2.41 2.38 2.35
2.42 2.40 2.38 2.36 2.34 2.32 2.30 2.28 2.27 2.26 2.25 2.24 2.22 2.21 2.20 2.19 2.18 2.17 2.15 2.14 2.13 2.11 2.10 2.09 2.08 2.07 2.07 2.07 2.08 2.10 2.12 2.13 2.15 2.16 2.18 2.21 2.21 2.22 2.23 2.25 2.26 2.27 2.27 2.27 2.27 2.26 2.25 2.24 2.22 2.21 2.20 2.18 2.16 2.14 2.12 2.10 2.09 2.07 2.06 2.05 2.05 2.05 2.05 2.05 2.07 2.08 2.11 2.14 2.17 2.21 2.24 2.28 2.32 2.36 2.38 2.41 2.43 2.46 2.47 2.47 2.47 2.47 2.48 2.48 2.48 2.47 2.47 2.46 2.45 2.44 2.42 2.40 2.38 2.36 2.33 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.14 2.13 2.12 2.10 2.09 2.08 2.07 2.06 2.05 2.05 2.06 2.06 2.06 2.06 2.07 2.07 2.08 2.08 2.09 2.11 2.13 2.15 2.17 2.20 2.22 2.25 2.28 2.32 2.36 2.39 2.43 2.47 2.51 2.55 2.58 2.62 2.66 2.69 2.71 2.74 2.77 2.79 2.81 2.83 2.85 2.87 2.89 2.91 2.93 2.94 2.95 2.96 2.96 2.97 2.97 2.98 2.98 2.98 2.98 2.98 2.97 2.95 2.94 2.92 2.89 2.86 2.83 2.79 2.74 2.71 2.67 2.63 2.59 2.56 2.53 2.50 2.47 2.45 2.42
2.46 2.45 2.43 2.41 2.39 2.38 2.36 2.34 2.33 2.32 2.31 2.30 2.29 2.28 2.26 2.24 2.22 2.20 2.18 2.17 2.15 2.15 2.14 2.14 2.13 2.12 2.12 2.12 2.12 2.13 2.14 2.14 2.15 2.15 2.16 2.17 2.18 2.18 2.19 2.20 2.21 2.21 2.22 2.22 2.22 2.22 2.21 2.20 2.19 2.18 2.16 2.15 2.13 2.12 2.10 2.08 2.08 2.07 2.06 2.05 2.05 2.05 2.05 2.06 2.08 2.10 2.13 2.17 2.21 2.24 2.28 2.33 2.37 2.41 2.44 2.47 2.49 2.52 2.54 2.54 2.55 2.56 2.57 2.58 2.58 2.59 2.59 2.59 2.59 2.58 2.57 2.55 2.53 2.51 2.49 2.46 2.44 2.41 2.39 2.36 2.33 2.31 2.29 2.27 2.25 2.23 2.22 2.20 2.18 2.17 2.15 2.14 2.13 2.13 2.12 2.11 2.11 2.10 2.10 2.10 2.10 2.11 2.13 2.14 2.15 2.17 2.19 2.22 2.24 2.27 2.30 2.33 2.36 2.40 2.43 2.46 2.49 2.52 2.55 2.57 2.59 2.61 2.64 2.66 2.68 2.70 2.71 2.73 2.74 2.76 2.77 2.79 2.80 2.81 2.82 2.82 2.83 2.84 2.85 2.86 2.87 2.86 2.86 2.85 2.84 2.83 2.81 2.80 2.78 2.75 2.72 2.70 2.67 2.64 2.61 2.58 2.55 2.53 2.51 2.49 2.46
2.48 2.46 2.45 2.43 2.42 2.40 2.38 2.36 2.35 2.34 2.33 2.32 2.31 2.29 2.27 2.24 2.22 2.20 2.18 2.16 2.15 2.14 2.14 2.14 2.13 2.12 2.12 2.12 2.12 2.12 2.12 2.12 2.12 2.12 2.11 2.11 2.11 2.12 2.12 2.13 2.13 2.14 2.14 2.15 2.15 2.16 2.16 2.16 2.15 2.14 2.13 2.12 2.11 2.11 2.10 2.08 2.08 2.08 2.07 2.07 2.06 2.07 2.07 2.09 2.11 2.14 2.18 2.22 2.26 2.30 2.34 2.39 2.43 2.47 2.50 2.53 2.56 2.58 2.61 2.63 2.64 2.66 2.68 2.69 2.70 2.71 2.72 2.72 2.73 2.72 2.72 2.70 2.69 2.67 2.65 2.63 2.61 2.59 2.56 2.54 2.51 2.48 2.46 2.43 2.41 2.39 2.37 2.35 2.33 2.31 2.29 2.27 2.26 2.24 2.22 2.21 2.20 2.18 2.18 2.17 2.16 2.17 2.17 2.18 2.18 2.19 2.21 2.23 2.25 2.26 2.28 2.31 2.33 2.36 2.38 2.40 2.42 2.44 2.46 2.48 2.50 2.51 2.53 2.54 2.56 2.58 2.59 2.60 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.72 2.73 2.74 2.76 2.76 2.76 2.75 2.75 2.74 2.73 2.72 2.71 2.70 2.68 2.66 2.64 2.62 2.59 2.57 2.55 2.53 2.52 2.50 2.48
2.47 2.45 2.43 2.42 2.40 2.39 2.37 2.35 2.34 2.32 2.31 2.29 2.27 2.25 2.23 2.21 2.19 2.17 2.15 2.13 2.11 2.11 2.10 2.09 2.08 2.08 2.08 2.08 2.08 2.08 2.08 2.07 2.06 2.06 2.05 2.04 2.04 2.04 2.04 2.04 2.04 2.04 2.05 2.06 2.07 2.09 2.10 2.11 2.11 2.11 2.10 2.11 2.11 2.11 2.10 2.10 2.10 2.10 2.10 2.10 2.09 2.10 2.11 2.13 2.16 2.20 2.24 2.29 2.33 2.38 2.42 2.46 2.50 2.53 2.56 2.59 2.62 2.65 2.68 2.70 2.73 2.76 2.79 2.81 2.82 2.84 2.84 2.85 2.85 2.85 2.85 2.85 2.84 2.83 2.81 2.80 2.78 2.76 2.74 2.71 2.69 2.66 2.64 2.61 2.59 2.56 2.54 2.52 2.50 2.47 2.45 2.42 2.40 2.38 2.36 2.34 2.33 2.31 2.29 2.28 2.27 2.27 2.26 2.25 2.25 2.25 2.26 2.27 2.28 2.30 2.31 2.32 2.34 2.35 2.36 2.37 2.39 2.40 2.41 2.43 2.45 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.61 2.63 2.64 2.66 2.66 2.66 2.66 2.66 2.66 2.65 2.65 2.64 2.62 2.61 2.60 2.58 2.57 2.55 2.53 2.52 2.51 2.50 2.48 2.47
2.42 2.41 2.39 2.38 2.36 2.34 2.33 2.31 2.29 2.27 2.24 2.22 2.20 2.19 2.17 2.16 2.14 2.12 2.10 2.08 2.06 2.05 2.04 2.03 2.02 2.01 2.01 2.00 2.00 2.00 2.00 2.00 2.00 1.99 1.98 1.96 1.96 1.95 1.95 1.94 1.93 1.94 1.95 1.96 1.98 2.00 2.02 2.05 2.07 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.13 2.13 2.13 2.14 2.14 2.16 2.17 2.20 2.23 2.27 2.32 2.36 2.41 2.45 2.48 2.52 2.56 2.59 2.62 2.65 2.68 2.71 2.74 2.78 2.81 2.84 2.87 2.90 2.92 2.94 2.95 2.96 2.97 2.97 2.97 2.97 2.97 2.96 2.95 2.94 2.92 2.90 2.88 2.86 2.84 2.82 2.80 2.78 2.76 2.74 2.72 2.69 2.67 2.64 2.61 2.59 2.56 2.54 2.52 2.50 2.48 2.46 2.44 2.42 2.40 2.39 2.38 2.37 2.37 2.37 2.37 2.37 2.37 2.37 2.37 2.38 2.38 2.39 2.39 2.39 2.40 2.41 2.42 2.42 2.42 2.43 2.43 2.44 2.44 2.44 2.44 2.45 2.45 2.45 2.45 2.46 2.46 2.47 2.48 2.48 2.50 2.52 2.54 2.55 2.57 2.57 2.58 2.58 2.57 2.57 2.57 2.57 2.56 2.55 2.53 2.52 2.51 2.49 2.48 2.47 2.46 2.46 2.45 2.43 2.42
2.33 2.32 2.30 2.29 2.27 2.26 2.25 2.23 2.21 2.18 2.15 2.14 2.12 2.10 2.09 2.07 2.05 2.03 2.01 1.99 1.97 1.96 1.95 1.94 1.93 1.92 1.92 1.91 1.90 1.90 1.89 1.90 1.90 1.90 1.90 1.90 1.89 1.89 1.88 1.86 1.84 1.84 1.84 1.86 1.88 1.91 1.94 1.97 2.00 2.03 2.07 2.08 2.10 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.18 2.21 2.24 2.27 2.30 2.34 2.38 2.43 2.47 2.50 2.54 2.56 2.59 2.62 2.65 2.67 2.71 2.74 2.78 2.82 2.86 2.90 2.94 2.96 2.98 3.00 3.02 3.03 3.04 3.04 3.05 3.05 3.05 3.05 3.05 3.04 3.03 3.01 3.00 2.99 2.97 2.95 2.93 2.91 2.89 2.87 2.85 2.83 2.81 2.79 2.77 2.74 2.72 2.70 2.68 2.66 2.63 2.61 2.59 2.57 2.56 2.55 2.53 2.52 2.50 2.49 2.49 2.49 2.48 2.48 2.47 2.47 2.47 2.46 2.46 2.45 2.45 2.46 2.46 2.45 2.45 2.44 2.44 2.44 2.44 2.44 2.44 2.44 2.44 2.43 2.43 2.43 2.43 2.43 2.43 2.43 2.44 2.44 2.45 2.46 2.47 2.47 2.47 2.46 2.46 2.46 2.46 2.46 2.46 2.45 2.44 2.43 2.42 2.41 2.40 2.39 2.38 2.37 2.35 2.34 2.33
2.23 2.21 2.19 2.18 2.17 2.16 2.15 2.13 2.11 2.09 2.07 2.05 2.03 2.01 1.99 1.98 1.96 1.94 1.92 1.91 1.89 1.88 1.86 1.85 1.84 1.83 1.83 1.82 1.81 1.81 1.80 1.80 1.80 1.81 1.81 1.81 1.81 1.81 1.81 1.80 1.80 1.80 1.79 1.79 1.80 1.81 1.84 1.87 1.91 1.96 2.01 2.05 2.08 2.11 2.13 2.15 2.17 2.18 2.19 2.21 2.22 2.26 2.29 2.33 2.36 2.39 2.43 2.46 2.49 2.51 2.54 2.57 2.59 2.62 2.66 2.69 2.73 2.77 2.81 2.85 2.89 2.92 2.96 2.98 3.01 3.03 3.04 3.06 3.07 3.08 3.09 3.09 3.10 3.10 3.09 3.09 3.08 3.08 3.07 3.06 3.05 3.04 3.02 3.01 3.00 2.98 2.96 2.95 2.93 2.91 2.89 2.86 2.84 2.82 2.81 2.79 2.77 2.74 2.72 2.71 2.70 2.68 2.67 2.65 2.63 2.61 2.61 2.60 2.59 2.58 2.58 2.57 2.56 2.55 2.54 2.53 2.52 2.51 2.50 2.50 2.49 2.48 2.47 2.46 2.45 2.45 2.44 2.44 2.43 2.43 2.42 2.41 2.40 2.40 2.39 2.39 2.39 2.38 2.38 2.39 2.39 2.39 2.38 2.38 2.38 2.38 2.37 2.37 2.36 2.35 2.34 2.33 2.32 2.31 2.30 2.29 2.28 2.26 2.25 2.24 2.23
2.12 2.10 2.08 2.07 2.06 2.05 2.04 2.02 2.00 1.99 1.98 1.96 1.94 1.92 1.91 1.89 1.87 1.85 1.84 1.83 1.81 1.80 1.79 1.78 1.77 1.76 1.75 1.74 1.73 1.73 1.72 1.72 1.72 1.72 1.72 1.71 1.72 1.73 1.73 1.74 1.75 1.75 1.75 1.75 1.75 1.75 1.77 1.80 1.83 1.87 1.91 1.97 2.02 2.07 2.10 2.14 2.16 2.17 2.19 2.22 2.25 2.28 2.31 2.34 2.37 2.39 2.42 2.45 2.48 2.51 2.53 2.56 2.59 2.62 2.66 2.70 2.74 2.79 2.83 2.86 2.89 2.92 2.94 2.96 2.99 3.02 3.03 3.04 3.06 3.07 3.08 3.09 3.09 3.10 3.10 3.09 3.09 3.09 3.09 3.09 3.09 3.07 3.06 3.05 3.05 3.04 3.02 3.01 2.99 2.98 2.96 2.94 2.92 2.91 2.90 2.88 2.86 2.84 2.83 2.81 2.80 2.79 2.77 2.76 2.74 2.72 2.71 2.69 2.68 2.67 2.66 2.65 2.63 2.62 2.60 2.59 2.58 2.57 2.55 2.55 2.54 2.52 2.51 2.50 2.48 2.46 2.46 2.45 2.44 2.43 2.42 2.41 2.39 2.38 2.37 2.36 2.35 2.34 2.33 2.33 2.33 2.32 2.32 2.31 2.30 2.30 2.29 2.29 2.28 2.26 2.25 2.23 2.22 2.21 2.19 2.18 2.17 2.15 2.14 2.13 2.12
2.01 2.00 1.98 1.97 1.96 1.95 1.93 1.92 1.91 1.89 1.88 1.87 1.86 1.84 1.83 1.82 1.80 1.78 1.77 1.76 1.75 1.74 1.73 1.73 1.72 1.71 1.70 1.69 1.69 1.68 1.67 1.67 1.67 1.67 1.67 1.66 1.67 1.68 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.78 1.79 1.81 1.83 1.89 1.96 2.02 2.07 2.12 2.14 2.15 2.18 2.23 2.27 2.29 2.30 2.32 2.34 2.36 2.40 2.43 2.46 2.50 2.54 2.57 2.60 2.63 2.67 2.70 2.74 2.78 2.81 2.83 2.86 2.88 2.90 2.92 2.94 2.97 2.98 2.99 3.01 3.02 3.03 3.04 3.05 3.05 3.06 3.06 3.06 3.06 3.07 3.07 3.07 3.06 3.06 3.05 3.04 3.02 3.02 3.01 3.00 2.99 2.98 2.97 2.95 2.94 2.93 2.92 2.90 2.89 2.88 2.87 2.86 2.84 2.83 2.82 2.80 2.78 2.77 2.75 2.74 2.73 2.71 2.70 2.68 2.66 2.65 2.64 2.62 2.61 2.59 2.58 2.57 2.56 2.54 2.53 2.51 2.48 2.47 2.46 2.44 2.43 2.42 2.40 2.39 2.37 2.35 2.33 2.32 2.31 2.30 2.29 2.28 2.27 2.26 2.25 2.24 2.23 2.22 2.21 2.20 2.19 2.17 2.15 2.13 2.11 2.10 2.08 2.06 2.05 2.04 2.02 2.01
1.93 1.92 1.91 1.89 1.88 1.87 1.85 1.84 1.83 1.82 1.80 1.80 1.79 1.78 1.77 1.75 1.75 1.74 1.73 1.73 1.72 1.71 1.70 1.69 1.69 1.69 1.69 1.69 1.69 1.68 1.67 1.67 1.67 1.68 1.68 1.69 1.70 1.70 1.71 1.71 1.72 1.73 1.73 1.75 1.77 1.79 1.81 1.83 1.85 1.89 1.93 1.98 2.02 2.07 2.11 2.16 2.17 2.17 2.19 2.22 2.26 2.29 2.32 2.35 2.38 2.40 2.44 2.47 2.50 2.52 2.55 2.58 2.60 2.62 2.64 2.66 2.70 2.73 2.75 2.77 2.79 2.81 2.83 2.85 2.87 2.89 2.90 2.92 2.93 2.94 2.96 2.96 2.97 2.97 2.97 2.97 2.98 2.98 2.99 2.99 2.99 2.99 2.99 2.98 2.98 2.97 2.96 2.96 2.95 2.95 2.94 2.93 2.93 2.92 2.90 2.89 2.88 2.88 2.87 2.86 2.86 2.84 2.83 2.82 2.81 2.79 2.78 2.77 2.75 2.74 2.73 2.71 2.70 2.69 2.68 2.66 2.64 2.62 2.61 2.59 2.58 2.57 2.56 2.54 2.51 2.48 2.47 2.46 2.44 2.42 2.40 2.39 2.38 2.36 2.34 2.32 2.31 2.30 2.28 2.26 2.24 2.23 2.21 2.20 2.19 2.18 2.16 2.15 2.13 2.11 2.09 2.08 2.06 2.04 2.03 2.01 1.99 1.97 1.96 1.95 1.93
1.92 1.90 1.88 1.87 1.85 1.84 1.83 1.82 1.80 1.80 1.79 1.78 1.78 1.77 1.76 1.75 1.75 1.74 1.74 1.74 1.73 1.73 1.72 1.72 1.72 1.73 1.73 1.73 1.73 1.73 1.72 1.73 1.73 1.74 1.75 1.77 1.77 1.78 1.79 1.81 1.83 1.84 1.85 1.87 1.89 1.92 1.94 1.97 1.99 2.03 2.06 2.09 2.12 2.15 2.17 2.20 2.21 2.22 2.23 2.25 2.26 2.28 2.30 2.32 2.34 2.35 2.38 2.41 2.43 2.45 2.47 2.49 2.51 2.53 2.55 2.57 2.59 2.62 2.64 2.66 2.68 2.69 2.70 2.72 2.73 2.75 2.76 2.77 2.79 2.80 2.81 2.81 2.81 2.82 2.82 2.83 2.83 2.83 2.83 2.84 2.84 2.84 2.84 2.84 2.84 2.84 2.84 2.84 2.84 2.83 2.82 2.82 2.82 2.82 2.81 2.81 2.81 2.80 2.79 2.79 2.78 2.77 2.76 2.76 2.75 2.74 2.73 2.71 2.70 2.69 2.68 2.67 2.66 2.65 2.64 2.62 2.61 2.60 2.58 2.56 2.54 2.53 2.52 2.50 2.49 2.47 2.46 2.45 2.43 2.41 2.39 2.37 2.35 2.34 2.32 2.31 2.29 2.27 2.26 2.24 2.23 2.21 2.19 2.17 2.16 2.14 2.12 2.11 2.10 2.08 2.07 2.05 2.03 2.01 1.99 1.97 1.96 1.95 1.93 1.93 1.92
1.95 1.93 1.92 1.91 1.90 1.88 1.87 1.86 1.85 1.85 1.84 1.84 1.83 1.83 1.82 1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.82 1.82 1.82 1.82 1.82 1.83 1.83 1.84 1.85 1.85 1.86 1.88 1.89 1.91 1.92 1.93 1.94 1.95 1.97 1.99 2.02 2.03 2.05 2.07 2.09 2.11 2.13 2.15 2.17 2.19 2.21 2.22 2.23 2.25 2.26 2.27 2.28 2.29 2.30 2.32 2.33 2.34 2.36 2.37 2.39 2.40 2.41 2.43 2.44 2.46 2.47 2.49 2.50 2.52 2.53 2.55 2.55 2.56 2.57 2.58 2.59 2.60 2.61 2.62 2.63 2.64 2.64 2.65 2.65 2.66 2.67 2.67 2.67 2.67 2.68 2.68 2.69 2.69 2.69 2.69 2.69 2.69 2.69 2.69 2.69 2.70 2.70 2.70 2.70 2.70 2.70 2.69 2.68 2.68 2.68 2.67 2.67 2.67 2.66 2.66 2.65 2.64 2.63 2.62 2.61 2.60 2.59 2.58 2.57 2.56 2.56 2.54 2.53 2.52 2.50 2.48 2.48 2.47 2.46 2.44 2.43 2.42 2.40 2.39 2.37 2.36 2.34 2.33 2.31 2.30 2.29 2.27 2.26 2.24 2.23 2.21 2.20 2.18 2.17 2.15 2.14 2.12 2.10 2.09 2.08 2.07 2.06 2.04 2.03 2.02 2.00 1.99 1.97 1.96 1.95 1.95
2.01 2.01 2.00 2.00 1.98 1.97 1.97 1.96 1.95 1.95 1.95 1.94 1.94 1.94 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.94 1.94 1.94 1.94 1.94 1.94 1.95 1.95 1.95 1.96 1.97 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.03 2.04 2.05 2.07 2.08 2.09 2.10 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.21 2.22 2.23 2.25 2.26 2.27 2.27 2.28 2.29 2.30 2.31 2.32 2.32 2.33 2.34 2.35 2.36 2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.44 2.44 2.45 2.46 2.47 2.48 2.48 2.48 2.49 2.50 2.50 2.51 2.52 2.52 2.52 2.52 2.52 2.53 2.53 2.54 2.54 2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.56 2.55 2.55 2.55 2.55 2.55 2.54 2.54 2.54 2.53 2.53 2.52 2.51 2.51 2.50 2.49 2.48 2.48 2.47 2.46 2.45 2.44 2.43 2.42 2.41 2.41 2.40 2.39 2.38 2.37 2.36 2.35 2.34 2.33 2.32 2.31 2.30 2.29 2.28 2.27 2.26 2.25 2.23 2.22 2.21 2.20 2.19 2.18 2.17 2.16 2.15 2.13 2.12 2.11 2.10 2.09 2.08 2.08 2.07 2.06 2.05 2.03 2.02 2.02 2.01
nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
];
rms=[
nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
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0.13 0.15 0.17 0.18 0.20 0.22 0.23 0.24 0.24 0.25 0.25 0.25 0.26 0.26 0.26 0.26 0.27 0.27 0.28 0.29 0.29 0.29 0.29 0.30 0.30 0.30 0.28 0.27 0.26 0.25 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.17 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.17 0.18 0.18 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.17 0.17 0.16 0.14 0.13 0.12 0.12 0.11 0.11 0.11 0.12 0.12 0.13 0.14 0.14 0.16 0.17 0.18 0.19 0.19 0.19 0.19 0.19 0.18 0.18 0.18 0.18 0.17 0.17 0.16 0.17 0.19 0.20 0.21 0.22 0.23 0.23 0.23 0.23 0.22 0.21 0.19 0.18 0.17 0.16 0.14 0.13 0.11 0.11 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.11 0.12 0.13 0.13 0.15 0.16 0.17 0.18 0.18 0.19 0.19 0.20 0.21 0.21 0.21 0.21 0.21 0.20 0.20 0.19 0.19 0.18 0.16 0.15 0.14 0.14 0.13 0.13 0.13 0.14 0.14 0.15 0.16 0.17 0.19 0.20 0.21 0.21 0.22 0.23 0.23 0.24 0.24 0.24 0.23 0.23 0.22 0.21 0.21 0.20 0.19 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.13 0.12 0.11 0.11 0.10 0.10 0.10 0.11 0.11 0.12 0.13
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0.15 0.16 0.17 0.19 0.20 0.22 0.23 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.26 0.27 0.27 0.28 0.29 0.29 0.29 0.28 0.28 0.27 0.26 0.24 0.23 0.21 0.19 0.18 0.17 0.16 0.15 0.14 0.14 0.14 0.14 0.14 0.13 0.13 0.13 0.13 0.13 0.13 0.14 0.15 0.15 0.16 0.17 0.17 0.18 0.18 0.18 0.19 0.18 0.18 0.17 0.17 0.16 0.15 0.14 0.14 0.14 0.14 0.14 0.14 0.15 0.16 0.17 0.18 0.19 0.21 0.21 0.22 0.23 0.24 0.24 0.23 0.22 0.22 0.22 0.22 0.21 0.21 0.21 0.22 0.23 0.24 0.25 0.26 0.26 0.27 0.27 0.27 0.27 0.26 0.25 0.23 0.21 0.19 0.17 0.15 0.14 0.13 0.12 0.11 0.10 0.10 0.10 0.11 0.11 0.11 0.11 0.11 0.13 0.14 0.15 0.17 0.18 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.17 0.16 0.15 0.15 0.14 0.14 0.14 0.14 0.14 0.15 0.17 0.18 0.20 0.21 0.23 0.25 0.26 0.27 0.27 0.27 0.27 0.26 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.20 0.19 0.19 0.18 0.18 0.17 0.17 0.17 0.16 0.15 0.15 0.14 0.13 0.13 0.13 0.13 0.13 0.13 0.14 0.15
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0.29 0.29 0.28 0.27 0.27 0.26 0.25 0.23 0.22 0.21 0.19 0.20 0.21 0.22 0.25 0.28 0.31 0.34 0.37 0.39 0.41 0.41 0.42 0.42 0.40 0.39 0.37 0.35 0.33 0.30 0.28 0.25 0.22 0.21 0.21 0.21 0.22 0.22 0.23 0.25 0.26 0.26 0.26 0.25 0.23 0.21 0.21 0.21 0.22 0.23 0.24 0.27 0.30 0.32 0.34 0.36 0.36 0.37 0.37 0.36 0.36 0.37 0.38 0.39 0.39 0.39 0.36 0.34 0.32 0.31 0.29 0.31 0.32 0.34 0.36 0.39 0.41 0.43 0.44 0.45 0.45 0.44 0.43 0.41 0.38 0.36 0.32 0.29 0.26 0.24 0.21 0.20 0.19 0.18 0.19 0.19 0.19 0.18 0.18 0.18 0.18 0.17 0.17 0.18 0.21 0.24 0.28 0.31 0.33 0.34 0.36 0.36 0.36 0.35 0.34 0.32 0.30 0.27 0.25 0.23 0.21 0.20 0.19 0.18 0.17 0.16 0.16 0.15 0.15 0.15 0.15 0.15 0.16 0.17 0.17 0.18 0.18 0.18 0.18 0.18 0.18 0.20 0.22 0.25 0.29 0.32 0.33 0.34 0.34 0.33 0.32 0.32 0.32 0.33 0.34 0.36 0.38 0.40 0.42 0.44 0.45 0.46 0.47 0.47 0.48 0.49 0.48 0.47 0.46 0.45 0.44 0.43 0.41 0.40 0.39 0.37 0.35 0.33 0.32 0.31 0.29
0.29 0.29 0.28 0.27 0.27 0.26 0.25 0.23 0.22 0.22 0.21 0.21 0.22 0.24 0.26 0.29 0.32 0.34 0.36 0.38 0.39 0.39 0.40 0.39 0.38 0.36 0.34 0.32 0.30 0.28 0.26 0.25 0.23 0.22 0.22 0.21 0.22 0.23 0.24 0.25 0.26 0.26 0.26 0.25 0.24 0.22 0.22 0.21 0.22 0.24 0.26 0.28 0.31 0.33 0.34 0.36 0.36 0.36 0.35 0.34 0.33 0.35 0.37 0.39 0.41 0.43 0.40 0.38 0.36 0.34 0.33 0.33 0.34 0.35 0.37 0.39 0.41 0.43 0.44 0.45 0.45 0.44 0.43 0.41 0.38 0.36 0.32 0.28 0.25 0.22 0.20 0.20 0.20 0.21 0.22 0.23 0.23 0.23 0.23 0.23 0.23 0.22 0.21 0.20 0.20 0.20 0.23 0.25 0.27 0.28 0.29 0.30 0.30 0.30 0.29 0.27 0.26 0.24 0.23 0.21 0.20 0.20 0.19 0.19 0.18 0.18 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.18 0.20 0.23 0.25 0.28 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.32 0.33 0.34 0.36 0.37 0.39 0.42 0.44 0.46 0.48 0.48 0.49 0.49 0.49 0.48 0.47 0.46 0.44 0.43 0.41 0.40 0.38 0.36 0.34 0.32 0.31 0.30 0.29
0.27 0.27 0.27 0.26 0.26 0.26 0.25 0.23 0.22 0.21 0.20 0.21 0.21 0.23 0.26 0.28 0.30 0.33 0.34 0.35 0.36 0.36 0.35 0.34 0.33 0.32 0.30 0.29 0.27 0.26 0.24 0.24 0.23 0.23 0.22 0.22 0.23 0.24 0.25 0.26 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.21 0.22 0.24 0.26 0.28 0.30 0.32 0.33 0.35 0.34 0.33 0.32 0.31 0.30 0.32 0.33 0.36 0.38 0.41 0.39 0.38 0.37 0.37 0.36 0.36 0.35 0.36 0.37 0.39 0.40 0.41 0.42 0.43 0.44 0.42 0.41 0.39 0.38 0.36 0.32 0.29 0.26 0.23 0.20 0.22 0.23 0.25 0.27 0.29 0.29 0.29 0.29 0.27 0.26 0.25 0.23 0.22 0.21 0.20 0.22 0.24 0.25 0.26 0.28 0.28 0.29 0.29 0.27 0.26 0.25 0.23 0.22 0.22 0.21 0.22 0.22 0.22 0.21 0.21 0.21 0.21 0.21 0.20 0.20 0.19 0.19 0.18 0.17 0.16 0.16 0.15 0.16 0.17 0.19 0.21 0.24 0.26 0.28 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.31 0.32 0.33 0.34 0.36 0.37 0.40 0.42 0.44 0.47 0.48 0.48 0.49 0.48 0.47 0.46 0.44 0.43 0.41 0.39 0.38 0.36 0.35 0.33 0.31 0.29 0.28 0.27
0.25 0.25 0.25 0.25 0.25 0.25 0.23 0.22 0.21 0.20 0.19 0.19 0.20 0.22 0.24 0.27 0.29 0.30 0.31 0.32 0.32 0.31 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.22 0.22 0.22 0.22 0.23 0.23 0.24 0.25 0.26 0.27 0.26 0.25 0.24 0.22 0.21 0.21 0.21 0.21 0.23 0.25 0.26 0.28 0.30 0.31 0.32 0.31 0.30 0.28 0.27 0.26 0.27 0.29 0.31 0.33 0.36 0.36 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.38 0.38 0.38 0.39 0.39 0.40 0.39 0.38 0.36 0.35 0.34 0.31 0.28 0.25 0.24 0.22 0.24 0.27 0.30 0.32 0.35 0.35 0.35 0.34 0.31 0.28 0.26 0.24 0.23 0.23 0.22 0.24 0.26 0.27 0.29 0.30 0.31 0.32 0.31 0.30 0.29 0.27 0.26 0.25 0.25 0.25 0.25 0.26 0.26 0.26 0.25 0.25 0.25 0.24 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.16 0.19 0.21 0.23 0.25 0.27 0.28 0.29 0.29 0.29 0.29 0.29 0.29 0.28 0.28 0.28 0.29 0.30 0.31 0.33 0.35 0.37 0.40 0.42 0.45 0.46 0.47 0.47 0.46 0.45 0.43 0.42 0.40 0.39 0.37 0.35 0.34 0.32 0.30 0.28 0.27 0.26 0.25
0.22 0.23 0.23 0.23 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.18 0.19 0.20 0.23 0.25 0.27 0.28 0.28 0.28 0.28 0.26 0.24 0.23 0.22 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.22 0.23 0.23 0.24 0.25 0.25 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.20 0.20 0.21 0.22 0.23 0.24 0.26 0.27 0.28 0.27 0.25 0.24 0.23 0.22 0.22 0.23 0.25 0.27 0.30 0.32 0.34 0.36 0.37 0.37 0.37 0.37 0.37 0.36 0.35 0.35 0.34 0.34 0.34 0.34 0.34 0.33 0.32 0.31 0.30 0.27 0.25 0.24 0.23 0.23 0.27 0.31 0.34 0.37 0.39 0.39 0.39 0.37 0.34 0.30 0.27 0.25 0.24 0.25 0.25 0.27 0.29 0.32 0.34 0.36 0.37 0.37 0.37 0.36 0.34 0.33 0.32 0.31 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.20 0.19 0.18 0.19 0.21 0.24 0.25 0.27 0.28 0.29 0.29 0.29 0.29 0.29 0.28 0.28 0.26 0.25 0.24 0.25 0.25 0.27 0.29 0.32 0.35 0.38 0.40 0.42 0.43 0.44 0.44 0.43 0.42 0.41 0.39 0.37 0.35 0.34 0.32 0.30 0.28 0.26 0.25 0.23 0.23 0.22
0.19 0.20 0.21 0.21 0.20 0.19 0.19 0.18 0.17 0.16 0.15 0.16 0.17 0.19 0.21 0.23 0.24 0.25 0.25 0.24 0.23 0.21 0.20 0.19 0.18 0.16 0.17 0.18 0.18 0.19 0.19 0.20 0.21 0.21 0.22 0.23 0.23 0.24 0.24 0.24 0.24 0.24 0.23 0.22 0.22 0.21 0.20 0.20 0.19 0.19 0.18 0.19 0.20 0.21 0.22 0.23 0.21 0.20 0.19 0.19 0.18 0.19 0.19 0.21 0.23 0.26 0.29 0.32 0.35 0.36 0.37 0.37 0.36 0.35 0.33 0.31 0.30 0.30 0.29 0.28 0.28 0.28 0.28 0.27 0.26 0.24 0.23 0.22 0.22 0.23 0.24 0.29 0.33 0.37 0.39 0.41 0.40 0.39 0.38 0.35 0.32 0.30 0.27 0.27 0.28 0.29 0.32 0.34 0.37 0.41 0.44 0.44 0.45 0.45 0.44 0.42 0.40 0.38 0.37 0.37 0.37 0.37 0.36 0.35 0.35 0.34 0.33 0.33 0.32 0.30 0.28 0.27 0.26 0.26 0.25 0.24 0.24 0.24 0.25 0.26 0.28 0.28 0.28 0.28 0.29 0.29 0.29 0.29 0.29 0.28 0.28 0.26 0.24 0.22 0.22 0.21 0.23 0.25 0.28 0.31 0.34 0.36 0.38 0.39 0.40 0.41 0.40 0.39 0.38 0.36 0.34 0.32 0.29 0.27 0.26 0.24 0.23 0.22 0.21 0.20 0.19
0.17 0.18 0.18 0.18 0.18 0.18 0.18 0.17 0.16 0.16 0.16 0.16 0.16 0.17 0.18 0.19 0.20 0.21 0.21 0.20 0.19 0.18 0.16 0.15 0.14 0.14 0.15 0.16 0.17 0.19 0.21 0.21 0.22 0.23 0.23 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.23 0.22 0.21 0.20 0.19 0.17 0.17 0.16 0.15 0.16 0.17 0.18 0.18 0.19 0.18 0.17 0.17 0.16 0.15 0.15 0.16 0.17 0.20 0.23 0.27 0.31 0.34 0.36 0.37 0.36 0.35 0.33 0.31 0.28 0.27 0.25 0.24 0.23 0.22 0.22 0.21 0.21 0.21 0.20 0.21 0.21 0.22 0.23 0.24 0.28 0.32 0.35 0.37 0.39 0.39 0.38 0.37 0.35 0.32 0.30 0.28 0.28 0.29 0.31 0.35 0.39 0.43 0.47 0.50 0.51 0.52 0.51 0.49 0.47 0.47 0.46 0.45 0.44 0.44 0.43 0.43 0.42 0.40 0.39 0.38 0.36 0.34 0.31 0.29 0.27 0.26 0.24 0.23 0.22 0.22 0.23 0.23 0.23 0.24 0.24 0.24 0.24 0.25 0.27 0.27 0.27 0.26 0.25 0.24 0.22 0.20 0.19 0.19 0.19 0.20 0.22 0.25 0.27 0.30 0.32 0.33 0.34 0.35 0.35 0.35 0.34 0.33 0.31 0.29 0.27 0.26 0.24 0.23 0.22 0.20 0.19 0.18 0.18 0.17
0.16 0.17 0.18 0.19 0.19 0.19 0.18 0.18 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.18 0.19 0.19 0.17 0.16 0.15 0.13 0.13 0.13 0.13 0.15 0.16 0.18 0.20 0.22 0.23 0.24 0.25 0.24 0.24 0.24 0.24 0.23 0.23 0.22 0.22 0.22 0.21 0.20 0.19 0.17 0.16 0.15 0.14 0.14 0.14 0.15 0.15 0.16 0.16 0.16 0.15 0.15 0.14 0.14 0.14 0.14 0.15 0.18 0.21 0.25 0.29 0.32 0.33 0.35 0.34 0.33 0.31 0.28 0.26 0.23 0.21 0.19 0.19 0.18 0.18 0.18 0.18 0.18 0.19 0.20 0.21 0.22 0.22 0.23 0.27 0.31 0.34 0.35 0.37 0.36 0.36 0.34 0.32 0.31 0.29 0.28 0.28 0.30 0.32 0.36 0.41 0.45 0.49 0.53 0.54 0.55 0.55 0.53 0.52 0.51 0.50 0.49 0.49 0.49 0.48 0.47 0.46 0.44 0.43 0.41 0.39 0.36 0.33 0.30 0.27 0.25 0.22 0.20 0.17 0.17 0.16 0.16 0.16 0.17 0.18 0.18 0.20 0.22 0.24 0.24 0.24 0.23 0.21 0.19 0.17 0.15 0.14 0.15 0.16 0.18 0.20 0.22 0.24 0.26 0.27 0.28 0.29 0.29 0.29 0.29 0.29 0.28 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.18 0.17 0.16
0.18 0.19 0.20 0.21 0.20 0.20 0.19 0.19 0.18 0.18 0.18 0.18 0.18 0.18 0.17 0.16 0.17 0.17 0.17 0.15 0.14 0.13 0.12 0.12 0.12 0.13 0.15 0.17 0.20 0.22 0.24 0.25 0.26 0.26 0.25 0.24 0.24 0.23 0.22 0.21 0.20 0.20 0.20 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.13 0.12 0.12 0.12 0.13 0.15 0.18 0.21 0.25 0.28 0.31 0.33 0.32 0.30 0.28 0.26 0.23 0.21 0.18 0.17 0.16 0.16 0.16 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.22 0.23 0.27 0.31 0.33 0.34 0.34 0.34 0.33 0.31 0.30 0.28 0.28 0.27 0.28 0.30 0.32 0.36 0.41 0.45 0.49 0.52 0.53 0.54 0.54 0.53 0.52 0.52 0.51 0.51 0.51 0.51 0.51 0.50 0.48 0.46 0.44 0.43 0.41 0.38 0.35 0.32 0.28 0.25 0.22 0.18 0.15 0.13 0.11 0.11 0.12 0.12 0.14 0.15 0.17 0.19 0.21 0.21 0.21 0.20 0.18 0.16 0.15 0.13 0.13 0.14 0.15 0.17 0.19 0.20 0.22 0.23 0.24 0.24 0.25 0.24 0.24 0.24 0.24 0.23 0.22 0.21 0.21 0.21 0.21 0.21 0.20 0.20 0.20 0.19 0.19 0.18
0.21 0.22 0.22 0.22 0.22 0.21 0.20 0.20 0.19 0.18 0.18 0.18 0.18 0.17 0.17 0.17 0.16 0.16 0.16 0.14 0.13 0.13 0.12 0.12 0.13 0.14 0.16 0.19 0.21 0.24 0.26 0.27 0.27 0.27 0.26 0.24 0.23 0.23 0.21 0.20 0.19 0.19 0.19 0.19 0.18 0.18 0.17 0.16 0.16 0.15 0.15 0.14 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.12 0.11 0.10 0.09 0.10 0.12 0.14 0.17 0.21 0.24 0.27 0.30 0.29 0.27 0.26 0.23 0.21 0.19 0.17 0.16 0.16 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.21 0.22 0.24 0.25 0.28 0.31 0.32 0.32 0.32 0.30 0.29 0.28 0.26 0.25 0.25 0.25 0.26 0.28 0.30 0.35 0.39 0.43 0.46 0.49 0.50 0.50 0.50 0.49 0.48 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.47 0.45 0.43 0.41 0.39 0.36 0.32 0.29 0.26 0.23 0.19 0.15 0.13 0.11 0.11 0.11 0.11 0.12 0.13 0.15 0.16 0.17 0.18 0.19 0.19 0.18 0.17 0.16 0.16 0.16 0.16 0.17 0.18 0.19 0.19 0.20 0.20 0.21 0.21 0.21 0.21 0.20 0.20 0.19 0.19 0.19 0.19 0.20 0.21 0.21 0.22 0.22 0.22 0.23 0.23 0.22 0.21
0.23 0.23 0.23 0.22 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.16 0.16 0.17 0.17 0.18 0.17 0.17 0.16 0.15 0.15 0.14 0.13 0.14 0.15 0.16 0.19 0.21 0.24 0.26 0.28 0.28 0.28 0.27 0.26 0.24 0.22 0.20 0.19 0.19 0.18 0.18 0.18 0.18 0.17 0.16 0.16 0.16 0.16 0.15 0.15 0.14 0.13 0.13 0.14 0.15 0.14 0.13 0.12 0.10 0.08 0.07 0.07 0.07 0.08 0.10 0.13 0.16 0.19 0.22 0.24 0.24 0.24 0.24 0.22 0.21 0.20 0.19 0.18 0.18 0.18 0.19 0.20 0.21 0.22 0.23 0.23 0.23 0.24 0.26 0.28 0.30 0.32 0.32 0.31 0.29 0.27 0.25 0.24 0.22 0.21 0.21 0.21 0.22 0.25 0.28 0.32 0.35 0.39 0.41 0.44 0.44 0.44 0.43 0.42 0.41 0.41 0.42 0.43 0.44 0.45 0.45 0.45 0.45 0.44 0.44 0.42 0.40 0.38 0.35 0.32 0.30 0.27 0.24 0.21 0.18 0.16 0.14 0.13 0.12 0.11 0.11 0.11 0.12 0.13 0.15 0.16 0.17 0.18 0.19 0.19 0.20 0.21 0.21 0.20 0.19 0.19 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.17 0.16 0.16 0.16 0.17 0.18 0.19 0.21 0.22 0.23 0.24 0.24 0.24 0.24 0.24 0.23 0.23
0.21 0.22 0.22 0.22 0.20 0.19 0.18 0.17 0.16 0.16 0.16 0.17 0.17 0.18 0.19 0.19 0.19 0.19 0.18 0.17 0.16 0.15 0.15 0.15 0.16 0.18 0.20 0.23 0.25 0.27 0.29 0.29 0.29 0.28 0.25 0.23 0.21 0.19 0.18 0.17 0.17 0.17 0.17 0.16 0.16 0.16 0.16 0.16 0.16 0.15 0.15 0.14 0.13 0.13 0.14 0.15 0.15 0.15 0.14 0.11 0.08 0.07 0.06 0.06 0.07 0.08 0.10 0.12 0.14 0.17 0.19 0.20 0.20 0.20 0.20 0.20 0.20 0.19 0.19 0.19 0.19 0.20 0.21 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.29 0.29 0.30 0.29 0.28 0.27 0.25 0.23 0.21 0.21 0.20 0.19 0.19 0.20 0.22 0.24 0.27 0.30 0.33 0.34 0.36 0.36 0.36 0.35 0.34 0.33 0.33 0.34 0.35 0.36 0.38 0.39 0.40 0.40 0.40 0.40 0.39 0.37 0.35 0.33 0.31 0.29 0.26 0.23 0.21 0.18 0.16 0.14 0.13 0.12 0.11 0.11 0.11 0.12 0.14 0.16 0.17 0.19 0.19 0.20 0.21 0.21 0.22 0.22 0.21 0.19 0.18 0.17 0.16 0.16 0.15 0.15 0.15 0.15 0.16 0.16 0.17 0.17 0.18 0.19 0.21 0.22 0.23 0.24 0.23 0.23 0.23 0.23 0.23 0.22 0.21
0.19 0.19 0.20 0.19 0.18 0.17 0.16 0.16 0.15 0.16 0.16 0.17 0.18 0.19 0.20 0.20 0.20 0.20 0.20 0.19 0.18 0.18 0.17 0.17 0.19 0.20 0.22 0.24 0.26 0.28 0.29 0.29 0.28 0.27 0.24 0.22 0.20 0.19 0.18 0.17 0.16 0.17 0.17 0.17 0.17 0.17 0.17 0.16 0.16 0.15 0.15 0.14 0.13 0.13 0.13 0.14 0.15 0.16 0.15 0.13 0.10 0.09 0.07 0.07 0.07 0.07 0.08 0.09 0.11 0.13 0.15 0.16 0.17 0.18 0.19 0.19 0.20 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.22 0.23 0.24 0.25 0.27 0.29 0.30 0.30 0.30 0.29 0.28 0.26 0.24 0.21 0.20 0.19 0.19 0.19 0.20 0.20 0.19 0.19 0.22 0.25 0.27 0.28 0.29 0.28 0.28 0.27 0.26 0.25 0.26 0.26 0.27 0.29 0.31 0.32 0.33 0.34 0.34 0.34 0.33 0.32 0.31 0.29 0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.13 0.12 0.11 0.12 0.12 0.13 0.15 0.16 0.18 0.19 0.20 0.20 0.21 0.21 0.22 0.21 0.20 0.19 0.17 0.16 0.15 0.15 0.15 0.15 0.15 0.16 0.17 0.18 0.18 0.19 0.19 0.20 0.21 0.22 0.23 0.23 0.22 0.22 0.21 0.21 0.20 0.20 0.19
0.17 0.16 0.16 0.16 0.15 0.15 0.15 0.14 0.14 0.15 0.16 0.18 0.19 0.21 0.21 0.22 0.22 0.22 0.22 0.21 0.20 0.19 0.19 0.19 0.20 0.20 0.22 0.24 0.26 0.27 0.29 0.28 0.27 0.26 0.23 0.21 0.20 0.19 0.18 0.17 0.17 0.18 0.18 0.19 0.19 0.19 0.18 0.17 0.16 0.16 0.15 0.14 0.13 0.13 0.13 0.13 0.14 0.15 0.15 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.08 0.08 0.09 0.11 0.12 0.14 0.15 0.16 0.18 0.19 0.21 0.22 0.23 0.23 0.23 0.22 0.22 0.21 0.22 0.23 0.24 0.25 0.27 0.29 0.31 0.31 0.31 0.30 0.28 0.26 0.23 0.21 0.19 0.19 0.18 0.19 0.20 0.20 0.18 0.16 0.17 0.19 0.20 0.21 0.21 0.20 0.19 0.18 0.18 0.17 0.18 0.19 0.20 0.22 0.24 0.25 0.26 0.27 0.28 0.28 0.28 0.27 0.26 0.25 0.24 0.23 0.21 0.20 0.18 0.17 0.15 0.13 0.12 0.11 0.11 0.11 0.11 0.12 0.13 0.15 0.16 0.18 0.19 0.19 0.20 0.20 0.20 0.19 0.18 0.17 0.16 0.15 0.15 0.15 0.15 0.16 0.17 0.18 0.19 0.19 0.20 0.20 0.20 0.21 0.21 0.21 0.22 0.22 0.21 0.20 0.19 0.19 0.18 0.17 0.17
0.14 0.13 0.13 0.12 0.12 0.12 0.13 0.13 0.14 0.15 0.17 0.19 0.20 0.22 0.23 0.25 0.25 0.24 0.24 0.22 0.21 0.20 0.20 0.19 0.19 0.19 0.21 0.23 0.25 0.26 0.27 0.26 0.26 0.24 0.23 0.21 0.20 0.19 0.18 0.18 0.18 0.19 0.19 0.20 0.20 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.14 0.13 0.14 0.14 0.14 0.15 0.15 0.14 0.14 0.13 0.12 0.11 0.10 0.09 0.09 0.08 0.09 0.10 0.11 0.12 0.14 0.15 0.16 0.18 0.20 0.23 0.24 0.24 0.24 0.23 0.22 0.22 0.23 0.23 0.24 0.26 0.27 0.29 0.31 0.31 0.31 0.30 0.28 0.26 0.23 0.21 0.19 0.18 0.18 0.19 0.20 0.19 0.17 0.15 0.14 0.14 0.14 0.14 0.13 0.12 0.11 0.10 0.10 0.10 0.12 0.13 0.14 0.16 0.18 0.19 0.19 0.20 0.21 0.23 0.22 0.21 0.21 0.20 0.20 0.19 0.18 0.17 0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.09 0.08 0.09 0.11 0.12 0.14 0.15 0.16 0.17 0.17 0.17 0.17 0.17 0.15 0.14 0.14 0.14 0.15 0.15 0.16 0.17 0.19 0.19 0.19 0.19 0.20 0.21 0.21 0.21 0.21 0.21 0.21 0.20 0.19 0.17 0.17 0.16 0.15 0.15 0.14
0.11 0.11 0.10 0.10 0.10 0.10 0.11 0.12 0.14 0.16 0.18 0.20 0.22 0.23 0.25 0.26 0.25 0.25 0.24 0.22 0.21 0.20 0.19 0.18 0.19 0.19 0.21 0.22 0.23 0.24 0.24 0.24 0.23 0.22 0.21 0.19 0.19 0.19 0.19 0.19 0.19 0.20 0.21 0.21 0.21 0.21 0.20 0.19 0.18 0.17 0.16 0.16 0.15 0.15 0.16 0.16 0.17 0.18 0.18 0.18 0.18 0.17 0.15 0.14 0.13 0.11 0.10 0.09 0.08 0.09 0.10 0.11 0.12 0.14 0.16 0.18 0.20 0.22 0.23 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.26 0.27 0.28 0.30 0.31 0.31 0.31 0.30 0.28 0.26 0.23 0.21 0.19 0.18 0.16 0.17 0.18 0.17 0.16 0.15 0.13 0.12 0.11 0.09 0.08 0.07 0.06 0.05 0.06 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.16 0.15 0.15 0.15 0.15 0.15 0.15 0.14 0.14 0.13 0.12 0.10 0.09 0.08 0.06 0.06 0.06 0.07 0.08 0.10 0.11 0.12 0.13 0.14 0.15 0.15 0.15 0.14 0.14 0.13 0.13 0.13 0.14 0.15 0.16 0.17 0.18 0.18 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.13 0.12 0.11
0.10 0.09 0.09 0.09 0.09 0.10 0.12 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.26 0.27 0.27 0.26 0.25 0.23 0.21 0.20 0.19 0.18 0.17 0.17 0.18 0.18 0.19 0.20 0.20 0.20 0.20 0.20 0.19 0.18 0.19 0.19 0.20 0.20 0.21 0.21 0.22 0.22 0.22 0.22 0.21 0.20 0.19 0.18 0.18 0.17 0.16 0.16 0.17 0.18 0.19 0.20 0.20 0.20 0.20 0.19 0.18 0.16 0.14 0.13 0.11 0.10 0.09 0.10 0.10 0.12 0.13 0.15 0.17 0.19 0.21 0.23 0.24 0.25 0.26 0.26 0.27 0.27 0.27 0.27 0.28 0.29 0.29 0.30 0.31 0.30 0.30 0.29 0.27 0.26 0.23 0.21 0.19 0.18 0.16 0.15 0.14 0.14 0.14 0.15 0.14 0.13 0.12 0.11 0.09 0.08 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.07 0.08 0.09 0.10 0.10 0.09 0.10 0.10 0.11 0.11 0.11 0.11 0.11 0.10 0.10 0.09 0.08 0.08 0.08 0.08 0.08 0.08 0.10 0.11 0.12 0.13 0.13 0.14 0.15 0.15 0.15 0.14 0.14 0.13 0.14 0.14 0.15 0.15 0.16 0.16 0.16 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.16 0.16 0.15 0.14 0.13 0.13 0.12 0.11 0.11 0.10
0.10 0.09 0.09 0.09 0.09 0.10 0.12 0.14 0.16 0.18 0.20 0.23 0.25 0.27 0.28 0.30 0.29 0.27 0.26 0.24 0.22 0.20 0.18 0.16 0.15 0.14 0.15 0.15 0.15 0.16 0.17 0.17 0.17 0.18 0.18 0.18 0.19 0.19 0.20 0.20 0.21 0.22 0.22 0.23 0.23 0.23 0.22 0.21 0.20 0.19 0.18 0.18 0.17 0.18 0.18 0.19 0.21 0.22 0.23 0.23 0.23 0.22 0.20 0.19 0.17 0.15 0.13 0.12 0.11 0.11 0.11 0.12 0.14 0.16 0.18 0.19 0.21 0.23 0.25 0.26 0.27 0.28 0.28 0.29 0.29 0.30 0.30 0.30 0.30 0.31 0.31 0.30 0.30 0.28 0.27 0.25 0.22 0.20 0.18 0.17 0.15 0.15 0.14 0.14 0.16 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.04 0.03 0.04 0.04 0.05 0.06 0.06 0.06 0.05 0.06 0.06 0.07 0.07 0.07 0.07 0.07 0.08 0.08 0.08 0.08 0.08 0.08 0.09 0.09 0.10 0.10 0.11 0.12 0.13 0.13 0.14 0.15 0.15 0.15 0.14 0.14 0.14 0.15 0.15 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.15 0.15 0.15 0.15 0.15 0.14 0.14 0.14 0.13 0.13 0.12 0.11 0.11 0.10 0.10 0.10
0.10 0.10 0.10 0.10 0.10 0.10 0.13 0.15 0.17 0.20 0.22 0.25 0.28 0.30 0.31 0.32 0.31 0.29 0.27 0.25 0.23 0.20 0.17 0.15 0.13 0.12 0.12 0.12 0.12 0.12 0.13 0.14 0.14 0.15 0.16 0.17 0.18 0.19 0.19 0.20 0.20 0.21 0.22 0.22 0.22 0.22 0.21 0.21 0.20 0.19 0.18 0.18 0.18 0.18 0.19 0.20 0.22 0.24 0.24 0.25 0.25 0.24 0.22 0.21 0.19 0.17 0.15 0.13 0.12 0.12 0.11 0.13 0.14 0.16 0.18 0.19 0.21 0.23 0.25 0.26 0.28 0.29 0.29 0.30 0.31 0.32 0.32 0.32 0.31 0.31 0.31 0.30 0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.17 0.15 0.16 0.16 0.16 0.17 0.19 0.18 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.09 0.08 0.06 0.05 0.04 0.02 0.03 0.04 0.04 0.04 0.05 0.04 0.04 0.03 0.04 0.04 0.04 0.04 0.05 0.05 0.06 0.06 0.07 0.08 0.08 0.09 0.09 0.10 0.10 0.11 0.11 0.12 0.13 0.13 0.14 0.15 0.15 0.15 0.15 0.15 0.15 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.15 0.15 0.14 0.14 0.14 0.13 0.13 0.13 0.13 0.13 0.12 0.12 0.11 0.11 0.10 0.10 0.10
0.12 0.11 0.11 0.11 0.11 0.12 0.15 0.17 0.20 0.22 0.25 0.28 0.30 0.32 0.33 0.34 0.32 0.30 0.28 0.25 0.22 0.19 0.16 0.13 0.11 0.09 0.09 0.09 0.09 0.10 0.10 0.11 0.12 0.13 0.14 0.16 0.16 0.17 0.18 0.19 0.19 0.20 0.20 0.21 0.21 0.21 0.20 0.19 0.19 0.18 0.18 0.18 0.18 0.19 0.20 0.21 0.22 0.24 0.25 0.25 0.26 0.25 0.24 0.22 0.20 0.18 0.16 0.14 0.13 0.12 0.12 0.13 0.14 0.16 0.17 0.19 0.21 0.23 0.25 0.26 0.28 0.29 0.30 0.31 0.32 0.32 0.32 0.32 0.32 0.31 0.31 0.30 0.29 0.28 0.26 0.25 0.23 0.21 0.19 0.18 0.17 0.17 0.18 0.19 0.19 0.20 0.20 0.20 0.20 0.19 0.19 0.17 0.16 0.15 0.13 0.12 0.10 0.08 0.07 0.05 0.04 0.04 0.05 0.05 0.05 0.05 0.05 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.05 0.06 0.06 0.07 0.08 0.09 0.10 0.10 0.10 0.11 0.11 0.12 0.12 0.13 0.14 0.14 0.14 0.15 0.15 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.15 0.15 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.13 0.12 0.12 0.12 0.12
0.13 0.13 0.13 0.13 0.14 0.15 0.17 0.20 0.22 0.25 0.28 0.30 0.33 0.34 0.34 0.34 0.32 0.30 0.28 0.24 0.21 0.18 0.15 0.12 0.10 0.08 0.08 0.08 0.08 0.09 0.10 0.10 0.11 0.12 0.12 0.13 0.14 0.16 0.17 0.17 0.18 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.18 0.18 0.18 0.18 0.18 0.19 0.20 0.21 0.22 0.24 0.25 0.25 0.26 0.25 0.25 0.23 0.21 0.19 0.18 0.16 0.14 0.14 0.13 0.14 0.14 0.15 0.17 0.18 0.20 0.22 0.24 0.26 0.28 0.29 0.30 0.31 0.32 0.32 0.32 0.31 0.31 0.31 0.31 0.30 0.28 0.27 0.27 0.26 0.24 0.22 0.21 0.20 0.19 0.19 0.19 0.20 0.20 0.21 0.21 0.21 0.21 0.21 0.21 0.20 0.19 0.17 0.16 0.15 0.13 0.12 0.10 0.08 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.06 0.06 0.07 0.07 0.08 0.09 0.09 0.10 0.10 0.10 0.10 0.11 0.12 0.12 0.13 0.14 0.14 0.15 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.15 0.14 0.14 0.13
0.17 0.17 0.17 0.17 0.18 0.19 0.21 0.24 0.26 0.28 0.30 0.32 0.34 0.35 0.35 0.34 0.32 0.29 0.26 0.23 0.20 0.17 0.14 0.13 0.12 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.12 0.12 0.12 0.13 0.14 0.15 0.15 0.16 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.18 0.19 0.20 0.21 0.22 0.24 0.24 0.25 0.25 0.24 0.23 0.22 0.20 0.18 0.17 0.15 0.14 0.14 0.13 0.14 0.14 0.15 0.17 0.18 0.20 0.22 0.23 0.25 0.26 0.27 0.28 0.29 0.29 0.30 0.30 0.30 0.29 0.29 0.28 0.27 0.27 0.26 0.25 0.25 0.24 0.23 0.22 0.21 0.21 0.20 0.20 0.20 0.20 0.21 0.21 0.21 0.21 0.20 0.20 0.20 0.19 0.19 0.18 0.17 0.16 0.15 0.13 0.12 0.10 0.09 0.08 0.08 0.08 0.08 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.05 0.06 0.06 0.07 0.07 0.07 0.07 0.07 0.07 0.08 0.09 0.11 0.12 0.13 0.14 0.14 0.15 0.16 0.16 0.16 0.16 0.16 0.17 0.18 0.18 0.18 0.18 0.18 0.19 0.19 0.19 0.19 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.19 0.19 0.18 0.18 0.17
0.22 0.22 0.22 0.22 0.23 0.23 0.25 0.26 0.28 0.29 0.31 0.32 0.33 0.33 0.33 0.32 0.30 0.28 0.25 0.22 0.19 0.17 0.15 0.14 0.14 0.13 0.13 0.13 0.13 0.13 0.12 0.12 0.12 0.11 0.11 0.11 0.12 0.12 0.12 0.13 0.14 0.14 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.23 0.23 0.23 0.23 0.23 0.22 0.21 0.19 0.17 0.16 0.15 0.14 0.14 0.13 0.13 0.14 0.14 0.16 0.17 0.19 0.21 0.22 0.24 0.25 0.26 0.26 0.27 0.27 0.27 0.27 0.26 0.26 0.26 0.26 0.25 0.24 0.24 0.24 0.23 0.23 0.22 0.21 0.21 0.20 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.17 0.15 0.14 0.13 0.12 0.11 0.11 0.10 0.09 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.05 0.05 0.05 0.06 0.05 0.05 0.05 0.05 0.06 0.07 0.09 0.10 0.12 0.13 0.14 0.15 0.16 0.17 0.17 0.18 0.18 0.19 0.19 0.20 0.19 0.19 0.19 0.20 0.20 0.21 0.21 0.22 0.22 0.23 0.23 0.23 0.23 0.24 0.24 0.24 0.24 0.23 0.22 0.22
0.27 0.27 0.26 0.26 0.27 0.27 0.28 0.28 0.29 0.29 0.30 0.31 0.31 0.31 0.30 0.29 0.27 0.25 0.23 0.21 0.19 0.17 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.15 0.14 0.14 0.13 0.13 0.12 0.12 0.12 0.12 0.12 0.13 0.14 0.14 0.14 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.22 0.22 0.21 0.21 0.19 0.18 0.16 0.15 0.14 0.14 0.13 0.12 0.13 0.13 0.14 0.15 0.16 0.18 0.20 0.21 0.22 0.23 0.23 0.24 0.24 0.24 0.24 0.24 0.23 0.23 0.23 0.22 0.22 0.22 0.21 0.21 0.21 0.21 0.21 0.21 0.20 0.19 0.19 0.18 0.18 0.17 0.17 0.16 0.15 0.14 0.14 0.14 0.14 0.14 0.15 0.15 0.16 0.15 0.14 0.14 0.13 0.13 0.12 0.12 0.11 0.10 0.10 0.09 0.08 0.08 0.08 0.07 0.07 0.07 0.07 0.07 0.07 0.06 0.06 0.05 0.05 0.05 0.04 0.04 0.04 0.05 0.06 0.07 0.09 0.11 0.12 0.14 0.15 0.16 0.17 0.17 0.18 0.19 0.20 0.20 0.21 0.21 0.21 0.20 0.20 0.21 0.21 0.22 0.22 0.23 0.24 0.25 0.26 0.26 0.27 0.27 0.28 0.28 0.28 0.28 0.27 0.27
0.32 0.31 0.31 0.30 0.30 0.30 0.30 0.29 0.29 0.29 0.29 0.29 0.29 0.28 0.27 0.25 0.24 0.22 0.21 0.20 0.20 0.19 0.19 0.19 0.19 0.19 0.18 0.18 0.18 0.17 0.17 0.16 0.16 0.15 0.14 0.14 0.14 0.14 0.14 0.14 0.15 0.15 0.15 0.15 0.16 0.17 0.16 0.15 0.15 0.16 0.17 0.17 0.18 0.19 0.20 0.21 0.22 0.22 0.22 0.22 0.21 0.20 0.20 0.18 0.16 0.14 0.14 0.13 0.12 0.12 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.19 0.19 0.20 0.21 0.21 0.21 0.21 0.21 0.20 0.20 0.20 0.20 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.20 0.20 0.21 0.20 0.19 0.19 0.18 0.17 0.16 0.15 0.13 0.12 0.11 0.09 0.08 0.09 0.09 0.10 0.11 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.11 0.11 0.10 0.10 0.09 0.09 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.05 0.06 0.07 0.08 0.10 0.11 0.13 0.15 0.15 0.16 0.17 0.17 0.18 0.19 0.20 0.21 0.21 0.20 0.20 0.20 0.21 0.21 0.21 0.22 0.22 0.23 0.25 0.26 0.27 0.29 0.30 0.31 0.32 0.32 0.32 0.32 0.32 0.32
0.36 0.35 0.34 0.34 0.33 0.32 0.31 0.30 0.29 0.28 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.20 0.20 0.19 0.19 0.18 0.18 0.18 0.18 0.17 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.17 0.17 0.18 0.17 0.17 0.17 0.17 0.18 0.19 0.19 0.20 0.21 0.23 0.23 0.23 0.22 0.22 0.21 0.20 0.19 0.17 0.15 0.13 0.12 0.12 0.11 0.11 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.18 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.17 0.17 0.16 0.16 0.16 0.16 0.16 0.17 0.17 0.18 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.17 0.16 0.15 0.13 0.11 0.08 0.06 0.03 0.03 0.03 0.04 0.05 0.06 0.07 0.08 0.08 0.09 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.09 0.08 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.15 0.16 0.17 0.17 0.18 0.19 0.19 0.19 0.19 0.18 0.18 0.18 0.18 0.19 0.19 0.21 0.22 0.23 0.25 0.26 0.28 0.30 0.32 0.33 0.34 0.35 0.35 0.36 0.36 0.36
0.38 0.38 0.37 0.36 0.35 0.34 0.32 0.31 0.30 0.29 0.28 0.26 0.25 0.24 0.24 0.23 0.23 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.21 0.21 0.21 0.20 0.20 0.20 0.20 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.20 0.20 0.20 0.20 0.20 0.21 0.22 0.22 0.23 0.24 0.24 0.24 0.24 0.24 0.23 0.22 0.21 0.20 0.19 0.17 0.15 0.13 0.12 0.12 0.11 0.11 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.18 0.18 0.18 0.18 0.18 0.16 0.15 0.15 0.14 0.14 0.14 0.14 0.14 0.15 0.15 0.16 0.17 0.18 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.17 0.15 0.13 0.11 0.09 0.07 0.06 0.04 0.03 0.03 0.03 0.03 0.04 0.05 0.06 0.07 0.08 0.08 0.09 0.09 0.10 0.10 0.10 0.10 0.11 0.11 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.13 0.13 0.14 0.14 0.13 0.13 0.14 0.14 0.14 0.15 0.16 0.16 0.16 0.17 0.17 0.18 0.18 0.18 0.18 0.18 0.18 0.17 0.17 0.17 0.17 0.17 0.18 0.18 0.19 0.21 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.35 0.36 0.37 0.37 0.38 0.38
0.39 0.38 0.38 0.37 0.36 0.35 0.33 0.32 0.30 0.29 0.28 0.26 0.25 0.24 0.23 0.23 0.23 0.24 0.24 0.23 0.23 0.23 0.23 0.23 0.23 0.23 0.22 0.22 0.22 0.23 0.24 0.24 0.24 0.24 0.24 0.23 0.23 0.23 0.23 0.23 0.22 0.23 0.23 0.24 0.24 0.24 0.24 0.24 0.24 0.25 0.26 0.26 0.27 0.27 0.27 0.27 0.27 0.26 0.26 0.24 0.23 0.22 0.20 0.19 0.17 0.15 0.14 0.12 0.11 0.11 0.11 0.12 0.13 0.14 0.15 0.16 0.16 0.17 0.18 0.18 0.19 0.18 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.12 0.12 0.12 0.13 0.13 0.14 0.15 0.17 0.18 0.19 0.19 0.19 0.20 0.20 0.20 0.19 0.19 0.18 0.16 0.15 0.14 0.12 0.10 0.09 0.08 0.07 0.06 0.05 0.05 0.06 0.07 0.07 0.08 0.09 0.09 0.10 0.11 0.11 0.11 0.11 0.12 0.12 0.13 0.14 0.14 0.14 0.14 0.14 0.15 0.15 0.16 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.17 0.16 0.16 0.16 0.16 0.17 0.17 0.18 0.19 0.21 0.23 0.25 0.27 0.29 0.31 0.33 0.35 0.36 0.38 0.38 0.39 0.39
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0.28 0.28 0.28 0.27 0.27 0.26 0.26 0.25 0.25 0.25 0.25 0.24 0.24 0.23 0.22 0.22 0.21 0.21 0.20 0.19 0.19 0.18 0.18 0.17 0.17 0.18 0.17 0.16 0.16 0.16 0.16 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.16 0.16 0.17 0.17 0.17 0.17 0.18 0.19 0.20 0.23 0.26 0.29 0.31 0.34 0.34 0.34 0.35 0.36 0.38 0.38 0.38 0.38 0.39 0.40 0.41 0.42 0.43 0.45 0.46 0.47 0.48 0.50 0.52 0.53 0.54 0.55 0.56 0.56 0.57 0.58 0.59 0.60 0.61 0.61 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.61 0.61 0.60 0.60 0.60 0.60 0.59 0.59 0.58 0.57 0.57 0.55 0.54 0.53 0.53 0.52 0.51 0.49 0.48 0.47 0.46 0.44 0.43 0.42 0.42 0.40 0.39 0.37 0.35 0.34 0.33 0.32 0.32 0.30 0.29 0.28 0.27 0.25 0.25 0.24 0.23 0.23 0.22 0.22 0.22 0.21 0.21 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.21 0.21 0.22 0.22 0.22 0.22 0.22 0.22 0.23 0.23 0.23 0.23 0.23 0.24 0.25 0.25 0.26 0.26 0.26 0.27 0.27 0.27 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28
0.29 0.29 0.28 0.28 0.28 0.28 0.28 0.28 0.27 0.27 0.26 0.26 0.26 0.26 0.25 0.24 0.24 0.24 0.24 0.23 0.23 0.22 0.21 0.21 0.22 0.23 0.22 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.22 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.39 0.38 0.36 0.36 0.38 0.39 0.40 0.40 0.41 0.42 0.42 0.44 0.45 0.46 0.46 0.47 0.48 0.50 0.51 0.51 0.52 0.53 0.55 0.56 0.56 0.57 0.57 0.58 0.59 0.59 0.60 0.61 0.61 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.61 0.61 0.60 0.59 0.59 0.58 0.57 0.57 0.56 0.54 0.53 0.53 0.52 0.51 0.49 0.48 0.48 0.47 0.46 0.44 0.43 0.42 0.41 0.39 0.38 0.37 0.35 0.34 0.33 0.33 0.32 0.31 0.29 0.29 0.28 0.27 0.27 0.26 0.25 0.25 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.25 0.25 0.26 0.26 0.26 0.26 0.26 0.26 0.27 0.27 0.28 0.28 0.28 0.28 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29
0.31 0.30 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.29 0.28 0.28 0.28 0.29 0.30 0.30 0.30 0.31 0.32 0.33 0.33 0.34 0.35 0.37 0.39 0.40 0.41 0.42 0.43 0.44 0.43 0.43 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.49 0.50 0.51 0.51 0.52 0.53 0.54 0.54 0.55 0.56 0.56 0.57 0.57 0.58 0.58 0.58 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.59 0.58 0.57 0.57 0.56 0.56 0.56 0.56 0.55 0.54 0.53 0.53 0.52 0.51 0.49 0.48 0.48 0.47 0.46 0.46 0.44 0.43 0.42 0.41 0.40 0.39 0.38 0.37 0.36 0.35 0.35 0.34 0.33 0.32 0.32 0.31 0.31 0.30 0.30 0.30 0.29 0.29 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.29 0.29 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.31 0.31 0.32 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31
0.34 0.34 0.33 0.33 0.33 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.33 0.33 0.33 0.33 0.33 0.34 0.34 0.34 0.34 0.33 0.33 0.33 0.33 0.34 0.34 0.34 0.34 0.34 0.34 0.35 0.35 0.36 0.36 0.37 0.37 0.38 0.38 0.39 0.39 0.40 0.40 0.41 0.42 0.42 0.43 0.43 0.44 0.44 0.45 0.45 0.45 0.45 0.45 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.45 0.45 0.46 0.46 0.46 0.47 0.47 0.48 0.48 0.49 0.49 0.49 0.50 0.51 0.51 0.51 0.52 0.52 0.53 0.53 0.53 0.53 0.53 0.54 0.54 0.54 0.55 0.55 0.55 0.55 0.55 0.54 0.54 0.53 0.53 0.52 0.52 0.52 0.52 0.52 0.51 0.50 0.50 0.49 0.49 0.48 0.47 0.47 0.46 0.45 0.45 0.44 0.43 0.42 0.42 0.42 0.41 0.40 0.39 0.39 0.38 0.38 0.37 0.37 0.36 0.36 0.35 0.35 0.35 0.34 0.34 0.34 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.33 0.33 0.34 0.33 0.33 0.33 0.33 0.33 0.33 0.34 0.33 0.33 0.33 0.33 0.33 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34
0.37 0.37 0.37 0.37 0.36 0.36 0.36 0.37 0.37 0.36 0.36 0.36 0.37 0.37 0.37 0.37 0.37 0.37 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.40 0.41 0.41 0.42 0.42 0.42 0.43 0.43 0.43 0.43 0.43 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.46 0.46 0.46 0.46 0.46 0.47 0.47 0.47 0.47 0.47 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.47 0.47 0.46 0.46 0.46 0.46 0.46 0.45 0.45 0.44 0.44 0.44 0.43 0.43 0.43 0.42 0.42 0.42 0.41 0.41 0.40 0.40 0.40 0.40 0.39 0.39 0.39 0.39 0.39 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.37 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.37 0.37 0.36 0.36 0.36 0.36 0.36 0.37 0.37 0.37 0.36 0.36 0.36 0.36 0.36 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.37
nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
];
% test data 1 for tec
function tec=testdata2
tec=[
0 nan nan
30 nan nan
60 nan nan
90 nan nan
120 nan nan
150 nan nan
180 nan nan
210 nan nan
240 nan nan
270 nan nan
300 nan nan
330 nan nan
360 nan nan
390 nan nan
420 nan nan
450 nan nan
480 nan nan
510 nan nan
540 nan nan
570 nan nan
600 nan nan
630 nan nan
660 nan nan
690 nan nan
720 nan nan
750 nan nan
780 nan nan
810 nan nan
840 nan nan
870 nan nan
900 nan nan
930 nan nan
960 nan nan
990 nan nan
1020 nan nan
1050 nan nan
1080 nan nan
1110 nan nan
1140 nan nan
1170 nan nan
1200 nan nan
1230 nan nan
1260 nan nan
1290 nan nan
1320 nan nan
1350 nan nan
1380 nan nan
1410 nan nan
1440 nan nan
1470 nan nan
1500 nan nan
1530 nan nan
1560 nan nan
1590 nan nan
1620 nan nan
1650 nan nan
1680 nan nan
1710 nan nan
1740 nan nan
1770 nan nan
1800 nan nan
1830 nan nan
1860 nan nan
1890 nan nan
1920 nan nan
1950 nan nan
1980 nan nan
2010 nan nan
2040 nan nan
2070 nan nan
2100 nan nan
2130 nan nan
2160 nan nan
2190 nan nan
2220 nan nan
2250 nan nan
2280 nan nan
2310 nan nan
2340 nan nan
2370 nan nan
2400 nan nan
2430 nan nan
2460 nan nan
2490 nan nan
2520 nan nan
2550 nan nan
2580 nan nan
2610 nan nan
2640 nan nan
2670 nan nan
2700 nan nan
2730 nan nan
2760 nan nan
2790 nan nan
2820 nan nan
2850 nan nan
2880 nan nan
2910 nan nan
2940 nan nan
2970 nan nan
3000 nan nan
3030 nan nan
3060 nan nan
3090 nan nan
3120 nan nan
3150 nan nan
3180 nan nan
3210 nan nan
3240 nan nan
3270 nan nan
3300 nan nan
3330 nan nan
3360 nan nan
3390 nan nan
3420 nan nan
3450 nan nan
3480 nan nan
3510 nan nan
3540 nan nan
3570 nan nan
3600 nan nan
3630 nan nan
3660 nan nan
3690 nan nan
3720 nan nan
3750 nan nan
3780 nan nan
3810 nan nan
3840 nan nan
3870 nan nan
3900 nan nan
3930 nan nan
3960 nan nan
3990 nan nan
4020 nan nan
4050 nan nan
4080 nan nan
4110 nan nan
4140 nan nan
4170 nan nan
4200 nan nan
4230 nan nan
4260 nan nan
4290 nan nan
4320 nan nan
4350 nan nan
4380 nan nan
4410 nan nan
4440 nan nan
4470 nan nan
4500 nan nan
4530 nan nan
4560 nan nan
4590 nan nan
4620 nan nan
4650 nan nan
4680 nan nan
4710 nan nan
4740 nan nan
4770 nan nan
4800 nan nan
4830 nan nan
4860 nan nan
4890 nan nan
4920 nan nan
4950 nan nan
4980 nan nan
5010 nan nan
5040 nan nan
5070 nan nan
5100 nan nan
5130 nan nan
5160 nan nan
5190 nan nan
5220 nan nan
5250 nan nan
5280 nan nan
5310 nan nan
5340 nan nan
5370 nan nan
5400 nan nan
5430 nan nan
5460 nan nan
5490 nan nan
5520 nan nan
5550 nan nan
5580 nan nan
5610 nan nan
5640 nan nan
5670 nan nan
5700 nan nan
5730 nan nan
5760 nan nan
5790 nan nan
5820 nan nan
5850 nan nan
5880 nan nan
5910 nan nan
5940 nan nan
5970 nan nan
6000 nan nan
6030 nan nan
6060 nan nan
6090 nan nan
6120 nan nan
6150 nan nan
6180 nan nan
6210 nan nan
6240 nan nan
6270 nan nan
6300 nan nan
6330 nan nan
6360 nan nan
6390 nan nan
6420 nan nan
6450 nan nan
6480 nan nan
6510 nan nan
6540 nan nan
6570 nan nan
6600 nan nan
6630 nan nan
6660 nan nan
6690 nan nan
6720 nan nan
6750 nan nan
6780 nan nan
6810 nan nan
6840 nan nan
6870 nan nan
6900 nan nan
6930 nan nan
6960 nan nan
6990 nan nan
7020 nan nan
7050 nan nan
7080 nan nan
7110 nan nan
7140 nan nan
7170 nan nan
7200 nan nan
7230 nan nan
7260 nan nan
7290 nan nan
7320 nan nan
7350 nan nan
7380 nan nan
7410 nan nan
7440 nan nan
7470 nan nan
7500 nan nan
7530 nan nan
7560 nan nan
7590 nan nan
7620 nan nan
7650 nan nan
7680 nan nan
7710 nan nan
7740 nan nan
7770 nan nan
7800 nan nan
7830 nan nan
7860 nan nan
7890 nan nan
7920 nan nan
7950 nan nan
7980 nan nan
8010 nan nan
8040 nan nan
8070 nan nan
8100 nan nan
8130 nan nan
8160 nan nan
8190 nan nan
8220 nan nan
8250 nan nan
8280 nan nan
8310 nan nan
8340 nan nan
8370 nan nan
8400 nan nan
8430 nan nan
8460 nan nan
8490 nan nan
8520 nan nan
8550 nan nan
8580 nan nan
8610 nan nan
8640 nan nan
8670 nan nan
8700 nan nan
8730 nan nan
8760 nan nan
8790 nan nan
8820 nan nan
8850 nan nan
8880 nan nan
8910 nan nan
8940 nan nan
8970 nan nan
9000 nan nan
9030 nan nan
9060 nan nan
9090 nan nan
9120 nan nan
9150 nan nan
9180 nan nan
9210 nan nan
9240 nan nan
9270 nan nan
9300 nan nan
9330 nan nan
9360 nan nan
9390 nan nan
9420 nan nan
9450 nan nan
9480 nan nan
9510 nan nan
9540 nan nan
9570 nan nan
9600 nan nan
9630 nan nan
9660 nan nan
9690 nan nan
9720 nan nan
9750 nan nan
9780 nan nan
9810 nan nan
9840 nan nan
9870 nan nan
9900 nan nan
9930 nan nan
9960 nan nan
9990 nan nan
10020 nan nan
10050 nan nan
10080 nan nan
10110 nan nan
10140 nan nan
10170 nan nan
10200 nan nan
10230 nan nan
10260 nan nan
10290 nan nan
10320 nan nan
10350 nan nan
10380 nan nan
10410 nan nan
10440 nan nan
10470 nan nan
10500 nan nan
10530 nan nan
10560 nan nan
10590 nan nan
10620 nan nan
10650 nan nan
10680 nan nan
10710 nan nan
10740 nan nan
10770 nan nan
10800 nan nan
10830 nan nan
10860 nan nan
10890 nan nan
10920 nan nan
10950 nan nan
10980 nan nan
11010 nan nan
11040 nan nan
11070 nan nan
11100 nan nan
11130 nan nan
11160 nan nan
11190 nan nan
11220 nan nan
11250 nan nan
11280 nan nan
11310 nan nan
11340 nan nan
11370 nan nan
11400 nan nan
11430 nan nan
11460 nan nan
11490 nan nan
11520 nan nan
11550 nan nan
11580 nan nan
11610 nan nan
11640 nan nan
11670 nan nan
11700 nan nan
11730 nan nan
11760 nan nan
11790 nan nan
11820 nan nan
11850 nan nan
11880 nan nan
11910 nan nan
11940 nan nan
11970 nan nan
12000 nan nan
12030 nan nan
12060 nan nan
12090 nan nan
12120 nan nan
12150 nan nan
12180 nan nan
12210 nan nan
12240 nan nan
12270 nan nan
12300 nan nan
12330 nan nan
12360 nan nan
12390 nan nan
12420 nan nan
12450 nan nan
12480 nan nan
12510 nan nan
12540 nan nan
12570 nan nan
12600 nan nan
12630 nan nan
12660 nan nan
12690 nan nan
12720 nan nan
12750 nan nan
12780 nan nan
12810 nan nan
12840 nan nan
12870 nan nan
12900 nan nan
12930 nan nan
12960 nan nan
12990 nan nan
13020 nan nan
13050 nan nan
13080 nan nan
13110 nan nan
13140 nan nan
13170 nan nan
13200 nan nan
13230 nan nan
13260 nan nan
13290 nan nan
13320 nan nan
13350 nan nan
13380 nan nan
13410 nan nan
13440 nan nan
13470 nan nan
13500 nan nan
13530 nan nan
13560 nan nan
13590 nan nan
13620 nan nan
13650 nan nan
13680 nan nan
13710 nan nan
13740 nan nan
13770 nan nan
13800 nan nan
13830 nan nan
13860 nan nan
13890 nan nan
13920 nan nan
13950 nan nan
13980 nan nan
14010 nan nan
14040 nan nan
14070 nan nan
14100 nan nan
14130 nan nan
14160 nan nan
14190 nan nan
14220 nan nan
14250 nan nan
14280 nan nan
14310 nan nan
14340 nan nan
14370 nan nan
14400 nan nan
14430 nan nan
14460 nan nan
14490 nan nan
14520 nan nan
14550 nan nan
14580 nan nan
14610 nan nan
14640 nan nan
14670 nan nan
14700 nan nan
14730 nan nan
14760 nan nan
14790 nan nan
14820 nan nan
14850 nan nan
14880 nan nan
14910 nan nan
14940 nan nan
14970 nan nan
15000 nan nan
15030 nan nan
15060 nan nan
15090 nan nan
15120 nan nan
15150 nan nan
15180 nan nan
15210 nan nan
15240 nan nan
15270 nan nan
15300 nan nan
15330 nan nan
15360 nan nan
15390 nan nan
15420 nan nan
15450 nan nan
15480 nan nan
15510 nan nan
15540 nan nan
15570 nan nan
15600 nan nan
15630 nan nan
15660 nan nan
15690 nan nan
15720 nan nan
15750 nan nan
15780 nan nan
15810 nan nan
15840 nan nan
15870 nan nan
15900 nan nan
15930 nan nan
15960 nan nan
15990 nan nan
16020 nan nan
16050 nan nan
16080 nan nan
16110 nan nan
16140 nan nan
16170 nan nan
16200 nan nan
16230 nan nan
16260 nan nan
16290 nan nan
16320 nan nan
16350 nan nan
16380 nan nan
16410 nan nan
16440 nan nan
16470 nan nan
16500 nan nan
16530 nan nan
16560 nan nan
16590 nan nan
16620 nan nan
16650 nan nan
16680 nan nan
16710 nan nan
16740 nan nan
16770 nan nan
16800 nan nan
16830 nan nan
16860 nan nan
16890 nan nan
16920 nan nan
16950 nan nan
16980 nan nan
17010 nan nan
17040 nan nan
17070 nan nan
17100 nan nan
17130 nan nan
17160 nan nan
17190 nan nan
17220 nan nan
17250 nan nan
17280 nan nan
17310 nan nan
17340 nan nan
17370 nan nan
17400 nan nan
17430 nan nan
17460 nan nan
17490 nan nan
17520 nan nan
17550 nan nan
17580 nan nan
17610 nan nan
17640 nan nan
17670 nan nan
17700 nan nan
17730 nan nan
17760 nan nan
17790 nan nan
17820 nan nan
17850 nan nan
17880 nan nan
17910 nan nan
17940 nan nan
17970 nan nan
18000 nan nan
18030 nan nan
18060 nan nan
18090 nan nan
18120 nan nan
18150 nan nan
18180 nan nan
18210 nan nan
18240 nan nan
18270 nan nan
18300 nan nan
18330 nan nan
18360 nan nan
18390 nan nan
18420 nan nan
18450 nan nan
18480 nan nan
18510 nan nan
18540 nan nan
18570 nan nan
18600 nan nan
18630 nan nan
18660 nan nan
18690 nan nan
18720 nan nan
18750 nan nan
18780 nan nan
18810 nan nan
18840 nan nan
18870 nan nan
18900 nan nan
18930 nan nan
18960 nan nan
18990 nan nan
19020 nan nan
19050 nan nan
19080 nan nan
19110 nan nan
19140 nan nan
19170 nan nan
19200 nan nan
19230 nan nan
19260 nan nan
19290 nan nan
19320 nan nan
19350 nan nan
19380 nan nan
19410 nan nan
19440 nan nan
19470 nan nan
19500 nan nan
19530 nan nan
19560 nan nan
19590 nan nan
19620 nan nan
19650 nan nan
19680 nan nan
19710 nan nan
19740 nan nan
19770 nan nan
19800 nan nan
19830 nan nan
19860 nan nan
19890 nan nan
19920 nan nan
19950 nan nan
19980 nan nan
20010 nan nan
20040 nan nan
20070 nan nan
20100 nan nan
20130 nan nan
20160 nan nan
20190 nan nan
20220 nan nan
20250 nan nan
20280 nan nan
20310 nan nan
20340 nan nan
20370 nan nan
20400 nan nan
20430 nan nan
20460 nan nan
20490 nan nan
20520 nan nan
20550 nan nan
20580 nan nan
20610 nan nan
20640 nan nan
20670 nan nan
20700 nan nan
20730 nan nan
20760 nan nan
20790 nan nan
20820 nan nan
20850 nan nan
20880 nan nan
20910 nan nan
20940 nan nan
20970 nan nan
21000 nan nan
21030 nan nan
21060 nan nan
21090 nan nan
21120 nan nan
21150 nan nan
21180 nan nan
21210 nan nan
21240 nan nan
21270 nan nan
21300 nan nan
21330 nan nan
21360 nan nan
21390 nan nan
21420 nan nan
21450 nan nan
21480 nan nan
21510 nan nan
21540 nan nan
21570 nan nan
21600 nan nan
21630 nan nan
21660 nan nan
21690 nan nan
21720 nan nan
21750 nan nan
21780 nan nan
21810 nan nan
21840 nan nan
21870 nan nan
21900 nan nan
21930 nan nan
21960 nan nan
21990 nan nan
22020 nan nan
22050 nan nan
22080 nan nan
22110 nan nan
22140 nan nan
22170 nan nan
22200 nan nan
22230 nan nan
22260 nan nan
22290 nan nan
22320 nan nan
22350 nan nan
22380 nan nan
22410 nan nan
22440 nan nan
22470 nan nan
22500 nan nan
22530 nan nan
22560 nan nan
22590 nan nan
22620 nan nan
22650 nan nan
22680 nan nan
22710 nan nan
22740 nan nan
22770 nan nan
22800 nan nan
22830 nan nan
22860 nan nan
22890 nan nan
22920 nan nan
22950 nan nan
22980 nan nan
23010 nan nan
23040 nan nan
23070 nan nan
23100 nan nan
23130 nan nan
23160 nan nan
23190 nan nan
23220 nan nan
23250 nan nan
23280 nan nan
23310 nan nan
23340 nan nan
23370 nan nan
23400 nan nan
23430 nan nan
23460 nan nan
23490 nan nan
23520 nan nan
23550 nan nan
23580 nan nan
23610 nan nan
23640 nan nan
23670 nan nan
23700 nan nan
23730 nan nan
23760 nan nan
23790 nan nan
23820 nan nan
23850 nan nan
23880 nan nan
23910 nan nan
23940 nan nan
23970 nan nan
24000 nan nan
24030 nan nan
24060 nan nan
24090 nan nan
24120 nan nan
24150 nan nan
24180 nan nan
24210 nan nan
24240 nan nan
24270 nan nan
24300 nan nan
24330 nan nan
24360 nan nan
24390 nan nan
24420 nan nan
24450 nan nan
24480 nan nan
24510 nan nan
24540 nan nan
24570 nan nan
24600 nan nan
24630 nan nan
24660 nan nan
24690 nan nan
24720 nan nan
24750 nan nan
24780 nan nan
24810 nan nan
24840 nan nan
24870 nan nan
24900 nan nan
24930 nan nan
24960 nan nan
24990 nan nan
25020 nan nan
25050 nan nan
25080 nan nan
25110 nan nan
25140 nan nan
25170 nan nan
25200 nan nan
25230 nan nan
25260 nan nan
25290 nan nan
25320 nan nan
25350 nan nan
25380 nan nan
25410 nan nan
25440 nan nan
25470 nan nan
25500 nan nan
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258990 nan nan
259020 nan nan
259050 nan nan
259080 nan nan
259110 nan nan
259140 nan nan
259170 nan nan
259200 nan nan
];
|
github
|
JuXinCheng/rtklib_2.4.2-master
|
testionppp.m
|
.m
|
rtklib_2.4.2-master/test/utest/testionppp.m
| 1,136 |
utf_8
|
7023ec339e81fd7fa267d573c3d2d588
|
function testionppp
%
% test RTCA/DO229C bug (A.4.4.10.1 A-22,23)
%
az=0:0.1:360;
figure, axes, hold on, box on, grid on;
pos=[80,0];
for i=1:length(az), posp(i,:)=ionppp(pos,[az(i),0]); end
plot(posp(:,2),posp(:,1),'.');
pos=[-75,170];
for i=1:length(az), posp(i,:)=ionppp(pos,[az(i),0]); end
plot(posp(:,2),posp(:,1),'.');
xlim([-180,180]);
ylim([-90,90]);
% pierce point -----------------------------------------------------------------
function posp=ionppp(pos,azel)
re=6378; hion=350;
pos=pos*pi/180;
azel=azel*pi/180;
rp=re/(re+hion)*cos(azel(2));
ap=pi/2-azel(2)-asin(rp);
posp(1)=asin(sin(pos(1))*cos(ap)+cos(pos(1))*sin(ap)*cos(azel(1)));
%if (pos(1)> 70.0*pi/180& tan(ap)*cos(azel(1))>tan(pi/2-pos(1)))|...
% (pos(1)>-70.0*pi/180&-tan(ap)*cos(azel(1))>tan(pi/2-pos(1))) % DO229C
if (pos(1)> 70.0*pi/180& tan(ap)*cos(azel(1))>tan(pi/2-pos(1)))|...
(pos(1)<-70.0*pi/180&-tan(ap)*cos(azel(1))>tan(pi/2+pos(1))) % corrected
posp(2)=pos(2)+pi-asin(sin(ap)*sin(azel(1))/cos(posp(1)));
else
posp(2)=pos(2)+asin(sin(ap)*sin(azel(1))/cos(posp(1)));
end
posp=posp*180/pi;
if posp(2)>180, posp(2)=posp(2)-360; end
|
github
|
JuXinCheng/rtklib_2.4.2-master
|
plotigp.m
|
.m
|
rtklib_2.4.2-master/test/utest/plotigp.m
| 1,278 |
utf_8
|
bf7bb7d90d3221bbc76f008e0c03363f
|
function plotigp
figure
mesh=readmesh;
gmt('mmap','proj','eq','cent',[135,35],'scale',10,'pos',[0,0,1,1]);
gmt('mcoast');
gmt('mgrid','gint',2,'lint',10,'color',[.5 .5 .5]);
for i=1:size(mesh,1)
gmt('mplot',mesh(i,1),mesh(i,2),'r','marker','.','markersize',10);
end
plotarea([36,138],15);
% plot ipp area ----------------------------------------------------------------
function plotarea(pos,elmask)
posp=[];
for az=0:3:360
posp=[posp;igppos(pos*pi/180,[az,elmask]*pi/180)*180/pi];
end
gmt('mplot',pos(2),pos(1),'b','marker','.','markersize',10);
gmt('mplot',posp(:,2),posp(:,1),'b');
% read mesh data ---------------------------------------------------------------
function mesh=readmesh
mesh=[];
fp=fopen('../../nicttec/vtec/2011/001.txt','r');
while 1
s=fgets(fp); if ~ischar(s), break; end
v=sscanf(s,' Mesh %d: (%f, %f)');
if length(v)<2, continue; end
mesh=[mesh;v(2:3)'];
end
fclose(fp);
% igp position -----------------------------------------------------------------
function posp=igppos(pos,azel)
re=6380; hion=350;
rp=re/(re+hion)*cos(azel(2));
ap=pi/2-azel(2)-asin(rp);
sinap=sin(ap);
tanap=tan(ap);
cosaz=cos(azel(1));
posp(1)=asin(sin(pos(1))*cos(ap)+cos(pos(1))*sinap*cosaz);
posp(2)=pos(2)+asin(sinap*sin(azel(1))/cos(posp(1)));
|
github
|
moskante/OpenTouch_Matlab-master
|
pause_arduino.m
|
.m
|
OpenTouch_Matlab-master/CODE/pause_arduino.m
| 955 |
utf_8
|
bdb85291db86dfe46bafa65788160ecc
|
%Function pause_arduino make a pause (blocking!) for a time interval equal
%to interval and read the voltage specified by aruino_board/pin and write
%it to an output file.
%interval: time interval in seconds
%arduino_board: an arduino object
%pin: the pin to read
%fileout: serial object to write on.
%
%Example
%myresults = fopen('test.txt', 'wt');
%uno = arduino('COM4', 'uno')
%test = pause_arduino(2, uno, 'A3', myresults)
%fclose(myresults)
%clear all
function [out] = pause_arduino(interval, arduino_board, pin, serial_id)
myformat = '%4f\t %4f\n';
%Start the stopwatch timer
tstart = tic;
%enter the main loop
while(1)
%read serial port
voltage = readVoltage(arduino_board, pin);
%save current time
ti = toc(tstart);
%output to serial object serial_id
fprintf(serial_id, myformat, [voltage ti]);
if(ti >= interval)
break;
end
end
out = ti;
|
github
|
douyouzhe/Machine-Learning-for-Signal-Processing-master
|
pdco.m
|
.m
|
Machine-Learning-for-Signal-Processing-master/LDA_language_classification/Code/pdco.m
| 54,596 |
utf_8
|
9b21477124e43a4155c4b72508dccb8f
|
function [x,y,z,inform,PDitns,CGitns,time] = ...
pdco( Fname,Aname,b,bl,bu,d1,d2,options,x0,y0,z0,xsize,zsize )
%-----------------------------------------------------------------------
% pdco.m: Primal-Dual Barrier Method for Convex Objectives (23 Sep 2003)
%-----------------------------------------------------------------------
% [x,y,z,inform,PDitns,CGitns,time] = ...
% pdco(Fname,Aname,b,bl,bu,d1,d2,options,x0,y0,z0,xsize,zsize);
%
% solves optimization problems of the form
%
% minimize phi(x) + 1/2 norm(D1*x)^2 + 1/2 norm(r)^2
% x,r
% subject to A*x + D2*r = b, bl <= x <= bu, r unconstrained,
%
% where
% phi(x) is a smooth convex function defined by function Fname;
% A is an m x n matrix defined by matrix or function Aname;
% b is a given m-vector;
% D1, D2 are positive-definite diagonal matrices defined from d1, d2.
% In particular, d2 indicates the accuracy required for
% satisfying each row of Ax = b.
%
% D1 and D2 (via d1 and d2) provide primal and dual regularization
% respectively. They ensure that the primal and dual solutions
% (x,r) and (y,z) are unique and bounded.
%
% A scalar d1 is equivalent to d1 = ones(n,1), D1 = diag(d1).
% A scalar d2 is equivalent to d2 = ones(m,1), D2 = diag(d2).
% Typically, d1 = d2 = 1e-4.
% These values perturb phi(x) only slightly (by about 1e-8) and request
% that A*x = b be satisfied quite accurately (to about 1e-8).
% Set d1 = 1e-4, d2 = 1 for least-squares problems with bound constraints.
% The problem is then equivalent to
%
% minimize phi(x) + 1/2 norm(d1*x)^2 + 1/2 norm(A*x - b)^2
% subject to bl <= x <= bu.
%
% More generally, d1 and d2 may be n and m vectors containing any positive
% values (preferably not too small, and typically no larger than 1).
% Bigger elements of d1 and d2 improve the stability of the solver.
%
% At an optimal solution, if x(j) is on its lower or upper bound,
% the corresponding z(j) is positive or negative respectively.
% If x(j) is between its bounds, z(j) = 0.
% If bl(j) = bu(j), x(j) is fixed at that value and z(j) may have
% either sign.
%
% Also, r and y satisfy r = D2 y, so that Ax + D2^2 y = b.
% Thus if d2(i) = 1e-4, the i-th row of Ax = b will be satisfied to
% approximately 1e-8. This determines how large d2(i) can safely be.
%
%
% EXTERNAL FUNCTIONS:
% options = pdcoSet; provided with pdco.m
% [obj,grad,hess] = Fname( x ); provided by user
% y = Aname( name,mode,m,n,x ); provided by user if pdMat
% is a string, not a matrix
%
% INPUT ARGUMENTS:
% Fname may be an explicit n x 1 column vector c,
% or a string containing the name of a function Fname.m
%%%!!! Revised 12/16/04 !!!
% (Fname cannot be a function handle)
% such that [obj,grad,hess] = Fname(x) defines
% obj = phi(x) : a scalar,
% grad = gradient of phi(x) : an n-vector,
% hess = diag(Hessian of phi): an n-vector.
% Examples:
% If phi(x) is the linear function c'*x, Fname could be
% be the vector c, or the name or handle of a function
% that returns
% [obj,grad,hess] = [c'*x, c, zeros(n,1)].
% If phi(x) is the entropy function E(x) = sum x(j) log x(j),
% Fname should return
% [obj,grad,hess] = [E(x), log(x)+1, 1./x].
% Aname may be an explicit m x n matrix A (preferably sparse!),
% or a string containing the name of a function Aname.m
%%%!!! Revised 12/16/04 !!!
% (Aname cannot be a function handle)
% such that y = aname( name,mode,m,n,x )
% returns y = A*x (mode=1) or y = A'*x (mode=2).
% The input parameter "name" will be the string 'Aname'
% or whatever the name of the actual function is.
% b is an m-vector.
% bl is an n-vector of lower bounds. Non-existent bounds
% may be represented by bl(j) = -Inf or bl(j) <= -1e+20.
% bu is an n-vector of upper bounds. Non-existent bounds
% may be represented by bu(j) = Inf or bu(j) >= 1e+20.
% d1, d2 may be positive scalars or positive vectors (see above).
% options is a structure that may be set and altered by pdcoSet
% (type help pdcoSet).
% x0, y0, z0 provide an initial solution.
% xsize, zsize are estimates of the biggest x and z at the solution.
% They are used to scale (x,y,z). Good estimates
% should improve the performance of the barrier method.
%
%
% OUTPUT ARGUMENTS:
% x is the primal solution.
% y is the dual solution associated with Ax + D2 r = b.
% z is the dual solution associated with bl <= x <= bu.
% inform = 0 if a solution is found;
% = 1 if too many iterations were required;
% = 2 if the linesearch failed too often.
% = 3 if the step lengths became too small.
% PDitns is the number of Primal-Dual Barrier iterations required.
% CGitns is the number of Conjugate-Gradient iterations required
% if an iterative solver is used (LSQR).
% time is the cpu time used.
%----------------------------------------------------------------------
% PRIVATE FUNCTIONS:
% pdxxxbounds
% pdxxxdistrib
% pdxxxlsqr
% pdxxxlsqrmat
% pdxxxmat
% pdxxxmerit
% pdxxxresid1
% pdxxxresid2
% pdxxxstep
%
% GLOBAL VARIABLES:
% global pdDDD1 pdDDD2 pdDDD3
%
%
% NOTES:
% The matrix A should be reasonably well scaled: norm(A,inf) =~ 1.
% The vector b and objective phi(x) may be of any size, but ensure that
% xsize and zsize are reasonably close to norm(x,inf) and norm(z,inf)
% at the solution.
%
% The files defining Fname and Aname
% must not be called Fname.m or Aname.m!!
%
%
% AUTHOR:
% Michael Saunders, Systems Optimization Laboratory (SOL),
% Stanford University, Stanford, California, USA.
% [email protected]
%
% CONTRIBUTORS:
% Byunggyoo Kim, Samsung, Seoul, Korea.
% [email protected]
%
% DEVELOPMENT:
% 20 Jun 1997: Original version of pdsco.m derived from pdlp0.m.
% 29 Sep 2002: Original version of pdco.m derived from pdsco.m.
% Introduced D1, D2 in place of gamma*I, delta*I
% and allowed for general bounds bl <= x <= bu.
% 06 Oct 2002: Allowed for fixed variabes: bl(j) = bu(j) for any j.
% 15 Oct 2002: Eliminated some work vectors (since m, n might be LARGE).
% Modularized residuals, linesearch
% 16 Oct 2002: pdxxx..., pdDDD... names rationalized.
% pdAAA eliminated (global copy of A).
% Aname is now used directly as an explicit A or a function.
% NOTE: If Aname is a function, it now has an extra parameter.
% 23 Oct 2002: Fname and Aname can now be function handles.
% 01 Nov 2002: Bug fixed in feval in pdxxxmat.
% 19 Apr 2003: Bug fixed in pdxxxbounds.
% 07 Aug 2003: Let d1, d2 be scalars if input that way.
% 10 Aug 2003: z isn't needed except at the end for output.
% 10 Aug 2003: mu0 is now an absolute value -- the initial mu.
% 13 Aug 2003: Access only z1(low) and z2(upp) everywhere.
% stepxL, stepxU introduced to keep x within bounds.
% (With poor starting points, dx may take x outside,
% where phi(x) may not be defined.
% Entropy once gave complex values for the gradient!)
% 16 Sep 2003: Fname can now be a vector c, implying a linear obj c'*x.
% 19 Sep 2003: Large system K4 dv = rhs implemented.
% 23 Sep 2003: Options LSproblem and LSmethod replaced by Method.
% 18 Nov 2003: stepxL, stepxU gave trouble on lptest (see 13 Aug 2003).
% Disabled them for now. Nonlinear problems need good x0.
% 19 Nov 2003: Bugs with x(fix) and z(fix).
% In particular, x(fix) = bl(fix) throughout, so Objective
% in iteration log is correct for LPs with explicit c vector.
%-----------------------------------------------------------------------
global pdDDD1 pdDDD2 pdDDD3
fprintf('\n --------------------------------------------------------')
fprintf('\n pdco.m Version of 19 Nov 2003')
fprintf('\n Primal-dual barrier method to minimize a convex function')
fprintf('\n subject to linear constraints Ax + r = b, bl <= x <= bu')
fprintf('\n --------------------------------------------------------\n')
m = length(b);
n = length(bl);
%---------------------------------------------------------------------
% Decode Fname.
%---------------------------------------------------------------------
%%%!!! Revised 12/16/04 !!!
% Fname cannot be a function handle
operator = ischar(Fname);
explicitF = ~operator;
if explicitF
fprintf('\n')
disp('The objective is linear')
else
fname = Fname;
fprintf('\n')
disp(['The objective function is named ' fname])
end
%---------------------------------------------------------------------
% Decode Aname.
%---------------------------------------------------------------------
%%%!!! Revised 12/16/04 !!!
% Aname cannot be a function handle
operator = ischar(Aname) || isa(Aname, 'function_handle');
explicitA = ~operator;
if explicitA % assume Aname is an explicit matrix A.
nnzA = nnz(Aname);
if issparse(Aname)
fprintf('The matrix A is an explicit sparse matrix')
else
fprintf('The matrix A is an explicit dense matrix' )
end
fprintf('\n\nm = %8g n = %8g nnz(A) =%9g', m,n,nnzA)
else
if ischar(Aname)
disp(['The matrix A is an operator defined by ' Aname])
end
fprintf('\nm = %8g n = %8g', m,n)
end
normb = norm(b ,inf); normx0 = norm(x0,inf);
normy0 = norm(y0,inf); normz0 = norm(z0,inf);
fprintf('\nmax |b | = %8g max |x0| = %8.1e', normb , normx0)
fprintf( ' xsize = %8.1e', xsize)
fprintf('\nmax |y0| = %8g max |z0| = %8.1e', normy0, normz0)
fprintf( ' zsize = %8.1e', zsize)
%---------------------------------------------------------------------
% Initialize.
%---------------------------------------------------------------------
true = 1;
false = 0;
zn = zeros(n,1);
nb = n + m;
nkkt = nb;
CGitns = 0;
inform = 0;
% 07 Aug 2003: No need for next lines.
%if length(d1)==1, d1 = d1*ones(n,1); end % Allow scalar d1, d2
%if length(d2)==1, d2 = d2*ones(m,1); end % to mean d1*e, d2*e
%---------------------------------------------------------------------
% Grab input options.
%---------------------------------------------------------------------
maxitn = options.MaxIter;
featol = options.FeaTol;
opttol = options.OptTol;
steptol = options.StepTol;
stepSame = options.StepSame; % 1 means stepx==stepz
x0min = options.x0min;
z0min = options.z0min;
mu0 = options.mu0;
Method = options.Method;
itnlim = options.LSQRMaxIter * min(m,n);
atol1 = options.LSQRatol1; % Initial atol
atol2 = options.LSQRatol2; % Smallest atol, unless atol1 is smaller
conlim = options.LSQRconlim;
wait = options.wait;
%---------------------------------------------------------------------
% Set other parameters.
%---------------------------------------------------------------------
kminor = 0; % 1 stops after each iteration
eta = 1e-4; % Linesearch tolerance for "sufficient descent"
maxf = 10; % Linesearch backtrack limit (function evaluations)
maxfail = 1; % Linesearch failure limit (consecutive iterations)
bigcenter = 1e+3; % mu is reduced if center < bigcenter
thresh = 1e-8; % For sparse LU with Method=41
% Parameters for LSQR.
atolmin = eps; % Smallest atol if linesearch back-tracks
btol = 0; % Should be small (zero is ok)
show = false; % Controls LSQR iteration log
gamma = max(d1);
delta = max(d2);
fprintf('\n\nx0min = %8g featol = %8.1e', x0min, featol)
fprintf( ' d1max = %8.1e', gamma)
fprintf( '\nz0min = %8g opttol = %8.1e', z0min, opttol)
fprintf( ' d2max = %8.1e', delta)
fprintf( '\nmu0 = %8.1e steptol = %8g', mu0 , steptol)
fprintf( ' bigcenter= %8g' , bigcenter)
fprintf('\n\nLSQR:')
fprintf('\natol1 = %8.1e atol2 = %8.1e', atol1 , atol2 )
fprintf( ' btol = %8.1e', btol )
fprintf('\nconlim = %8.1e itnlim = %8g' , conlim, itnlim)
fprintf( ' show = %8g' , show )
% Method = 3; %%% Hardwire LSQR
% Method = 41; %%% Hardwire K4 and sparse LU
fprintf('\n\nMethod = %8g (1=chol 2=QR 3=LSQR 41=K4)', Method)
if wait
fprintf('\n\nReview parameters... then type "return"\n')
keyboard
end
if eta < 0
fprintf('\n\nLinesearch disabled by eta < 0')
end
%---------------------------------------------------------------------
% All parameters have now been set.
% Check for valid Method.
%---------------------------------------------------------------------
time = cputime;
if operator
if Method==3
% relax
else
fprintf('\n\nWhen A is an operator, we have to use Method = 3')
Method = 3;
end
end
if Method== 1, solver = ' Chol'; head3 = ' Chol';
elseif Method== 2, solver = ' QR'; head3 = ' QR';
elseif Method== 3, solver = ' LSQR'; head3 = ' atol LSQR Inexact';
elseif Method==41, solver = ' LU'; head3 = ' L U res';
else error('Method must be 1, 2, 3, or 41')
end
%---------------------------------------------------------------------
% Categorize bounds and allow for fixed variables by modifying b.
%---------------------------------------------------------------------
[low,upp,fix] = pdxxxbounds( bl,bu );
nfix = length(fix);
if nfix > 0
x1 = zn; x1(fix) = bl(fix);
r1 = pdxxxmat( Aname, 1, m, n, x1 );
b = b - r1;
% At some stage, might want to look at normfix = norm(r1,inf);
end
%---------------------------------------------------------------------
% Scale the input data.
% The scaled variables are
% xbar = x/beta,
% ybar = y/zeta,
% zbar = z/zeta.
% Define
% theta = beta*zeta;
% The scaled function is
% phibar = ( 1 /theta) fbar(beta*xbar),
% gradient = (beta /theta) grad,
% Hessian = (beta2/theta) hess.
%---------------------------------------------------------------------
beta = xsize; if beta==0, beta = 1; end % beta scales b, x.
zeta = zsize; if zeta==0, zeta = 1; end % zeta scales y, z.
theta = beta*zeta; % theta scales obj.
% (theta could be anything, but theta = beta*zeta makes
% scaled grad = grad/zeta = 1 approximately if zeta is chosen right.)
bl(fix)= bl(fix)/beta;
bu(fix)= bu(fix)/beta;
bl(low)= bl(low)/beta;
bu(upp)= bu(upp)/beta;
d1 = d1*( beta/sqrt(theta) );
d2 = d2*( sqrt(theta)/beta );
beta2 = beta^2;
b = b /beta; y0 = y0/zeta;
x0 = x0/beta; z0 = z0/zeta;
%---------------------------------------------------------------------
% Initialize vectors that are not fully used if bounds are missing.
%---------------------------------------------------------------------
rL = zn; rU = zn;
cL = zn; cU = zn;
x1 = zn; x2 = zn;
z1 = zn; z2 = zn;
dx1 = zn; dx2 = zn;
dz1 = zn; dz2 = zn;
clear zn
%---------------------------------------------------------------------
% Initialize x, y, z1, z2, objective, etc.
% 10 Aug 2003: z isn't needed here -- just at end for output.
%---------------------------------------------------------------------
x = x0;
y = y0;
x(fix) = bl(fix);
x(low) = max( x(low) , bl(low));
x(upp) = min( x(upp) , bu(upp));
x1(low)= max( x(low) - bl(low), x0min );
x2(upp)= max(bu(upp) - x(upp), x0min );
z1(low)= max( z0(low) , z0min );
z2(upp)= max(-z0(upp) , z0min );
clear x0 y0 z0
%%%%%%%%%% Assume hess is diagonal for now. %%%%%%%%%%%%%%%%%%
if explicitF
obj = (Fname'*x)*beta; grad = Fname; hess = zeros(n,1);
else
[obj,grad,hess] = feval( Fname, (x*beta) );
end
obj = obj /theta; % Scaled obj.
grad = grad*(beta /theta) + (d1.^2).*x;% grad includes x regularization.
H = hess*(beta2/theta) + (d1.^2); % H includes x regularization.
%---------------------------------------------------------------------
% Compute primal and dual residuals:
% r1 = b - A*x - d2.^2*y
% r2 = grad - A'*y + (z2-z1)
% rL = bl - x + x1
% rU = -bu + x + x2
%---------------------------------------------------------------------
[r1,r2,rL,rU,Pinf,Dinf] = ...
pdxxxresid1( Aname,fix,low,upp, ...
b,bl,bu,d1,d2,grad,rL,rU,x,x1,x2,y,z1,z2 );
%---------------------------------------------------------------------
% Initialize mu and complementarity residuals:
% cL = mu*e - X1*z1.
% cU = mu*e - X2*z2.
%
% 25 Jan 2001: Now that b and obj are scaled (and hence x,y,z),
% we should be able to use mufirst = mu0 (absolute value).
% 0.1 worked poorly on StarTest1 with x0min = z0min = 0.1.
% 29 Jan 2001: We might as well use mu0 = x0min * z0min;
% so that most variables are centered after a warm start.
% 29 Sep 2002: Use mufirst = mu0*(x0min * z0min),
% regarding mu0 as a scaling of the initial center.
% 07 Aug 2003: mulast is controlled by opttol.
% mufirst should not be smaller.
% 10 Aug 2003: Revert to mufirst = mu0 (absolute value).
%---------------------------------------------------------------------
% mufirst = mu0*(x0min * z0min);
mufirst = mu0;
mulast = 0.1 * opttol;
mufirst = max( mufirst, mulast );
mu = mufirst;
[cL,cU,center,Cinf,Cinf0] = ...
pdxxxresid2( mu,low,upp,cL,cU,x1,x2,z1,z2 );
fmerit = pdxxxmerit( low,upp,r1,r2,rL,rU,cL,cU );
% Initialize other things.
PDitns = 0;
converged = 0;
atol = atol1;
atol2 = max( atol2, atolmin );
atolmin = atol2;
pdDDD2 = d2; % Global vector for diagonal matrix D2
% Iteration log.
stepx = 0;
stepz = 0;
nf = 0;
itncg = 0;
nfail = 0;
head1 = '\n\nItn mu stepx stepz Pinf Dinf';
head2 = ' Cinf Objective nf center';
fprintf([ head1 head2 head3 ])
regterm = norm(d1.*x)^2 + norm(d2.*y)^2;
objreg = obj + 0.5*regterm;
objtrue = objreg * theta;
fprintf('\n%3g ', PDitns )
fprintf('%6.1f%6.1f' , log10(Pinf ), log10(Dinf))
fprintf('%6.1f%15.7e', log10(Cinf0), objtrue )
fprintf(' %8.1f' , center )
if Method==41
fprintf(' thresh=%7.1e', thresh)
end
if kminor
fprintf('\n\nStart of first minor itn...\n')
keyboard
end
%---------------------------------------------------------------------
% Main loop.
%---------------------------------------------------------------------
while ~converged
PDitns = PDitns + 1;
% 31 Jan 2001: Set atol according to progress, a la Inexact Newton.
% 07 Feb 2001: 0.1 not small enough for Satellite problem. Try 0.01.
% 25 Apr 2001: 0.01 seems wasteful for Star problem.
% Now that starting conditions are better, go back to 0.1.
r3norm = max([Pinf Dinf Cinf]);
atol = min([atol r3norm*0.1]);
atol = max([atol atolmin ]);
if Method<=3
%-----------------------------------------------------------------
% Solve (*) for dy.
%-----------------------------------------------------------------
% Define a damped Newton iteration for solving f = 0,
% keeping x1, x2, z1, z2 > 0. We eliminate dx1, dx2, dz1, dz2
% to obtain the system
%
% [-H2 A' ] [dx] = [w ], H2 = H + D1^2 + X1inv Z1 + X2inv Z2,
% [ A D2^2] [dy] = [r1] w = r2 - X1inv(cL + Z1 rL)
% + X2inv(cU + Z2 rU),
%
% which is equivalent to the least-squares problem
%
% min || [ D A']dy - [ D w ] ||, D = H2^{-1/2}. (*)
% || [ D2 ] [D2inv r1] ||
%-----------------------------------------------------------------
H(low) = H(low) + z1(low)./x1(low);
H(upp) = H(upp) + z2(upp)./x2(upp);
w = r2;
w(low) = w(low) - (cL(low) + z1(low).*rL(low))./x1(low);
w(upp) = w(upp) + (cU(upp) + z2(upp).*rU(upp))./x2(upp);
H = 1./H; % H is now Hinv (NOTE!)
H(fix) = 0;
D = sqrt(H);
pdDDD1 = D;
rw = [explicitA Method m n 0 0 0]; % Passed to LSQR.
if Method==1
% --------------------------------------------------------------
% Use chol to get dy
% --------------------------------------------------------------
AD = Aname*sparse( 1:n, 1:n, D, n, n );
ADDA = AD*AD' + sparse( 1:m, 1:m, (d2.^2), m, m );
if PDitns==1, P = symamd(ADDA); end % Do ordering only once.
[R,indef] = chol(ADDA(P,P));
if indef
fprintf('\n\n chol says AD^2A'' is not pos def')
fprintf('\n Use bigger d2, or set options.Method = 2 or 3')
break
end
% dy = ADDA \ rhs;
rhs = Aname*(H.*w) + r1;
dy = R \ (R'\rhs(P));
dy(P) = dy;
elseif Method==2
% --------------------------------------------------------------
% Use QR to get dy
% --------------------------------------------------------------
DAt = sparse( 1:n, 1:n, D, n, n )*(Aname');
if PDitns==1, P = colamd(DAt); end % Do ordering only once.
if length(d2)==1
D2 = d2*speye(m);
else
D2 = spdiags(d2,0,m,m);
end
DAt = [ DAt; D2 ];
rhs = [ D.*w; r1./d2 ];
% dy = DAt \ rhs;
[rhs,R] = qr(DAt(:,P),rhs,0);
dy = R \ rhs;
dy(P) = dy;
else
% --------------------------------------------------------------
% Method=3. Use LSQR (iterative solve) to get dy
% --------------------------------------------------------------
rhs = [ D.*w; r1./d2 ];
damp = 0;
if explicitA % A is a sparse matrix.
precon = true;
if precon % Construct diagonal preconditioner for LSQR
AD = Aname*sparse( 1:n, 1:n, D, n, n );
AD = AD.^2;
wD = sum(AD,2); % Sum of sqrs of each row of AD. %(Sparse)
wD = sqrt( full(wD) + (d2.^2) ); %(Dense)
pdDDD3 = 1./wD;
clear AD wD
end
else % A is an operator
precon = false;
end
rw(7) = precon;
info.atolmin = atolmin;
info.r3norm = fmerit; % Must be the 2-norm here.
[ dy, istop, itncg, outfo ] = ...
pdxxxlsqr( nb,m,'pdxxxlsqrmat',Aname,rw,rhs,damp, ...
atol,btol,conlim,itnlim,show,info );
if precon, dy = pdDDD3 .* dy; end
if istop==3 | istop==7 % conlim or itnlim
fprintf('\n LSQR stopped early: istop = %3d', istop)
end
atolold = outfo.atolold;
atol = outfo.atolnew;
r3ratio = outfo.r3ratio;
CGitns = CGitns + itncg;
end % computation of dy
% dy is now known. Get dx, dx1, dx2, dz1, dz2.
grad = pdxxxmat( Aname,2,m,n,dy ); % grad = A'dy
grad(fix) = 0; % Is this needed? % grad is a work vector
dx = H .* (grad - w);
dx1(low) = - rL(low) + dx(low);
dx2(upp) = - rU(upp) - dx(upp);
dz1(low) = (cL(low) - z1(low).*dx1(low)) ./ x1(low);
dz2(upp) = (cU(upp) - z2(upp).*dx2(upp)) ./ x2(upp);
elseif Method==41
%-----------------------------------------------------------------
% Solve the symmetric-structure 4 x 4 system K4 dv = t:
%
% ( X1 Z1 ) [dz1] = [tL], tL = cL + Z1 rL
% ( X2 -Z2 ) [dz2] [tU] tU = cU + Z2 rU
% ( I -I -H1 A' ) [dx ] [r2]
% ( A D2^2 ) [dy ] [r1]
%-----------------------------------------------------------------
X1 = ones(n,1); X1(low) = x1(low);
X2 = ones(n,1); X2(upp) = x2(upp);
if length(d2)==1
D22 = d2^2*speye(m);
else
D22 = spdiags(d2.^2,0,m,m);
end
Onn = sparse(n,n);
Omn = sparse(m,n);
if PDitns==1 % First time through: Choose LU ordering from
% lower-triangular part of dummy K4
I1 = sparse( low, low, ones(length(low),1), n, n );
I2 = sparse( upp, upp, ones(length(upp),1), n, n );
K4 =[speye(n) Onn Onn Omn'
Onn speye(n) Onn Omn'
I1 I2 speye(n) Omn'
Omn Omn Aname speye(m)];
p = symamd(K4);
% disp(' '); keyboard
end
K4 = [spdiags(X1,0,n,n) Onn spdiags(z1,0,n,n) Omn'
Onn spdiags(X2,0,n,n) -spdiags(z2,0,n,n) Omn'
I1 -I2 -spdiags(H,0,n,n) Aname'
Omn Omn Aname D22];
tL = zeros(n,1); tL(low) = cL(low) + z1(low).*rL(low);
tU = zeros(n,1); tU(upp) = cU(upp) + z2(upp).*rU(upp);
rhs = [tL; tU; r2; r1];
% dv = K4 \ rhs; % BIG SYSTEM!
[L,U,P] = lu( K4(p,p), thresh ); % P K4 = L U
dv = U \ (L \ (P*rhs(p)));
dv(p) = dv;
resK4 = norm((K4*dv - rhs),inf) / norm(rhs,inf);
dz1 = dv( 1: n);
dz2 = dv( n+1:2*n);
dx = dv(2*n+1:3*n);
dy = dv(3*n+1:3*n+m);
dx1(low) = - rL(low) + dx(low);
dx2(upp) = - rU(upp) - dx(upp);
end
%-------------------------------------------------------------------
% Find the maximum step.
% 13 Aug 2003: We need stepxL, stepxU also to keep x feasible
% so that nonlinear functions are defined.
% 18 Nov 2003: But this gives stepx = 0 for lptest. (??)
%--------------------------------------------------------------------
stepx1 = pdxxxstep( x1(low), dx1(low) );
stepx2 = pdxxxstep( x2(upp), dx2(upp) );
stepz1 = pdxxxstep( z1(low), dz1(low) );
stepz2 = pdxxxstep( z2(upp), dz2(upp) );
% stepxL = pdxxxstep( x(low), dx(low) );
% stepxU = pdxxxstep( x(upp), dx(upp) );
% stepx = min( [stepx1, stepx2, stepxL, stepxU] );
stepx = min( [stepx1, stepx2] );
stepz = min( [stepz1, stepz2] );
stepx = min( [steptol*stepx, 1] );
stepz = min( [steptol*stepz, 1] );
if stepSame % For NLPs, force same step
stepx = min( stepx, stepz ); % (true Newton method)
stepz = stepx;
end
%-------------------------------------------------------------------
% Backtracking linesearch.
%-------------------------------------------------------------------
fail = true;
nf = 0;
while nf < maxf
nf = nf + 1;
x = x + stepx * dx;
y = y + stepz * dy;
x1(low) = x1(low) + stepx * dx1(low);
x2(upp) = x2(upp) + stepx * dx2(upp);
z1(low) = z1(low) + stepz * dz1(low);
z2(upp) = z2(upp) + stepz * dz2(upp);
if explicitF
obj = (Fname'*x)*beta; grad = Fname; hess = zeros(n,1);
else
[obj,grad,hess] = feval( Fname, (x*beta) );
end
obj = obj /theta;
grad = grad*(beta /theta) + (d1.^2).*x;
H = hess*(beta2/theta) + (d1.^2);
[r1,r2,rL,rU,Pinf,Dinf] = ...
pdxxxresid1( Aname,fix,low,upp, ...
b,bl,bu,d1,d2,grad,rL,rU,x,x1,x2,y,z1,z2 );
[cL,cU,center,Cinf,Cinf0] = ...
pdxxxresid2( mu,low,upp,cL,cU,x1,x2,z1,z2 );
fmeritnew = pdxxxmerit( low,upp,r1,r2,rL,rU,cL,cU );
step = min( stepx, stepz );
if fmeritnew <= (1 - eta*step)*fmerit
fail = false;
break;
end
% Merit function didn't decrease.
% Restore variables to previous values.
% (This introduces a little error, but save lots of space.)
x = x - stepx * dx;
y = y - stepz * dy;
x1(low) = x1(low) - stepx * dx1(low);
x2(upp) = x2(upp) - stepx * dx2(upp);
z1(low) = z1(low) - stepz * dz1(low);
z2(upp) = z2(upp) - stepz * dz2(upp);
% Back-track.
% If it's the first time,
% make stepx and stepz the same.
if nf==1 & stepx~=stepz
stepx = step;
elseif nf < maxf
stepx = stepx/2;
end;
stepz = stepx;
end
if fail
fprintf('\n Linesearch failed (nf too big)');
nfail = nfail + 1;
else
nfail = 0;
end
%-------------------------------------------------------------------
% Set convergence measures.
%--------------------------------------------------------------------
regterm = norm(d1.*x)^2 + norm(d2.*y)^2;
objreg = obj + 0.5*regterm;
objtrue = objreg * theta;
primalfeas = Pinf <= featol;
dualfeas = Dinf <= featol;
complementary = Cinf0 <= opttol;
enough = PDitns>= 4; % Prevent premature termination.
converged = primalfeas & dualfeas & complementary & enough;
%-------------------------------------------------------------------
% Iteration log.
%-------------------------------------------------------------------
str1 = sprintf('\n%3g%5.1f' , PDitns , log10(mu) );
str2 = sprintf('%6.3f%6.3f' , stepx , stepz );
if stepx < 0.001 | stepz < 0.001
str2 = sprintf('%6.1e%6.1e' , stepx , stepz );
end
str3 = sprintf('%6.1f%6.1f' , log10(Pinf) , log10(Dinf));
str4 = sprintf('%6.1f%15.7e', log10(Cinf0), objtrue );
str5 = sprintf('%3g%8.1f' , nf , center );
if center > 99999
str5 = sprintf('%3g%8.1e' , nf , center );
end
fprintf([str1 str2 str3 str4 str5])
if Method== 1
if PDitns==1, fprintf(' %8g', nnz(R)); end
elseif Method== 2
if PDitns==1, fprintf(' %8g', nnz(R)); end
elseif Method== 3
fprintf(' %5.1f%7g%7.3f', log10(atolold), itncg, r3ratio)
elseif Method==41,
resK4 = max( resK4, 1e-99 );
fprintf(' %8g%8g%6.1f', nnz(L),nnz(U),log10(resK4))
end
%-------------------------------------------------------------------
% Test for termination.
%-------------------------------------------------------------------
if kminor
fprintf( '\nStart of next minor itn...\n')
keyboard
end
if converged
fprintf('\nConverged')
elseif PDitns >= maxitn
fprintf('\nToo many iterations')
inform = 1;
break
elseif nfail >= maxfail
fprintf('\nToo many linesearch failures')
inform = 2;
break
elseif step <= 1e-10
fprintf('\nStep lengths too small')
inform = 3;
break
else
% Reduce mu, and reset certain residuals.
stepmu = min( stepx , stepz );
stepmu = min( stepmu, steptol );
muold = mu;
mu = mu - stepmu * mu;
if center >= bigcenter, mu = muold; end
% mutrad = mu0*(sum(Xz)/n); % 24 May 1998: Traditional value, but
% mu = min(mu,mutrad ); % it seemed to decrease mu too much.
mu = max(mu,mulast); % 13 Jun 1998: No need for smaller mu.
[cL,cU,center,Cinf,Cinf0] = ...
pdxxxresid2( mu,low,upp,cL,cU,x1,x2,z1,z2 );
fmerit = pdxxxmerit( low,upp,r1,r2,rL,rU,cL,cU );
% Reduce atol for LSQR (and SYMMLQ).
% NOW DONE AT TOP OF LOOP.
atolold = atol;
% if atol > atol2
% atolfac = (mu/mufirst)^0.25;
% atol = max( atol*atolfac, atol2 );
% end
% atol = min( atol, mu ); % 22 Jan 2001: a la Inexact Newton.
% atol = min( atol, 0.5*mu ); % 30 Jan 2001: A bit tighter
% If the linesearch took more than one function (nf > 1),
% we assume the search direction needed more accuracy
% (though this may be true only for LPs).
% 12 Jun 1998: Ask for more accuracy if nf > 2.
% 24 Nov 2000: Also if the steps are small.
% 30 Jan 2001: Small steps might be ok with warm start.
% 06 Feb 2001: Not necessarily. Reinstated tests in next line.
if nf > 2 | step <= 0.01
atol = atolold*0.1;
end
end
end
%---------------------------------------------------------------------
% End of main loop.
%---------------------------------------------------------------------
% Print statistics.
x(fix) = 0; % Exclude x(fix) temporarily from |x|.
z = zeros(n,1); % Exclude z(fix) also.
z(low) = z1(low);
z(upp) = z(upp) - z2(upp);
fprintf('\n\nmax |x| =%10.3f', norm(x,inf))
fprintf(' max |y| =%10.3f', norm(y,inf))
fprintf(' max |z| =%10.3f', norm(z,inf)) % excludes z(fix)
fprintf(' scaled')
bl(fix)= bl(fix)*beta; % Unscale bl, bu, x, y, z.
bu(fix)= bu(fix)*beta;
bl(low)= bl(low)*beta;
bu(upp)= bu(upp)*beta;
x = x*beta; y = y*zeta; z = z*zeta;
fprintf( '\nmax |x| =%10.3f', norm(x,inf))
fprintf(' max |y| =%10.3f', norm(y,inf))
fprintf(' max |z| =%10.3f', norm(z,inf)) % excludes z(fix)
fprintf(' unscaled')
x(fix) = bl(fix); % Reinstate x(fix).
% Reconstruct b.
b = b *beta;
if nfix > 0
x1 = zeros(n,1); x1(fix) = bl(fix);
r1 = pdxxxmat( Aname, 1, m, n, x1 );
b = b + r1;
fprintf('\nmax |x| and max |z| exclude fixed variables')
end
% Evaluate function at final point.
% Reconstruct z. This finally defines z(fix).
if explicitF
obj = (Fname'*x); grad = Fname; hess = zeros(n,1);
else
[obj,grad,hess] = feval( Fname, x );
end
z = grad - pdxxxmat( Aname,2,m,n,y ); % z = grad - A'y
time = cputime - time;
str1 = sprintf('\nPDitns =%10g', PDitns );
str2 = sprintf( 'itns =%10g', CGitns );
fprintf( [str1 ' ' solver str2] )
fprintf(' time =%10.1f', time);
pdxxxdistrib( abs(x),abs(z) ); % Private function
if wait
keyboard
end
%-----------------------------------------------------------------------
% End function pdco.m
%-----------------------------------------------------------------------
function [low,upp,fix] = pdxxxbounds( bl,bu )
% Categorize various types of bounds.
% pos overlaps with low.
% neg overlaps with upp.
% two overlaps with low and upp.
% fix and free are disjoint from all other sets.
bigL = -9.9e+19;
bigU = 9.9e+19;
pos = find( bl==0 & bu>=bigU );
neg = find( bl<=bigL & bu==0 );
low = find( bl> bigL & bl< bu );
upp = find( bu< bigU & bl< bu );
two = find( bl> bigL & bu< bigU & bl< bu );
fix = find( bl==bu );
free = find( bl<=bigL & bu>=bigU );
fprintf('\n\nBounds:\n [0,inf] [-inf,0]')
fprintf(' Finite bl Finite bu Two bnds Fixed Free')
fprintf('\n %8g %9g %10g %10g %9g %7g %7g', ...
length(pos), length(neg), length(low), ...
length(upp), length(two), length(fix), length(free))
%-----------------------------------------------------------------------
% End private function pdxxxbounds
%-----------------------------------------------------------------------
function pdxxxdistrib( x,z )
% pdxxxdistrib(x) or pdxxxdistrib(x,z) prints the
% distribution of 1 or 2 vectors.
%
% 18 Dec 2000. First version with 2 vectors.
two = nargin > 1;
fprintf('\n\nDistribution of vector x')
if two, fprintf(' z'); end
x1 = 10^(floor(log10(max(x)+eps)) + 1);
z1 = 10^(floor(log10(max(z)+eps)) + 1);
x1 = max(x1,z1);
kmax = 10;
for k = 1:kmax
x2 = x1; x1 = x1/10;
if k==kmax, x1 = 0; end
nx = length(find(x>=x1 & x<x2));
fprintf('\n[%7.3g,%7.3g )%10g', x1, x2, nx);
if two
nz = length(find(z>=x1 & z<x2));
fprintf('%10g', nz);
end
end
disp(' ')
%-----------------------------------------------------------------------
% End private function pdxxxdistrib
%-----------------------------------------------------------------------
function [ x, istop, itn, outfo ] = ...
pdxxxlsqr( m, n, aprodname, iw, rw, b, damp, ...
atol, btol, conlim, itnlim, show, info )
% Special version of LSQR for use with pdco.m.
% It continues with a reduced atol if a pdco-specific test isn't
% satisfied with the input atol.
%
% LSQR solves Ax = b or min ||b - Ax||_2 if damp = 0,
% or min || (b) - ( A )x || otherwise.
% || (0) (damp I) ||2
% A is an m by n matrix defined by y = aprod( mode,m,n,x,iw,rw ),
% where the parameter 'aprodname' refers to a function 'aprod' that
% performs the matrix-vector operations.
% If mode = 1, aprod must return y = Ax without altering x.
% If mode = 2, aprod must return y = A'x without altering x.
% WARNING: The file containing the function 'aprod'
% must not be called aprodname.m !!!!
%-----------------------------------------------------------------------
% LSQR uses an iterative (conjugate-gradient-like) method.
% For further information, see
% 1. C. C. Paige and M. A. Saunders (1982a).
% LSQR: An algorithm for sparse linear equations and sparse least squares,
% ACM TOMS 8(1), 43-71.
% 2. C. C. Paige and M. A. Saunders (1982b).
% Algorithm 583. LSQR: Sparse linear equations and least squares problems,
% ACM TOMS 8(2), 195-209.
% 3. M. A. Saunders (1995). Solution of sparse rectangular systems using
% LSQR and CRAIG, BIT 35, 588-604.
%
% Input parameters:
% iw, rw are not used by lsqr, but are passed to aprod.
% atol, btol are stopping tolerances. If both are 1.0e-9 (say),
% the final residual norm should be accurate to about 9 digits.
% (The final x will usually have fewer correct digits,
% depending on cond(A) and the size of damp.)
% conlim is also a stopping tolerance. lsqr terminates if an estimate
% of cond(A) exceeds conlim. For compatible systems Ax = b,
% conlim could be as large as 1.0e+12 (say). For least-squares
% problems, conlim should be less than 1.0e+8.
% Maximum precision can be obtained by setting
% atol = btol = conlim = zero, but the number of iterations
% may then be excessive.
% itnlim is an explicit limit on iterations (for safety).
% show = 1 gives an iteration log,
% show = 0 suppresses output.
% info is a structure special to pdco.m, used to test if
% was small enough, and continuing if necessary with smaller atol.
%
%
% Output parameters:
% x is the final solution.
% istop gives the reason for termination.
% istop = 1 means x is an approximate solution to Ax = b.
% = 2 means x approximately solves the least-squares problem.
% rnorm = norm(r) if damp = 0, where r = b - Ax,
% = sqrt( norm(r)**2 + damp**2 * norm(x)**2 ) otherwise.
% xnorm = norm(x).
% var estimates diag( inv(A'A) ). Omitted in this special version.
% outfo is a structure special to pdco.m, returning information
% about whether atol had to be reduced.
%
% Other potential output parameters:
% anorm, acond, arnorm, xnorm
%
% 1990: Derived from Fortran 77 version of LSQR.
% 22 May 1992: bbnorm was used incorrectly. Replaced by anorm.
% 26 Oct 1992: More input and output parameters added.
% 01 Sep 1994: Matrix-vector routine is now a parameter 'aprodname'.
% Print log reformatted.
% 14 Jun 1997: show added to allow printing or not.
% 30 Jun 1997: var added as an optional output parameter.
% It returns an estimate of diag( inv(A'A) ).
% 12 Feb 2001: atol can now be reduced and iterations continued if necessary.
% info, outfo are new problem-dependent parameters for such purposes.
% In this version they are specialized for pdco.m.
%-----------------------------------------------------------------------
% Initialize.
msg=['The exact solution is x = 0 '
'Ax - b is small enough, given atol, btol '
'The least-squares solution is good enough, given atol '
'The estimate of cond(Abar) has exceeded conlim '
'Ax - b is small enough for this machine '
'The least-squares solution is good enough for this machine'
'Cond(Abar) seems to be too large for this machine '
'The iteration limit has been reached '];
% wantvar= nargout >= 6;
% if wantvar, var = zeros(n,1); end
itn = 0; istop = 0; nstop = 0;
ctol = 0; if conlim > 0, ctol = 1/conlim; end;
anorm = 0; acond = 0;
dampsq = damp^2; ddnorm = 0; res2 = 0;
xnorm = 0; xxnorm = 0; z = 0;
cs2 = -1; sn2 = 0;
% Set up the first vectors u and v for the bidiagonalization.
% These satisfy beta*u = b, alfa*v = A'u.
u = b(1:m); x = zeros(n,1);
alfa = 0; beta = norm( u );
if beta > 0
u = (1/beta) * u; v = feval( aprodname, 2, m, n, u, iw, rw );
alfa = norm( v );
end
if alfa > 0
v = (1/alfa) * v; w = v;
end
arnorm = alfa * beta; if arnorm==0, disp(msg(1,:)); return, end
rhobar = alfa; phibar = beta; bnorm = beta;
rnorm = beta;
head1 = ' Itn x(1) Function';
head2 = ' Compatible LS Norm A Cond A';
if show
disp(' ')
disp([head1 head2])
test1 = 1; test2 = alfa / beta;
str1 = sprintf( '%6g %12.5e %10.3e', itn, x(1), rnorm );
str2 = sprintf( ' %8.1e %8.1e', test1, test2 );
disp([str1 str2])
end
%----------------------------------------------------------------
% Main iteration loop.
%----------------------------------------------------------------
while itn < itnlim
itn = itn + 1;
% Perform the next step of the bidiagonalization to obtain the
% next beta, u, alfa, v. These satisfy the relations
% beta*u = A*v - alfa*u,
% alfa*v = A'*u - beta*v.
u = feval( aprodname, 1, m, n, v, iw, rw ) - alfa*u;
beta = norm( u );
if beta > 0
u = (1/beta) * u;
anorm = norm([anorm alfa beta damp]);
v = feval( aprodname, 2, m, n, u, iw, rw ) - beta*v;
alfa = norm( v );
if alfa > 0, v = (1/alfa) * v; end
end
% Use a plane rotation to eliminate the damping parameter.
% This alters the diagonal (rhobar) of the lower-bidiagonal matrix.
rhobar1 = norm([rhobar damp]);
cs1 = rhobar / rhobar1;
sn1 = damp / rhobar1;
psi = sn1 * phibar;
phibar = cs1 * phibar;
% Use a plane rotation to eliminate the subdiagonal element (beta)
% of the lower-bidiagonal matrix, giving an upper-bidiagonal matrix.
rho = norm([rhobar1 beta]);
cs = rhobar1/ rho;
sn = beta / rho;
theta = sn * alfa;
rhobar = - cs * alfa;
phi = cs * phibar;
phibar = sn * phibar;
tau = sn * phi;
% Update x and w.
t1 = phi /rho;
t2 = - theta/rho;
dk = (1/rho)*w;
x = x + t1*w;
w = v + t2*w;
ddnorm = ddnorm + norm(dk)^2;
% if wantvar, var = var + dk.*dk; end
% Use a plane rotation on the right to eliminate the
% super-diagonal element (theta) of the upper-bidiagonal matrix.
% Then use the result to estimate norm(x).
delta = sn2 * rho;
gambar = - cs2 * rho;
rhs = phi - delta * z;
zbar = rhs / gambar;
xnorm = sqrt(xxnorm + zbar^2);
gamma = norm([gambar theta]);
cs2 = gambar / gamma;
sn2 = theta / gamma;
z = rhs / gamma;
xxnorm = xxnorm + z^2;
% Test for convergence.
% First, estimate the condition of the matrix Abar,
% and the norms of rbar and Abar'rbar.
acond = anorm * sqrt( ddnorm );
res1 = phibar^2;
res2 = res2 + psi^2;
rnorm = sqrt( res1 + res2 );
arnorm = alfa * abs( tau );
% Now use these norms to estimate certain other quantities,
% some of which will be small near a solution.
test1 = rnorm / bnorm;
test2 = arnorm/( anorm * rnorm );
test3 = 1 / acond;
t1 = test1 / (1 + anorm * xnorm / bnorm);
rtol = btol + atol * anorm * xnorm / bnorm;
% The following tests guard against extremely small values of
% atol, btol or ctol. (The user may have set any or all of
% the parameters atol, btol, conlim to 0.)
% The effect is equivalent to the normal tests using
% atol = eps, btol = eps, conlim = 1/eps.
if itn >= itnlim, istop = 7; end
if 1 + test3 <= 1, istop = 6; end
if 1 + test2 <= 1, istop = 5; end
if 1 + t1 <= 1, istop = 4; end
% Allow for tolerances set by the user.
if test3 <= ctol, istop = 3; end
if test2 <= atol, istop = 2; end
if test1 <= rtol, istop = 1; end
%-------------------------------------------------------------------
% SPECIAL TEST THAT DEPENDS ON pdco.m.
% Aname in pdco is iw in lsqr.
% dy is x
% Other stuff is in info.
% We allow for diagonal preconditioning in pdDDD3.
%-------------------------------------------------------------------
if istop > 0
r3new = arnorm;
r3ratio = r3new / info.r3norm;
atolold = atol;
atolnew = atol;
if atol > info.atolmin
if r3ratio <= 0.1 % dy seems good
% Relax
elseif r3ratio <= 0.5 % Accept dy but make next one more accurate.
atolnew = atolnew * 0.1;
else % Recompute dy more accurately
fprintf('\n ')
fprintf(' ')
fprintf(' %5.1f%7g%7.3f', log10(atolold), itn, r3ratio)
atol = atol * 0.1;
atolnew = atol;
istop = 0;
end
end
outfo.atolold = atolold;
outfo.atolnew = atolnew;
outfo.r3ratio = r3ratio;
end
%-------------------------------------------------------------------
% See if it is time to print something.
%-------------------------------------------------------------------
prnt = 0;
if n <= 40 , prnt = 1; end
if itn <= 10 , prnt = 1; end
if itn >= itnlim-10, prnt = 1; end
if rem(itn,10)==0 , prnt = 1; end
if test3 <= 2*ctol , prnt = 1; end
if test2 <= 10*atol , prnt = 1; end
if test1 <= 10*rtol , prnt = 1; end
if istop ~= 0 , prnt = 1; end
if prnt==1
if show
str1 = sprintf( '%6g %12.5e %10.3e', itn, x(1), rnorm );
str2 = sprintf( ' %8.1e %8.1e', test1, test2 );
str3 = sprintf( ' %8.1e %8.1e', anorm, acond );
disp([str1 str2 str3])
end
end
if istop > 0, break, end
end
% End of iteration loop.
% Print the stopping condition.
if show
disp(' ')
disp('LSQR finished')
disp(msg(istop+1,:))
disp(' ')
str1 = sprintf( 'istop =%8g itn =%8g', istop, itn );
str2 = sprintf( 'anorm =%8.1e acond =%8.1e', anorm, acond );
str3 = sprintf( 'rnorm =%8.1e arnorm =%8.1e', rnorm, arnorm );
str4 = sprintf( 'bnorm =%8.1e xnorm =%8.1e', bnorm, xnorm );
disp([str1 ' ' str2])
disp([str3 ' ' str4])
disp(' ')
end
%-----------------------------------------------------------------------
% End private function pdxxxlsqr
%-----------------------------------------------------------------------
function y = pdxxxlsqrmat( mode, mlsqr, nlsqr, x, Aname, rw )
% pdxxxlsqrmat is required by pdco.m (when it calls pdxxxlsqr.m).
% It forms Mx or M'x for some operator M that depends on Method below.
%
% mlsqr, nlsqr are the dimensions of the LS problem that lsqr is solving.
%
% Aname is pdco's Aname.
%
% rw contains parameters [explicitA Method LSdamp]
% from pdco.m to say which least-squares subproblem is being solved.
%
% global pdDDD1 pdDDD3 provides various diagonal matrices
% for each value of Method.
%-----------------------------------------------------------------------
% 17 Mar 1998: First version to go with pdsco.m and lsqr.m.
% 01 Apr 1998: global pdDDD1 pdDDD3 now used for efficiency.
% 11 Feb 2000: Added diagonal preconditioning for LSQR, solving for dy.
% 14 Dec 2000: Added diagonal preconditioning for LSQR, solving for dx.
% 12 Feb 2001: Included in pdco.m as private function.
% Specialized to allow only solving for dy.
% 03 Oct 2002: First version to go with pdco.m with general H2 and D2.
% 16 Oct 2002: Aname is now the user's Aname.
%-----------------------------------------------------------------------
global pdDDD1 pdDDD2 pdDDD3
Method = rw(2);
precon = rw(7);
if Method==3
% The operator M is [ D1 A'; D2 ].
m = nlsqr;
n = mlsqr - m;
if mode==1
if precon, x = pdDDD3.*x; end
t = pdxxxmat( Aname, 2, m, n, x ); % Ask 'aprod' to form t = A'x.
y = [ (pdDDD1.*t); (pdDDD2.*x) ];
else
t = pdDDD1.*x(1:n);
y = pdxxxmat( Aname, 1, m, n, t ); % Ask 'aprod' to form y = A t.
y = y + pdDDD2.*x(n+1:mlsqr);
if precon, y = pdDDD3.*y; end
end
else
error('Error in pdxxxlsqrmat: Only Method = 3 is allowed at present')
end
%-----------------------------------------------------------------------
% End private function pdxxxlsqrmat
%-----------------------------------------------------------------------
function y = pdxxxmat( Aname, mode, m, n, x )
% y = pdxxxmat( Aname, mode, m, n, x )
% computes y = Ax (mode=1) or A'x (mode=2)
% for a matrix A defined by pdco's input parameter Aname.
%-----------------------------------------------------------------------
% 04 Apr 1998: Default A*x and A'*y function for pdco.m.
% Assumed A was a global matrix pdAAA created by pdco.m
% from the user's input parameter A.
% 16 Oct 2002: pdAAA eliminated to save storage.
% User's parameter Aname is now passed thru to here.
% 01 Nov 2002: Bug: feval had one too many parameters.
%-----------------------------------------------------------------------
if (ischar(Aname) || isa(Aname, 'function_handle'))
y = feval( Aname, mode, m, n, x );
else
if mode==1, y = Aname*x; else y = Aname'*x; end
end
%-----------------------------------------------------------------------
% End private function pdxxxmat
%-----------------------------------------------------------------------
function fmerit = pdxxxmerit( low,upp,r1,r2,rL,rU,cL,cU )
% Evaluate the merit function for Newton's method.
% It is the 2-norm of the three sets of residuals.
f = [norm(r1)
norm(r2)
norm(rL(low))
norm(rU(upp))
norm(cL(low))
norm(cU(upp))];
fmerit = norm(f);
%-----------------------------------------------------------------------
% End private function pdxxxmerit
%-----------------------------------------------------------------------
function [r1,r2,rL,rU,Pinf,Dinf] = ...
pdxxxresid1( Aname,fix,low,upp, ...
b,bl,bu,d1,d2,grad,rL,rU,x,x1,x2,y,z1,z2 )
% Form residuals for the primal and dual equations.
% rL, rU are output, but we input them as full vectors
% initialized (permanently) with any relevant zeros.
% 13 Aug 2003: z2-z1 coded more carefully
% (although MATLAB was doing the right thing).
% 19 Nov 2003: r2(fix) = 0 has to be done after r2 = grad - r2;
m = length(b);
n = length(bl);
x(fix) = 0;
r1 = pdxxxmat( Aname, 1, m, n, x );
r2 = pdxxxmat( Aname, 2, m, n, y );
r1 = b - r1 - (d2.^2).*y;
r2 = grad - r2; % + (z2-z1); % grad includes (d1.^2)*x
r2(fix) = 0;
r2(upp) = r2(upp) + z2(upp);
r2(low) = r2(low) - z1(low);
rL(low) = ( bl(low) - x(low)) + x1(low);
rU(upp) = (- bu(upp) + x(upp)) + x2(upp);
Pinf = max([norm(r1,inf) norm(rL(low),inf) norm(rU(upp),inf)]);
Dinf = norm(r2,inf);
Pinf = max( Pinf, 1e-99 );
Dinf = max( Dinf, 1e-99 );
%-----------------------------------------------------------------------
% End private function pdxxxresid1
%-----------------------------------------------------------------------
function [cL,cU,center,Cinf,Cinf0] = ...
pdxxxresid2( mu,low,upp,cL,cU,x1,x2,z1,z2 )
% Form residuals for the complementarity equations.
% cL, cU are output, but we input them as full vectors
% initialized (permanently) with any relevant zeros.
% Cinf is the complementarity residual for X1 z1 = mu e, etc.
% Cinf0 is the same for mu=0 (i.e., for the original problem).
x1z1 = x1(low).*z1(low);
x2z2 = x2(upp).*z2(upp);
cL(low) = mu - x1z1;
cU(upp) = mu - x2z2;
maxXz = max( [max(x1z1) max(x2z2)] );
minXz = min( [min(x1z1) min(x2z2)] );
maxXz = max( maxXz, 1e-99 );
minXz = max( minXz, 1e-99 );
center = maxXz / minXz;
Cinf = max([norm(cL(low),inf) norm(cU(upp),inf)]);
Cinf0 = maxXz;
%-----------------------------------------------------------------------
% End private function pdxxxresid2
%-----------------------------------------------------------------------
function step = pdxxxstep( x, dx )
% Assumes x > 0.
% Finds the maximum step such that x + step*dx >= 0.
step = 1e+20;
blocking = find( dx < 0 );
if length( blocking ) > 0
steps = x(blocking) ./ (- dx(blocking));
step = min( steps );
end
%
% Copyright (c) 2006. Michael Saunders
%
%
% Part of SparseLab Version:100
% Created Tuesday March 28, 2006
% This is Copyrighted Material
% For Copying permissions see COPYING.m
% Comments? e-mail [email protected]
%
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
dvar4abcdk.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/dvar4abcdk.m
| 3,316 |
utf_8
|
058df17bcb6eb4c30e5c5e4b0a78d9bf
|
function [P,sigma,dA,dB,dC,dD,dK] = dvar4abcdk(x,u,y,f,p,A,B,C,D,K,U,Zps)
%DVAR4ABCDK Asymptotic variance of the PBSIDopt (VARX only) estimation
% P=dvar4abck(x,u,f,p,A,B,C,DK,U,Zps) returns the covariance of the
% estimated state space matrices and acts as a pre-processor for dvar2frd.
% The latter is used to calculate the probalistic error bounds around the
% identified bode diagrams. The data matrices U and Zps can be obtained
% from dordvarx.
%
% [P,sigma]=dvar4abck(x,u,f,p,A,B,C,D,K,U,Zps) also returns the covariance
% matrix of the innovation noise.
%
% [P,sigma,dA,dB,dC,dD,dK]=dvar4abck(x,u,f,p,A,B,C,D,K,U,Zps)
% Ivo Houtzager
% Delft Center of Systems and Control
% Delft University of Technology
% The Netherlands, 2010
% check number if input arguments
if nargin < 12
error('DVAR4ABCDK requires twelve input arguments.')
end
% check the size of the windows
if f > p
error('Future window size f must equal or smaller then past window p. (f <= p)')
end
% check dimensions of inputs
if size(y,2) < size(y,1)
y = y';
end
if size(x,2) < size(x,1)
x = x';
end
N = size(y,2);
l = size(y,1);
n = size(x,1);
if isempty(u);
r = 0;
u = zeros(0,N);
else
if size(u,2) < size(u,1)
u = u';
end
r = size(u,1);
if ~isequal(N,length(u))
error('The number of rows of vectors/matrices u and y must be the same.')
end
end
if l == 0
error('DVAR4ABCDK requires an output vector y.')
end
% store the past and future vectors
m = r+l;
z = [u; y];
Z = zeros(p*m,N-p);
for i = 1:p
Z((i-1)*m+1:i*m,:) = z(:,i:N+i-p-1);
end
% select the only the system order
U = U(1:n,:);
if size(Zps,2)/m > p
Zps = Zps(:,1:p*m);
end
% remove the window sizes from input and output vector
u = u(:,p+1:p+size(x,2));
y = y(:,p+1:p+size(x,2));
% calculate the innovation sequence
e = y - C*x - D*u;
sigma = (e*e')/length(e);
%% Asymptotic variance
LL = pinv([x(:,1:end-1); u(:,1:end-1); e(:,1:end-1)]);
LL2 = pinv([x; u]);
Term1 = ObsContSum(Zps,speye(N-p),l,r,f,p,LL,Z(:,2:end),U);
Term2 = ObsContSum(Zps,speye(N-p),l,r,f,p,LL,Z(:,1:end-1),A*U);
Term3 = ObsContSum(Zps,speye(N-p),l,r,f,p,LL2,Z,-C*U);
alpha1P = Term1-Term2;
alpha2P = Term3;
beta1 = -kron(LL'*Z(:,1:end-1)'*Zps',K);
beta2 = -kron(LL2'*Z'*Zps',eye(l));
if l==1
P = sigma*[alpha1P+beta1;alpha2P+beta2]*[alpha1P+beta1;alpha2P+beta2]';
else
P = [alpha1P+beta1;alpha2P+beta2]*sparse(kron(speye(N-p),sigma))*[alpha1P+beta1;alpha2P+beta2]';
end
if nargout > 2
dTh = [alpha1P+beta1;alpha2P+beta2]*e(:);
dA = reshape(dTh(1:n*n,1),n,n);
dB = reshape(dTh(n*n+1:n*n+n*r,1),n,r);
dK = reshape(dTh(n*n+n*r+1:n*n+n*r+n*l,1),n,l);
dC = reshape(dTh(n*n+n*r+n*l+1:n*n+n*r+n*l+n*l,1),l,n);
dD = reshape(dTh(n*n+n*r+n*l+n*l+1:n*n+n*r+n*l+n*l+l*r,1),l,r);
end
end
function SumKron = ObsContSum(Zps,Y,l,r,f,p,LL,Z,S)
q = size(Y,1);
for i = 1:p
CK(:,1+(l+r)*(i-1):(l+r)*i) = Y*Zps(:,1+(l+r)*(i-1):(l+r)*i);
end
SumKron = zeros(size(LL,2)*size(S,1),size(Y,2)*l);
for i = 1:f
GammaK = zeros(q,(l+r)*p);
GammaK(:,1+(l+r)*(i-1):(l+r)*p) = CK(:,1:(l+r)*(p+1-i));
SumKron = SumKron + kron(LL'*Z'*GammaK',S(:,1+l*(i-1):l*i));
end
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
lordvarxydist.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/lordvarxydist.m
| 6,288 |
utf_8
|
bd92ed5b69652e03f4a87777bf0d9b4d
|
function [S,X] = lordvarxydist(u,d,y,mu,f,p,reg,opt,c,noD,ObsMatPoint)
%LORDVARXYDIST is LORDVARX for a special case with output disturbances
% x(k+1) = A kron(mu(k),x(k)) + B kron(mu(k),u(k)) + K kron(mu(k),e(k))
% y(k) = C x(k) + [Du, Dd] [u(k); kron(mu(k),d(k))] + e(k)
%
% if c(4)=1 then Dd is not varying with the scheduling mu
%
% See also: lordvarx.m and lx2abcdkydist.m.
%
% References:
% [1] J.W. van Wingerden, and M. Verhaegen, ``Subspace identification
% of Bilinear and LPV systems for open- and closed-loop data'',
% Automatica 45, pp 372--381, 2009.
% Pieter Gebraad
% Delft Center of Systems and Control
% Delft University of Technology
% The Netherlands, 2011
% check number if input arguments
if nargin < 6
error('LORDVARX requires four or five input arguments.')
end
% assign default values to unspecified parameters
if (nargin < 11) || isempty(ObsMatPoint)
ObsMatPoint = 0;
end
if (nargin < 10) || isempty(noD)
noD = 0;
end
if (nargin < 9) || isempty(c)
c = [0 0 0 0];
end
if (nargin < 8) || isempty(opt)
opt = 'gcv';
end
if (nargin < 7) || isempty(reg)
reg = 'none';
end
% check the size of the windows
if f > p
error('Future window size f must equal or smaller then past window p. (f <= p)')
end
% check dimensions of inputs
if size(y,2) < size(y,1)
y = y';
end
if size(mu,2) < size(mu,1)
mu = mu';
end
if size(d,2) < size(d,1)
d = d';
end
N = size(y,2);
l = size(y,1);
rd = size(d,1);
if ~isequal(N,length(d))
error('The number of rows of vectors/matrices d and y must be the same.')
end
s = size(mu,1);
if isempty(u);
r = 0;
u = zeros(0,N);
else
if size(u,2) < size(u,1)
u = u';
end
r = size(u,1);
if ~isequal(N,length(u))
error('The number of rows of vectors/matrices u and y must be the same.')
end
end
if l == 0
error('LORDVARX requires an output vector y.')
end
if s == 0
error('LORDVARX requires a scheduling sequence mu, use DORDVARX for LTI systems.')
end
if c(4)==0
d = khatrirao(mu,d);
end
% determine sizes
k = r*s.^(1-c(2)+(1-c(1))*(p-1:-1:0))+ l*s.^(1-c(3)+(1-c(1))*(p-1:-1:0));
q = sum(k);
if q > (N-p)
if ~strcmpi(reg,'bpdn')
if ObsMatPoint == 1
warning('lordvarx:ObsMatPoint1ThenNoKernel','Taking the observability matrix for p = ones(1,m) is not implemented for the kernel method. LORDVARX continues with ObsMatPoint=1, without kernel method.')
kernel = 0;
elseif ObsMatPoint == 0
kernel = 1;
else
error('ObsMatPoint should be 0 or 1')
end
else
warning('lordvarx:BpdnThenNoKernel','The BPDN regularization is not implemented for the kernel method. LORDVARX continues with BPDN method, without kernel method.')
kernel = 0;
end
else
kernel = 0;
end
% store the past and future vectors
if kernel
Z = zeros(N-p,N-p);
for j = 0:p-1
Z = optkernel(Z,u,y,mu,p,c,0,j);
end
else
Z = zeros(q,N-p);
if (c(2) == 0) && (c(3) == 0)
z = [khatrirao(mu,u); khatrirao(mu,y)];
elseif (c(2) == 1) && (c(3) == 0)
z = [u; khatrirao(mu,y)];
elseif (c(2) == 0) && (c(3) == 1)
z = [khatrirao(mu,u); y];
elseif (c(2) == 1) && (c(3) == 1)
z = [u; y];
end
for i = 1:p
Z(sum(k(1:i-1))+1:sum(k(1:i-1))+k(p),:) = z(:,i:N+i-p-1);
if c(1) ~= 0
for j = (i+1):p
Z(sum(k(1:i-1))+1:sum(k(1:i-1))+k(p-j+i),:) = khatrirao(mu(:,j:N+j-p-1),Z(sum(k(1:i-1))+1:sum(k(1:i-1))+k(p-j+i+1),:));
end
end
end
end
Y = y(:,p+1:N);
U = u(:,p+1:N);
d = d(:,p+1:N);
% solve VARX/KERNEL problem
if kernel
if ~noD
Z = Z + U'*U + d'*d;
end
A = kernregress(Y,Z,reg,opt);
else
if ~noD
Z = [Z; U; d];
end
VARX = regress(Y,Z,reg,opt);
end
% construct LambdaKappaZ
if kernel
LKZ = zeros(f*l,N-p);
for i = 0:f-1
Z = zeros(N-p,N-p);
for j = i:p-1
Z = optkernel(Z,u,y,mu,p,c,i,j);
end
LKZ(i*l+1:(i+1)*l,:) = A*Z;
end
% singular value decomposition
[~,S,V] = svd(LKZ,'econ');
else
if c(1) == 0
if ObsMatPoint % consider the observability matrix in the operating point p = ones(1,m)
LKZ = zeros(f*l,N-p);
for i = 1:f
for j = i:p
for h = 1:s^(i-1)
LKZ((i-1)*l+1:i*l,:) = LKZ((i-1)*l+1:i*l,:) + VARX(:,sum(k(1:j-i))+((h-1)*k(j)+1:h*k(j)))*Z(sum(k(1:j-1))+1:sum(k(1:j)),:);
end
end
end
% singular value decomposition
[~,S,V] = svd(LKZ,'econ');
else % consider the observability matrix in the operating point p = [1,zeros(1,m-1)]
LK = zeros(f*l,q);
for i = 1:f
for j = i:p
LK((i-1)*l+1:i*l,sum(k(1:j-1))+1:sum(k(1:j))) = VARX(:,sum(k(1:j-i))+1:sum(k(1:j-i))+k(j));
end
end
% singular value decomposition
[~,S,V] = svd(LK*Z(1:q,:),'econ');
end
else
LK = zeros(f*l,q);
for i = 1:f
LK((i-1)*l+1:i*l,q-(p-i+1)*(q/p)+1:q) = VARX(:,1:(p-i+1)*(q/p));
end
% singular value decomposition
[~,S,V] = svd(LK*Z(1:q,:),'econ');
end
end
X = diag(sqrt(diag(S)))*V';
S = diag(S)';
end
function Z = optkernel(Z,u,y,mu,p,c,i,j)
N = size(y,2);
P = 1:1:N-p;
T = ones(N-p,N-p);
if all(c == 0)
for v = 0:p-j-1
T = T.*(mu(:,P+v+j-i)'*mu(:,P+v+j));
end
Z = Z + T.*([u(:,P+j-i); y(:,P+j-i)]'*[u(:,P+j); y(:,P+j)]);
else
for v = 1:(1-c(1))*(p-j-1)
T = T.*(mu(:,P+v+j-i)'*mu(:,P+v+j));
end
if c(2)
Z = Z + T.*(u(:,P+j-i)'*u(:,P+j));
else
Z = Z + T.*(mu(:,P+j-i)'*mu(:,P+j)).*(u(:,P+j-i)'*u(:,P+j));
end
if c(3)
Z = Z + T.*(y(:,P+j-i)'*y(:,P+j));
else
Z = Z + T.*(mu(:,P+j-i)'*mu(:,P+j)).*(y(:,P+j-i)'*y(:,P+j));
end
end
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
dvar2eig.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/dvar2eig.m
| 602 |
utf_8
|
34202c6ba1c8543cd49f5820c2b1e2ff
|
function [E,covE] = dvar2eig(P,A)
%DVAR2EIG Eigenvalues and its covariance estimation
% [E,covE]=dvar2frd(P,A) returns the estimated eigenvalues and its
% covariance for the state space matrix A and its covariance P.
% Ivo Houtzager
% Delft Center of Systems and Control
% Delft University of Technology
% The Netherlands, 2010
n = size(A,1);
P = P(1:n^2,1:n^2);
J = jacobianest(@(x) eigen(x,n),A(:));
covE = J*P*J';
E = eig(A);
end
function dE = eigen(A,n)
A = reshape(A,n,n);
E = eig(A);
dE = zeros(2*n,1);
dE(1:2:end) = real(E);
dE(2:2:end) = imag(E);
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
spaavf.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/spaavf.m
| 8,045 |
utf_8
|
45c5f4decdd81c84cde81c45ea634e7c
|
function [G,w,Coh] = spaavf(u,y,r,dt,Nband,Nfft,ZeroPadding,Wname)
%SPAAVF Spectral analysis with frequency averaging
% [G,W]=SPAAVF(U,Y,Ts,Nband) determines a frequency-domain estimate
% SYS=FRD(G,W) of the transfer function of the plant. The sample time is
% given in Ts. Nband is the number of frequency bands to average.
% Averaging in the frequency domain is used to get a smoother frequency
% response function. The spectrum is smoothed locally in the region of
% the target frequencies, as a weighted average of values to the right
% and left of a target frequency. The variance of the spectrum will
% decrease as the number of frequencies used in the smoothing increases.
% As the bandwidth increases, more spectral ordinates are averaged, and
% hence the resulting estimator becomes smoother, more stable and has
% smaller variance.
%
% [G,W]=SPAAVF(U,Y,R,Ts,Nband) determines a frequency-domain estimate
% SYS=FRD(G,W) of the transfer function of the plant operating in
% closed-loop. Because the conventional transfer function estimate, will
% give a biased estimate under closed-loop [2]. An unbiased alternative
% is to use cross-spectral between the input/output signals with an
% external excitation signal r [1]. Hence, we define the estimate:
% G(exp(j*omega)) = Phi_yr(omega)*inv(Phi_ur(omega))
%
% [G,W]=SPAAVF(...,Ts,Nband,Nfft) specifies the number of evaluated
% frequencies. For large data sequences it is wortwhile to choose the
% value Nfft as function of the power of two. In this case a faster
% method is used durring the FFT. Nfft <= length(u), unless zeros are
% added. See also, FFT.
%
% [G,W]=SPAAVF(...,Ts,Nband,Nfft,ZeroPading) adds additional zeros to
% the data sequences. Usefull for increasing the number of evaluated
% frequencies.
%
% [G,W]=SPAAVF(...,Ts,Nband,Nfft,ZeroPading,Wname) specifies and aplies
% an window to the data. Windowing weigths the data, it increases the
% importance of the data in the middle of the vector and decreases the
% importance of the data at the end and the beginning, thus reducing the
% effect of spectral leakage. See also, WINDOW.
%
% References:
% [1] Akaike, H., Some problems in the application of the cross-spectral
% method, In spectral analysis of time series, pp. 81-107, Wiley,
% New York, 1967.
% [2] van den Hof, P., System Identification, Lecture Notes, Delft, 2007.
% Revision 2: Now also works properly for MIMO cases.
% Ivo Houtzager
% Delft Center of Systems and Control
% Delft University of Technology
% The Netherlands, 2010
% Check closed-loop
if isequal(length(u),length(r));
clmode = 1;
else
clmode = 0;
if nargin == 7
Wname = ZeroPadding;
end
if nargin == 6
ZeroPadding = Nfft;
end
if nargin == 5
Nfft = Nband;
end
Nband = dt;
dt = r;
end
% Transpose vectors if needed
if size(u,2) < size(u,1);
u = u';
end
nu = size(u,1); % number of inputs
if size(y,2) < size(y,1);
y = y';
end
ny = size(y,1); % number of inputs
if clmode
if size(r,2) < size(r,1);
r = r';
end
nr = size(r,1); % number of inputs
end
% Apply window and zeros if needed
if nargin == (7+clmode)
u = u.*(ones(nu,1)*window(Wname,size(u,2))');
y = y.*(ones(ny,1)*window(Wname,size(u,2))');
if clmode
r = r.*(ones(nr,1)*window(Wname,length(u))');
end
end
if nargin == (6+clmode)
u = [u zeros(nu,ZeroPadding)];
y = [y zeros(ny,ZeroPadding)];
if clmode
r = [r zeros(nr,ZeroPadding)];
end
end
if nargin < (5+clmode)
Nfft = [];
end
% Some administration
Fs = 1./dt; % Sample frequency
N = length(u); % Number of samples
T = N*dt; % Total time
if isempty(Nfft)
f = (0:N-1)'/T; % Frequency vector (double sided)
else
f = (linspace(0,N-1,Nfft))'/T; % Frequency vector (double sided)
end
Nf = length(f);
% Determine Fourier transforms
U = zeros(nu,1,Nf);
Y = zeros(ny,1,Nf);
for i = 1:nu
ut = dt*fft(u(i,:),Nfft);
for k = 1:Nf
U(i,:,k) = ut(k);
end
end
for i = 1:ny
yt = dt*fft(y(i,:),Nfft);
for k = 1:Nf
Y(i,:,k) = yt(k);
end
end
if clmode
R = zeros(nr,1,Nf);
for i = 1:nr
rt = dt*fft(r(i,:),Nfft);
for k = 1:Nf
R(i,:,k) = rt(k);
end
end
end
% Do closed-loop or open-loop
if clmode
% Determine spectral densities
Sur = zeros(nu,nr,Nf);
Srr = zeros(nr,nr,Nf);
Syr = zeros(ny,nr,Nf);
Syy = zeros(ny,ny,Nf);
for k = 1:Nf
Sur(:,:,k) = (1/T).*U(:,:,k)*R(:,:,k)';
Srr(:,:,k) = (1/T).*R(:,:,k)*R(:,:,k)';
Syr(:,:,k) = (1/T).*Y(:,:,k)*R(:,:,k)';
Syy(:,:,k) = (1/T).*Y(:,:,k)*Y(:,:,k)';
end
% Apply frequency averaging
Nmod = floor(Nf/Nband);
mSur = zeros(nu,nr,Nmod);
mSrr = zeros(nr,nr,Nmod);
mSyr = zeros(ny,nr,Nmod);
mSyy = zeros(ny,ny,Nmod);
mf = freqAvg(f,Nband);
for i = 1:nr
for j = 1:nr
tSur = Sur(i,j,:);
tSur = freqAvg(tSur(:),Nband);
mSur(i,j,:) = tSur;
tSrr = Srr(i,j,:);
tSrr = freqAvg(tSrr(:),Nband);
mSrr(i,j,:) = tSrr;
end
end
for i = 1:ny
for j = 1:ny
tSyy = Syy(i,j,:);
tSyy = freqAvg(tSyy(:),Nband);
mSyy(i,j,:) = tSyy;
end
for j = 1:nr
tSyr = Syr(i,j,:);
tSyr = freqAvg(tSyr(:),Nband);
mSyr(i,j,:) = tSyr;
end
end
% Estimate transfer function
G = zeros(ny,nr,Nmod);
for k = 1:Nmod
G(:,:,k) = mSyr(:,:,k)/mSur(:,:,k);
end
% Squared coherence Cohuy function between r and y
if nr == ny && nargout == 3
Coh = zeros(ny,nr,Nmod);
for k = 1:Nmod
Coh(:,:,k) = sqrtm(abs(mSyr(:,:,k))*abs(mSyr(:,:,k))/(mSyy(:,:,k)*mSrr(:,:,k)));
end
end
else
% Determine spectral densities
Suu = zeros(nu,nu,Nf);
Syu = zeros(ny,nu,Nf);
Syy = zeros(ny,ny,Nf);
for k = 1:Nf
Suu(:,:,k) = (1/T).*U(:,:,k)*U(:,:,k)';
Syu(:,:,k) = (1/T).*Y(:,:,k)*U(:,:,k)';
Syy(:,:,k) = (1/T).*Y(:,:,k)*Y(:,:,k)';
end
% Apply frequency averaging
Nmod = floor(Nf/Nband);
mSuu = zeros(nu,nu,Nmod);
mSyu = zeros(ny,nu,Nmod);
mSyy = zeros(ny,ny,Nmod);
mf = freqAvg(f,Nband);
for i = 1:nu
for j = 1:nu
tSuu = Suu(i,j,:);
tSuu = freqAvg(tSuu(:),Nband);
mSuu(i,j,:) = tSuu;
end
end
for i = 1:ny
for j = 1:ny
tSyy = Syy(i,j,:);
tSyy = freqAvg(tSyy(:),Nband);
mSyy(i,j,:) = tSyy;
end
for j = 1:nu
tSyu = Syu(i,j,:);
tSyu = freqAvg(tSyu(:),Nband);
mSyu(i,j,:) = tSyu;
end
end
% Estimate transfer function
G = zeros(ny,nu,Nmod);
for k = 1:Nmod
G(:,:,k) = conj(mSyu(:,:,k)*pinv(mSuu(:,:,k)));
end
% Squared coherence Cohuy function between u and y
if nu == ny && nargout == 3
Coh = zeros(ny,nu,Nmod);
for k = 1:Nmod
Coh(:,:,k) = conj(sqrtm(abs(mSyu(:,:,k))*abs(mSyu(:,:,k))*pinv(mSyy(:,:,k)*mSuu(:,:,k))));
end
end
end
% Construct function output
fmax = Fs/2;
fi = find(mf <= fmax);
w = mf(fi).*2*pi;
G = G(:,:,fi);
if nu == ny && nargout == 3
Coh = Coh(:,:,fi);
end
end
function out = freqAvg(in,nrbands)
%FREQAVG Frequency averaging
% out=freqAvg(in,nrbands) averages the input 'in' over the number of
% frequency bands 'nrbands'.
N = length(in);
Nmod = floor((N/nrbands)); % number of remaining frequencies after averaging
tmp = zeros(nrbands,Nmod); % initialization of temporary matrix for averaging: nrband rows and nmod columns
tmp(:) = in(1:nrbands*Nmod); % arrange the samples of 'in' in the elements of tmp.
out = mean(tmp,1)'; % average over columns and make it a vector
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
dnyquistdetsd.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/dnyquistdetsd.m
| 2,246 |
utf_8
|
316ca202611ed3a61b547168e98175f9
|
function dnyquistdetsd(G,covG,sd,Greal)
%DNYQUISTDETSD Nyquist diagram with probalistic error bounds (det(G))
% dnyquistdetsd(G,covG,sd) plots the nyquist diagram using the determinant
% of the estimated frequency response given in G and the frequency bounds
% given in covG and sd. The value sd is the standard deviation and is
% larger than zero.
%
% dnyquistdetsd(G,covG,sd,Greal) includes the plot of the determinant of
% the frequency response given in Greal
if nargin > 3
if ~isequal(size(G,3),size(covG,3),size(Greal,3))
error('Number of frequencies points should be the same!')
end
else
if ~isequal(size(G,3),size(covG,3))
error('Number of frequencies points should be the same!')
end
end
l = size(G,1);
r = size(G,2);
Gdet = zeros(1,size(G,3));
covGdet = zeros(2,2,size(G,3));
if nargin > 3
Gdetreal = zeros(1,size(G,3));
end
for k = 1:size(G,3)
Gdet(1,k) = det(G(:,:,k));
if nargin > 3
Gdetreal(1,k) = det(Greal(:,:,k));
end
X = zeros(2*l,r);
X(1:2:end,:) = real(G(:,:,k));
X(2:2:end,:) = imag(G(:,:,k));
J = jacobianest(@(x) deter(x,l,r),X(:));
P = zeros(l*r*2);
for i = 1:l
for j = 1:r
P((i-1)*r*2+(j-1)*2+(1:2),(i-1)*r*2+(j-1)*2+(1:2)) = squeeze(covG(i,j,k,:,:));
end
end
covGdet(:,:,k) = J*P*J';
end
if l == r
figure
hold on;
[sdreal,sdimag] = pol2cart(unwrap(angle(Gdet)),max(squeeze(abs(Gdet)),1e-5));
for k = 1:size(G,3)
ellipsebnd(covGdet(:,:,k),[sdreal(k); sdimag(k)],'conf',erf(sd/sqrt(2)),'style','k')
end
plot(sdreal',sdimag','k','Linewidth',2);
if nargin > 3
[sdreal,sdimag] = pol2cart(unwrap(angle(Gdetreal)),max(squeeze(abs(Gdetreal)),1e-5));
plot(sdreal',sdimag','k--','Linewidth',1);
end
%axis equal;
vline(0,'k:');
hline(0,'k:');
xlabel('Real axis');
ylabel('Imaginary axis');
box on;
hold off;
else
error('Only for square MIMO systems!')
end
end
function dD = deter(G,l,r)
Gr = G(1:2:end,:);
Gi = G(2:2:end,:);
G = reshape(Gr,l,r) + 1i.*reshape(Gi,l,r);
D = det(G);
dD = zeros(2,1);
dD(1,1) = real(D);
dD(2,1) = imag(D);
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
lordvarx.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/lordvarx.m
| 8,773 |
utf_8
|
44ca081bff91cd9866ddbcc209c13735
|
function [S,X] = lordvarx(u,y,mu,f,p,reg,opt,c,noD,ObsMatPoint)
%LORDVARX Closed-loop LPV system identification using the PBSIDopt method.
% [S,X]=lordvarx(u,y,mu,f,p) delivers information about the order of the
% Linear Parameter Varing system and acts as a pre-processor for lmodx.
% The latter is used to identify an open-loop or closed-loop system for
% the N-by-r, N-by-l and N-by-m data matrices u, y and mu, where r, l and
% m are the number of inputs, outputs and scheduling parameters. The input
% matrix u, output matrix y and scheduling matrix mu must have the same
% number of observations but can have different numbers of variables. The
% past and future window size p and f must be higher then the expected
% system order n. The outputs are the singular values S, which can be used
% to determine the order of the identifiable system. Further the state
% matrix X is returned, which has to be forwarded to lmodx.
%
% The data u,y,mu can be supplied in batches to prevent out-of-memory
% errors when using a large number of samples: supply u,y,mu in a cell
% array with different parts of the data in each cell.
% See Example 5 ([PBSID toolbox path]/examples/ex05_lti_wts_batch.m)
%
% [S,X]=lordvarx(u,y,mu,f,p,reg,opt) adds a regularization to the
% identification problem. The additional inputs are the regularization
% method and selection parameters: reg = {'none', 'tikh', 'tsvd'} and opt
% = {'gcv', 'lcurve', or any regularisation value as scalar}. With
% regularisation, the solver can better deal with singular covariance
% matrices. (default reg='none' and opt='gcv')
% If reg = {'bpdn'} and opt = {'sv', or a scalar between 0 and 1}
% then sparse estimation through Basis Pursuit Denoising is used. The
% solver can then better deal with a past window that is chosen too
% large.
% See help private/regress for more details.
%
% [S,X]=lordvarx(u,y,mu,f,p,reg,opt,c) specifies which of the system
% matrices are constant and not parameter-varing. For each of the matrices
% A, B, and K an 1 or 0 can be given in the vector c.
%
% [S,X]=lordvarx(u,y,mu,f,p,reg,opt,c,noD) if noD=1, then the direct
% feedtrough term D is not considered during estimatoion. Note the direct
% feedtrough term D can improve the results when estimating low-order
% models from high-order models. (default noD=0)
%
% [S,X]=lordvarx(u,y,mu,f,p,reg,opt,c,noD,ObsMatPoint) with ObsMatPoint=1,
% then in estimation of the state sequence, we consider the observability
% matrix in the operating point p = ones(1,m), which may yield better
% results in some cases than with the default ObsMatPoint=0, which takes
% the observability matrix in the operating point p = [1,zeros(1,m-1)].
%
% See also: lmodx.m, lx2abcdk.m, and lx2abck.m.
%
% References:
% [1] J.W. van Wingerden, and M. Verhaegen, ``Subspace identification
% of Bilinear and LPV systems for open- and closed-loop data'',
% Automatica 45, pp 372--381, 2009.
% Ivo Houtzager
% Delft Center of Systems and Control
% Delft University of Technology
% The Netherlands, 2010
% Pieter Gebraad
% Delft Center of Systems and Control
% Delft University of Technology
% The Netherlands, 2011
% Jan-Willem van Wingerden
% Delft Center of Systems and Control
% Delft University of Technology
% The Netherlands, 2015
% check number if input arguments
if nargin < 4
error('LORDVARX requires four or five input arguments.')
end
% assign default values to unspecified parameters
if (nargin < 10) || isempty(ObsMatPoint)
ObsMatPoint = 0;
end
if (nargin < 9) || isempty(noD)
noD = 0;
end
if (nargin < 8) || isempty(c)
c = [0 0 0];
end
if (nargin < 7) || isempty(opt)
opt = 'gcv';
end
if (nargin < 6) || isempty(reg)
reg = 'none';
end
% check the size of the windows
if f > p
error('Future window size f must equal or smaller then past window p. (f <= p)')
end
% check dimensions of inputs
if size(y,2) < size(y,1)
y = y';
end
if size(mu,2) < size(mu,1)
mu = mu';
end
N = size(y,2);
l = size(y,1);
s = size(mu,1);
if isempty(u);
r = 0;
u = zeros(0,N);
else
if size(u,2) < size(u,1)
u = u';
end
r = size(u,1);
if ~isequal(N,length(u))
error('The number of rows of vectors/matrices u and y must be the same.')
end
end
if l == 0
error('LORDVARX requires an output vector y.')
end
if s == 0
error('LORDVARX requires a scheduling sequence mu, use DORDVARX for LTI systems.')
end
% determine sizes
m = r+l;
k = r*s.^(1-c(2)+(1-c(1))*(p-1:-1:0))+ l*s.^(1-c(3)+(1-c(1))*(p-1:-1:0));
q = sum(k);
if q > (N-p)
if ~strcmpi(reg,'bpdn')
if ObsMatPoint == 1
warning('lordvarx:ObsMatPoint1ThenNoKernel','Taking the observability matrix for p = ones(1,m) is not implemented for the kernel method. LORDVARX continues with ObsMatPoint=1, without kernel method.')
kernel = 0;
elseif ObsMatPoint == 0
kernel = 1;
else
error('ObsMatPoint should be 0 or 1')
end
else
warning('lordvarx:BpdnThenNoKernel','The BPDN regularization is not implemented for the kernel method. LORDVARX continues with BPDN method, without kernel method.')
kernel = 0;
end
else
kernel = 0;
end
% store the past and future vectors
if kernel
Z = zeros(N-p,N-p);
for j = 0:p-1
Z = optkernel(Z,u,y,mu,p,c,0,j);
end
else
Z = zeros(q,N-p);
if (c(2) == 0) && (c(3) == 0)
z = [khatrirao(mu,u); khatrirao(mu,y)];
elseif (c(2) == 1) && (c(3) == 0)
z = [u; khatrirao(mu,y)];
elseif (c(2) == 0) && (c(3) == 1)
z = [khatrirao(mu,u); y];
elseif (c(2) == 1) && (c(3) == 1)
z = [u; y];
end
for i = 1:p
Z(sum(k(1:i-1))+1:sum(k(1:i-1))+k(p),:) = z(:,i:N+i-p-1);
if c(1) == 0
for j = (i+1):p
Z(sum(k(1:i-1))+1:sum(k(1:i-1))+k(p-j+i),:) = khatrirao(mu(:,j:N+j-p-1),Z(sum(k(1:i-1))+1:sum(k(1:i-1))+k(p-j+i+1),:));
end
end
end
end
% solve VARX/KERNEL problem
if kernel
Y = y(:,p+1:N);
U = u(:,p+1:N);
if ~noD
Z = Z + U'*U;
end
A = kernregress(Y,Z,reg,opt);
else
Y = y(:,p+1:N);
U = u(:,p+1:N);
if ~noD
Z = [Z; U];
end
VARX = regress(Y,Z,reg,opt);
end
% construct LambdaKappaZ
if kernel
LKZ = zeros(f*l,N-p);
for i = 0:f-1
Z = zeros(N-p,N-p);
for j = i:p-1
Z = optkernel(Z,u,y,mu,p,c,i,j);
end
LKZ(i*l+1:(i+1)*l,:) = A*Z;
end
% singular value decomposition
[~,S,V] = svd(LKZ,'econ');
else
if c(1) == 0
if ObsMatPoint % consider the observability matrix in the operating point p = ones(1,m)
LKZ = zeros(f*l,N-p);
for i = 1:f
for j = i:p
for h = 1:s^(i-1)
LKZ((i-1)*l+1:i*l,:) = LKZ((i-1)*l+1:i*l,:) + VARX(:,sum(k(1:j-i))+((h-1)*k(j)+1:h*k(j)))*Z(sum(k(1:j-1))+1:sum(k(1:j)),:);
end
end
end
% singular value decomposition
[~,S,V] = svd(LKZ,'econ');
else % consider the observability matrix in the operating point p = [1,zeros(1,m-1)]
LK = zeros(f*l,q);
for i = 1:f
for j = i:p
LK((i-1)*l+1:i*l,sum(k(1:j-1))+1:sum(k(1:j))) = VARX(:,sum(k(1:j-i))+1:sum(k(1:j-i))+k(j));
end
end
% singular value decomposition
[~,S,V] = svd(LK*Z(1:q,:),'econ');
end
else
LK = zeros(f*l,q);
for i = 1:f
LK((i-1)*l+1:i*l,q-(p-i+1)*(q/p)+1:q) = VARX(:,1:(p-i+1)*(q/p));
end
% singular value decomposition
[~,S,V] = svd(LK*Z(1:q,:),'econ');
end
end
X = diag(sqrt(diag(S)))*V';
S = diag(S)';
end
function Z = optkernel(Z,u,y,mu,p,c,i,j)
N = size(y,2);
P = 1:1:N-p;
T = ones(N-p,N-p);
if all(c == 0)
for v = 0:p-j-1
T = T.*(mu(:,P+v+j-i)'*mu(:,P+v+j));
end
Z = Z + T.*([u(:,P+j-i); y(:,P+j-i)]'*[u(:,P+j); y(:,P+j)]);
else
for v = 1:(1-c(1))*(p-j-1)
T = T.*(mu(:,P+v+j-i)'*mu(:,P+v+j));
end
if c(2)
Z = Z + T.*(u(:,P+j-i)'*u(:,P+j));
else
Z = Z + T.*(mu(:,P+j-i)'*mu(:,P+j)).*(u(:,P+j-i)'*u(:,P+j));
end
if c(3)
Z = Z + T.*(y(:,P+j-i)'*y(:,P+j));
else
Z = Z + T.*(mu(:,P+j-i)'*mu(:,P+j)).*(y(:,P+j-i)'*y(:,P+j));
end
end
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
dvar4abck.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/dvar4abck.m
| 3,215 |
utf_8
|
02975a072981cc80129a641df8b8fd61
|
function [P,sigma,dA,dB,dC,dK] = dvar4abck(x,u,y,f,p,A,B,C,K,U,Zps)
%DVAR4ABCK Asymptotic variance of the PBSIDopt (VARX only) estimation
% P=dvar4abck(x,u,f,p,A,B,C,K,U,Zps) returns the covariance of the
% estimated state space matrices and acts as a pre-processor for dvar2frd.
% The latter is used to calculate the probalistic error bounds around the
% identified bode diagrams. The data matrices U and Zps can be obtained
% from dordvarx.
%
% [P,sigma]=dvar4abck(x,u,f,p,A,B,C,K,U,Zps) also returns the covariance
% matrix of the innovation noise.
%
% [P,sigma,dA,dB,dC,dK]=dvar4abck(x,u,f,p,A,B,C,K,U,Zps)
% Ivo Houtzager
% Delft Center of Systems and Control
% Delft University of Technology
% The Netherlands, 2010
% check number if input arguments
if nargin < 11
error('DVAR4ABCK requires eleven input arguments.')
end
% check the size of the windows
if f > p
error('Future window size f must equal or smaller then past window p. (f <= p)')
end
% check dimensions of inputs
if size(y,2) < size(y,1)
y = y';
end
if size(x,2) < size(x,1)
x = x';
end
N = size(y,2);
l = size(y,1);
n = size(x,1);
if isempty(u);
r = 0;
u = zeros(0,N);
else
if size(u,2) < size(u,1)
u = u';
end
r = size(u,1);
if ~isequal(N,length(u))
error('The number of rows of vectors/matrices u and y must be the same.')
end
end
if l == 0
error('DVAR4ABCK requires an output vector y.')
end
% store the past and future vectors
m = r+l;
z = [u; y];
Z = zeros(p*m,N-p);
for i = 1:p
Z((i-1)*m+1:i*m,:) = z(:,i:N+i-p-1);
end
% select the only the system order
U = U(1:n,:);
if size(Zps,2)/m > p
Zps = Zps(:,1:p*m);
end
% remove the window sizes from input and output vector
u = u(:,p+1:p+size(x,2));
y = y(:,p+1:p+size(x,2));
% calculate the innovation sequence
e = y - C*x;
sigma = (e*e')/length(e);
% asymptotic variance
LL = pinv([x(:,1:end-1); u(:,1:end-1); e(:,1:end-1)]);
LL2 = pinv(x);
Term1 = ObsContSum(Zps,speye(N-p),l,r,f,p,LL,Z(:,2:end),U);
Term2 = ObsContSum(Zps,speye(N-p),l,r,f,p,LL,Z(:,1:end-1),A*U);
Term3 = ObsContSum(Zps,speye(N-p),l,r,f,p,LL2,Z,-C*U);
alpha1P = Term1-Term2;
alpha2P = Term3;
beta1 = -kron(LL'*Z(:,1:end-1)'*Zps',K);
beta2 = -kron(LL2'*Z'*Zps',eye(l));
if l==1
P = sigma*[alpha1P+beta1;alpha2P+beta2]*[alpha1P+beta1;alpha2P+beta2]';
else
P = [alpha1P+beta1;alpha2P+beta2]*sparse(kron(speye(N-p),sigma))*[alpha1P+beta1;alpha2P+beta2]';
end
if nargout > 2
dTh = [alpha1P+beta1;alpha2P+beta2]*e(:);
dA = reshape(dTh(1:n*n,1),n,n);
dB = reshape(dTh(n*n+1:n*n+n*r,1),n,r);
dK = reshape(dTh(n*n+n*r+1:n*n+n*r+n*l,1),n,l);
dC = reshape(dTh(n*n+n*r+n*l+1:n*n+n*r+n*l+n*l,1),l,n);
end
end
function SumKron = ObsContSum(Zps,Y,l,r,f,p,LL,Z,S)
q = size(Y,1);
for i = 1:p
CK(:,1+(l+r)*(i-1):(l+r)*i) = Y*Zps(:,1+(l+r)*(i-1):(l+r)*i);
end
SumKron = zeros(size(LL,2)*size(S,1),size(Y,2)*l);
for i = 1:f
GammaK = zeros(q,(l+r)*p);
GammaK(:,1+(l+r)*(i-1):(l+r)*p) = CK(:,1:(l+r)*(p+1-i));
SumKron = SumKron + kron(LL'*Z'*GammaK',S(:,1+l*(i-1):l*i));
end
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
rpbsid.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/rpbsid.m
| 14,530 |
utf_8
|
c27e37236a8ce6a7bf4db32bc0bc7212
|
function [Ak,Bk,Ck,Dk,Kk,err,eigA,regA] = rpbsid(u,y,f,p,n,W,idopts,rlsopts,A,B,C,D,K,s)
%RPBSID Recursive Predictor-based Subspace IDentification
% [A,B,C,D,K]=rpbsid(u,y,f,p,n) reursively estimates the matrices A, B, C,
% D and K of the state space model:
%
% x(N) = A x(N-1) + B u(N-1) + K e(N-1)
% y(N-1) = C x(N-1) + D u(N-1) + e(N-1)
%
% where N is the number of observations. The input matrix u and output
% matrix y must have the same number of observations but can have
% different numbers of variables. The past and future window size p and f
% must be higher then the expected order n.
%
% [A,B,C,D,K]=rpbsid(u,y,f,p,n,S) specifies the n times f*l permutation
% matrix S. The default is S=[eye(n) zeros(n,n-f)].
%
% [A,B,C,D,K]=rpbsid(u,y,f,p,n,S,idopts) specifies the identification
% options. The default is idopts =
% struct('method','varx','weight',0,'ltv',0,'noD',0,'past',0,'Kalm',0);
%
% [A,B,C,D,K]=rpbsid(u,y,f,p,n,S,idopts,rlsopts) specified the recursive
% least squares options. The default is rlsopts = struct('lambda',[0.999
% 0.999 0.999],'ireg',[1e-6 1e-6 1e6],'reg',0);
%
% [A,B,C,D,K]=rpbsid(u,y,f,p,n,S,idopts,rlsopts,S,A0,B0,C0,D0,K0,Ts)
% specifies the initial state-space matrices and sampling time.
%
% [A,B,C,D,K,err]=rpbsid(u,y,f,p,n,S) resturns the prediction error of the
% recursive least squares solvers.
%
% [A,B,C,D,K,err,eigA]=rpbsid(u,y,f,p,n,S) returns the eigenvalue and
% damping vectors over time.
%
% [A,B,C,D,K,err,eigA,regA]=rpbsid(u,y,f,p,n,S) returns the regularisation
% over time.
%
% [A,B,C,D,K,err,eigA,regA]=rpbsid(u,y,f,p,n,S,idopts,rlsopts,A0,B0,C0,D0,K0,Ts)
% specifies the sampling of returned vector err, eigA. The default is Ts=0
% (off).
%
% References:
% [1] Ali H. Sayed, "Adaptive Filters", Wiley and Sons, 2008
% Ivo Houtzager
% Delft Center of Systems and Control
% The Netherlands, 2010
% check number if input arguments
if nargin < 5
error('RPBSID requires at least five input arguments.')
end
% Determine sizes
if size(u,1) > size(u,2)
u = u';
end
if size(y,1) > size(y,2)
y = y';
end
r = size(u,1);
l = size(y,1);
N = size(y,2);
% Assign known values to the parameters
if nargin < 14 || isempty(s)
s = 0;
end
if nargin < 13 || isempty(K)
K = zeros(n,l);
end
if nargin < 12 || isempty(D)
D = zeros(l,r);
end
if nargin < 11 || isempty(C)
C = zeros(l,n);
end
if nargin < 10 || isempty(B)
B = zeros(n,r);
end
if nargin < 9 || isempty(A)
A = zeros(n,n);
end
if nargin < 8 || isempty(idopts)
rlsopts = struct('ireg',[1e-6 1e-6 1e-6],'lambda',[0.999 0.999 0.999],'reg',0);
end
if nargin < 7 || isempty(idopts)
idopts = struct('method','varx','weight',0,'ltv',0,'noD',0,'past',0,'Kalm',0);
end
if nargin < 6 || isempty(W)
W = [eye(n) zeros(n,f*l-n)];
end
switch lower(idopts.method)
case 'fir'
m = r;
case 'varx'
m = r+l;
case 'varmax'
m = r+2*l;
otherwise
error('Unknown type.')
end
% Initialisation of recursive least squares iterations
% Estimation of the Markov parameters
Plk = rlsopts.ireg(1); % Initial inverse of sample covariance matrix
CKD = zeros(l,p*m+~idopts.noD*r); % Initial solution
CKDS = zeros(f*l,p*m+~idopts.noD*r); % Initial solutions
P = zeros((p+f-1)*m,1); % Initial regression vector
reg_min = rlsopts.reg(1);
% Estimation of the output matrices
Pcd = rlsopts.ireg(2);
if idopts.noD
CD = C;
else
CD = [C D];
end
% Estimation of the state matrices
Pabk = rlsopts.ireg(3);
if strcmpi(idopts.method,'fir')
ABK = [A B];
else
ABK = [A B K];
end
% Initialisation of forward Ricatti iterations (if selected)
if idopts.Kalm
Px = rlsopts.ireg(3).*eye(n);
Q = rlsopts.ireg(3).*eye(n);
R = rlsopts.ireg(3).*eye(l);
S = zeros(n,l);
end
% Allocate state-space matrices for return
if s > 0
Ak = zeros(n,n,floor(N/s));
Bk = zeros(n,r,floor(N/s));
Ck = zeros(l,n,floor(N/s));
Dk = zeros(l,r,floor(N/s));
Ak(:,:,1) = A;
Bk(:,:,1) = B;
Ck(:,:,1) = C;
Dk(:,:,1) = D;
if strcmpi(idopts.method,'varx') || strcmpi(idopts.method,'varmax')
Kk = zeros(n,l,floor(N/s));
Kk(:,:,1) = K;
end
end
if nargout > 5
err = zeros(2*l+n,N);
end
if nargout > 6
eigA = zeros(n,N);
end
if nargout > 7
regA = zeros(1,N);
end
if idopts.past
if norm(W) > 1;
Pw = 1/rlsopts.ireg(1);
else
Pw = rlsopts.ireg(1);
end
W = W';
end
% Store vectors for next iteration
start = 2;
U1 = u(:,start-1);
Y1 = y(:,start-1);
E1 = zeros(l,1);
Xf1 = zeros(n,1);
% Start recursive identification
h = 1;
startA = 3;
for k = start:1:N
% New signal vector
switch lower(idopts.method)
case 'fir'
P = [P(m+1:end,:); U1];
case 'varx'
P = [P(m+1:end,:); U1; Y1];
case 'varmax'
P = [P(m+1:end,:); U1; Y1; E1];
otherwise
error('Unknown type.')
end
Y = y(:,k);
U = u(:,k);
if idopts.noD
Z = P((f-1)*m+1:end,:);
else
Z = [P((f-1)*m+1:end,:); U];
end
if k >= p
% Solve Regression problem recursively
if reg_min ~= 0
[CKD,Plk] = rls_ew_track_reg(Z,Y,CKD,Plk,rlsopts.lambda(1),reg_min);
else
[CKD,Plk] = rls_ew_track(Z,Y,CKD,Plk,rlsopts.lambda(1));
end
if nargout > 5
err(1:l,k-f+1) = Y - CKD*Z;
end
if nargout > 7
regA(:,k) = reg_min;
end
CKDS(1:(f-1)*l,:) = CKDS(l+1:f*l,:);
CKDS((f-1)*l+1:f*l,:) = CKD;
end
if k >= startA*p
% Construction of observability times controllability
LK = zeros(l*f,m*p);
if idopts.ltv
if idopts.weight
for i = 0:f-1
LK(i*l+1:(i+1)*l,i*m+1:p*m) = CKDS(i*l+1:(i+1)*l,1:(p-i)*m);
if i ~= 0
for j = 0:i-1
LK(i*l+1:(i+1)*l,:) = LK(i*l+1:(i+1)*l,:) + CKDS(i*l+1:(i+1)*l,(p-i+j)*m+r+(1:l))*LK(j*l+1:(j+1)*l,:);
end
end
end
else
for i = 1:f
LK((i-1)*l+1:i*l,p*m-(p-i+1)*m+1:p*m) = CKDS((i-1)*l+1:i*l,1:(p-i+1)*m);
end
end
else
if idopts.weight
for i = 0:f-1
LK(i*l+1:(i+1)*l,i*m+1:p*m) = CKD(:,1:(p-i)*m);
if i ~= 0
for j = 0:i-1
LK(i*l+1:(i+1)*l,:) = LK(i*l+1:(i+1)*l,:) + CKD(:,(p-i+j)*m+r+(1:l))*LK(j*l+1:(j+1)*l,:);
end
end
end
else
for i = 1:f
LK((i-1)*l+1:i*l,p*m-(p-i+1)*m+1:p*m) = CKD(:,1:(p-i+1)*m);
end
end
end
% Predicte future signal vector (= state estimate X)
if idopts.ltv
Xf = LK*P(1:p*m,:);
Uf1 = P((p-1)*m+1:(p-1)*m+r,1);
Yf1 = P((p-1)*m+r+1:(p-1)*m+r+l,1);
else
Xf = LK*P((f-1)*m+1:(p+f-1)*m,:);
Uf1 = P((p+f-2)*m+1:(p+f-2)*m+r,1);
Yf1 = P((p+f-2)*m+r+1:(p+f-2)*m+r+l,1);
end
if idopts.past
[W,Pw] = rls_ew_track(W'*Xf,Xf,W,Pw,rlsopts.lambda(1));
Xf = W'*Xf;
else
Xf = W*Xf;
end
end
if k >= startA*p+1
% The estimation of the system matrices
if idopts.noD
[CD,Pcd] = rls_ew_track(Xf1,Yf1,CD,Pcd,rlsopts.lambda(2));
if nargout > 5
err(l+1:2*l,k) = Yf1 - CD*Xf1;
end
Ef1 = Yf1 - CD*Xf1;
else
[CD,Pcd] = rls_ew_track([Xf1; Uf1],Yf1,CD,Pcd,rlsopts.lambda(2));
if nargout > 5
err(l+1:2*l,k-f+1) = Yf1 - CD*[Xf1; Uf1];
end
Ef1 = Yf1 - CD*[Xf1; Uf1];
end
if strcmpi(idopts.method,'fir')
[ABK,Pabk] = rls_ew_track([Xf1; Uf1],Xf,ABK,Pabk,rlsopts.lambda(3));
if nargout > 5
err(2*l+1:2*l+n,k) = Xf - ABK*[Xf1; Uf1];
end
if nargout > 6
eigA(:,k) = sort(eig(ABK(:,1:n)));
end
else
[ABK,Pabk] = rls_ew_track([Xf1; Uf1; Ef1],Xf,ABK,Pabk,rlsopts.lambda(3));
if nargout > 5
err(2*l+1:2*l+n,k) = Xf - ABK*[Xf1; Uf1; Ef1];
end
if nargout > 6
eigA(:,k) = sort(eig(ABK(:,1:n)));
end
end
% Estimate stable Kalman gain by the forward Riccati iteration
if idopts.Kalm
VW = [Xf; Yf1] - [ABK(:,1:n+r); CD zeros(l,idopts.noD*r)]*[Xf1; Uf1];
VW = VW*VW';
Q = 0.5.*(VW(1:n,1:n) + rlsopts.lambda(3).*Q);
R = 0.5.*(VW(n+1:n+l,n+1:n+l) + rlsopts.lambda(3).*R);
S = 0.5.*(VW(1:n,n+1:n+l) + rlsopts.lambda(3).*S);
K = (ABK(:,1:n)*Px*CD(:,1:n)' + S)/(R + CD(:,1:n)*Px*CD(:,1:n)');
Px = ABK(:,1:n)*Px*ABK(:,1:n)' + Q - ABK(:,(n+r+1):(n+r+l))*(ABK(:,1:n)*Px*CD(:,1:n)' + S)';
end
end
% Store state-space matrices (if selected)
if k-f == h*s && s ~= 0
if k >= startA*p+1
Ak(:,:,h) = ABK(:,1:n);
Ck(:,:,h) = CD(:,1:n);
if strcmpi(idopts.method,'fir')
Bk(:,:,h) = ABK(:,n+1:n+r);
if idopts.noD
Dk(:,:,h) = zeros(l,r);
else
Dk(:,:,h) = CD(:,n+1:n+r);
end
else
if idopts.Kalm
Bk(:,:,h) = [ABK(:,n+1:n+r) K];
else
Bk(:,:,h) = ABK(:,n+1:end);
end
if idopts.noD
Dk(:,:,h) = [zeros(l,r) eye(l)];
else
Dk(:,:,h) = [CD(:,n+1:n+r) eye(l)];
end
end
end
h = h + 1;
end
% Store vectors and matrices for next iteration
if k >= startA*p
Xf1 = Xf;
end
if k < startA*p+1
E = E1;
end
Y1 = Y;
U1 = U;
E1 = E;
end
if nargout >= 1 && s == 0
% Store state-space matrices
Ak = ABK(:,1:n);
Ck = CD(:,1:n);
if strcmpi(idopts.method,'fir')
Bk = ABK(:,n+1:n+r);
if idopts.noD
Dk = zeros(l,r);
else
Dk = CD(:,n+1:n+r);
end
else
Bk = ABK(:,n+1:n+r);
if idopts.Kalm
Kk = K;
else
Kk = ABK(:,n+r+1:end);
end
if idopts.noD
Dk = zeros(l,r);
else
Dk = CD(:,n+1:n+r);
end
end
end
end % end of function RPBSID
function [theta,P] = rls_ew_track(z,y,theta,P,lambda)
%RLS_EW_TRACK Exponentially Weighted RLS iteration
% [THETA,P]=RLS_EW_TRACK(Z,Y,THETA,P,LAMBDA) applies one iteration of
% exponentially weighted regularized least-squares problem. In recursive
% least-squares, we deal with the issue of an increasing amount of date Z
% and Y. At each iteration, THETA is the solution. The scalar LAMBDA is
% called the forgetting factor since past data are exponentially weighted
% less heavily than more recent data.
% Ivo Houtzager
% Delft Center of Systems and Control
% The Netherlands, 2010
% Assign default values to unspecified parameters
mz = size(z,1);
if (nargin < 5) || isempty(lambda)
lambda = 1;
end
if (nargin < 4) || isempty(P)
P = zeros(mz);
elseif isscalar(P)
P = (1/P).*eye(mz);
end
if (nargin < 3) || isempty(theta)
theta = zeros(size(y,1),mz);
end
% Exponentially-Weighted, Tracking, Regularized, Least-Squares Iteration
P = (1/lambda).*(P - P*(z*z')*P./(lambda + z'*P*z));
P = 0.5.*(P+P'); % force symmetric
e = y - theta*z;
theta = theta + e*z'*P;
end % end of function RLS_EW_TRACK
function [theta,P] = rls_ew_track_reg(z,y,theta,P,lambda,reg_min)
%RLS_EW_TRACK_REG Exponentially Weighted and Regularized RLS iteration
% [THETA,P]=RLS_EW_TRACK_REG(Z,Y,THETA,P,LAMBDA,REG) applies one iteration
% of exponentially weighted regularized least-squares problem. In
% recursive least-squares, we deal with the issue of an inceasing amount
% of date Z and Y. At each iteration, THETA is the solution. The scalar
% LAMBDA is called the forgetting factor since past data are exponentially
% weighted less heavily than more recent data.
% Ivo Houtzager
% Delft Center of Systems and Control
% The Netherlands, 2010
% Assign default values to unspecified parameters
mz = size(z,1);
if (nargin < 6) || isempty(reg_min)
reg_min = 0;
end
if (nargin < 5) || isempty(lambda)
lambda = 1;
end
if (nargin < 4) || isempty(P)
P = zeros(mz);
elseif isscalar(P)
P = (1/P).*eye(mz);
end
if (nargin < 3) || isempty(theta)
theta = zeros(size(y,1),mz);
end
% Exponentially-Weighted, Tracking, Regularized, Least-Squares Iteration
P = (1/lambda).*(P - P*(z*z')*P./(lambda + z'*P*z));
P = 0.5.*(P+P'); % force symmetric
if isscalar(reg_min)
opts.SYM = true;
opts.POSDEF = true;
P1 = linsolve((eye(size(P)) + reg_min^2.*P),P,opts);
e = y - theta*z;
theta = theta + e*z'*P1;
elseif strcmpi(reg_min,'tikh')
[U,S,V] = svd(pinv(P));
s = diag(S);
YP = (y-theta*z)';
if isscalar(opt)
reg_min = opt;
elseif strcmpi(opt,'lcurve')
reg_min = reglcurve(YP,U,s);
elseif strcmpi(opt,'gcv')
reg_min = reggcv(YP,U,s);
end
theta = theta + (V*(diag(s./(s.^2 + reg_min^2)))*U'*YP)';
elseif strcmpi(reg_min,'tsvd')
[U,S,V] = svd(pinv(P));
s = diag(S);
YP = (y-theta*z)';
if isscalar(opt)
k_min = opt;
elseif strcmpi(opt,'lcurve')
k_min = reglcurve(YP,U,s,'tsvd');
elseif strcmpi(opt,'gcv')
k_min = reggcv(YP,U,s,'tsvd');
end
theta = theta + (V(:,1:k_min)*diag(1./s(1:k_min))*U(:,1:k_min)'*YP)';
end
end % end of function RLS_EW_TRACK
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
sim.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/@idafflpv/sim.m
| 3,203 |
utf_8
|
0b3bda0b8d5cab02080d8f3436ebb3d0
|
function [y,t,x] = sim(sys,u,t,p,e,x0)
%SIM Linear response simulation of affine LPV state-space model.
% [Y,T,X] = SIM(M,U,T,MU) returns the output response of the IDAFFLPV
% model M to the input and scheduling signal described by U, MU and T.
% The time vector T consists of regularly spaced time samples, U and MU is
% are matrices with as many columns as inputs and scheduling variables and
% whose i-th row specifies the input value at time T(i). For discrete-time
% models, U should be sampled at the same rate as M.
%
% [Y,T,X] = SIM(M,U,T,MU,E) adds the innovation noise to the simulation.
%
% [Y,T,X] = SIM(M,U,T,MU,E,X0) specifies the initial state vector X0 at
% time T(1). X0 is set to zero when omitted.
% Get the system matrices
[a b c d k] = getABCDK(sys);
% Define sizes
N = length(t);
Ny = size(c,1);
[Ns,nu] = size(u);
[Nn,np] = size(p);
if size(u,1) < size(u,2);
u = u';
end
if size(t,1) < size(t,2);
t = t';
end
if size(p,1) < size(p,2);
p = p';
end
if ~(nargin < 5 || isempty(e))
if size(e,1) < size(e,2);
e = e';
end
else
e = zeros(N,Ny);
end
[Ny,Nu,Nx,Np] = size(sys);
% Computability and consistency checks
if ~isequal(N,Ns,Nn)
error('Number of samples in vector T, U and P must be equal.')
end
if nu ~= Nu
error('Input data U must have as many columns as system inputs.')
end
if np ~= Np
error('Input data P must have as many columns as scheduling parameters.')
end
% Assign values to unspecified parameters
if nargin < 6 || isempty(x0)
x0 = zeros(Nx,1);
elseif length(x0)~=Nx
error('Length of initial condition X0 must match number of states.')
end
if isct(sys) % Continuous models
% Determine the states
tspan = [t(1) t(end)];
options = odeset('RelTol',1e-6,'AbsTol',1e-6);
[tc,x] = ode15s(@statesim,tspan,x0,options,t,u,p,e,a,b,k);
% Determine the output
y = zeros(length(tc),Ny);
uc = interp1q(t,u,tc)';
pc = interp1q(t,p,tc)';
ec = interp1q(t,e,tc)';
for i = 1:length(tc)
y(i,:) = c*(kron([1; pc(:,i)],x(i,:)')) + d*(kron([1; pc(:,i)],uc(:,i))) + ec(:,i);
end
% Format output arrays
y = y.';
t = tc;
else % Discrete models
Ts = sys.Ts;
if length(t)>1 && Ts>0 && abs(t(2)-t(1)-Ts)>1e-4*Ts
error('Time step must match sample time of discrete-time models.')
end
bt = zeros(Nx,(Np+1)*(Nu+Ny));
dt = zeros(Ny,(Np+1)*(Nu+Ny));
for i = 1:Np+1
bt(:,(i-1)*(Nu+Ny)+1:i*(Nu+Ny)) = [b(:,(i-1)*Nu+1:i*Nu) k(:,(i-1)*Ny+1:i*Ny)];
if i == 1
dt(:,(i-1)*(Nu+Ny)+1:i*(Nu+Ny)) = [d(:,(i-1)*Nu+1:i*Nu) eye(Ny)];
else
dt(:,(i-1)*(Nu+Ny)+1:i*(Nu+Ny)) = [d(:,(i-1)*Nu+1:i*Nu) zeros(Ny)];
end
end
[y,x] = lpvsim(a,bt,c,dt,p,[u e],[],x0);
end
end
% State-derivative function used for the simulation of continuous models
function dx = statesim(ts,xs,t,u,p,e,a,b,k)
ps = interp1q(t,p,ts)';
us = interp1q(t,u,ts)';
es = interp1q(t,e,ts)';
dx = a*(kron([1; ps],xs)) + b*(kron([1; ps],us))+ k*(kron([1; ps],es));
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
display.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/@idafflpv/display.m
| 7,346 |
utf_8
|
7b6e9ea522f9f96baae2cb837ea575e0
|
function display(sys)
%IDAFFLPV/DISPLAY Pretty-print for IDAFFLPV models.
% Get size
[Ny,Nu] = size(sys);
% Use ISSTATIC to account for delays
StaticFlag = isstatic(sys);
% Handle various types
if ((Ny==0 || Nu==0) && StaticFlag)
disp(xlate('Empty affine LPV state-space model.'))
else
% Display name
disp(xlate(sys.Name))
% Single IDAFFLPV model
dispsys(sys,'')
% Display IDAFFLPV properties (sample times)
if ~StaticFlag,
if sys.Ts<0,
disp(xlate('Sampling time: unspecified'))
elseif sys.Ts>0,
disp(sprintf('Sampling time: %0.5g',sys.Ts))
end
end
% Last line
if StaticFlag,
disp(xlate('Static gain.'))
elseif sys.Ts==0,
disp(xlate('Continuous-time affine LPV state-space model.'))
else
disp(xlate('Discrete-time affine LPV state-space model.'));
end
end
end
function dispsys(sys,LeftMargin)
%DISPLAY Pretty-print for affine LPV state-space models.
% Print matrices
printsys(sys.a,sys.b,sys.c,sys.d,sys.k,sys.InputName,sys.OutputName,sys.StateName,sys.SchedulingName,LeftMargin);
end
function printsys(a,b,c,d,k,ulabels,ylabels,xlabels,plabels,LeftMargin)
%PRINTSYS Print system in pretty format.
%
% PRINTSYS is used to print state space systems with labels to the
% right and above the system matrices.
%
% PRINTSYS(A,B,C,D,K,E,ULABELS,YLABELS,XLABELS) prints the state-space
% system with the input, output and state labels contained in the
% cellarrays ULABELS, YLABELS, and XLABELS, respectively.
%
% PRINTSYS(A,B,C,D,K) prints the system with numerical labels.
%
% See also: PRINTMAT
nx = size(a,1);
np = size(a,2)/nx;
[ny,nu] = size(d);
nu = nu/np;
if ((isempty(ulabels) || isequal('',ulabels{:})) && (isempty(plabels) || isequal('',plabels{:})))
for j=1:np
for i=1:nu,
if j == 1
ulabels{i} = sprintf('u%d',i);
else
ulabels{(j-1)*nu+i} = sprintf('u%d*p%d',i,j-1);
end
end
end
elseif (isempty(ulabels) || isequal('',ulabels{:}))
for j=1:np
if j ~= 1
if isempty(plabels{j-1})
plabels{j-1} = '?';
end
end
for i=1:nu,
if j == 1
ulabels{i} = sprintf('u%d',i);
else
ulabels{(j-1)*nu+i} = strcat(sprintf('u%d',i),'*',plabels{j-1});
end
end
end
else
for j=1:np
if j ~= 1
if isempty(plabels{j-1})
plabels{j-1} = '?';
end
end
for i=1:nu,
if j == 1
if isempty(ulabels{i})
ulabels{i} = '?';
end
else
ulabels{(j-1)*nu+i} = strcat(ulabels{i},'*',plabels{j-1});
end
end
end
end
if ((isempty(ylabels) || isequal('',ylabels{:})) && (isempty(plabels) || isequal('',plabels{:})))
for j=1:np
for i=1:ny,
if j == 1
ylabels{i} = sprintf('y%d',i);
else
ylabels{(j-1)*ny+i} = sprintf('y%d*p%d',i,j-1);
end
end
end
elseif (isempty(ylabels) || isequal('',ylabels{:}))
for j=1:np
if j ~= 1
if isempty(plabels{j-1})
plabels{j-1} = '?';
end
end
for i=1:ny,
if j == 1
ylabels{i} = sprintf('y%d',i);
else
ylabels{(j-1)*ny+i} = strcat(sprintf('y%d',i),'*',plabels{j-1});
end
end
end
else
for j=1:np
if j ~= 1
if isempty(plabels{j-1})
plabels{j-1} = '?';
end
end
for i=1:ny,
if j == 1
if isempty(ylabels{i})
ylabels{i} = '?';
end
else
ylabels{(j-1)*ny+i} = strcat(ylabels{i},'*',plabels{j-1});
end
end
end
end
if ((isempty(xlabels) || isequal('',xlabels{:})) && (isempty(plabels) || isequal('',plabels{:})))
for j=1:np
for i=1:nx,
if j == 1
xlabels{i} = sprintf('x%d',i);
else
xlabels{(j-1)*nx+i} = sprintf('x%d*p%d',i,j-1);
end
end
end
elseif (isempty(xlabels) || isequal('',xlabels{:}))
for j=1:np
if j ~= 1
if isempty(plabels{j-1})
plabels{j-1} = '?';
end
end
for i=1:nx,
if j == 1
xlabels{i} = sprintf('x%d',i);
else
xlabels{(j-1)*nx+i} = strcat(sprintf('x%d',i),'*',plabels{j-1});
end
end
end
else
for j=1:np
if j ~= 1
if isempty(plabels{j-1})
plabels{j-1} = '?';
end
end
for i=1:nx,
if j == 1
if isempty(xlabels{i})
xlabels{i} = '?';
end
else
xlabels{(j-1)*nx+i} = strcat(xlabels{i},'*',plabels{j-1});
end
end
end
end
disp(' ')
if isempty(a),
% Gain matrix
printmat(d,[LeftMargin 'd'],ylabels,ulabels);
else
printmat(a,[LeftMargin 'a'],xlabels,xlabels);
printmat(b,[LeftMargin 'b'],xlabels,ulabels);
printmat(c,[LeftMargin 'c'],ylabels,xlabels);
printmat(d,[LeftMargin 'd'],ylabels,ulabels);
printmat(k,[LeftMargin 'k'],xlabels,ylabels);
end
end
function printmat(a,name,rlab,clab)
%PRINTMAT Print matrix with labels.
% PRINTMAT(A,NAME,RLAB,CLAB) prints the matrix A with the row labels
% RLAB and column labels CLAB. NAME is a string used to name the
% matrix. RLAB and CLAB are cell vectors of strings.
%
% See also PRINTSYS.
CWS = get(0,'CommandWindowSize'); % max number of char. per line
MaxLength = round(.9*CWS(1));
[nrows,ncols] = size(a);
len = 12; % Max length of labels
Space = ' ';
% Print name
%disp(' ')
if ~isempty(name),
disp([name,' = ']),
end
% Empty case
if (nrows==0) || (ncols==0),
if (nrows==0) && (ncols==0),
disp(' []')
else
disp(sprintf(' Empty matrix: %d-by-%d',nrows,ncols));
end
disp(' ')
return
end
% Row labels
RowLabels = strjust(strvcat(' ',rlab{1:nrows}),'left');
RowLabels = RowLabels(:,1:min(len,end));
RowLabels = [Space(ones(nrows+1,1),ones(3,1)),RowLabels];
% Construct matrix display
Columns = cell(1,ncols);
prec = 3 + isreal(a);
for ct=1:ncols,
clab{ct} = clab{ct}(:,1:min(end,len));
col = [clab(ct); cellstr(deblank(num2str(a(:,ct),prec)))];
col = strrep(col,'+0i',''); % xx+0i->xx
Columns{ct} = strjust(strvcat(col{:}),'right');
end
% Equalize column width
lc = cellfun('size',Columns,2);
lcmax = max(lc)+2;
for ct=1:ncols,
Columns{ct} = [Space(ones(nrows+1,1),ones(lcmax-lc(ct),1)) , Columns{ct}];
end
% Display MAXCOL columns at a time
maxcol = max(1,round((MaxLength-size(RowLabels,2))/lcmax));
for ct=1:ceil(ncols/maxcol)
disp([RowLabels Columns{(ct-1)*maxcol+1:min(ct*maxcol,ncols)}]);
disp(' ');
end
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
findstates.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/@idafflpv/findstates.m
| 5,220 |
utf_8
|
80ca78bbdbfb3de9885bda4a8fe898d2
|
function x0 = findstates(sys,u,y,t,p,type,wn)
%FINDSTATES Estimate initial states of the model for a given data set.
% X0 = FINDSTATES(M,U,Y,T,MU) returns the residue response and initial
% state of the IDAFFLPV model M to the input and scheduling signal
% described by U, Y, MU and T. The time vector T consists of regularly
% spaced time samples, U, Y, and MU is are matrices with as many columns
% as inputs and scheduling variables and whose i-th row specifies the
% input value at time T(i). For discrete-time models, U, Y, and MU should
% be sampled at the same rate as M.
%
% X0 = FINDSTATES(M,U,Y,T,MU,'Type') specifies the type of LPV predictor.
% 'K' specifies that K is not dependent on scheduling, or 'CD' specifies
% that C and D is not dependent on scheduling. Default is Type = 'CD'.
%
% X0 = FINDSTATES(M,U,Y,T,MU,'Type',P) specifies the past window P. The
% default is P = 5*size(M.a,1).
%
% NOTE: This is only possible if K or C and D are not dependent on the
% scheduling sequence.
% Define sizes
N = length(t);
if size(u,1) < size(u,2);
u = u';
end
if size(y,1) < size(y,2);
y = y';
end
if size(t,1) < size(t,2);
t = t';
end
if size(p,1) < size(p,2);
p = p';
end
[Ns,nu] = size(u);
[Nn,np] = size(p);
[Ni,ny] = size(y);
[Ny,Nu,Nx,Np] = size(sys);
% Computability and consistency checks
if ~isequal(N,Ns,Nn,Ni)
error('Number of samples in vector T, U and P must be equal.')
end
if ny ~= Ny
error('Input data Y must have as many columns as system outputs.')
end
if nu ~= Nu
error('Input data U must have as many columns as system inputs.')
end
if np ~= Np
error('Input data P must have as many columns as scheduling parameters.')
end
% Assign values to unspecified parameters
if nargin < 7 || isempty(wn)
wn = 5*Nx;
end
if nargin < 6 || isempty(type)
type = 'CD';
end
% Get the system matrices
[a b c d k] = getABCDK(sys);
if strcmpi(type,'CD')
for i = 1:Np+1
a(:,(i-1)*Nx+1:i*Nx) = a(:,(i-1)*Nx+1:i*Nx) - k(:,(i-1)*Ny+1:i*Ny)*c(:,1:Nx);
b(:,(i-1)*Nu+1:i*Nu) = b(:,(i-1)*Nu+1:i*Nu) - k(:,(i-1)*Ny+1:i*Ny)*d(:,1:Nu);
end
elseif strcmpi(type,'K')
for i = 1:Np+1
a(:,(i-1)*Nx+1:i*Nx) = a(:,(i-1)*Nx+1:i*Nx) - k(:,1:Ny)*c(:,(i-1)*Nx+1:i*Nx);
b(:,(i-1)*Nu+1:i*Nu) = b(:,(i-1)*Nu+1:i*Nu) - k(:,1:Ny)*d(:,(i-1)*Nu+1:i*Nu);
end
else
error('Type not recognized!')
end
bt = zeros(Nx,(Np+1)*(Nu+Ny));
dt = zeros(Ny,(Np+1)*(Nu+Ny));
for i = 1:Np+1
bt(:,(i-1)*(Nu+Ny)+1:i*(Nu+Ny)) = [b(:,(i-1)*Nu+1:i*Nu) k(:,(i-1)*Ny+1:i*Ny)];
dt(:,(i-1)*(Nu+Ny)+1:i*(Nu+Ny)) = [d(:,(i-1)*Nu+1:i*Nu) zeros(ny)];
end
b = bt;
d = dt;
u = [u y];
if isct(sys) % Continuous models
yr = y;
options = optimset('Display','off');
x0 = lsqnonlin(@residue,ones(Nx,1),[],[],options,u(1:wn,:),yr(1:wn,:),t(1:wn,:),p(1:wn,:),a,b,c,d,type);
else
% Get the system matrices
[Altv,Bltv,Cltv,Dltv] = lpv2ltv(a,b,c,d,p(1:wn,:));
[HU,Gamma] = lift(Altv,Bltv,Cltv,Dltv,wn);
Y = y(1:wn,:)';
Y = Y(:);
U = u(1:wn,:)';
U = U(:);
Y = Y - HU*U;
x0 = pinv(Gamma)*Y;
end
end
% State-derivative function used for the simulation of continuous models
function cost = residue(x0,u,yr,t,p,a,b,c,d,type)
Ny = size(yr,2);
if strcmpi(type,'K')
error('Continuous models with varying C and D is not yet implemented.')
end
% Determine the states
tspan = [t(1) t(end)];
options = odeset('RelTol',1e-6,'AbsTol',1e-6);
[tc,x] = ode15s(@statesim,tspan,x0,options,t,u,p,a,b);
% Determine the output
y = zeros(length(tc),Ny);
uc = interp1q(t,u,tc)';
pc = interp1q(t,p,tc)';
for i = 1:length(tc)
y(i,:) = c*(kron([1; pc(:,i)],x(i,:)')) + d*(kron([1; pc(:,i)],uc(:,i)));
end
% Format output arrays
y = y.';
t = tc;
yc = interp1q(t,yr,tc)';
cost = pec(yc,y);
end
% State-derivative function used for the simulation of continuous models
function dx = statesim(ts,xs,t,u,p,a,b)
ps = interp1q(t,p,ts)';
us = interp1q(t,u,ts)';
dx = a*(kron([1; ps],xs)) + b*(kron([1; ps],us));
end
function [HU,Gamma] = lift(A,B,C,D,p)
%LIFT Lift state-spave matrices
% written by, I. Houtzager [2007]
% Delft Center of Systems and Control
% determine lpv system sizes
r = size(B{1},2); % The number of inputs
l = size(C{1},1); % The number of outputs
n = size(A{1},1); % The number of states
% build lifted impulse matrix
HU = zeros(p*l,p*r);
Gamma = zeros(p*l,n);
for i = 1:p
for j = 1:p
if i == j
if isempty(D)
HU((i-1)*l+1:i*l,(j-1)*r+1:j*r) = zeros(l,r);
else
HU((i-1)*l+1:i*l,(j-1)*r+1:j*r) = D{i};
end
elseif j < i
if j == i-1
HU((i-1)*l+1:i*l,(j-1)*r+1:j*r) = C{i}*B{j};
else
T = C{i};
for k = i-1:-1:j+1
T = T*A{k};
end
HU((i-1)*l+1:i*l,(j-1)*r+1:j*r) = T*B{j};
end
end
end
T = C{i};
for k = i-1:-1:1
T = T*A{k};
end
Gamma((i-1)*l+1:i*l,:) = T;
end
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
predict.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/@idafflpv/predict.m
| 3,900 |
utf_8
|
1550385bd044b2391df567f2e4ec70de
|
function [y,t,x] = predict(sys,u,y,t,p,x0,type)
%PREDICT Linear response simulation of affine LPV state-space predictor.
% [Y,T,X] = PREDICT(M,U,Y,T,MU) returns the predicted output response of
% the IDAFFLPV model M to the input and scheduling signal described by U,
% Y, MU and T. The time vector T consists of regularly spaced time
% samples, U, Y, and MU is are matrices with as many columns as inputs and
% scheduling variables and whose i-th row specifies the input value at
% time T(i). For discrete-time models, U, Y, and MU should be sampled at
% the same rate as M.
%
% [Y,T,X] = PREDICT(M,U,Y,T,MU,X0) specifies the initial state vector X0
% at time T(1). X0 is taken from model.
%
% [Y,T,X] = PREDICT(M,U,Y,T,MU,X0,'Type') specifies the type of LPV
% predictor. 'K' specifies that K does not dependent on scheduling, or
% 'CD' specifies that C and D does not dependent on scheduling. Default is
% Type = 'CD'.
%
% NOTE: This is only possible if K or C and D are not dependent on the
% scheduling sequence.
% Define sizes
N = length(t);
if size(u,1) < size(u,2);
u = u';
end
if size(y,1) < size(y,2);
y = y';
end
if size(t,1) < size(t,2);
t = t';
end
if size(p,1) < size(p,2);
p = p';
end
[Ns,nu] = size(u);
[Nn,np] = size(p);
[Ni,ny] = size(y);
[Ny,Nu,Nx,Np] = size(sys);
% Computability and consistency checks
if ~isequal(N,Ns,Nn,Ni)
error('Number of samples in vector T, U and P must be equal.')
end
if ny ~= Ny
error('Input data Y must have as many columns as system outputs.')
end
if nu ~= Nu
error('Input data U must have as many columns as system inputs.')
end
if np ~= Np
error('Input data P must have as many columns as scheduling parameters.')
end
% Assign values to unspecified parameters
if nargout < 7 || isempty(type)
type = 'CD';
end
if nargout < 5 || isempty(x0)
x0 = sys.x0;
elseif length(x0)~=Nx
error('Length of initial condition X0 must match number of states.')
end
% Get the system matrices
[a b c d k] = getABCDK(sys);
if strcmpi(type,'CD')
for i = 1:Np+1
a(:,(i-1)*Nx+1:i*Nx) = a(:,(i-1)*Nx+1:i*Nx) - k(:,(i-1)*Ny+1:i*Ny)*c(:,1:Nx);
b(:,(i-1)*Nu+1:i*Nu) = b(:,(i-1)*Nu+1:i*Nu) - k(:,(i-1)*Ny+1:i*Ny)*d(:,1:Nu);
end
elseif strcmpi(type,'K')
for i = 1:Np+1
a(:,(i-1)*Nx+1:i*Nx) = a(:,(i-1)*Nx+1:i*Nx) - k(:,1:Ny)*c(:,(i-1)*Nx+1:i*Nx);
b(:,(i-1)*Nu+1:i*Nu) = b(:,(i-1)*Nu+1:i*Nu) - k(:,1:Ny)*d(:,(i-1)*Nu+1:i*Nu);
end
else
error('Type not recognized!')
end
bt = zeros(Nx,(Np+1)*(Nu+Ny));
dt = zeros(Ny,(Np+1)*(Nu+Ny));
for i = 1:Np+1
bt(:,(i-1)*(Nu+Ny)+1:i*(Nu+Ny)) = [b(:,(i-1)*Nu+1:i*Nu) k(:,(i-1)*Ny+1:i*Ny)];
dt(:,(i-1)*(Nu+Ny)+1:i*(Nu+Ny)) = [d(:,(i-1)*Nu+1:i*Nu) zeros(ny)];
end
b = bt;
d = dt;
u = [u y];
if isct(sys) % Continuous models
% Determine the states
tspan = [t(1) t(end)];
options = odeset('RelTol',1e-6,'AbsTol',1e-6);
[tc,x] = ode15s(@statesim,tspan,x0,options,t,u,p,a,b);
% Determine the output
y = zeros(length(tc),Ny);
uc = interp1q(t,u,tc)';
pc = interp1q(t,p,tc)';
for i = 1:length(tc)
y(i,:) = c*(kron([1; pc(:,i)],x(i,:)')) + d*(kron([1; pc(:,i)],uc(:,i)));
end
% Format output arrays
y = y.';
t = tc;
else % Discrete models
Ts = sys.Ts;
if length(t)>1 && Ts>0 && abs(t(2)-t(1)-Ts)>1e-4*Ts
error('Time step must match sample time of discrete-time models.')
end
% Discrete simulation of LPV systems
[y,x] = lpvsim(a,b,c,d,p,u,[],x0);
end
end
% State-derivative function used for the simulation of continuous models
function dx = statesim(ts,xs,t,u,p,a,b)
ps = interp1q(t,p,ts)';
us = interp1q(t,u,ts)';
dx = a*(kron([1; ps],xs)) + b*(kron([1; ps],us));
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
resid.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/@idafflpv/resid.m
| 4,059 |
utf_8
|
7f209685ad0625e1c20982807dfff7e1
|
function [e,t] = resid(sys,u,y,t,p,x0,type)
%RESID Compute the residuals associated with an IDAFFLPV.
% E = RESID(M,U,Y,T,MU) returns the residue response of the IDAFFLPV model
% M to the input and scheduling signal described by U, Y, MU and T. The
% time vector T consists of regularly spaced time samples, U, Y, and MU is
% are matrices with as many columns as inputs and scheduling variables and
% whose i-th row specifies the input value at time T(i). For discrete-time
% models, U, Y, and MU should be sampled at the same rate as M.
%
% E = RESID(M,U,Y,T,MU,X0) specifies the initial state vector X0 at time
% T(1). X0 is taken from model.
%
% E = RESID(M,U,Y,T,MU,X0,'Type') specifies the type of LPV predictor. 'K'
% specifies that K does not dependent on scheduling, or 'CD' specifies
% that C and D does not dependent on scheduling. Default is Type = 'CD'.
%
% NOTE: This is only possible if K or C and D are not dependent on the
% scheduling sequence.
% Define sizes
N = length(t);
if size(u,1) < size(u,2);
u = u';
end
if size(y,1) < size(y,2);
y = y';
end
if size(t,1) < size(t,2);
t = t';
end
if size(p,1) < size(p,2);
p = p';
end
[Ns,nu] = size(u);
[Nn,np] = size(p);
[Ni,ny] = size(y);
[Ny,Nu,Nx,Np] = size(sys);
% Computability and consistency checks
if ~isequal(N,Ns,Nn,Ni)
error('Number of samples in vector T, U and P must be equal.')
end
if ny ~= Ny
error('Input data Y must have as many columns as system outputs.')
end
if nu ~= Nu
error('Input data U must have as many columns as system inputs.')
end
if np ~= Np
error('Input data P must have as many columns as scheduling parameters.')
end
% Assign values to unspecified parameters
if nargout < 7 || isempty(type)
type = 'CD';
end
if nargout < 5 || isempty(x0)
x0 = sys.x0;
elseif length(x0)~=Nx
error('Length of initial condition X0 must match number of states.')
end
% Get the system matrices
[a b c d k] = getABCDK(sys);
if strcmpi(type,'CD')
for i = 1:Np+1
a(:,(i-1)*Nx+1:i*Nx) = a(:,(i-1)*Nx+1:i*Nx) - k(:,(i-1)*Ny+1:i*Ny)*c(:,1:Nx);
b(:,(i-1)*Nu+1:i*Nu) = b(:,(i-1)*Nu+1:i*Nu) - k(:,(i-1)*Ny+1:i*Ny)*d(:,1:Nu);
end
elseif strcmpi(type,'K')
for i = 1:Np+1
a(:,(i-1)*Nx+1:i*Nx) = a(:,(i-1)*Nx+1:i*Nx) - k(:,1:Ny)*c(:,(i-1)*Nx+1:i*Nx);
b(:,(i-1)*Nu+1:i*Nu) = b(:,(i-1)*Nu+1:i*Nu) - k(:,1:Ny)*d(:,(i-1)*Nu+1:i*Nu);
end
else
error('Type not recognized!')
end
bt = zeros(Nx,(Np+1)*(Nu+Ny));
dt = zeros(Ny,(Np+1)*(Nu+Ny));
for i = 1:Np+1
bt(:,(i-1)*(Nu+Ny)+1:i*(Nu+Ny)) = [b(:,(i-1)*Nu+1:i*Nu) k(:,(i-1)*Ny+1:i*Ny)];
if i == 1
dt(:,(i-1)*(Nu+Ny)+1:i*(Nu+Ny)) = [d(:,(i-1)*Nu+1:i*Nu) -eye(Ny)];
else
dt(:,(i-1)*(Nu+Ny)+1:i*(Nu+Ny)) = [d(:,(i-1)*Nu+1:i*Nu) zeros(ny)];
end
end
b = bt;
d = dt;
u = [u y];
if isct(sys) % Continuous models
if strcmpi(type,'K')
error('Continuous models with varying C and D is not yet implemented.')
end
% Determine the states
tspan = [t(1) t(end)];
options = odeset('RelTol',1e-6,'AbsTol',1e-6);
[tc,x] = ode15s(@statesim,tspan,x0,options,t,u,p,a,b);
% Determine the output
e = zeros(length(tc),Ny);
uc = interp1q(t,u,tc)';
pc = interp1q(t,p,tc)';
for i = 1:length(tc)
e(i,:) = c*(kron([1; pc(:,i)],x(i,:)')) + d*(kron([1; pc(:,i)],uc(:,i)));
end
% Format output arrays
e = e.';
t = tc;
else % Discrete models
Ts = sys.Ts;
if length(t)>1 && Ts>0 && abs(t(2)-t(1)-Ts)>1e-4*Ts
error('Time step must match sample time of discrete-time models.')
end
% Discrete simulation of LPV systems
e = lpvsim(a,b,c,d,p,u,[],x0);
end
end
% State-derivative function used for the simulation of continuous models
function dx = statesim(ts,xs,t,u,p,a,b)
ps = interp1q(t,p,ts)';
us = interp1q(t,u,ts)';
dx = a*(kron([1; ps],xs)) + b*(kron([1; ps],us));
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
spg_mmv.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/spg_mmv.m
| 2,900 |
utf_8
|
fdede75f442cc17862d6d4252d0d7cfd
|
function [x,r,g,info] = spg_mmv( A, B, sigma, options, x0)
%SPG_MMV Solve multi-measurement basis pursuit denoise (BPDN)
%
% SPG_MMV is designed to solve the basis pursuit denoise problem
%
% (BPDN) minimize ||X||_1,2 subject to ||A X - B||_2,2 <= SIGMA,
%
% where A is an M-by-N matrix, B is an M-by-G matrix, and SIGMA is a
% nonnegative scalar. In all cases below, A can be an explicit M-by-N
% matrix or matrix-like object for which the operations A*x and A'*y
% are defined (i.e., matrix-vector multiplication with A and its
% adjoint.)
%
% Also, A can be a function handle that points to a function with the
% signature
%
% v = A(w,mode) which returns v = A *w if mode == 1;
% v = A'*w if mode == 2.
%
% X = SPG_MMV(A,B,SIGMA) solves the BPDN problem. If SIGMA=0 or
% SIGMA=[], then the basis pursuit (BP) problem is solved; i.e., the
% constraints in the BPDN problem are taken as AX=B.
%
% X = SPG_MMV(A,B,SIGMA,OPTIONS) specifies options that are set using
% SPGSETPARMS.
%
% [X,R,G,INFO] = SPG_BPDN(A,B,SIGMA,OPTIONS) additionally returns the
% residual R = B - A*X, the objective gradient G = A'*R, and an INFO
% structure. (See SPGL1 for a description of this last output argument.)
%
% See also spgl1, spgSetParms, spg_bp, spg_lasso.
% Copyright 2008, Ewout van den Berg and Michael P. Friedlander
% http://www.cs.ubc.ca/labs/scl/spgl1
% $Id$
if ~exist('options','var'), options = []; end
if ~exist('x0','var'), x0 = []; else x0 = x0(:); end
if ~exist('sigma','var'), sigma = 0; end
if ~exist('B','var') || isempty(B)
error('Second argument cannot be empty.');
end
if ~exist('A','var') || isempty(A)
error('First argument cannot be empty.');
end
groups = size(B,2);
if isa(A,'function_handle')
y = A(B(:,1),2); m = size(B,1); n = length(y);
A = @(x,mode) blockDiagonalImplicit(A,m,n,groups,x,mode);
else
m = size(A,1); n = size(A,2);
A = @(x,mode) blockDiagonalExplicit(A,m,n,groups,x,mode);
end
% Set projection specific functions
options.project = @(x,weight,tau) NormL12_project(groups,x,weight,tau);
options.primal_norm = @(x,weight ) NormL12_primal(groups,x,weight);
options.dual_norm = @(x,weight ) NormL12_dual(groups,x,weight);
tau = 0;
[x,r,g,info] = spgl1(A,B(:),tau,sigma,x0,options);
n = round(length(x) / groups);
m = size(B,1);
x = reshape(x,n,groups);
y = reshape(r,m,groups);
g = reshape(g,n,groups);
function y = blockDiagonalImplicit(A,m,n,g,x,mode)
if mode == 1
y = zeros(m*g,1);
for i=1:g
y(1+(i-1)*m:i*m) = A(x(1+(i-1)*n:i*n),mode);
end
else
y = zeros(n*g,1);
for i=1:g
y(1+(i-1)*n:i*n) = A(x(1+(i-1)*m:i*m),mode);
end
end
function y = blockDiagonalExplicit(A,m,n,g,x,mode)
if mode == 1
y = A * reshape(x,n,g);
y = y(:);
else
x = reshape(x,m,g);
y = (x' * A)';
y = y(:);
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
spg_mmv_stopvali.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/spg_mmv_stopvali.m
| 3,198 |
utf_8
|
1590321e24fb5c98666449eb424584dc
|
function [x,r,g,info] = spg_mmv_stopvali( A, B, A2, B2, sigma, options, x0)
%SPG_MMV Solve multi-measurement basis pursuit denoise (BPDN)
%
% SPG_MMV is designed to solve the basis pursuit denoise problem
%
% (BPDN) minimize ||X||_1,2 subject to ||A X - B||_2,2 <= SIGMA,
%
% where A is an M-by-N matrix, B is an M-by-G matrix, and SIGMA is a
% nonnegative scalar. In all cases below, A can be an explicit M-by-N
% matrix or matrix-like object for which the operations A*x and A'*y
% are defined (i.e., matrix-vector multiplication with A and its
% adjoint.)
%
% Also, A can be a function handle that points to a function with the
% signature
%
% v = A(w,mode) which returns v = A *w if mode == 1;
% v = A'*w if mode == 2.
%
% X = SPG_MMV(A,B,SIGMA) solves the BPDN problem. If SIGMA=0 or
% SIGMA=[], then the basis pursuit (BP) problem is solved; i.e., the
% constraints in the BPDN problem are taken as AX=B.
%
% X = SPG_MMV(A,B,SIGMA,OPTIONS) specifies options that are set using
% SPGSETPARMS.
%
% [X,R,G,INFO] = SPG_BPDN(A,B,SIGMA,OPTIONS) additionally returns the
% residual R = B - A*X, the objective gradient G = A'*R, and an INFO
% structure. (See SPGL1 for a description of this last output argument.)
%
% See also spgl1, spgSetParms, spg_bp, spg_lasso.
% Copyright 2008, Ewout van den Berg and Michael P. Friedlander
% http://www.cs.ubc.ca/labs/scl/spgl1
% $Id$
if ~exist('options','var'), options = []; end
if ~exist('x0','var'), x0 = []; else x0 = x0(:); end
if ~exist('sigma','var'), sigma = 0; end
if ~exist('B','var') || isempty(B)
error('Second argument cannot be empty.');
end
if ~exist('A','var') || isempty(A)
error('First argument cannot be empty.');
end
groups = size(B,2);
if isa(A,'function_handle')
y = A(B(:,1),2); m = size(B,1); n = length(y);
A = @(x,mode) blockDiagonalImplicit(A,m,n,groups,x,mode);
else
m = size(A,1); n = size(A,2);
A = @(x,mode) blockDiagonalExplicit(A,m,n,groups,x,mode);
end
if isa(A2,'function_handle')
y2 = A2(B2(:,1),2); m2 = size(B2,1); n2 = length(y2);
A2 = @(x,mode) blockDiagonalImplicit(A2,m2,n2,groups,x,mode);
else
m2 = size(A2,1); n2 = size(A2,2);
A2 = @(x,mode) blockDiagonalExplicit(A2,m2,n2,groups,x,mode);
end
% Set projection specific functions
options.project = @(x,weight,tau) NormL12_project(groups,x,weight,tau);
options.primal_norm = @(x,weight ) NormL12_primal(groups,x,weight);
options.dual_norm = @(x,weight ) NormL12_dual(groups,x,weight);
tau = 0;
[x,r,g,info] = spgl1_stopvali(A,B(:),A2,B2(:),tau,sigma,x0,options);
n = round(length(x) / groups);
m = size(B,1);
x = reshape(x,n,groups);
y = reshape(r,m,groups);
g = reshape(g,n,groups);
function y = blockDiagonalImplicit(A,m,n,g,x,mode)
if mode == 1
y = zeros(m*g,1);
for i=1:g
y(1+(i-1)*m:i*m) = A(x(1+(i-1)*n:i*n),mode);
end
else
y = zeros(n*g,1);
for i=1:g
y(1+(i-1)*n:i*n) = A(x(1+(i-1)*m:i*m),mode);
end
end
function y = blockDiagonalExplicit(A,m,n,g,x,mode)
if mode == 1
y = A * reshape(x,n,g);
y = y(:);
else
x = reshape(x,m,g);
y = (x' * A)';
y = y(:);
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
spgl1_stopvali.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/spgl1_stopvali.m
| 31,870 |
utf_8
|
10a3cc0feb68137df5dc691058289e6d
|
function [x,r,g,info] = spgl1_stopvali( A, b, A2, b2, tau, sigma, x, options )
%SPGL1 Solve basis pursuit, basis pursuit denoise, and LASSO
%
% [x, r, g, info] = spgl1(A, b, tau, sigma, x0, options)
%
% ---------------------------------------------------------------------
% Solve the basis pursuit denoise (BPDN) problem
%
% (BPDN) minimize ||x||_1 subj to ||Ax-b||_2 <= sigma,
%
% or the l1-regularized least-squares problem
%
% (LASSO) minimize ||Ax-b||_2 subj to ||x||_1 <= tau.
% ---------------------------------------------------------------------
%
% INPUTS
% ======
% A is an m-by-n matrix, explicit or an operator.
% If A is a function, then it must have the signature
%
% y = A(x,mode) if mode == 1 then y = A x (y is m-by-1);
% if mode == 2 then y = A'x (y is n-by-1).
%
% b is an m-vector.
% tau is a nonnegative scalar; see (LASSO).
% sigma if sigma != inf or != [], then spgl1 will launch into a
% root-finding mode to find the tau above that solves (BPDN).
% In this case, it's STRONGLY recommended that tau = 0.
% x0 is an n-vector estimate of the solution (possibly all
% zeros). If x0 = [], then SPGL1 determines the length n via
% n = length( A'b ) and sets x0 = zeros(n,1).
% options is a structure of options from spgSetParms. Any unset options
% are set to their default value; set options=[] to use all
% default values.
%
% OUTPUTS
% =======
% x is a solution of the problem
% r is the residual, r = b - Ax
% g is the gradient, g = -A'r
% info is a structure with the following information:
% .tau final value of tau (see sigma above)
% .rNorm two-norm of the optimal residual
% .rGap relative duality gap (an optimality measure)
% .gNorm Lagrange multiplier of (LASSO)
% .stat = 1 found a BPDN solution
% = 2 found a BP sol'n; exit based on small gradient
% = 3 found a BP sol'n; exit based on small residual
% = 4 found a LASSO solution
% = 5 error: too many iterations
% = 6 error: linesearch failed
% = 7 error: found suboptimal BP solution
% = 8 error: too many matrix-vector products
% .time total solution time (seconds)
% .nProdA number of multiplications with A
% .nProdAt number of multiplications with A'
%
% OPTIONS
% =======
% Use the options structure to control various aspects of the algorithm:
%
% options.fid File ID to direct log output
% .verbosity 0=quiet, 1=some output, 2=more output.
% .iterations Max. number of iterations (default if 10*m).
% .bpTol Tolerance for identifying a basis pursuit solution.
% .optTol Optimality tolerance (default is 1e-4).
% .decTol Larger decTol means more frequent Newton updates.
% .subspaceMin 0=no subspace minimization, 1=subspace minimization.
%
% EXAMPLE
% =======
% m = 120; n = 512; k = 20; % m rows, n cols, k nonzeros.
% p = randperm(n); x0 = zeros(n,1); x0(p(1:k)) = sign(randn(k,1));
% A = randn(m,n); [Q,R] = qr(A',0); A = Q';
% b = A*x0 + 0.005 * randn(m,1);
% opts = spgSetParms('optTol',1e-4);
% [x,r,g,info] = spgl1(A, b, 0, 1e-3, [], opts); % Find BP sol'n.
%
% AUTHORS
% =======
% Ewout van den Berg ([email protected])
% Michael P. Friedlander ([email protected])
% Scientific Computing Laboratory (SCL)
% University of British Columbia, Canada.
%
% BUGS
% ====
% Please send bug reports or comments to
% Michael P. Friedlander ([email protected])
% Ewout van den Berg ([email protected])
% 15 Apr 07: First version derived from spg.m.
% Michael P. Friedlander ([email protected]).
% Ewout van den Berg ([email protected]).
% 17 Apr 07: Added root-finding code.
% 18 Apr 07: sigma was being compared to 1/2 r'r, rather than
% norm(r), as advertised. Now immediately change sigma to
% (1/2)sigma^2, and changed log output accordingly.
% 24 Apr 07: Added quadratic root-finding code as an option.
% 24 Apr 07: Exit conditions need to guard against small ||r||
% (ie, a BP solution). Added test1,test2,test3 below.
% 15 May 07: Trigger to update tau is now based on relative difference
% in objective between consecutive iterations.
% 15 Jul 07: Added code to allow a limited number of line-search
% errors.
% 23 Feb 08: Fixed bug in one-norm projection using weights. Thanks
% to Xiangrui Meng for reporting this bug.
% 26 May 08: The simple call spgl1(A,b) now solves (BPDN) with sigma=0.
% spgl1.m
% $Id: spgl1.m 1225 2009-01-30 20:36:31Z ewout78 $
%
% ----------------------------------------------------------------------
% This file is part of SPGL1 (Spectral Projected-Gradient for L1).
%
% Copyright (C) 2007 Ewout van den Berg and Michael P. Friedlander,
% Department of Computer Science, University of British Columbia, Canada.
% All rights reserved. E-mail: <{ewout78,mpf}@cs.ubc.ca>.
%
% SPGL1 is free software; you can redistribute it and/or modify it
% under the terms of the GNU Lesser General Public License as
% published by the Free Software Foundation; either version 2.1 of the
% License, or (at your option) any later version.
%
% SPGL1 is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
% or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
% Public License for more details.
%
% You should have received a copy of the GNU Lesser General Public
% License along with SPGL1; if not, write to the Free Software
% Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301
% USA
% ----------------------------------------------------------------------
REVISION = '$Revision: 1017 $';
DATE = '$Date: 2008-06-16 22:43:07 -0700 (Mon, 16 Jun 2008) $';
REVISION = REVISION(11:end-1);
DATE = DATE(35:50);
tic; % Start your watches!
m = length(b);
%----------------------------------------------------------------------
% Check arguments.
%----------------------------------------------------------------------
if ~exist('options','var'), options = []; end
if ~exist('x','var'), x = []; end
if ~exist('sigma','var'), sigma = []; end
if ~exist('tau','var'), tau = []; end
if nargin < 2 || isempty(b) || isempty(A)
error('At least two arguments are required');
elseif isempty(tau) && isempty(sigma)
tau = 0;
sigma = 0;
singleTau = false;
elseif isempty(sigma) % && ~isempty(tau) <-- implied
singleTau = true;
else
if isempty(tau)
tau = 0;
end
singleTau = false;
end
%----------------------------------------------------------------------
% Grab input options and set defaults where needed.
%----------------------------------------------------------------------
defaultopts = spgSetParms(...
'fid' , 1 , ... % File ID for output
'verbosity' , 2 , ... % Verbosity level
'iterations' , 10*m , ... % Max number of iterations
'nPrevVals' , 3 , ... % Number previous func values for linesearch
'bpTol' , 1e-06 , ... % Tolerance for basis pursuit solution
'optTol' , 1e-04 , ... % Optimality tolerance
'decTol' , 1e-04 , ... % Req'd rel. change in primal obj. for Newton
'stepMin' , 1e-16 , ... % Minimum spectral step
'stepMax' , 1e+05 , ... % Maximum spectral step
'rootMethod' , 2 , ... % Root finding method: 2=quad,1=linear (not used).
'activeSetIt', Inf , ... % Exit with EXIT_ACTIVE_SET if nnz same for # its.
'subspaceMin', 0 , ... % Use subspace minimization
'iscomplex' , NaN , ... % Flag set to indicate complex problem
'maxMatvec' , Inf , ... % Maximum matrix-vector multiplies allowed
'weights' , 1 , ... % Weights W in ||Wx||_1
'project' , @NormL1_project , ...
'primal_norm', @NormL1_primal , ...
'dual_norm' , @NormL1_dual ...
);
options = spgSetParms(defaultopts, options);
fid = options.fid;
logLevel = options.verbosity;
maxIts = options.iterations;
nPrevVals = options.nPrevVals;
bpTol = options.bpTol;
optTol = options.optTol;
decTol = options.decTol;
stepMin = options.stepMin;
stepMax = options.stepMax;
activeSetIt = options.activeSetIt;
subspaceMin = options.subspaceMin;
maxMatvec = max(3,options.maxMatvec);
weights = options.weights;
maxLineErrors = 10; % Maximum number of line-search failures.
pivTol = 1e-12; % Threshold for significant Newton step.
%----------------------------------------------------------------------
% Initialize local variables.
%----------------------------------------------------------------------
iter = 0; itnTotLSQR = 0; % Total SPGL1 and LSQR iterations.
nProdA = 0; nProdAt = 0;
nProdA2 = 0; nProdA2t = 0;
lastFv = -inf(nPrevVals,1); % Last m function values.
nLineTot = 0; % Total no. of linesearch steps.
printTau = false;
nNewton = 0;
bNorm = norm(b,2);
stat = false;
timeProject = 0;
timeMatProd = 0;
nnzIter = 0; % No. of its with fixed pattern.
nnzIdx = []; % Active-set indicator.
subspace = false; % Flag if did subspace min in current itn.
stepG = 1; % Step length for projected gradient.
testUpdateTau = 0; % Previous step did not update tau
% Determine initial x, vector length n, and see if problem is complex
explicit = ~(isa(A,'function_handle'));
if isempty(x)
if isnumeric(A)
n = size(A,2);
realx = isreal(A) && isreal(b);
else
x = Aprod(b,2);
n = length(x);
realx = isreal(x) && isreal(b);
end
x = zeros(n,1);
else
n = length(x);
realx = isreal(x) && isreal(b);
end
if isnumeric(A), realx = realx && isreal(A); end;
% Override options when options.iscomplex flag is set
if (~isnan(options.iscomplex)), realx = (options.iscomplex == 0); end
% Check if all weights (if any) are strictly positive. In previous
% versions we also checked if the number of weights was equal to
% n. In the case of multiple measurement vectors, this no longer
% needs to apply, so the check was removed.
if ~isempty(weights)
if any(~isfinite(weights))
error('Entries in options.weights must be finite');
end
if any(weights <= 0)
error('Entries in options.weights must be strictly positive');
end
else
weights = 1;
end
% Quick exit if sigma >= ||b||. Set tau = 0 to short-circuit the loop.
if bNorm <= sigma
printf('W: sigma >= ||b||. Exact solution is x = 0.\n');
tau = 0; singleTau = true;
end
% Don't do subspace minimization if x is complex.
if ~realx && subspaceMin
printf('W: Subspace minimization disabled when variables are complex.\n');
subspaceMin = false;
end
% Pre-allocate iteration info vectors
xNorm1 = zeros(min(maxIts,10000),1);
rNorm2 = zeros(min(maxIts,10000),1);
r2Norm2 = zeros(min(maxIts,10000),1);
lambda = zeros(min(maxIts,10000),1);
% Exit conditions (constants).
EXIT_ROOT_FOUND = 1;
EXIT_BPSOL1_FOUND = 2;
EXIT_BPSOL2_FOUND = 3;
EXIT_OPTIMAL = 4;
EXIT_ITERATIONS = 5;
EXIT_LINE_ERROR = 6;
EXIT_SUBOPTIMAL_BP = 7;
EXIT_MATVEC_LIMIT = 8;
EXIT_ACTIVE_SET = 9; % [sic]
EXIT_ERRORVALINCR = 10;
%----------------------------------------------------------------------
% Log header.
%----------------------------------------------------------------------
printf('\n');
printf(' %s\n',repmat('=',1,80));
printf(' SPGL1 v.%s (%s)\n', REVISION, DATE);
printf(' %s\n',repmat('=',1,80));
printf(' %-22s: %8i %4s' ,'No. rows' ,m ,'');
printf(' %-22s: %8i\n' ,'No. columns' ,n );
printf(' %-22s: %8.2e %4s' ,'Initial tau' ,tau ,'');
printf(' %-22s: %8.2e\n' ,'Two-norm of b' ,bNorm );
printf(' %-22s: %8.2e %4s' ,'Optimality tol' ,optTol ,'');
if singleTau
printf(' %-22s: %8.2e\n' ,'Target one-norm of x' ,tau );
else
printf(' %-22s: %8.2e\n','Target objective' ,sigma );
end
printf(' %-22s: %8.2e %4s' ,'Basis pursuit tol' ,bpTol ,'');
printf(' %-22s: %8i\n' ,'Maximum iterations',maxIts );
printf('\n');
if singleTau
logB = ' %5i %13.7e %13.7e %9.2e %6.1f %6i %6i';
logH = ' %5s %13s %13s %9s %6s %6s %6s\n';
printf(logH,'Iter','Objective','Relative Gap','gNorm','stepG','nnzX','nnzG');
else
logB = ' %5i %13.7e %13.7e %9.2e %9.3e %6.1f %6i %6i';
logH = ' %5s %13s %13s %9s %9s %6s %6s %6s %13s\n';
printf(logH,'Iter','Objective','Relative Gap','Rel Error',...
'gNorm','stepG','nnzX','nnzG','tau');
end
% Project the starting point and evaluate function and gradient.
x = project(x,tau);
r = b - Aprod(x,1); % r = b - Ax
r2 = b2 - A2prod(x,1); % r = b2 - A2x
g = - Aprod(r,2); % g = -A'r
f = r'*r / 2;
f2 = r2'*r2 / 2;
% Required for nonmonotone strategy.
lastFv(1) = f;
fBest = f;
f2Best = f2;
xBest = x;
fOld = f;
f2Old = f2;
% Compute projected gradient direction and initial steplength.
dx = project(x - g, tau) - x;
dxNorm = norm(dx,inf);
if dxNorm < (1 / stepMax)
gStep = stepMax;
else
gStep = min( stepMax, max(stepMin, 1/dxNorm) );
end
%----------------------------------------------------------------------
% MAIN LOOP.
%----------------------------------------------------------------------
while 1
%------------------------------------------------------------------
% Test exit conditions.
%------------------------------------------------------------------
% Compute quantities needed for log and exit conditions.
gNorm = options.dual_norm(-g,weights);
rNorm = norm(r, 2);
r2Norm = norm(r2, 2);
gap = r'*(r-b) + tau*gNorm;
rGap = abs(gap) / max(1,f);
aError1 = rNorm - sigma;
aError2 = f - sigma^2 / 2;
rError1 = abs(aError1) / max(1,rNorm);
rError2 = abs(aError2) / max(1,f);
% Count number of consecutive iterations with identical support.
nnzOld = nnzIdx;
[nnzX,nnzG,nnzIdx,nnzDiff] = activeVars(x,g,nnzIdx,options);
if nnzDiff
nnzIter = 0;
else
nnzIter = nnzIter + 1;
if nnzIter >= activeSetIt, stat=EXIT_ACTIVE_SET; end
end
% Single tau: Check if we're optimal.
% The 2nd condition is there to guard against large tau.
if singleTau
if rGap <= optTol || rNorm < optTol*bNorm
stat = EXIT_OPTIMAL;
end
% Multiple tau: Check if found root and/or if tau needs updating.
else
if rGap <= max(optTol, rError2) || rError1 <= optTol
% The problem is nearly optimal for the current tau.
% Check optimality of the current root.
test1 = rNorm <= bpTol * bNorm;
test2 = gNorm <= bpTol * rNorm;
test3 = rError1 <= optTol;
test4 = rNorm <= sigma;
test5 = f2 > f2Old;
if test4, stat=EXIT_SUBOPTIMAL_BP;end % Found suboptimal BP sol.
if test3, stat=EXIT_ROOT_FOUND; end % Found approx root.
if test2, stat=EXIT_BPSOL2_FOUND; end % Gradient zero -> BP sol.
if test1, stat=EXIT_BPSOL1_FOUND; end % Resid minim'zd -> BP sol.
if test5, stat=EXIT_ERRORVALINCR; end
end
testRelChange1 = (abs(f - fOld) <= decTol * f);
testRelChange2 = (abs(f - fOld) <= 1e-1 * f * (abs(rNorm - sigma)));
testUpdateTau = ((testRelChange1 && rNorm > 2 * sigma) || ...
(testRelChange2 && rNorm <= 2 * sigma)) && ...
~stat && ~testUpdateTau;
if testUpdateTau
% Update tau.
tauOld = tau;
tau = max(0,tau + (rNorm * aError1) / gNorm);
nNewton = nNewton + 1;
printTau = abs(tauOld - tau) >= 1e-6 * tau; % For log only.
if tau < tauOld
% The one-norm ball has decreased. Need to make sure that the
% next iterate if feasible, which we do by projecting it.
x = project(x,tau);
end
% Remember the residual norm on validation data at tau update
f2Old = r2'*r2 / 2;
end
end
% Too many its and not converged.
if ~stat && iter >= maxIts
stat = EXIT_ITERATIONS;
end
%------------------------------------------------------------------
% Print log, update history and act on exit conditions.
%------------------------------------------------------------------
if logLevel >= 2 || singleTau || printTau || iter == 0 || stat
tauFlag = ' '; subFlag = '';
if printTau, tauFlag = sprintf(' %13.7e',tau); end
if subspace, subFlag = sprintf(' S %2i',itnLSQR); end
if singleTau
printf(logB,iter,rNorm,rGap,gNorm,log10(stepG),nnzX,nnzG);
if subspace
printf(' %s',subFlag);
end
else
printf(logB,iter,rNorm,rGap,rError1,gNorm,log10(stepG),nnzX,nnzG);
if printTau || subspace
printf(' %s',[tauFlag subFlag]);
end
end
printf('\n');
end
printTau = false;
subspace = false;
% Update history info
xNorm1(iter+1) = options.primal_norm(x,weights);
rNorm2(iter+1) = rNorm;
r2Norm2(iter+1) = r2Norm;
lambda(iter+1) = gNorm;
if stat, break; end % Act on exit conditions.
%==================================================================
% Iterations begin here.
%==================================================================
iter = iter + 1;
xOld = x; fOld = f; gOld = g; rOld = r;
try
%---------------------------------------------------------------
% Projected gradient step and linesearch.
%---------------------------------------------------------------
[f,x,r,nLine,stepG,lnErr] = ...
spgLineCurvy(x,gStep*g,max(lastFv),@Aprod,b,@project,tau);
r2 = b2 - A2prod(x,1); f2 = r2'*r2 / 2;
nLineTot = nLineTot + nLine;
if lnErr
% Projected backtrack failed. Retry with feasible dir'n linesearch.
x = xOld;
dx = project(x - gStep*g, tau) - x;
gtd = g'*dx;
[f,x,r,nLine,lnErr] = spgLine(f,x,dx,gtd,max(lastFv),@Aprod,b);
r2 = b2 - A2prod(x,1); f2 = r2'*r2 / 2;
nLineTot = nLineTot + nLine;
end
if lnErr
% Failed again. Revert to previous iterates and damp max BB step.
if maxLineErrors <= 0
stat = EXIT_LINE_ERROR;
else
stepMax = stepMax / 10;
printf(['W: Linesearch failed with error %i. '...
'Damping max BB scaling to %6.1e.\n'],lnErr,stepMax);
maxLineErrors = maxLineErrors - 1;
end
end
%---------------------------------------------------------------
% Subspace minimization (only if active-set change is small).
%---------------------------------------------------------------
doSubspaceMin = false;
if subspaceMin
g = - Aprod(r,2);
[nnzX,nnzG,nnzIdx,nnzDiff] = activeVars(x,g,nnzOld,options);
if ~nnzDiff
if nnzX == nnzG, itnMaxLSQR = 20;
else itnMaxLSQR = 5;
end
nnzIdx = abs(x) >= optTol;
doSubspaceMin = true;
end
end
if doSubspaceMin
% LSQR parameters
damp = 1e-5;
aTol = 1e-1;
bTol = 1e-1;
conLim = 1e12;
showLSQR = 0;
ebar = sign(x(nnzIdx));
nebar = length(ebar);
Sprod = @(y,mode)LSQRprod(@Aprod,nnzIdx,ebar,n,y,mode);
[dxbar, istop, itnLSQR] = ...
lsqr(m,nebar,Sprod,r,damp,aTol,bTol,conLim,itnMaxLSQR,showLSQR);
itnTotLSQR = itnTotLSQR + itnLSQR;
if istop ~= 4 % LSQR iterations successful. Take the subspace step.
% Push dx back into full space: dx = Z dx.
dx = zeros(n,1);
dx(nnzIdx) = dxbar - (1/nebar)*(ebar'*dxbar)*dxbar;
% Find largest step to a change in sign.
block1 = nnzIdx & x < 0 & dx > +pivTol;
block2 = nnzIdx & x > 0 & dx < -pivTol;
alpha1 = Inf; alpha2 = Inf;
if any(block1), alpha1 = min(-x(block1) ./ dx(block1)); end
if any(block2), alpha2 = min(-x(block2) ./ dx(block2)); end
alpha = min([1 alpha1 alpha2]);
ensure(alpha >= 0);
ensure(ebar'*dx(nnzIdx) <= optTol);
% Update variables.
x = x + alpha*dx;
r = b - Aprod(x,1);
f = r'*r / 2;
r2 = b2 - A2prod(x,1);
f2 = r2'*r2 / 2;
subspace = true;
end
end
ensure(options.primal_norm(x,weights) <= tau+optTol);
%---------------------------------------------------------------
% Update gradient and compute new Barzilai-Borwein scaling.
%---------------------------------------------------------------
g = - Aprod(r,2);
s = x - xOld;
y = g - gOld;
sts = s'*s;
sty = s'*y;
if sty <= 0, gStep = stepMax;
else gStep = min( stepMax, max(stepMin, sts/sty) );
end
catch % Detect matrix-vector multiply limit error
err = lasterror;
if strcmp(err.identifier,'SPGL1:MaximumMatvec')
stat = EXIT_MATVEC_LIMIT;
iter = iter - 1;
x = xOld; f = fOld; g = gOld; r = rOld;
break;
else
rethrow(err);
end
end
%------------------------------------------------------------------
% Update function history.
%------------------------------------------------------------------
if singleTau || f > sigma^2 / 2 % Don't update if superoptimal.
lastFv(mod(iter,nPrevVals)+1) = f;
if fBest > f
fBest = f;
if f2Best >= f2
f2Best = f2;
xBest = x;
BestIter = iter;
end
end
end
end % while 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Restore best solution (only if solving single problem).
if f2 >= f2Best || f >= fBest
rNorm = sqrt(2*fBest);
x = xBest;
r = b - Aprod(x,1);
r2 = b2 - A2prod(x,1);
g = - Aprod(r,2);
gNorm = options.dual_norm(g,weights);
rNorm = norm(r, 2);
r2Norm = norm(r2, 2);
printf('\n Restoring best iterate to objective ||r|| = %13.7e\n',rNorm);
printf('\n ................................... ||r2|| = %13.7e\n',r2Norm);
end
% Final cleanup before exit.
info.tau = tau;
info.rNorm = rNorm;
info.r2Norm = r2Norm;
info.rGap = rGap;
info.gNorm = gNorm;
info.rGap = rGap;
info.stat = stat;
info.iter = iter;
info.nProdA = nProdA;
info.nProdAt = nProdAt;
info.nNewton = nNewton;
info.timeProject = timeProject;
info.timeMatProd = timeMatProd;
info.itnLSQR = itnTotLSQR;
info.options = options;
info.timeTotal = toc;
info.xNorm1 = xNorm1(1:iter);
info.rNorm2 = rNorm2(1:iter);
info.r2Norm2 = r2Norm2(1:iter);
info.lambda = lambda(1:iter);
% Print final output.
switch (stat)
case EXIT_OPTIMAL
printf('\n EXIT -- Optimal solution found\n')
case EXIT_ITERATIONS
printf('\n ERROR EXIT -- Too many iterations\n');
case EXIT_ROOT_FOUND
printf('\n EXIT -- Found a root\n');
case {EXIT_BPSOL1_FOUND, EXIT_BPSOL2_FOUND}
printf('\n EXIT -- Found a BP solution\n');
case EXIT_LINE_ERROR
printf('\n ERROR EXIT -- Linesearch error (%i)\n',lnErr);
case EXIT_SUBOPTIMAL_BP
printf('\n EXIT -- Found a suboptimal BP solution\n');
case EXIT_MATVEC_LIMIT
printf('\n EXIT -- Maximum matrix-vector operations reached\n');
case EXIT_ACTIVE_SET
printf('\n EXIT -- Found a possible active set\n');
case EXIT_ERRORVALINCR
printf('\n EXIT -- Increase of prediction error on validation data\n');
otherwise
error('Unknown termination condition\n');
end
printf('\n');
printf(' %-20s: %6i %6s %-20s: %6.1f\n',...
'Products with A',nProdA,'','Total time (secs)',info.timeTotal);
printf(' %-20s: %6i %6s %-20s: %6.1f\n',...
'Products with A''',nProdAt,'','Project time (secs)',timeProject);
printf(' %-20s: %6i %6s %-20s: %6.1f\n',...
'Newton iterations',nNewton,'','Mat-vec time (secs)',timeMatProd);
printf(' %-20s: %6i %6s %-20s: %6i\n', ...
'Line search its',nLineTot,'','Subspace iterations',itnTotLSQR);
printf('\n');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NESTED FUNCTIONS. These share some vars with workspace above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function z = Aprod(x,mode)
if (nProdA + nProdAt >= maxMatvec)
error('SPGL1:MaximumMatvec','');
end
tStart = toc;
if mode == 1
nProdA = nProdA + 1;
if explicit, z = A*x;
else z = A(x,1);
end
elseif mode == 2
nProdAt = nProdAt + 1;
if explicit, z = A'*x;
else z = A(x,2);
end
else
error('Wrong mode!');
end
timeMatProd = timeMatProd + (toc - tStart);
end % function Aprod
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function z = A2prod(x,mode)
if (nProdA2 + nProdA2t >= maxMatvec)
error('SPGL1:MaximumMatvec','');
end
tStart = toc;
if mode == 1
nProdA2 = nProdA2 + 1;
if explicit, z = A2*x;
else z = A2(x,1);
end
elseif mode == 2
nProdA2t = nProdA2t + 1;
if explicit, z = A2'*x;
else z = A2(x,2);
end
else
error('Wrong mode!');
end
timeMatProd = timeMatProd + (toc - tStart);
end % function Aprod
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function printf(varargin)
if logLevel > 0
fprintf(fid,varargin{:});
end
end % function printf
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function x = project(x, tau)
tStart = toc;
x = options.project(x,weights,tau);
timeProject = timeProject + (toc - tStart);
end % function project
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% End of nested functions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
end % function spg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PRIVATE FUNCTIONS.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [nnzX,nnzG,nnzIdx,nnzDiff] = activeVars(x,g,nnzIdx,options)
% Find the current active set.
% nnzX is the number of nonzero x.
% nnzG is the number of elements in nnzIdx.
% nnzIdx is a vector of primal/dual indicators.
% nnzDiff is the no. of elements that changed in the support.
xTol = min(.1,10*options.optTol);
gTol = min(.1,10*options.optTol);
gNorm = options.dual_norm(g,options.weights);
nnzOld = nnzIdx;
% Reduced costs for postive & negative parts of x.
z1 = gNorm + g;
z2 = gNorm - g;
% Primal/dual based indicators.
xPos = x > xTol & z1 < gTol; %g < gTol;%
xNeg = x < -xTol & z2 < gTol; %g > gTol;%
nnzIdx = xPos | xNeg;
% Count is based on simple primal indicator.
nnzX = sum(abs(x) >= xTol);
nnzG = sum(nnzIdx);
if isempty(nnzOld)
nnzDiff = inf;
else
nnzDiff = sum(nnzIdx ~= nnzOld);
end
end % function spgActiveVars
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function z = LSQRprod(Aprod,nnzIdx,ebar,n,dx,mode)
% Matrix multiplication for subspace minimization.
% Only called by LSQR.
nbar = length(ebar);
if mode == 1
y = zeros(n,1);
y(nnzIdx) = dx - (1/nbar)*(ebar'*dx)*ebar; % y(nnzIdx) = Z*dx
z = Aprod(y,1); % z = S Z dx
else
y = Aprod(dx,2);
z = y(nnzIdx) - (1/nbar)*(ebar'*y(nnzIdx))*ebar;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [fNew,xNew,rNew,iter,err] = spgLine(f,x,d,gtd,fMax,Aprod,b)
% Nonmonotone linesearch.
EXIT_CONVERGED = 0;
EXIT_ITERATIONS = 1;
maxIts = 10;
step = 1;
iter = 0;
gamma = 1e-4;
gtd = -abs(gtd); % 03 Aug 07: If gtd is complex,
% then should be looking at -abs(gtd).
while 1
% Evaluate trial point and function value.
xNew = x + step*d;
rNew = b - Aprod(xNew,1);
fNew = rNew'*rNew / 2;
% Check exit conditions.
if fNew < fMax + gamma*step*gtd % Sufficient descent condition.
err = EXIT_CONVERGED;
break
elseif iter >= maxIts % Too many linesearch iterations.
err = EXIT_ITERATIONS;
break
end
% New linesearch iteration.
iter = iter + 1;
% Safeguarded quadratic interpolation.
if step <= 0.1
step = step / 2;
else
tmp = (-gtd*step^2) / (2*(fNew-f-step*gtd));
if tmp < 0.1 || tmp > 0.9*step || isnan(tmp)
tmp = step / 2;
end
step = tmp;
end
end % while 1
end % function spgLine
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [fNew,xNew,rNew,iter,step,err] = ...
spgLineCurvy(x,g,fMax,Aprod,b,project,tau)
% Projected backtracking linesearch.
% On entry,
% g is the (possibly scaled) steepest descent direction.
EXIT_CONVERGED = 0;
EXIT_ITERATIONS = 1;
EXIT_NODESCENT = 2;
gamma = 1e-4;
maxIts = 10;
step = 1;
sNorm = 0;
scale = 1; % Safeguard scaling. (See below.)
nSafe = 0; % No. of safeguarding steps.
iter = 0;
debug = false; % Set to true to enable log.
n = length(x);
if debug
fprintf(' %5s %13s %13s %13s %8s\n',...
'LSits','fNew','step','gts','scale');
end
while 1
% Evaluate trial point and function value.
xNew = project(x - step*scale*g, tau);
rNew = b - Aprod(xNew,1);
fNew = rNew'*rNew / 2;
s = xNew - x;
gts = scale * g' * s;
if gts >= 0 % Should we check real and complex parts individually?
err = EXIT_NODESCENT;
break
end
if debug
fprintf(' LS %2i %13.7e %13.7e %13.6e %8.1e\n',...
iter,fNew,step,gts,scale);
end
% 03 Aug 07: If gts is complex, then should be looking at -abs(gts).
if fNew < fMax - gamma*step*abs(gts) % Sufficient descent condition.
err = EXIT_CONVERGED;
break
elseif iter >= maxIts % Too many linesearch iterations.
err = EXIT_ITERATIONS;
break
end
% New linesearch iteration.
iter = iter + 1;
step = step / 2;
% Safeguard: If stepMax is huge, then even damped search
% directions can give exactly the same point after projection. If
% we observe this in adjacent iterations, we drastically damp the
% next search direction.
% 31 May 07: Damp consecutive safeguarding steps.
sNormOld = sNorm;
sNorm = norm(s) / sqrt(n);
% if sNorm >= sNormOld
if abs(sNorm - sNormOld) <= 1e-6 * sNorm
gNorm = norm(g) / sqrt(n);
scale = sNorm / gNorm / (2^nSafe);
nSafe = nSafe + 1;
end
end % while 1
end % function spgLineCurvy
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
kernmatrix.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/kernmatrix.m
| 3,842 |
utf_8
|
f5ac9d877d0e03e735508333491c3fcb
|
function omega = kernmatrix(Xtrain,kernel_type,kernel_pars,Xt)
%KERNMATRIX Construct the positive (semi-) definite and symmetric kernel matrix
%
% Omega = kernel_matrix(X, kernel_fct, sig2)
%
% This matrix should be positive definite if the kernel function
% satisfies the Mercer condition. Construct the kernel values for
% all test data points in the rows of Xt, relative to the points of X.
%
% Omega_Xt = kernel_matrix(X, kernel_fct, sig2, Xt)
%
%
% Full syntax:
%
% Omega = kernel_matrix(X, kernel_fct, sig2)
% Omega = kernel_matrix(X, kernel_fct, sig2, Xt)
%
% Outputs:
% Omega : N x N (N x Nt) kernel matrix
% Inputs:
% X : N x d matrix with the inputs of the training data
% kernel : Kernel type (by default 'RBF_kernel')
% sig2 : Kernel parameter (bandwidth in the case of the 'RBF_kernel')
% Xt(*) : Nt x d matrix with the inputs of the test data
% Copyright (c) 2002, KULeuven-ESAT-SCD,
% License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlab
nb_data = size(Xtrain,1);
if nb_data> 3000,
error('Too memory intensive, the kernel matrix is restricted to size 3000 x 3000 ');
end
if strcmpi(kernel_type,'rbf'),
if nargin<4,
XXh = sum(Xtrain.^2,2)*ones(1,nb_data);
omega = (XXh+XXh') - 2*(Xtrain*Xtrain');
omega = exp(-omega./kernel_pars(1));
else
XXh1 = sum(Xtrain.^2,2)*ones(1,size(Xt,1));
XXh2 = sum(Xt.^2,2)*ones(1,nb_data);
omega = XXh1+XXh2' - 2*Xtrain*Xt';
omega = exp(-omega./kernel_pars(1));
end
else
if nargin<4,
omega = zeros(nb_data,nb_data);
for i=1:nb_data,
omega(i:end,i) = feval(lower(kernel_type),Xtrain(i,:),Xtrain(i:end,:),kernel_pars);
omega(i,i:end) = omega(i:end,i)';
end
else
if size(Xt,2)~=size(Xtrain,2),
error('dimension test data not equal to dimension traindata;');
end
omega = zeros(nb_data, size(Xt,1));
for i=1:size(Xt,1),
omega(:,i) = feval(lower(kernel_type),Xt(i,:),Xtrain,kernel_pars);
end
end
end
end
function x = lin(a,b,c)
% kernel function for implicit higher dimension mapping, based on
% the standard inner-product
%
% x = lin_kernel(a,b)
%
% 'a' can only contain one datapoint in a row, 'b' can contain N
% datapoints of the same dimension as 'a'.
%
% see also:
% poly_kernel, RBF_kernel, MLP_kernel, trainlssvm, simlssvm
% Copyright (c) 2002, KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlab
x = zeros(size(b,1),1);
for i=1:size(b,1),
x(i,1) = a*b(i,:)';
end
end
function x = poly(a,b,d)
% polynomial kernel function for implicit higher dimension mapping
%
% X = poly_kernel(a,b,[t,degree])
%
% 'a' can only contain one datapoint in a row, 'b' can contain N
% datapoints of the same dimension as 'a'.
%
% x = (a*b'+t^2).^degree;
%
% see also:
% RBF_kernel, lin_kernel, MLP_kernel, trainlssvm, simlssvm
%
% Copyright (c) 2002, KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlab
if length(d)>1, d=d(2); t=d(1); else d = d(1);t=1; end
d = (abs(d)>=1)*abs(d)+(abs(d)<1); % >=1 !!
x = zeros(size(b,1),1);
for i=1:size(b,1),
x(i,1) = (a*b(i,:)'+t^2).^d;
end
end
function x = mlp(a,b, par)
% Multi Layer Perceptron kernel function for implicit higher dimension mapping
%
% x = MLP_kernel(a,b,[s,t])
%
% 'a' can only contain one datapoint in a row, 'b' can contain N
% datapoints of the same dimension as 'a'.
%
% x = tanh(s*a'b+t^2)
%
% see also:
% poly_kernel, lin_kernel, RBF_kernel, trainlssvm, simlssvm
% Copyright (c) 2002, KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlab
if length(par)==1, par(2) = 1; end
x = zeros(size(b,1),1);
for i=1:size(b,1),
dp = a*b(i,:)';
x(i,1) = tanh(par(1)*dp + par(2)^2);
end
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
ellipsebnd.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/ellipsebnd.m
| 8,035 |
utf_8
|
888067d692a8d486bbb999fcfaa52bf7
|
function h=ellipsebnd(varargin)
% ELLIPSEBND - plot an error ellipse, or ellipsoid, defining confidence region
% ELLIPSEBND(C22) - Given a 2x2 covariance matrix, plot the
% associated error ellipse, at the origin. It returns a graphics handle
% of the ellipse that was drawn.
%
% ELLIPSEBND(C33) - Given a 3x3 covariance matrix, plot the
% associated error ellipsoid, at the origin, as well as its projections
% onto the three axes. Returns a vector of 4 graphics handles, for the
% three ellipses (in the X-Y, Y-Z, and Z-X planes, respectively) and for
% the ellipsoid.
%
% ELLIPSEBND(C,MU) - Plot the ellipse, or ellipsoid, centered at MU,
% a vector whose length should match that of C (which is 2x2 or 3x3).
%
% ELLIPSEBND(...,'Property1',Value1,'Name2',Value2,...) sets the
% values of specified properties, including:
% 'C' - Alternate method of specifying the covariance matrix
% 'mu' - Alternate method of specifying the ellipse (-oid) center
% 'conf' - A value betwen 0 and 1 specifying the confidence interval.
% the default is 0.5 which is the 50% error ellipse.
% 'scale' - Allow the plot the be scaled to difference units.
% 'style' - A plotting style used to format ellipses.
% 'clip' - specifies a clipping radius. Portions of the ellipse, -oid,
% outside the radius will not be shown.
%
% NOTES: C must be positive definite for this function to work properly.
default_properties = struct(...
'C', [], ... % The covaraince matrix (required)
'mu', [], ... % Center of ellipse (optional)
'conf', 0.5, ... % Percent confidence/100
'scale', 1, ... % Scale factor, e.g. 1e-3 to plot m as km
'style', '', ... % Plot style
'clip', inf); % Clipping radius
if length(varargin) >= 1 && isnumeric(varargin{1})
default_properties.C = varargin{1};
varargin(1) = [];
end
if length(varargin) >= 1 && isnumeric(varargin{1})
default_properties.mu = varargin{1};
varargin(1) = [];
end
if length(varargin) >= 1 && isnumeric(varargin{1})
default_properties.conf = varargin{1};
varargin(1) = [];
end
if length(varargin) >= 1 && isnumeric(varargin{1})
default_properties.scale = varargin{1};
varargin(1) = [];
end
if length(varargin) >= 1 && ~ischar(varargin{1})
error('Invalid parameter/value pair arguments.')
end
prop = getopt(default_properties, varargin{:});
C = prop.C;
if isempty(prop.mu)
mu = zeros(length(C),1);
else
mu = prop.mu;
end
conf = prop.conf;
scale = prop.scale;
style = prop.style;
if conf <= 0 || conf >= 1
error('conf parameter must be in range 0 to 1, exclusive')
end
[r,c] = size(C);
if r ~= c || (r ~= 2 && r ~= 3)
error(['Don''t know what to do with ',num2str(r),'x',num2str(c),' matrix'])
end
x0=mu(1);
y0=mu(2);
% Compute quantile for the desired percentile
k = sqrt(qchisq(conf,r)); % r is the number of dimensions (degrees of freedom)
hold_state = get(gca,'nextplot');
if r==3 && c==3
z0=mu(3);
% Make the matrix has positive eigenvalues - else it's not a valid covariance matrix!
if any(eig(C) <=0)
error('The covariance matrix must be positive definite (it has non-positive eigenvalues)')
end
% C is 3x3; extract the 2x2 matricies, and plot the associated error
% ellipses. They are drawn in space, around the ellipsoid; it may be
% preferable to draw them on the axes.
Cxy = C(1:2,1:2);
Cyz = C(2:3,2:3);
Czx = C([3 1],[3 1]);
[x,y,z] = getpoints(Cxy,prop.clip);
h1=plot3(x0+k*x,y0+k*y,z0+k*z,prop.style);hold on
[y,z,x] = getpoints(Cyz,prop.clip);
h2=plot3(x0+k*x,y0+k*y,z0+k*z,prop.style);hold on
[z,x,y] = getpoints(Czx,prop.clip);
h3=plot3(x0+k*x,y0+k*y,z0+k*z,prop.style);hold on
[eigvec,eigval] = eig(C);
[X,Y,Z] = ellipsoid(0,0,0,1,1,1);
XYZ = [X(:),Y(:),Z(:)]*sqrt(eigval)*eigvec';
X(:) = scale*(k*XYZ(:,1)+x0);
Y(:) = scale*(k*XYZ(:,2)+y0);
Z(:) = scale*(k*XYZ(:,3)+z0);
h4=surf(X,Y,Z);
colormap gray
alpha(0.3)
camlight
if nargout
h=[h1 h2 h3 h4];
end
elseif r==2 && c==2
% Make the matrix has positive eigenvalues - else it's not a valid covariance matrix!
if any(eig(C) <=0)
error('The covariance matrix must be positive definite (it has non-positive eigenvalues)')
end
[x,y,z] = getpoints(C,prop.clip);
h1=plot(scale*(x0+k*x),scale*(y0+k*y),prop.style);
set(h1,'zdata',z+1)
if nargout
h=h1;
end
else
error('C (covaraince matrix) must be specified as a 2x2 or 3x3 matrix)')
end
%axis equal
set(gca,'nextplot',hold_state);
end
% getpoints - Generate x and y points that define an ellipse, given a 2x2
% covariance matrix, C. z, if requested, is all zeros with same shape as
% x and y.
function [x,y,z] = getpoints(C,clipping_radius)
n=100; % Number of points around ellipse
p=0:pi/n:2*pi; % angles around a circle
[eigvec,eigval] = eig(C); % Compute eigen-stuff
xy = [cos(p'),sin(p')] * sqrt(eigval) * eigvec'; % Transformation
x = xy(:,1);
y = xy(:,2);
z = zeros(size(x));
% Clip data to a bounding radius
if nargin >= 2
r = sqrt(sum(xy.^2,2)); % Euclidian distance (distance from center)
x(r > clipping_radius) = nan;
y(r > clipping_radius) = nan;
z(r > clipping_radius) = nan;
end
end
function x=qchisq(P,n)
% QCHISQ(P,N) - quantile of the chi-square distribution.
if nargin<2
n=1;
end
s0 = P==0;
s1 = P==1;
s = P>0 & P<1;
x = 0.5*ones(size(P));
x(s0) = -inf;
x(s1) = inf;
x(~(s0|s1|s))=nan;
for ii=1:14
dx = -(pchisq(x(s),n)-P(s))./dchisq(x(s),n);
x(s) = x(s)+dx;
if all(abs(dx) < 1e-6)
break;
end
end
end
function F=pchisq(x,n)
% PCHISQ(X,N) - Probability function of the chi-square distribution.
if nargin<2
n=1;
end
F=zeros(size(x));
if rem(n,2) == 0
s = x>0;
k = 0;
for jj = 0:n/2-1;
k = k + (x(s)/2).^jj/factorial(jj);
end
F(s) = 1-exp(-x(s)/2).*k;
else
for ii=1:numel(x)
if x(ii) > 0
F(ii) = quadl(@dchisq,0,x(ii),1e-6,0,n);
else
F(ii) = 0;
end
end
end
end
function f=dchisq(x,n)
% DCHISQ(X,N) - Density function of the chi-square distribution.
if nargin<2
n=1;
end
f=zeros(size(x));
s = x>=0;
f(s) = x(s).^(n/2-1).*exp(-x(s)/2)./(2^(n/2)*gamma(n/2));
end
function properties = getopt(properties,varargin)
%GETOPT - Process paired optional arguments as 'prop1',val1,'prop2',val2,...
%
% getopt(properties,varargin) returns a modified properties structure,
% given an initial properties structure, and a list of paired arguments.
% Each argumnet pair should be of the form property_name,val where
% property_name is the name of one of the field in properties, and val is
% the value to be assigned to that structure field.
%
% No validation of the values is performed.
%
% EXAMPLE:
% properties = struct('zoom',1.0,'aspect',1.0,'gamma',1.0,'file',[],'bg',[]);
% properties = getopt(properties,'aspect',0.76,'file','mydata.dat')
% would return:
% properties =
% zoom: 1
% aspect: 0.7600
% gamma: 1
% file: 'mydata.dat'
% bg: []
%
% Typical usage in a function:
% properties = getopt(properties,varargin{:})
% Process the properties (optional input arguments)
prop_names = fieldnames(properties);
TargetField = [];
for ii=1:length(varargin)
arg = varargin{ii};
if isempty(TargetField)
if ~ischar(arg)
error('Propery names must be character strings');
end
f = find(strcmp(prop_names, arg));
if length(f) == 0
error('%s ',['invalid property ''',arg,'''; must be one of:'],prop_names{:});
end
TargetField = arg;
else
% properties.(TargetField) = arg; % Ver 6.5 and later only
properties = setfield(properties, TargetField, arg); % Ver 6.1 friendly
TargetField = '';
end
end
if ~isempty(TargetField)
error('Property names and values must be specified in pairs.');
end
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
reggcv.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/reggcv.m
| 4,183 |
utf_8
|
96e102b790fbfa5f01c26445d60a36fc
|
function reg_min=reggcv(Y,Vn,Sn,method,show)
%REGGCV Compute regularization using generalized cross validation.
% Determine the regularization parameter for ordkernel
% using Generalized Cross-Validation (GCV). It plots the
% GCV function as a function of the regularization
% parameter and finds its minimum.
%
% Syntax:
% reg=reggcv(Y,V,S)
% reg=reggcv(Y,V,S,method,show)
%
% Input:
% Y,V,S Data matrices from lpvkernel or bilkernel.
% method Regularization method to be used.
% 'Tikh' - Tikhonov regularization (default).
% 'tsvd' - Truncated singular value decomposition.
% show Display intermediate steps of the algorithm.
%
% Output:
% reg Regularization paramater for the kernel
% subspace identification method of ordkernel.
% Written by Vincent Verdult, May 2004.
% Based on Regularization Tools by P. C. Hansen
% default method
if nargin<4
method='Tikh';
end
if nargin<5
show=0;
end
if size(Y,1)~=size(Vn,1)
error('The number of rows in Y must equal the number of rows in V.')
end
if size(Vn,1)~=size(Vn,2)
error('V must be a square matrix.')
end
if size(Sn,2)~=1
error('S must be a column vector.')
end
if size(Vn,1)~=size(Sn,1)
error('The number of rows in S must equal the number of rows in V.')
end
% Initialization.
N = size(Sn,1);
beta = Vn'*Y;
% Tikhonov regularization
if (strncmp(method,'Tikh',4) || strncmp(method,'tikh',4))
npoints = 200; % Number of points on the curve.
smin_ratio = 16*eps; % Smallest regularization parameter.
reg_param = zeros(npoints,1);
G = zeros(npoints,1);
s = sqrt(Sn);
reg_param(npoints) = max([s(N),s(1)*smin_ratio]);
ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));
for i=npoints-1:-1:1
reg_param(i) = ratio*reg_param(i+1);
end
if show==1
disp('Calculating GCV curve.')
end
% Vector of GCV-function values.
for i=1:npoints
G(i) = reggcvfun(reg_param(i),Sn,beta);
end
% Plot GCV function.
if show==1
loglog(reg_param,G,'-'), xlabel('\lambda'), ylabel('G(\lambda)')
title('GCV function')
end
% Find minimum
if show==1
disp('Searching GCV minimum')
OPT=optimset('Display','iter');
else
OPT=optimset('Display','off');
end
[minG,minGi] = min(G); % Initial guess.
reg_min = fminbnd(@reggcvfun,...
reg_param(min(minGi+1,npoints)),...
reg_param(max(minGi-1,1)),OPT,Sn,beta); % Minimizer.
minG = reggcvfun(reg_min,Sn,beta); % Minimum of GCV function.
if show==1
ax = axis;
HoldState = ishold; hold on;
loglog(reg_min,minG,'*r',[reg_min,reg_min],[minG/1000,minG],':r')
title(['GCV function, minimum at \lambda = ',num2str(reg_min)])
axis(ax)
if (~HoldState)
hold off
end
end
% Truncated SVD
elseif (strncmp(method,'tsvd',4) || strncmp(method,'TSVD',4))
rho=zeros(1,N-1);
G=zeros(1,N-1);
rho(N-1) = sum(beta(N,:).^2);
G(N-1) = rho(N-1);
for k=N-2:-1:1
rho(k) = rho(k+1) + sum(beta(k+1,:).^2);
G(k) = rho(k)/((N - k)^2);
end
reg_param = (1:N-1)';
% Plot GCV function.
if show==1
semilogy(reg_param,G,'o'), xlabel('k'), ylabel('G(k)')
title('GCV function')
end
% Find minimum
[minG,reg_min] = min(G);
if show==1
ax = axis;
HoldState = ishold; hold on;
semilogy(reg_min,minG,'*r',[reg_min,reg_min],[minG/1000,minG],':r')
title(['GCV function, minimum at k = ',num2str(reg_min)])
axis(ax);
if (~HoldState)
hold off
end
end
else
error('Illegal method.')
end
end
function G=reggcvfun(lam,s2,beta)
% reggcvfun Computes GCV function for reggcv.
% Auxiliary function for reggcv.
%
% Written by Vincent Verdult, May 2004.
% Based on Regularization Tools by P. C. Hansen
f=lam^2./(s2+lam^2);
G=norm((f*ones(1,size(beta,2))).*beta,'fro')^2/(sum(f)^2);
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
jacobianest.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/jacobianest.m
| 5,842 |
utf_8
|
46460a027f39b6c92c0f0b818d2e1721
|
function [jac,err] = jacobianest(fun,x0)
% gradest: estimate of the Jacobian matrix of a vector valued function of n variables
% usage: [jac,err] = jacobianest(fun,x0)
%
%
% arguments: (input)
% fun - (vector valued) analytical function to differentiate.
% fun must be a function of the vector or array x0.
%
% x0 - vector location at which to differentiate fun
% If x0 is an nxm array, then fun is assumed to be
% a function of n*m variables.
%
%
% arguments: (output)
% jac - array of first partial derivatives of fun.
% Assuming that x0 is a vector of length p
% and fun returns a vector of length n, then
% jac will be an array of size (n,p)
%
% err - vector of error estimates corresponding to
% each partial derivative in jac.
%
%
% Example: (nonlinear least squares)
% xdata = (0:.1:1)';
% ydata = 1+2*exp(0.75*xdata);
% fun = @(c) ((c(1)+c(2)*exp(c(3)*xdata)) - ydata).^2;
%
% [jac,err] = jacobianest(fun,[1 1 1])
%
% jac =
% -2 -2 0
% -2.1012 -2.3222 -0.23222
% -2.2045 -2.6926 -0.53852
% -2.3096 -3.1176 -0.93528
% -2.4158 -3.6039 -1.4416
% -2.5225 -4.1589 -2.0795
% -2.629 -4.7904 -2.8742
% -2.7343 -5.5063 -3.8544
% -2.8374 -6.3147 -5.0518
% -2.9369 -7.2237 -6.5013
% -3.0314 -8.2403 -8.2403
%
% err =
% 5.0134e-15 5.0134e-15 0
% 5.0134e-15 0 2.8211e-14
% 5.0134e-15 8.6834e-15 1.5804e-14
% 0 7.09e-15 3.8227e-13
% 5.0134e-15 5.0134e-15 7.5201e-15
% 5.0134e-15 1.0027e-14 2.9233e-14
% 5.0134e-15 0 6.0585e-13
% 5.0134e-15 1.0027e-14 7.2673e-13
% 5.0134e-15 1.0027e-14 3.0495e-13
% 5.0134e-15 1.0027e-14 3.1707e-14
% 5.0134e-15 2.0053e-14 1.4013e-12
%
% (At [1 2 0.75], jac should be numerically zero)
%
%
% See also: derivest, gradient, gradest
%
%
% Author: John D'Errico
% e-mail: [email protected]
% Release: 1.0
% Release date: 3/6/2007
% get the length of x0 for the size of jac
nx = numel(x0);
MaxStep = 100;
StepRatio = 2;
% was a string supplied?
if ischar(fun)
fun = str2func(fun);
end
% get fun at the center point
f0 = fun(x0);
f0 = f0(:);
n = length(f0);
if n==0
% empty begets empty
jac = zeros(0,nx);
err = jac;
return
end
relativedelta = MaxStep*StepRatio .^(0:-1:-25);
nsteps = length(relativedelta);
% total number of derivatives we will need to take
jac = zeros(n,nx);
err = jac;
for i = 1:nx
x0_i = x0(i);
if x0_i ~= 0
delta = x0_i*relativedelta;
else
delta = relativedelta;
end
% evaluate at each step, centered around x0_i
% difference to give a second order estimate
fdel = zeros(n,nsteps);
for j = 1:nsteps
fdif = fun(swapelement(x0,i,x0_i + delta(j))) - ...
fun(swapelement(x0,i,x0_i - delta(j)));
fdel(:,j) = fdif(:);
end
% these are pure second order estimates of the
% first derivative, for each trial delta.
derest = fdel.*repmat(0.5 ./ delta,n,1);
% The error term on these estimates has a second order
% component, but also some 4th and 6th order terms in it.
% Use Romberg exrapolation to improve the estimates to
% 6th order, as well as to provide the error estimate.
% loop here, as rombextrap coupled with the trimming
% will get complicated otherwise.
for j = 1:n
[der_romb,errest] = rombextrap(StepRatio,derest(j,:),[2 4]);
% trim off 3 estimates at each end of the scale
nest = length(der_romb);
trim = [1:3, nest+(-2:0)];
[der_romb,tags] = sort(der_romb);
der_romb(trim) = [];
tags(trim) = [];
errest = errest(tags);
% now pick the estimate with the lowest predicted error
[err(j,i),ind] = min(errest);
jac(j,i) = der_romb(ind);
end
end
end % mainline function end
% =======================================
% sub-functions
% =======================================
function vec = swapelement(vec,ind,val)
% swaps val as element ind, into the vector vec
vec(ind) = val;
end % sub-function end
% ============================================
% subfunction - romberg extrapolation
% ============================================
function [der_romb,errest] = rombextrap(StepRatio,der_init,rombexpon)
% do romberg extrapolation for each estimate
%
% StepRatio - Ratio decrease in step
% der_init - initial derivative estimates
% rombexpon - higher order terms to cancel using the romberg step
%
% der_romb - derivative estimates returned
% errest - error estimates
% amp - noise amplification factor due to the romberg step
srinv = 1/StepRatio;
% do nothing if no romberg terms
nexpon = length(rombexpon);
rmat = ones(nexpon+2,nexpon+1);
% two romberg terms
rmat(2,2:3) = srinv.^rombexpon;
rmat(3,2:3) = srinv.^(2*rombexpon);
rmat(4,2:3) = srinv.^(3*rombexpon);
% qr factorization used for the extrapolation as well
% as the uncertainty estimates
[qromb,rromb] = qr(rmat,0);
% the noise amplification is further amplified by the Romberg step.
% amp = cond(rromb);
% this does the extrapolation to a zero step size.
ne = length(der_init);
rhs = vec2mat(der_init,nexpon+2,ne - (nexpon+2));
rombcoefs = rromb\(qromb'*rhs);
der_romb = rombcoefs(1,:)';
% uncertainty estimate of derivative prediction
s = sqrt(sum((rhs - rmat*rombcoefs).^2,1));
rinv = rromb\eye(nexpon+1);
cov1 = sum(rinv.^2,2); % 1 spare dof
errest = s'*12.7062047361747*sqrt(cov1(1));
end % rombextrap
% ============================================
% subfunction - vec2mat
% ============================================
function mat = vec2mat(vec,n,m)
% forms the matrix M, such that M(i,j) = vec(i+j-1)
[i,j] = ndgrid(1:n,0:m-1);
ind = i+j;
mat = vec(ind);
if n==1
mat = mat';
end
end % vec2mat
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
spgl1.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/spgl1.m
| 30,253 |
utf_8
|
03d7576dbae19e8aa54fe160c192f10a
|
function [x,r,g,info] = spgl1( A, b, tau, sigma, x, options )
%SPGL1 Solve basis pursuit, basis pursuit denoise, and LASSO
%
% [x, r, g, info] = spgl1(A, b, tau, sigma, x0, options)
%
% ---------------------------------------------------------------------
% Solve the basis pursuit denoise (BPDN) problem
%
% (BPDN) minimize ||x||_1 subj to ||Ax-b||_2 <= sigma,
%
% or the l1-regularized least-squares problem
%
% (LASSO) minimize ||Ax-b||_2 subj to ||x||_1 <= tau.
% ---------------------------------------------------------------------
%
% INPUTS
% ======
% A is an m-by-n matrix, explicit or an operator.
% If A is a function, then it must have the signature
%
% y = A(x,mode) if mode == 1 then y = A x (y is m-by-1);
% if mode == 2 then y = A'x (y is n-by-1).
%
% b is an m-vector.
% tau is a nonnegative scalar; see (LASSO).
% sigma if sigma != inf or != [], then spgl1 will launch into a
% root-finding mode to find the tau above that solves (BPDN).
% In this case, it's STRONGLY recommended that tau = 0.
% x0 is an n-vector estimate of the solution (possibly all
% zeros). If x0 = [], then SPGL1 determines the length n via
% n = length( A'b ) and sets x0 = zeros(n,1).
% options is a structure of options from spgSetParms. Any unset options
% are set to their default value; set options=[] to use all
% default values.
%
% OUTPUTS
% =======
% x is a solution of the problem
% r is the residual, r = b - Ax
% g is the gradient, g = -A'r
% info is a structure with the following information:
% .tau final value of tau (see sigma above)
% .rNorm two-norm of the optimal residual
% .rGap relative duality gap (an optimality measure)
% .gNorm Lagrange multiplier of (LASSO)
% .stat = 1 found a BPDN solution
% = 2 found a BP sol'n; exit based on small gradient
% = 3 found a BP sol'n; exit based on small residual
% = 4 found a LASSO solution
% = 5 error: too many iterations
% = 6 error: linesearch failed
% = 7 error: found suboptimal BP solution
% = 8 error: too many matrix-vector products
% .time total solution time (seconds)
% .nProdA number of multiplications with A
% .nProdAt number of multiplications with A'
%
% OPTIONS
% =======
% Use the options structure to control various aspects of the algorithm:
%
% options.fid File ID to direct log output
% .verbosity 0=quiet, 1=some output, 2=more output.
% .iterations Max. number of iterations (default if 10*m).
% .bpTol Tolerance for identifying a basis pursuit solution.
% .optTol Optimality tolerance (default is 1e-4).
% .decTol Larger decTol means more frequent Newton updates.
% .subspaceMin 0=no subspace minimization, 1=subspace minimization.
%
% EXAMPLE
% =======
% m = 120; n = 512; k = 20; % m rows, n cols, k nonzeros.
% p = randperm(n); x0 = zeros(n,1); x0(p(1:k)) = sign(randn(k,1));
% A = randn(m,n); [Q,R] = qr(A',0); A = Q';
% b = A*x0 + 0.005 * randn(m,1);
% opts = spgSetParms('optTol',1e-4);
% [x,r,g,info] = spgl1(A, b, 0, 1e-3, [], opts); % Find BP sol'n.
%
% AUTHORS
% =======
% Ewout van den Berg ([email protected])
% Michael P. Friedlander ([email protected])
% Scientific Computing Laboratory (SCL)
% University of British Columbia, Canada.
%
% BUGS
% ====
% Please send bug reports or comments to
% Michael P. Friedlander ([email protected])
% Ewout van den Berg ([email protected])
% 15 Apr 07: First version derived from spg.m.
% Michael P. Friedlander ([email protected]).
% Ewout van den Berg ([email protected]).
% 17 Apr 07: Added root-finding code.
% 18 Apr 07: sigma was being compared to 1/2 r'r, rather than
% norm(r), as advertised. Now immediately change sigma to
% (1/2)sigma^2, and changed log output accordingly.
% 24 Apr 07: Added quadratic root-finding code as an option.
% 24 Apr 07: Exit conditions need to guard against small ||r||
% (ie, a BP solution). Added test1,test2,test3 below.
% 15 May 07: Trigger to update tau is now based on relative difference
% in objective between consecutive iterations.
% 15 Jul 07: Added code to allow a limited number of line-search
% errors.
% 23 Feb 08: Fixed bug in one-norm projection using weights. Thanks
% to Xiangrui Meng for reporting this bug.
% 26 May 08: The simple call spgl1(A,b) now solves (BPDN) with sigma=0.
% spgl1.m
% $Id: spgl1.m 1225 2009-01-30 20:36:31Z ewout78 $
%
% ----------------------------------------------------------------------
% This file is part of SPGL1 (Spectral Projected-Gradient for L1).
%
% Copyright (C) 2007 Ewout van den Berg and Michael P. Friedlander,
% Department of Computer Science, University of British Columbia, Canada.
% All rights reserved. E-mail: <{ewout78,mpf}@cs.ubc.ca>.
%
% SPGL1 is free software; you can redistribute it and/or modify it
% under the terms of the GNU Lesser General Public License as
% published by the Free Software Foundation; either version 2.1 of the
% License, or (at your option) any later version.
%
% SPGL1 is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
% or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
% Public License for more details.
%
% You should have received a copy of the GNU Lesser General Public
% License along with SPGL1; if not, write to the Free Software
% Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301
% USA
% ----------------------------------------------------------------------
REVISION = '$Revision: 1017 $';
DATE = '$Date: 2008-06-16 22:43:07 -0700 (Mon, 16 Jun 2008) $';
REVISION = REVISION(11:end-1);
DATE = DATE(35:50);
tic; % Start your watches!
m = length(b);
%----------------------------------------------------------------------
% Check arguments.
%----------------------------------------------------------------------
if ~exist('options','var'), options = []; end
if ~exist('x','var'), x = []; end
if ~exist('sigma','var'), sigma = []; end
if ~exist('tau','var'), tau = []; end
if nargin < 2 || isempty(b) || isempty(A)
error('At least two arguments are required');
elseif isempty(tau) && isempty(sigma)
tau = 0;
sigma = 0;
singleTau = false;
elseif isempty(sigma) % && ~isempty(tau) <-- implied
singleTau = true;
else
if isempty(tau)
tau = 0;
end
singleTau = false;
end
%----------------------------------------------------------------------
% Grab input options and set defaults where needed.
%----------------------------------------------------------------------
defaultopts = spgSetParms(...
'fid' , 1 , ... % File ID for output
'verbosity' , 2 , ... % Verbosity level
'iterations' , 10*m , ... % Max number of iterations
'nPrevVals' , 3 , ... % Number previous func values for linesearch
'bpTol' , 1e-06 , ... % Tolerance for basis pursuit solution
'optTol' , 1e-04 , ... % Optimality tolerance
'decTol' , 1e-04 , ... % Req'd rel. change in primal obj. for Newton
'stepMin' , 1e-16 , ... % Minimum spectral step
'stepMax' , 1e+05 , ... % Maximum spectral step
'rootMethod' , 2 , ... % Root finding method: 2=quad,1=linear (not used).
'activeSetIt', Inf , ... % Exit with EXIT_ACTIVE_SET if nnz same for # its.
'subspaceMin', 0 , ... % Use subspace minimization
'iscomplex' , NaN , ... % Flag set to indicate complex problem
'maxMatvec' , Inf , ... % Maximum matrix-vector multiplies allowed
'weights' , 1 , ... % Weights W in ||Wx||_1
'project' , @NormL1_project , ...
'primal_norm', @NormL1_primal , ...
'dual_norm' , @NormL1_dual ...
);
options = spgSetParms(defaultopts, options);
fid = options.fid;
logLevel = options.verbosity;
maxIts = options.iterations;
nPrevVals = options.nPrevVals;
bpTol = options.bpTol;
optTol = options.optTol;
decTol = options.decTol;
stepMin = options.stepMin;
stepMax = options.stepMax;
activeSetIt = options.activeSetIt;
subspaceMin = options.subspaceMin;
maxMatvec = max(3,options.maxMatvec);
weights = options.weights;
maxLineErrors = 10; % Maximum number of line-search failures.
pivTol = 1e-12; % Threshold for significant Newton step.
%----------------------------------------------------------------------
% Initialize local variables.
%----------------------------------------------------------------------
iter = 0; itnTotLSQR = 0; % Total SPGL1 and LSQR iterations.
nProdA = 0; nProdAt = 0;
lastFv = -inf(nPrevVals,1); % Last m function values.
nLineTot = 0; % Total no. of linesearch steps.
printTau = false;
nNewton = 0;
bNorm = norm(b,2);
stat = false;
timeProject = 0;
timeMatProd = 0;
nnzIter = 0; % No. of its with fixed pattern.
nnzIdx = []; % Active-set indicator.
subspace = false; % Flag if did subspace min in current itn.
stepG = 1; % Step length for projected gradient.
testUpdateTau = 0; % Previous step did not update tau
% Determine initial x, vector length n, and see if problem is complex
explicit = ~(isa(A,'function_handle'));
if isempty(x)
if isnumeric(A)
n = size(A,2);
realx = isreal(A) && isreal(b);
else
x = Aprod(b,2);
n = length(x);
realx = isreal(x) && isreal(b);
end
x = zeros(n,1);
else
n = length(x);
realx = isreal(x) && isreal(b);
end
if isnumeric(A), realx = realx && isreal(A); end;
% Override options when options.iscomplex flag is set
if (~isnan(options.iscomplex)), realx = (options.iscomplex == 0); end
% Check if all weights (if any) are strictly positive. In previous
% versions we also checked if the number of weights was equal to
% n. In the case of multiple measurement vectors, this no longer
% needs to apply, so the check was removed.
if ~isempty(weights)
if any(~isfinite(weights))
error('Entries in options.weights must be finite');
end
if any(weights <= 0)
error('Entries in options.weights must be strictly positive');
end
else
weights = 1;
end
% Quick exit if sigma >= ||b||. Set tau = 0 to short-circuit the loop.
if bNorm <= sigma
printf('W: sigma >= ||b||. Exact solution is x = 0.\n');
tau = 0; singleTau = true;
end
% Don't do subspace minimization if x is complex.
if ~realx && subspaceMin
printf('W: Subspace minimization disabled when variables are complex.\n');
subspaceMin = false;
end
% Pre-allocate iteration info vectors
xNorm1 = zeros(min(maxIts,10000),1);
rNorm2 = zeros(min(maxIts,10000),1);
lambda = zeros(min(maxIts,10000),1);
% Exit conditions (constants).
EXIT_ROOT_FOUND = 1;
EXIT_BPSOL1_FOUND = 2;
EXIT_BPSOL2_FOUND = 3;
EXIT_OPTIMAL = 4;
EXIT_ITERATIONS = 5;
EXIT_LINE_ERROR = 6;
EXIT_SUBOPTIMAL_BP = 7;
EXIT_MATVEC_LIMIT = 8;
EXIT_ACTIVE_SET = 9; % [sic]
%----------------------------------------------------------------------
% Log header.
%----------------------------------------------------------------------
printf('\n');
printf(' %s\n',repmat('=',1,80));
printf(' SPGL1 v.%s (%s)\n', REVISION, DATE);
printf(' %s\n',repmat('=',1,80));
printf(' %-22s: %8i %4s' ,'No. rows' ,m ,'');
printf(' %-22s: %8i\n' ,'No. columns' ,n );
printf(' %-22s: %8.2e %4s' ,'Initial tau' ,tau ,'');
printf(' %-22s: %8.2e\n' ,'Two-norm of b' ,bNorm );
printf(' %-22s: %8.2e %4s' ,'Optimality tol' ,optTol ,'');
if singleTau
printf(' %-22s: %8.2e\n' ,'Target one-norm of x' ,tau );
else
printf(' %-22s: %8.2e\n','Target objective' ,sigma );
end
printf(' %-22s: %8.2e %4s' ,'Basis pursuit tol' ,bpTol ,'');
printf(' %-22s: %8i\n' ,'Maximum iterations',maxIts );
printf('\n');
if singleTau
logB = ' %5i %13.7e %13.7e %9.2e %6.1f %6i %6i';
logH = ' %5s %13s %13s %9s %6s %6s %6s\n';
printf(logH,'Iter','Objective','Relative Gap','gNorm','stepG','nnzX','nnzG');
else
logB = ' %5i %13.7e %13.7e %9.2e %9.3e %6.1f %6i %6i';
logH = ' %5s %13s %13s %9s %9s %6s %6s %6s %13s\n';
printf(logH,'Iter','Objective','Relative Gap','Rel Error',...
'gNorm','stepG','nnzX','nnzG','tau');
end
% Project the starting point and evaluate function and gradient.
x = project(x,tau);
r = b - Aprod(x,1); % r = b - Ax
g = - Aprod(r,2); % g = -A'r
f = r'*r / 2;
% Required for nonmonotone strategy.
lastFv(1) = f;
fBest = f;
xBest = x;
fOld = f;
% Compute projected gradient direction and initial steplength.
dx = project(x - g, tau) - x;
dxNorm = norm(dx,inf);
if dxNorm < (1 / stepMax)
gStep = stepMax;
else
gStep = min( stepMax, max(stepMin, 1/dxNorm) );
end
%----------------------------------------------------------------------
% MAIN LOOP.
%----------------------------------------------------------------------
while 1
%------------------------------------------------------------------
% Test exit conditions.
%------------------------------------------------------------------
% Compute quantities needed for log and exit conditions.
gNorm = options.dual_norm(-g,weights);
rNorm = norm(r, 2);
gap = r'*(r-b) + tau*gNorm;
rGap = abs(gap) / max(1,f);
aError1 = rNorm - sigma;
aError2 = f - sigma^2 / 2;
rError1 = abs(aError1) / max(1,rNorm);
rError2 = abs(aError2) / max(1,f);
% Count number of consecutive iterations with identical support.
nnzOld = nnzIdx;
[nnzX,nnzG,nnzIdx,nnzDiff] = activeVars(x,g,nnzIdx,options);
if nnzDiff
nnzIter = 0;
else
nnzIter = nnzIter + 1;
if nnzIter >= activeSetIt, stat=EXIT_ACTIVE_SET; end
end
% Single tau: Check if we're optimal.
% The 2nd condition is there to guard against large tau.
if singleTau
if rGap <= optTol || rNorm < optTol*bNorm
stat = EXIT_OPTIMAL;
end
% Multiple tau: Check if found root and/or if tau needs updating.
else
if rGap <= max(optTol, rError2) || rError1 <= optTol
% The problem is nearly optimal for the current tau.
% Check optimality of the current root.
test1 = rNorm <= bpTol * bNorm;
test2 = gNorm <= bpTol * rNorm;
test3 = rError1 <= optTol;
test4 = rNorm <= sigma;
if test4, stat=EXIT_SUBOPTIMAL_BP;end % Found suboptimal BP sol.
if test3, stat=EXIT_ROOT_FOUND; end % Found approx root.
if test2, stat=EXIT_BPSOL2_FOUND; end % Gradient zero -> BP sol.
if test1, stat=EXIT_BPSOL1_FOUND; end % Resid minim'zd -> BP sol.
end
testRelChange1 = (abs(f - fOld) <= decTol * f);
testRelChange2 = (abs(f - fOld) <= 1e-1 * f * (abs(rNorm - sigma)));
testUpdateTau = ((testRelChange1 && rNorm > 2 * sigma) || ...
(testRelChange2 && rNorm <= 2 * sigma)) && ...
~stat && ~testUpdateTau;
if testUpdateTau
% Update tau.
tauOld = tau;
tau = max(0,tau + (rNorm * aError1) / gNorm);
nNewton = nNewton + 1;
printTau = abs(tauOld - tau) >= 1e-6 * tau; % For log only.
if tau < tauOld
% The one-norm ball has decreased. Need to make sure that the
% next iterate if feasible, which we do by projecting it.
x = project(x,tau);
end
end
end
% Too many its and not converged.
if ~stat && iter >= maxIts
stat = EXIT_ITERATIONS;
end
%------------------------------------------------------------------
% Print log, update history and act on exit conditions.
%------------------------------------------------------------------
if logLevel >= 2 || singleTau || printTau || iter == 0 || stat
tauFlag = ' '; subFlag = '';
if printTau, tauFlag = sprintf(' %13.7e',tau); end
if subspace, subFlag = sprintf(' S %2i',itnLSQR); end
if singleTau
printf(logB,iter,rNorm,rGap,gNorm,log10(stepG),nnzX,nnzG);
if subspace
printf(' %s',subFlag);
end
else
printf(logB,iter,rNorm,rGap,rError1,gNorm,log10(stepG),nnzX,nnzG);
if printTau || subspace
printf(' %s',[tauFlag subFlag]);
end
end
printf('\n');
end
printTau = false;
subspace = false;
% Update history info
xNorm1(iter+1) = options.primal_norm(x,weights);
rNorm2(iter+1) = rNorm;
lambda(iter+1) = gNorm;
if stat, break; end % Act on exit conditions.
%==================================================================
% Iterations begin here.
%==================================================================
iter = iter + 1;
xOld = x; fOld = f; gOld = g; rOld = r;
try
%---------------------------------------------------------------
% Projected gradient step and linesearch.
%---------------------------------------------------------------
[f,x,r,nLine,stepG,lnErr] = ...
spgLineCurvy(x,gStep*g,max(lastFv),@Aprod,b,@project,tau);
nLineTot = nLineTot + nLine;
if lnErr
% Projected backtrack failed. Retry with feasible dir'n linesearch.
x = xOld;
dx = project(x - gStep*g, tau) - x;
gtd = g'*dx;
[f,x,r,nLine,lnErr] = spgLine(f,x,dx,gtd,max(lastFv),@Aprod,b);
nLineTot = nLineTot + nLine;
end
if lnErr
% Failed again. Revert to previous iterates and damp max BB step.
if maxLineErrors <= 0
stat = EXIT_LINE_ERROR;
else
stepMax = stepMax / 10;
printf(['W: Linesearch failed with error %i. '...
'Damping max BB scaling to %6.1e.\n'],lnErr,stepMax);
maxLineErrors = maxLineErrors - 1;
end
end
%---------------------------------------------------------------
% Subspace minimization (only if active-set change is small).
%---------------------------------------------------------------
doSubspaceMin = false;
if subspaceMin
g = - Aprod(r,2);
[nnzX,nnzG,nnzIdx,nnzDiff] = activeVars(x,g,nnzOld,options);
if ~nnzDiff
if nnzX == nnzG, itnMaxLSQR = 20;
else itnMaxLSQR = 5;
end
nnzIdx = abs(x) >= optTol;
doSubspaceMin = true;
end
end
if doSubspaceMin
% LSQR parameters
damp = 1e-5;
aTol = 1e-1;
bTol = 1e-1;
conLim = 1e12;
showLSQR = 0;
ebar = sign(x(nnzIdx));
nebar = length(ebar);
Sprod = @(y,mode)LSQRprod(@Aprod,nnzIdx,ebar,n,y,mode);
[dxbar, istop, itnLSQR] = ...
lsqr(m,nebar,Sprod,r,damp,aTol,bTol,conLim,itnMaxLSQR,showLSQR);
itnTotLSQR = itnTotLSQR + itnLSQR;
if istop ~= 4 % LSQR iterations successful. Take the subspace step.
% Push dx back into full space: dx = Z dx.
dx = zeros(n,1);
dx(nnzIdx) = dxbar - (1/nebar)*(ebar'*dxbar)*dxbar;
% Find largest step to a change in sign.
block1 = nnzIdx & x < 0 & dx > +pivTol;
block2 = nnzIdx & x > 0 & dx < -pivTol;
alpha1 = Inf; alpha2 = Inf;
if any(block1), alpha1 = min(-x(block1) ./ dx(block1)); end
if any(block2), alpha2 = min(-x(block2) ./ dx(block2)); end
alpha = min([1 alpha1 alpha2]);
ensure(alpha >= 0);
ensure(ebar'*dx(nnzIdx) <= optTol);
% Update variables.
x = x + alpha*dx;
r = b - Aprod(x,1);
f = r'*r / 2;
subspace = true;
end
end
ensure(options.primal_norm(x,weights) <= tau+optTol);
%---------------------------------------------------------------
% Update gradient and compute new Barzilai-Borwein scaling.
%---------------------------------------------------------------
g = - Aprod(r,2);
s = x - xOld;
y = g - gOld;
sts = s'*s;
sty = s'*y;
if sty <= 0, gStep = stepMax;
else gStep = min( stepMax, max(stepMin, sts/sty) );
end
catch % Detect matrix-vector multiply limit error
err = lasterror;
if strcmp(err.identifier,'SPGL1:MaximumMatvec')
stat = EXIT_MATVEC_LIMIT;
iter = iter - 1;
x = xOld; f = fOld; g = gOld; r = rOld;
break;
else
rethrow(err);
end
end
%------------------------------------------------------------------
% Update function history.
%------------------------------------------------------------------
if singleTau || f > sigma^2 / 2 % Don't update if superoptimal.
lastFv(mod(iter,nPrevVals)+1) = f;
if fBest > f
fBest = f;
xBest = x;
end
end
end % while 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Restore best solution (only if solving single problem).
if singleTau && f > fBest
rNorm = sqrt(2*fBest);
printf('\n Restoring best iterate to objective %13.7e\n',rNorm);
x = xBest;
r = b - Aprod(x,1);
g = - Aprod(r,2);
gNorm = options.dual_norm(g,weights);
rNorm = norm(r, 2);
end
% Final cleanup before exit.
info.tau = tau;
info.rNorm = rNorm;
info.rGap = rGap;
info.gNorm = gNorm;
info.rGap = rGap;
info.stat = stat;
info.iter = iter;
info.nProdA = nProdA;
info.nProdAt = nProdAt;
info.nNewton = nNewton;
info.timeProject = timeProject;
info.timeMatProd = timeMatProd;
info.itnLSQR = itnTotLSQR;
info.options = options;
info.timeTotal = toc;
info.xNorm1 = xNorm1(1:iter);
info.rNorm2 = rNorm2(1:iter);
info.lambda = lambda(1:iter);
% Print final output.
switch (stat)
case EXIT_OPTIMAL
printf('\n EXIT -- Optimal solution found\n')
case EXIT_ITERATIONS
printf('\n ERROR EXIT -- Too many iterations\n');
case EXIT_ROOT_FOUND
printf('\n EXIT -- Found a root\n');
case {EXIT_BPSOL1_FOUND, EXIT_BPSOL2_FOUND}
printf('\n EXIT -- Found a BP solution\n');
case EXIT_LINE_ERROR
printf('\n ERROR EXIT -- Linesearch error (%i)\n',lnErr);
case EXIT_SUBOPTIMAL_BP
printf('\n EXIT -- Found a suboptimal BP solution\n');
case EXIT_MATVEC_LIMIT
printf('\n EXIT -- Maximum matrix-vector operations reached\n');
case EXIT_ACTIVE_SET
printf('\n EXIT -- Found a possible active set\n');
otherwise
error('Unknown termination condition\n');
end
printf('\n');
printf(' %-20s: %6i %6s %-20s: %6.1f\n',...
'Products with A',nProdA,'','Total time (secs)',info.timeTotal);
printf(' %-20s: %6i %6s %-20s: %6.1f\n',...
'Products with A''',nProdAt,'','Project time (secs)',timeProject);
printf(' %-20s: %6i %6s %-20s: %6.1f\n',...
'Newton iterations',nNewton,'','Mat-vec time (secs)',timeMatProd);
printf(' %-20s: %6i %6s %-20s: %6i\n', ...
'Line search its',nLineTot,'','Subspace iterations',itnTotLSQR);
printf('\n');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NESTED FUNCTIONS. These share some vars with workspace above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function z = Aprod(x,mode)
if (nProdA + nProdAt >= maxMatvec)
error('SPGL1:MaximumMatvec','');
end
tStart = toc;
if mode == 1
nProdA = nProdA + 1;
if explicit, z = A*x;
else z = A(x,1);
end
elseif mode == 2
nProdAt = nProdAt + 1;
if explicit, z = A'*x;
else z = A(x,2);
end
else
error('Wrong mode!');
end
timeMatProd = timeMatProd + (toc - tStart);
end % function Aprod
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function printf(varargin)
if logLevel > 0
fprintf(fid,varargin{:});
end
end % function printf
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function x = project(x, tau)
tStart = toc;
x = options.project(x,weights,tau);
timeProject = timeProject + (toc - tStart);
end % function project
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% End of nested functions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
end % function spg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PRIVATE FUNCTIONS.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [nnzX,nnzG,nnzIdx,nnzDiff] = activeVars(x,g,nnzIdx,options)
% Find the current active set.
% nnzX is the number of nonzero x.
% nnzG is the number of elements in nnzIdx.
% nnzIdx is a vector of primal/dual indicators.
% nnzDiff is the no. of elements that changed in the support.
xTol = min(.1,10*options.optTol);
gTol = min(.1,10*options.optTol);
gNorm = options.dual_norm(g,options.weights);
nnzOld = nnzIdx;
% Reduced costs for postive & negative parts of x.
z1 = gNorm + g;
z2 = gNorm - g;
% Primal/dual based indicators.
xPos = x > xTol & z1 < gTol; %g < gTol;%
xNeg = x < -xTol & z2 < gTol; %g > gTol;%
nnzIdx = xPos | xNeg;
% Count is based on simple primal indicator.
nnzX = sum(abs(x) >= xTol);
nnzG = sum(nnzIdx);
if isempty(nnzOld)
nnzDiff = inf;
else
nnzDiff = sum(nnzIdx ~= nnzOld);
end
end % function spgActiveVars
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function z = LSQRprod(Aprod,nnzIdx,ebar,n,dx,mode)
% Matrix multiplication for subspace minimization.
% Only called by LSQR.
nbar = length(ebar);
if mode == 1
y = zeros(n,1);
y(nnzIdx) = dx - (1/nbar)*(ebar'*dx)*ebar; % y(nnzIdx) = Z*dx
z = Aprod(y,1); % z = S Z dx
else
y = Aprod(dx,2);
z = y(nnzIdx) - (1/nbar)*(ebar'*y(nnzIdx))*ebar;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [fNew,xNew,rNew,iter,err] = spgLine(f,x,d,gtd,fMax,Aprod,b)
% Nonmonotone linesearch.
EXIT_CONVERGED = 0;
EXIT_ITERATIONS = 1;
maxIts = 10;
step = 1;
iter = 0;
gamma = 1e-4;
gtd = -abs(gtd); % 03 Aug 07: If gtd is complex,
% then should be looking at -abs(gtd).
while 1
% Evaluate trial point and function value.
xNew = x + step*d;
rNew = b - Aprod(xNew,1);
fNew = rNew'*rNew / 2;
% Check exit conditions.
if fNew < fMax + gamma*step*gtd % Sufficient descent condition.
err = EXIT_CONVERGED;
break
elseif iter >= maxIts % Too many linesearch iterations.
err = EXIT_ITERATIONS;
break
end
% New linesearch iteration.
iter = iter + 1;
% Safeguarded quadratic interpolation.
if step <= 0.1
step = step / 2;
else
tmp = (-gtd*step^2) / (2*(fNew-f-step*gtd));
if tmp < 0.1 || tmp > 0.9*step || isnan(tmp)
tmp = step / 2;
end
step = tmp;
end
end % while 1
end % function spgLine
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [fNew,xNew,rNew,iter,step,err] = ...
spgLineCurvy(x,g,fMax,Aprod,b,project,tau)
% Projected backtracking linesearch.
% On entry,
% g is the (possibly scaled) steepest descent direction.
EXIT_CONVERGED = 0;
EXIT_ITERATIONS = 1;
EXIT_NODESCENT = 2;
gamma = 1e-4;
maxIts = 10;
step = 1;
sNorm = 0;
scale = 1; % Safeguard scaling. (See below.)
nSafe = 0; % No. of safeguarding steps.
iter = 0;
debug = false; % Set to true to enable log.
n = length(x);
if debug
fprintf(' %5s %13s %13s %13s %8s\n',...
'LSits','fNew','step','gts','scale');
end
while 1
% Evaluate trial point and function value.
xNew = project(x - step*scale*g, tau);
rNew = b - Aprod(xNew,1);
fNew = rNew'*rNew / 2;
s = xNew - x;
gts = scale * g' * s;
if gts >= 0 % Should we check real and complex parts individually?
err = EXIT_NODESCENT;
break
end
if debug
fprintf(' LS %2i %13.7e %13.7e %13.6e %8.1e\n',...
iter,fNew,step,gts,scale);
end
% 03 Aug 07: If gts is complex, then should be looking at -abs(gts).
if fNew < fMax - gamma*step*abs(gts) % Sufficient descent condition.
err = EXIT_CONVERGED;
break
elseif iter >= maxIts % Too many linesearch iterations.
err = EXIT_ITERATIONS;
break
end
% New linesearch iteration.
iter = iter + 1;
step = step / 2;
% Safeguard: If stepMax is huge, then even damped search
% directions can give exactly the same point after projection. If
% we observe this in adjacent iterations, we drastically damp the
% next search direction.
% 31 May 07: Damp consecutive safeguarding steps.
sNormOld = sNorm;
sNorm = norm(s) / sqrt(n);
% if sNorm >= sNormOld
if abs(sNorm - sNormOld) <= 1e-6 * sNorm
gNorm = norm(g) / sqrt(n);
scale = sNorm / gNorm / (2^nSafe);
nSafe = nSafe + 1;
end
end % while 1
end % function spgLineCurvy
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
exls.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/exls.m
| 9,640 |
utf_8
|
41bd2c44a083af85a79ab91715bffc82
|
function [VARMAX,Z] = exls(Y,Z,p,r,method,tol,reg,opt,VARMAX0)
%EXLS Extended Least Squares
% [VARMAX,Z] = EXLS(Y,Z,P,R,METHOD,TOL,REG,OPT,VARMAX0) computes the
% extended least squares regression for the VARMAX estimation problem
% using recursive least squares. This function is intended for DORDVARMAX.
% Ivo Houtzager
% Delft Center of Systems and Control
% Delft University of Technology
% The Netherlands, 2010
% assign default values to unspecified parameters
if (nargin < 8) || isempty(opt)
opt = 'gcv';
end
if (nargin < 7) || isempty(reg)
reg = 'tikh';
end
if (nargin < 6) || isempty(tol)
tol = 1e-4;
end
if (nargin < 5) || isempty(method)
method = 'els';
end
% if strcmpi(method,'gradient') || strcmpi(method,'grad')
% if ~strcmpi(reg,'none')
% error('Gradient method does not support regularisation!')
% end
% end
ireg = sqrt(tol);
% determine size
N = size(Y,2)+p;
l = size(Y,1);
m = r+2*l;
% calculate initial VARMAX solution
if nargin < 9 || isempty(VARMAX0)
if ~strcmpi(reg,'none')
if strcmpi(method,'gradient') || strcmpi(method,'grad')
[VARMAX,reg_min] = regress(Y,Z,reg,opt);
else
VARMAX = zeros(l,size(Z,1));
end
else
VARMAX = zeros(l,size(Z,1));
end
else
if ~strcmpi(reg,'none')
[~,reg_min] = regress(Y,Z,reg,opt,VARMAX0);
end
VARMAX = VARMAX0;
end
cost1 = 1e10;
switch lower(method)
case {'grad','gradient'}
% do the VARMAX whitening iterations
lambda = 1;
if isscalar('opt')
reg_min = opt;
else
reg_min = 0;
end
maxit = 100;
ok = true;
k = 1;
while ok == true && k <= maxit
% evaluate function
E = eval_varmax(Z,Y,VARMAX,r,l,p,N);
if strcmpi(reg,'nuclear') || strcmpi(reg,'nuc')
cost = norm(E','fro')^2 + reg_min^2*sum(svd(VARMAX));
else
cost = norm(E','fro')^2 + reg_min^2*norm(VARMAX,'fro')^2;
end
% check residue
if abs(cost1 - cost) <= tol^2*N
ok = false;
end
% store the past and future vectors
e = zeros(l,N);
e(:,p+1:end) = E;
for i = 1:p
Z((i-1)*m+r+l+1:i*m,:) = e(:,i:N+i-p-1);
end
% compute regularization
if ~(strcmpi(reg,'none') || strcmpi(reg,'nuclear') || strcmpi(reg,'nuc'))
[~,reg_min] = regress(Y,Z,reg,opt);
end
% recalculate gradient step
k = k + 1;
if strcmpi(reg,'nuclear') || strcmpi(reg,'nuc')
lambda = fminbnd(@(x) (norm(eval_varmax(Z,Y,VARMAX + x.*(E*Z'),r,l,p,N)','fro')^2 + reg_min^2*sum(svd(VARMAX + x.*(E*Z'))))/N,0,lambda);
else
lambda = fminbnd(@(x) (norm(eval_varmax(Z,Y,VARMAX + x.*(E*Z'),r,l,p,N)','fro')^2 + reg_min^2*norm(VARMAX + x.*(E*Z'),'fro')^2)/N,0,lambda);
end
% step update
VARMAX = VARMAX + lambda.*(E*Z');
if strcmpi(reg,'nuclear') || strcmpi(reg,'nuc')
% step update with tresholding
VARMAX = VARMAX + lambda.*(E*Z');
[U,S,V] = svd(VARMAX);
if size(Y,1) == 1
VARMAX = U*diag(max(S(1,1)-reg_min^2*lambda,0))*V(:,1)';
else
VARMAX = U*diag(max(diag(S)-reg_min^2*lambda,0))*V(:,1:length(diag(S)))';
end
else
% apply shrinkage
VARMAX = (1/(1+2*lambda*reg_min^2))*VARMAX;
end
% swap
cost1 = cost;
end
case 'els'
% do the VARMAX whitening iterations
PS = ireg;
if isscalar('opt')
reg_min = opt;
else
reg_min = 0;
end
lambda = tol^(1/N);
maxit = 10;
ok = true;
k = 1;
PS0 = PS;
while ok == true && k <= maxit
if ~strcmpi(reg,'none') && ~isscalar('opt')
Y1 = Y.*(ones(size(Y,1),1)*lambda.^(length(Y)-1:-1:0));
Z1 = Z.*(ones(size(Z,1),1)*lambda.^(length(Z)-1:-1:0));
[~,reg_min] = regress(Y1,Z1,reg,opt,VARMAX);
end
PS = PS0;
for i = 1:N-p
if ~strcmpi(reg,'none')
[VARMAX,PS] = rls_ew_track_reg(Z(:,i),Y(:,i),VARMAX,PS,lambda,reg_min);
else
[VARMAX,PS] = rls_ew_track(Z(:,i),Y(:,i),VARMAX,PS,lambda);
end
E = Y(:,i) - VARMAX*Z(:,i);
if i ~= N-p
for j = 1:l
Z((1:p-1)*(l+r)+(0:p-2)*l+j,i+1) = Z((2:p)*(l+r)+(1:p-1)*l+j,i);
Z(p*(l+r)+(p-1)*l+j,i+1) = E(j);
end
end
end
% pre-allocate matrices
E = Y - VARMAX*Z;
% check residue
cost = norm(E','fro')^2 + reg_min^2*norm(VARMAX,'fro')^2;
if abs(cost1 - cost) <= tol^2*N
ok = false;
end
% store the past and future vectors
e = zeros(l,N);
e(:,p+1:end) = E;
for i = 1:p
Z((i-1)*m+r+l+1:i*m,:) = e(:,i:N+i-p-1);
end
% recalculate forgetting factor
k = k + 1;
lambda = (lambda^N)^(1/(k*N));
% swap
cost1 = cost;
end
otherwise
disp('Unknown method.')
end
end
function E = eval_varmax(Z,Y,VARMAX,r,l,p,N)
%EVAL_VARMAX Evaluate VARMAX Markov parameters
m = 2*l+r;
q = tf([0 1],[1 0],1,'Variable','z^-1');
e = zeros(l,N);
H = eye(l);
for i = 1:p
H = H - VARMAX(:,(i-1)*m+r+l+1:i*m).*q^i;
Z((i-1)*m+r+l+1:i*m,:) = e(:,i:N+i-p-1);
end
N = inv(N);
E = Y - VARMAX*Z;
T = 0:N-1;
E = lsim(H,E',T')';
end
function [theta,P] = rls_ew_track(z,y,theta,P,lambda)
%RLS_EW_TRACK Exponentially Weighted RLS iteration
% [THETA,P]=RLS_EW_TRACK(Z,Y,THETA,P,LAMBDA) applies one iteration of
% exponentially weighted regularized least-squares problem. In recursive
% least-squares, we deal with the issue of an inceasing amount of date Z
% and Y. At each iteration, THETA is the solution. The scalar LAMBDA is
% called the forgetting factor since past data are exponentially weighted
% less heavily than more recent data.
% Ivo Houtzager
% Delft Center of Systems and Control
% The Netherlands, 2010
% Assign default values to unspecified parameters
mz = size(z,1);
if (nargin < 5) || isempty(lambda)
lambda = 1;
end
if (nargin < 4) || isempty(P)
P = zeros(mz);
elseif isscalar(P)
P = (1/P).*eye(mz);
end
if (nargin < 3) || isempty(theta)
theta = zeros(size(y,1),mz);
end
% Exponentially-Weighted, Tracking, Regularized, Least-Squares Iteration
P = (1/lambda).*(P - P*(z*z')*P./(lambda + z'*P*z));
P = 0.5.*(P+P'); % force symmetric
e = y - theta*z;
theta = theta + e*z'*P;
end % end of function RLS_EW_TRACK
function [theta,P] = rls_ew_track_reg(z,y,theta,P,lambda,reg_min)
%RLS_EW_TRACK_REG Exponentially Weighted and Regularized RLS iteration
% [THETA,P]=RLS_EW_TRACK_REG(Z,Y,THETA,P,LAMBDA,REG) applies one iteration
% of exponentially weighted regularized least-squares problem. In
% recursive least-squares, we deal with the issue of an inceasing amount
% of date Z and Y. At each iteration, THETA is the solution. The scalar
% LAMBDA is called the forgetting factor since past data are exponentially
% weighted less heavily than more recent data.
% Ivo Houtzager
% Delft Center of Systems and Control
% The Netherlands, 2010
% Assign default values to unspecified parameters
mz = size(z,1);
if (nargin < 6) || isempty(reg_min)
reg_min = 0;
end
if (nargin < 5) || isempty(lambda)
lambda = 1;
end
if (nargin < 4) || isempty(P)
P = zeros(mz);
elseif isscalar(P)
P = (1/P).*eye(mz);
end
if (nargin < 3) || isempty(theta)
theta = zeros(size(y,1),mz);
end
% Exponentially-Weighted, Tracking, Regularized, Least-Squares Iteration
P = (1/lambda).*(P - P*(z*z')*P./(lambda + z'*P*z));
P = 0.5.*(P+P'); % force symmetric
if isscalar(reg_min)
opts.SYM = true;
opts.POSDEF = true;
P1 = linsolve((eye(size(P)) + reg_min^2.*P),P,opts);
e = y - theta*z;
theta = theta + e*z'*P1;
elseif strcmpi(reg_min,'tikh')
[U,S,V] = svd(pinv(P));
s = diag(S);
YP = (y-theta*z)';
if isscalar(opt)
reg_min = opt;
elseif strcmpi(opt,'lcurve')
reg_min = reglcurve(YP,U,s);
elseif strcmpi(opt,'gcv')
reg_min = reggcv(YP,U,s);
end
theta = theta + (V*(diag(s./(s.^2 + reg_min^2)))*U'*YP)';
elseif strcmpi(reg_min,'tsvd')
[U,S,V] = svd(pinv(P));
s = diag(S);
YP = (y-theta*z)';
if isscalar(opt)
k_min = opt;
elseif strcmpi(opt,'lcurve')
k_min = reglcurve(YP,U,s,'tsvd');
elseif strcmpi(opt,'gcv')
k_min = reggcv(YP,U,s,'tsvd');
end
theta = theta + (V(:,1:k_min)*diag(1./s(1:k_min))*U(:,1:k_min)'*YP)';
end
end % end of function RLS_EW_TRACK
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
oneProjectorMex.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/oneProjectorMex.m
| 3,797 |
utf_8
|
df5afe507062bc6b713674d862bf73cd
|
function [x, itn] = oneProjectorMex(b,d,tau)
% [x, itn] = oneProjectorMex(b,d,tau)
% Return the orthogonal projection of the vector b >=0 onto the
% (weighted) L1 ball. In case vector d is specified, matrix D is
% defined as diag(d), otherwise the identity matrix is used.
%
% On exit,
% x solves minimize ||b-x||_2 st ||Dx||_1 <= tau.
% itn is the number of elements of b that were thresholded.
%
% See also spgl1, oneProjector.
% oneProjectorMex.m
% $Id: oneProjectorMex.m 1200 2008-11-21 19:58:28Z mpf $
%
% ----------------------------------------------------------------------
% This file is part of SPGL1 (Spectral Projected Gradient for L1).
%
% Copyright (C) 2007 Ewout van den Berg and Michael P. Friedlander,
% Department of Computer Science, University of British Columbia, Canada.
% All rights reserved. E-mail: <{ewout78,mpf}@cs.ubc.ca>.
%
% SPGL1 is free software; you can redistribute it and/or modify it
% under the terms of the GNU Lesser General Public License as
% published by the Free Software Foundation; either version 2.1 of the
% License, or (at your option) any later version.
%
% SPGL1 is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
% or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
% Public License for more details.
%
% You should have received a copy of the GNU Lesser General Public
% License along with SPGL1; if not, write to the Free Software
% Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301
% USA
% ----------------------------------------------------------------------
if nargin < 3
tau = d;
d = 1;
end
if isscalar(d)
[x,itn] = oneProjectorMex_I(b,tau/abs(d));
else
[x,itn] = oneProjectorMex_D(b,d,tau);
end
end % function oneProjectorMex
% ----------------------------------------------------------------------
function [x,itn] = oneProjectorMex_I(b,tau)
% ----------------------------------------------------------------------
% Initialization
n = length(b);
x = zeros(n,1);
bNorm = norm(b,1);
% Check for quick exit.
if (tau >= bNorm), x = b; itn = 0; return; end
if (tau < eps ), itn = 0; return; end
% Preprocessing (b is assumed to be >= 0)
[b,idx] = sort(b,'descend'); % Descending.
csb = -tau;
alphaPrev = 0;
for j= 1:n
csb = csb + b(j);
alpha = csb / j;
% We are done as soon as the constraint can be satisfied
% without exceeding the current minimum value of b
if alpha >= b(j)
break;
end
alphaPrev = alpha;
end
% Set the solution by applying soft-thresholding with
% the previous value of alpha
x(idx) = max(0,b - alphaPrev);
% Set number of iterations
itn = j;
end
% ----------------------------------------------------------------------
function [x,itn] = oneProjectorMex_D(b,d,tau)
% ----------------------------------------------------------------------
% Initialization
n = length(b);
x = zeros(n,1);
% Check for quick exit.
if (tau >= norm(d.*b,1)), x = b; itn = 0; return; end
if (tau < eps ), itn = 0; return; end
% Preprocessing (b is assumed to be >= 0)
[bd,idx] = sort(b ./ d,'descend'); % Descending.
b = b(idx);
d = d(idx);
% Optimize
csdb = 0; csd2 = 0;
soft = 0; alpha1 = 0; i = 1;
while (i <= n)
csdb = csdb + d(i).*b(i);
csd2 = csd2 + d(i).*d(i);
alpha1 = (csdb - tau) / csd2;
alpha2 = bd(i);
if alpha1 >= alpha2
break;
end
soft = alpha1; i = i + 1;
end
x(idx(1:i-1)) = b(1:i-1) - d(1:i-1) * max(0,soft);
% Set number of iterations
itn = i;
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
lsqr.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/lsqr.m
| 11,849 |
utf_8
|
b60925c5944249161e00049c67d30868
|
function [ x, istop, itn, r1norm, r2norm, anorm, acond, arnorm, xnorm, var ]...
= lsqr( m, n, A, b, damp, atol, btol, conlim, itnlim, show )
%
% [ x, istop, itn, r1norm, r2norm, anorm, acond, arnorm, xnorm, var ]...
% = lsqr( m, n, A, b, damp, atol, btol, conlim, itnlim, show );
%
% LSQR solves Ax = b or min ||b - Ax||_2 if damp = 0,
% or min || (b) - ( A )x || otherwise.
% || (0) (damp I) ||2
% A is an m by n matrix defined or a function handle of aprod( mode,x ),
% that performs the matrix-vector operations.
% If mode = 1, aprod must return y = Ax without altering x.
% If mode = 2, aprod must return y = A'x without altering x.
%-----------------------------------------------------------------------
% LSQR uses an iterative (conjugate-gradient-like) method.
% For further information, see
% 1. C. C. Paige and M. A. Saunders (1982a).
% LSQR: An algorithm for sparse linear equations and sparse least squares,
% ACM TOMS 8(1), 43-71.
% 2. C. C. Paige and M. A. Saunders (1982b).
% Algorithm 583. LSQR: Sparse linear equations and least squares problems,
% ACM TOMS 8(2), 195-209.
% 3. M. A. Saunders (1995). Solution of sparse rectangular systems using
% LSQR and CRAIG, BIT 35, 588-604.
%
% Input parameters:
% atol, btol are stopping tolerances. If both are 1.0e-9 (say),
% the final residual norm should be accurate to about 9 digits.
% (The final x will usually have fewer correct digits,
% depending on cond(A) and the size of damp.)
% conlim is also a stopping tolerance. lsqr terminates if an estimate
% of cond(A) exceeds conlim. For compatible systems Ax = b,
% conlim could be as large as 1.0e+12 (say). For least-squares
% problems, conlim should be less than 1.0e+8.
% Maximum precision can be obtained by setting
% atol = btol = conlim = zero, but the number of iterations
% may then be excessive.
% itnlim is an explicit limit on iterations (for safety).
% show = 1 gives an iteration log,
% show = 0 suppresses output.
%
% Output parameters:
% x is the final solution.
% istop gives the reason for termination.
% istop = 1 means x is an approximate solution to Ax = b.
% = 2 means x approximately solves the least-squares problem.
% r1norm = norm(r), where r = b - Ax.
% r2norm = sqrt( norm(r)^2 + damp^2 * norm(x)^2 )
% = r1norm if damp = 0.
% anorm = estimate of Frobenius norm of Abar = [ A ].
% [damp*I]
% acond = estimate of cond(Abar).
% arnorm = estimate of norm(A'*r - damp^2*x).
% xnorm = norm(x).
% var (if present) estimates all diagonals of (A'A)^{-1} (if damp=0)
% or more generally (A'A + damp^2*I)^{-1}.
% This is well defined if A has full column rank or damp > 0.
% (Not sure what var means if rank(A) < n and damp = 0.)
%
%
% 1990: Derived from Fortran 77 version of LSQR.
% 22 May 1992: bbnorm was used incorrectly. Replaced by anorm.
% 26 Oct 1992: More input and output parameters added.
% 01 Sep 1994: Matrix-vector routine is now a parameter 'aprodname'.
% Print log reformatted.
% 14 Jun 1997: show added to allow printing or not.
% 30 Jun 1997: var added as an optional output parameter.
% 07 Aug 2002: Output parameter rnorm replaced by r1norm and r2norm.
% Michael Saunders, Systems Optimization Laboratory,
% Dept of MS&E, Stanford University.
% 03 Jul 2007: Modified 'aprodname' to A, which can either be an m by n
% matrix, or a function handle.
% Ewout van den Berg, University of British Columbia
% 03 Jul 2007: Modified 'test2' condition, omitted 'test1'.
% Ewout van den Berg, University of British Columbia
%-----------------------------------------------------------------------
% Initialize.
msg=['The exact solution is x = 0 '
'Ax - b is small enough, given atol, btol '
'The least-squares solution is good enough, given atol '
'The estimate of cond(Abar) has exceeded conlim '
'Ax - b is small enough for this machine '
'The least-squares solution is good enough for this machine'
'Cond(Abar) seems to be too large for this machine '
'The iteration limit has been reached '];
wantvar= nargout >= 6;
if wantvar, var = zeros(n,1); end
if show
disp(' ')
disp('LSQR Least-squares solution of Ax = b')
str1 = sprintf('The matrix A has %8g rows and %8g cols', m, n);
str2 = sprintf('damp = %20.14e wantvar = %8g', damp,wantvar);
str3 = sprintf('atol = %8.2e conlim = %8.2e', atol, conlim);
str4 = sprintf('btol = %8.2e itnlim = %8g' , btol, itnlim);
disp(str1); disp(str2); disp(str3); disp(str4);
end
itn = 0; istop = 0; nstop = 0;
ctol = 0; if conlim > 0, ctol = 1/conlim; end;
anorm = 0; acond = 0;
dampsq = damp^2; ddnorm = 0; res2 = 0;
xnorm = 0; xxnorm = 0; z = 0;
cs2 = -1; sn2 = 0;
% Set up the first vectors u and v for the bidiagonalization.
% These satisfy beta*u = b, alfa*v = A'u.
u = b(1:m); x = zeros(n,1);
alfa = 0; beta = norm( u );
if beta > 0
u = (1/beta) * u; v = Aprod(u,2);
alfa = norm( v );
end
if alfa > 0
v = (1/alfa) * v; w = v;
end
arnorm = alfa * beta;
if arnorm == 0
if show, disp(msg(1,:)); end
return
end
arnorm0= arnorm;
rhobar = alfa; phibar = beta; bnorm = beta;
rnorm = beta;
r1norm = rnorm;
r2norm = rnorm;
head1 = ' Itn x(1) r1norm r2norm ';
head2 = ' Compatible LS Norm A Cond A';
if show
disp(' ')
disp([head1 head2])
test1 = 1; test2 = alfa / beta;
str1 = sprintf( '%6g %12.5e', itn, x(1) );
str2 = sprintf( ' %10.3e %10.3e', r1norm, r2norm );
str3 = sprintf( ' %8.1e %8.1e', test1, test2 );
disp([str1 str2 str3])
end
%------------------------------------------------------------------
% Main iteration loop.
%------------------------------------------------------------------
while itn < itnlim
itn = itn + 1;
% Perform the next step of the bidiagonalization to obtain the
% next beta, u, alfa, v. These satisfy the relations
% beta*u = a*v - alfa*u,
% alfa*v = A'*u - beta*v.
u = Aprod(v,1) - alfa*u;
beta = norm( u );
if beta > 0
u = (1/beta) * u;
anorm = norm([anorm alfa beta damp]);
v = Aprod(u, 2) - beta*v;
alfa = norm( v );
if alfa > 0, v = (1/alfa) * v; end
end
% Use a plane rotation to eliminate the damping parameter.
% This alters the diagonal (rhobar) of the lower-bidiagonal matrix.
rhobar1 = norm([rhobar damp]);
cs1 = rhobar / rhobar1;
sn1 = damp / rhobar1;
psi = sn1 * phibar;
phibar = cs1 * phibar;
% Use a plane rotation to eliminate the subdiagonal element (beta)
% of the lower-bidiagonal matrix, giving an upper-bidiagonal matrix.
rho = norm([rhobar1 beta]);
cs = rhobar1/ rho;
sn = beta / rho;
theta = sn * alfa;
rhobar = - cs * alfa;
phi = cs * phibar;
phibar = sn * phibar;
tau = sn * phi;
% Update x and w.
t1 = phi /rho;
t2 = - theta/rho;
dk = (1/rho)*w;
x = x + t1*w;
w = v + t2*w;
ddnorm = ddnorm + norm(dk)^2;
if wantvar, var = var + dk.*dk; end
% Use a plane rotation on the right to eliminate the
% super-diagonal element (theta) of the upper-bidiagonal matrix.
% Then use the result to estimate norm(x).
delta = sn2 * rho;
gambar = - cs2 * rho;
rhs = phi - delta * z;
zbar = rhs / gambar;
xnorm = sqrt(xxnorm + zbar^2);
gamma = norm([gambar theta]);
cs2 = gambar / gamma;
sn2 = theta / gamma;
z = rhs / gamma;
xxnorm = xxnorm + z^2;
% Test for convergence.
% First, estimate the condition of the matrix Abar,
% and the norms of rbar and Abar'rbar.
acond = anorm * sqrt( ddnorm );
res1 = phibar^2;
res2 = res2 + psi^2;
rnorm = sqrt( res1 + res2 );
arnorm = alfa * abs( tau );
% 07 Aug 2002:
% Distinguish between
% r1norm = ||b - Ax|| and
% r2norm = rnorm in current code
% = sqrt(r1norm^2 + damp^2*||x||^2).
% Estimate r1norm from
% r1norm = sqrt(r2norm^2 - damp^2*||x||^2).
% Although there is cancellation, it might be accurate enough.
r1sq = rnorm^2 - dampsq * xxnorm;
r1norm = sqrt( abs(r1sq) ); if r1sq < 0, r1norm = - r1norm; end
r2norm = rnorm;
% Now use these norms to estimate certain other quantities,
% some of which will be small near a solution.
test1 = rnorm / bnorm;
test2 = arnorm / arnorm0;
% test2 = arnorm/( anorm * rnorm );
test3 = 1 / acond;
t1 = test1 / (1 + anorm * xnorm / bnorm);
rtol = btol + atol * anorm * xnorm / bnorm;
% The following tests guard against extremely small values of
% atol, btol or ctol. (The user may have set any or all of
% the parameters atol, btol, conlim to 0.)
% The effect is equivalent to the normal tests using
% atol = eps, btol = eps, conlim = 1/eps.
if itn >= itnlim, istop = 7; end
if 1 + test3 <= 1, istop = 6; end
if 1 + test2 <= 1, istop = 5; end
if 1 + t1 <= 1, istop = 4; end
% Allow for tolerances set by the user.
if test3 <= ctol, istop = 3; end
if test2 <= atol, istop = 2; end
% if test1 <= rtol, istop = 1; end
% See if it is time to print something.
prnt = 0;
if n <= 40 , prnt = 1; end
if itn <= 10 , prnt = 1; end
if itn >= itnlim-10, prnt = 1; end
if rem(itn,10) == 0 , prnt = 1; end
if test3 <= 2*ctol , prnt = 1; end
if test2 <= 10*atol , prnt = 1; end
% if test1 <= 10*rtol , prnt = 1; end
if istop ~= 0 , prnt = 1; end
if prnt == 1
if show
str1 = sprintf( '%6g %12.5e', itn, x(1) );
str2 = sprintf( ' %10.3e %10.3e', r1norm, r2norm );
str3 = sprintf( ' %8.1e %8.1e', test1, test2 );
str4 = sprintf( ' %8.1e %8.1e', anorm, acond );
disp([str1 str2 str3 str4])
end
end
if istop > 0, break, end
end
% End of iteration loop.
% Print the stopping condition.
if show
disp(' ')
disp('LSQR finished')
disp(msg(istop+1,:))
disp(' ')
str1 = sprintf( 'istop =%8g r1norm =%8.1e', istop, r1norm );
str2 = sprintf( 'anorm =%8.1e arnorm =%8.1e', anorm, arnorm );
str3 = sprintf( 'itn =%8g r2norm =%8.1e', itn, r2norm );
str4 = sprintf( 'acond =%8.1e xnorm =%8.1e', acond, xnorm );
disp([str1 ' ' str2])
disp([str3 ' ' str4])
disp(' ')
end
%-----------------------------------------------------------------------
% End of lsqr.m
%-----------------------------------------------------------------------
function z = Aprod(x,mode)
if mode == 1
if isnumeric(A), z = A*x;
else z = A(x,1);
end
else
if isnumeric(A), z = (x'*A)';
else z = A(x,2);
end
end
end % function Aprod
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
reglcurve.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/private/reglcurve.m
| 9,669 |
utf_8
|
4ba1982fb3c44a326be1ff4bec03edef
|
function reg_c=reglcurve(Y,Vn,Sn,method,show)
%REGLCURVE Compute regularization using L-curve criterion.
% Determine the regularization parameter for ordkernel
% using L-curve criterion. It plots the L-curve and
% find its corner. If the regularization method is
% 'tsvd' then the Spline Toolbox is needed to determine
% the corner. If this toolbox is not available NaN is
% returned.
%
% Syntax:
% reg=reglcurve(Y,V,S)
% reg=reglcurve(Y,V,S,method,show)
%
% Input:
% Y,V,S Data matrices from lpvkernel or bilkernel.
% method Regularization method to be used.
% 'Tikh' - Tikhonov regularization (default).
% 'tsvd' - Truncated singular value decomposition.
% show Display intermediate steps of the algorithm.
%
% Output:
% reg Regularization paramater for the kernel
% subspace identification method of ordkernel.
% Written by Vincent Verdult, May 2004.
% Based on Regularization Tools by P. C. Hansen
% default method
if nargin<4
method='Tikh';
end
if nargin<5
show=0;
end
if size(Y,1)~=size(Vn,1)
error('The number of rows in Y must equal the number of rows in V.')
end
if size(Vn,1)~=size(Vn,2)
error('V must be a square matrix.')
end
if size(Sn,2)~=1
error('S must be a column vector.')
end
if size(Vn,1)~=size(Sn,1)
error('The number of rows in S must equal the number of rows in V.')
end
% Initialization.
N = size(Sn,1);
s=sqrt(Sn);
beta = Vn'*Y;
xi = diag(1./s)*beta;
%%%%%%%%%%%%%%%%
% Tikhonov regularization
if (strncmp(method,'Tikh',4) || strncmp(method,'tikh',4))
SkipCorner=0;
txt = 'Tikh.';
marker='-';
npoints = 200; % Number of points on the curve.
smin_ratio = 16*eps; % Smallest regularization parameter.
eta = zeros(npoints,1);
rho = zeros(npoints,1);
reg_param = zeros(npoints,1);
reg_param(npoints) = max([s(N),s(1)*smin_ratio]);
ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));
if show==1
disp('Calculation points on L-curve')
end
for i=npoints-1:-1:1
reg_param(i) = ratio*reg_param(i+1);
end
n=size(xi,2);
for i=1:npoints
f = Sn./(Sn + reg_param(i)^2);
eta(i) = norm((f*ones(1,n)).*xi,'fro');
rho(i) = norm(((1-f)*ones(1,n)).*beta,'fro');
end
% locate corner
if show==1
disp('Calculating curvature of L-curve')
end
% The L-curve is differentiable; computation of curvature in
% log-log scale is easy.
% Compute g = - curvature of L-curve.
g = reglcfun(reg_param,Sn,beta,xi);
% Locate the corner. If the curvature is negative everywhere,
% then define the leftmost point of the L-curve as the corner.
if show==1
disp('Searching for corner in L-curve')
OPT=optimset('Display','iter');
else
OPT=optimset('Display','off');
end
[gmin,gi] = min(g);
reg_c = fminbnd(@reglcfun,...
reg_param(min(gi+1,length(g))),reg_param(max(gi-1,1)),...
OPT,Sn,beta,xi); % Minimizer.
kappa_max = - reglcfun(reg_c,Sn,beta,xi); % Maximum curvature.
if (kappa_max < 0)
lr = length(rho);
reg_c = reg_param(lr);
rho_c = rho(lr);
eta_c = eta(lr);
else
f = Sn./(Sn + reg_c^2);
eta_c = norm((f*ones(1,n)).*xi,'fro');
rho_c = norm(((1-f)*ones(1,n)).*beta,'fro');
end
%%%%%%%%%%%%%%%%
% Truncated SVD
elseif (strncmp(method,'tsvd',4) || strncmp(method,'TSVD',4))
% spline toolbox needed for determination of the corner.
SkipCorner = exist('splines','dir')~=7;
txt = 'TSVD';
marker='o';
eta = zeros(N,1);
rho = zeros(N,1);
eta(1) = sum(xi(1,:).^2);
for k=2:N
eta(k) = eta(k-1) + sum(xi(k,:).^2);
end
eta = sqrt(eta);
rho(N) = eps^2;
for k=N-1:-1:1
rho(k) = rho(k+1) + sum(beta(k+1,:).^2);
end
rho = sqrt(rho);
reg_param = (1:N)';
% Determine corner using Splines
if (SkipCorner)
reg_c = NaN;
else
% The L-curve is discrete and may include unwanted fine-grained
% corners. Use local smoothing, followed by fitting a 2-D spline
% curve to the smoothed discrete L-curve.
% Set default parameters for treatment of discrete L-curve.
deg = 2; % Degree of local smooting polynomial.
q = 2; % Half-width of local smoothing interval.
order = 4; % Order of fitting 2-D spline curve.
% Neglect singular values less than s_thr.
s_thr = eps;
index = find(s > s_thr);
rho_t = rho(index);
eta_t = eta(index);
reg_param_t = reg_param(index);
% Convert to logarithms.
lr = length(rho_t);
lrho = log(rho_t);
leta = log(eta_t);
slrho = lrho;
sleta = leta;
% For all interior points k = q+1:length(rho)-q-1 on the discrete
% L-curve, perform local smoothing with a polynomial of degree deg
% to the points k-q:k+q.
v = (-q:q)';
A = zeros(2*q+1,deg+1);
A(:,1) = ones(length(v),1);
for j = 2:deg+1
A(:,j) = A(:,j-1).*v;
end
for k = q+1:lr-q-1
cr = A\lrho(k+v); slrho(k) = cr(1);
ce = A\leta(k+v); sleta(k) = ce(1);
end
% Fit a 2-D spline curve to the smoothed discrete L-curve.
sp = spmak(1:lr+order,[slrho';sleta']);
pp = ppbrk(sp2pp(sp),[4,lr+1]);
% Extract abscissa and ordinate splines and differentiate them.
% Compute as many function values as default in spleval.
P = spleval(pp); dpp = fnder(pp);
D = spleval(dpp); ddpp = fnder(pp,2);
DD = spleval(ddpp);
ppx = P(1,:); ppy = P(2,:);
dppx = D(1,:); dppy = D(2,:);
ddppx = DD(1,:); ddppy = DD(2,:);
% Compute the corner of the discretized .spline curve via max. curvature.
% No need to refine this corner, since the final regularization
% parameter is discrete anyway.
% Define curvature = 0 where both dppx and dppy are zero.
k1 = dppx.*ddppy - ddppx.*dppy;
k2 = (dppx.^2 + dppy.^2).^(1.5);
I_nz = find(k2 ~= 0);
kappa = zeros(1,length(dppx));
kappa(I_nz) = -k1(I_nz)./k2(I_nz);
[kmax,ikmax] = max(kappa);
x_corner = ppx(ikmax); y_corner = ppy(ikmax);
% Locate the point on the discrete L-curve which is closest to the
% corner of the spline curve. Prefer a point below and to the
% left of the corner. If the curvature is negative everywhere,
% then define the leftmost point of the L-curve as the corner.
if (kmax < 0)
reg_c = reg_param_t(lr);
rho_c = rho_t(lr);
eta_c = eta_t(lr);
else
index = find(lrho < x_corner & leta < y_corner);
if ~isempty(index)
[dummy,rpi] = min((lrho(index)-x_corner).^2 + (leta(index)-y_corner).^2);
rpi = index(rpi);
else
[dummy,rpi] = min((lrho-x_corner).^2 + (leta-y_corner).^2);
end
reg_c = reg_param_t(rpi); rho_c = rho_t(rpi); eta_c = eta_t(rpi);
end
end
else
error('Illegal method')
end
%%%%%%%%%%%%%%%%
% Plot
if show==1
N=length(rho);
loglog(rho(2:end-1),eta(2:end-1))
ax = axis;
ni = round(N/10);
if (max(eta)/min(eta) > 10 || max(rho)/min(rho) > 10)
loglog(rho,eta,marker,rho(ni:ni:N),eta(ni:ni:N),'x')
else
plot(rho,eta,marker,rho(ni:ni:N),eta(ni:ni:N),'x')
end
HoldState = ishold;
hold on;
for k = ni:ni:N
text(rho(k),eta(k),num2str(reg_param(k)));
end
if ~(SkipCorner)
loglog([min(rho)/100,rho_c],[eta_c,eta_c],':r',...
[rho_c,rho_c],[min(eta)/100,eta_c],':r')
title(['L-curve, ',txt,' corner at ',num2str(reg_c)]);
else
title('L-curve')
end
axis(ax)
if (~HoldState)
hold off
end
xlabel('residual norm || A x - b ||_2')
ylabel('solution norm || x ||_2')
end
end
function g = reglcfun(lambda,Sn,beta,xi)
% reglcfun Computes L-curve for reglcurve.
% Auxiliary function for reglcurve.
%
% Written by Vincent Verdult, May 2004.
% Based on Regularization Tools by P. C. Hansen
% Initialization.
L=size(lambda,1);
n=size(xi,2);
phi = zeros(L,1);
dphi = zeros(L,1);
psi = zeros(L,1);
dpsi = zeros(L,1);
eta = zeros(L,1);
rho = zeros(L,1);
% Compute some intermediate quantities.
for i = 1:L
f = Sn./(Sn + lambda(i)^2);
cf = 1 - f;
eta(i) = norm((f*ones(1,n)).*xi,'fro');
rho(i) = norm((cf*ones(1,n)).*beta,'fro');
f1 = -2*f.*cf/lambda(i);
f2 = -f1.*(3-4*f)/lambda(i);
phi(i) = sum(f.*f1.*sum(xi.^2,2));
psi(i) = sum(cf.*f1.*sum(beta.^2,2));
dphi(i) = sum((f1.^2 + f.*f2).*sum(xi.^2,2));
dpsi(i) = sum((-f1.^2 + cf.*f2).*sum(beta.^2,2));
end
% Now compute the first and second derivatives of eta and rho
% with respect to lambda;
deta = phi./eta;
drho = -psi./rho;
ddeta = dphi./eta - deta.*(deta./eta);
ddrho = -dpsi./rho - drho.*(drho./rho);
% Convert to derivatives of log(eta) and log(rho).
dlogeta = deta./eta;
dlogrho = drho./rho;
ddlogeta = ddeta./eta - (dlogeta).^2;
ddlogrho = ddrho./rho - (dlogrho).^2;
% Let g = curvature.
g = - (dlogrho.*ddlogeta - ddlogrho.*dlogeta)./...
(dlogrho.^2 + dlogeta.^2).^(1.5);
end
|
github
|
TUDelft-DataDrivenControl/Predictor-Based-Subspace-IDentification-toolbox-master
|
kernmatrix.m
|
.m
|
Predictor-Based-Subspace-IDentification-toolbox-master/extra/backwards/private/kernmatrix.m
| 3,842 |
utf_8
|
f5ac9d877d0e03e735508333491c3fcb
|
function omega = kernmatrix(Xtrain,kernel_type,kernel_pars,Xt)
%KERNMATRIX Construct the positive (semi-) definite and symmetric kernel matrix
%
% Omega = kernel_matrix(X, kernel_fct, sig2)
%
% This matrix should be positive definite if the kernel function
% satisfies the Mercer condition. Construct the kernel values for
% all test data points in the rows of Xt, relative to the points of X.
%
% Omega_Xt = kernel_matrix(X, kernel_fct, sig2, Xt)
%
%
% Full syntax:
%
% Omega = kernel_matrix(X, kernel_fct, sig2)
% Omega = kernel_matrix(X, kernel_fct, sig2, Xt)
%
% Outputs:
% Omega : N x N (N x Nt) kernel matrix
% Inputs:
% X : N x d matrix with the inputs of the training data
% kernel : Kernel type (by default 'RBF_kernel')
% sig2 : Kernel parameter (bandwidth in the case of the 'RBF_kernel')
% Xt(*) : Nt x d matrix with the inputs of the test data
% Copyright (c) 2002, KULeuven-ESAT-SCD,
% License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlab
nb_data = size(Xtrain,1);
if nb_data> 3000,
error('Too memory intensive, the kernel matrix is restricted to size 3000 x 3000 ');
end
if strcmpi(kernel_type,'rbf'),
if nargin<4,
XXh = sum(Xtrain.^2,2)*ones(1,nb_data);
omega = (XXh+XXh') - 2*(Xtrain*Xtrain');
omega = exp(-omega./kernel_pars(1));
else
XXh1 = sum(Xtrain.^2,2)*ones(1,size(Xt,1));
XXh2 = sum(Xt.^2,2)*ones(1,nb_data);
omega = XXh1+XXh2' - 2*Xtrain*Xt';
omega = exp(-omega./kernel_pars(1));
end
else
if nargin<4,
omega = zeros(nb_data,nb_data);
for i=1:nb_data,
omega(i:end,i) = feval(lower(kernel_type),Xtrain(i,:),Xtrain(i:end,:),kernel_pars);
omega(i,i:end) = omega(i:end,i)';
end
else
if size(Xt,2)~=size(Xtrain,2),
error('dimension test data not equal to dimension traindata;');
end
omega = zeros(nb_data, size(Xt,1));
for i=1:size(Xt,1),
omega(:,i) = feval(lower(kernel_type),Xt(i,:),Xtrain,kernel_pars);
end
end
end
end
function x = lin(a,b,c)
% kernel function for implicit higher dimension mapping, based on
% the standard inner-product
%
% x = lin_kernel(a,b)
%
% 'a' can only contain one datapoint in a row, 'b' can contain N
% datapoints of the same dimension as 'a'.
%
% see also:
% poly_kernel, RBF_kernel, MLP_kernel, trainlssvm, simlssvm
% Copyright (c) 2002, KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlab
x = zeros(size(b,1),1);
for i=1:size(b,1),
x(i,1) = a*b(i,:)';
end
end
function x = poly(a,b,d)
% polynomial kernel function for implicit higher dimension mapping
%
% X = poly_kernel(a,b,[t,degree])
%
% 'a' can only contain one datapoint in a row, 'b' can contain N
% datapoints of the same dimension as 'a'.
%
% x = (a*b'+t^2).^degree;
%
% see also:
% RBF_kernel, lin_kernel, MLP_kernel, trainlssvm, simlssvm
%
% Copyright (c) 2002, KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlab
if length(d)>1, d=d(2); t=d(1); else d = d(1);t=1; end
d = (abs(d)>=1)*abs(d)+(abs(d)<1); % >=1 !!
x = zeros(size(b,1),1);
for i=1:size(b,1),
x(i,1) = (a*b(i,:)'+t^2).^d;
end
end
function x = mlp(a,b, par)
% Multi Layer Perceptron kernel function for implicit higher dimension mapping
%
% x = MLP_kernel(a,b,[s,t])
%
% 'a' can only contain one datapoint in a row, 'b' can contain N
% datapoints of the same dimension as 'a'.
%
% x = tanh(s*a'b+t^2)
%
% see also:
% poly_kernel, lin_kernel, RBF_kernel, trainlssvm, simlssvm
% Copyright (c) 2002, KULeuven-ESAT-SCD, License & help @ http://www.esat.kuleuven.ac.be/sista/lssvmlab
if length(par)==1, par(2) = 1; end
x = zeros(size(b,1),1);
for i=1:size(b,1),
dp = a*b(i,:)';
x(i,1) = tanh(par(1)*dp + par(2)^2);
end
end
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