venv/lib/python3.10/site-packages/scipy/interpolate/_fitpack_impl.py

"""
fitpack (dierckx in netlib) --- A Python-C wrapper to FITPACK (by P. Dierckx).
        FITPACK is a collection of FORTRAN programs for curve and surface
        fitting with splines and tensor product splines.

See
 https://web.archive.org/web/20010524124604/http://www.cs.kuleuven.ac.be:80/cwis/research/nalag/research/topics/fitpack.html
or
 http://www.netlib.org/dierckx/

Copyright 2002 Pearu Peterson all rights reserved,
Pearu Peterson <[email protected]>
Permission to use, modify, and distribute this software is given under the
terms of the SciPy (BSD style) license. See LICENSE.txt that came with
this distribution for specifics.

NO WARRANTY IS EXPRESSED OR IMPLIED.  USE AT YOUR OWN RISK.

TODO: Make interfaces to the following fitpack functions:
    For univariate splines: cocosp, concon, fourco, insert
    For bivariate splines: profil, regrid, parsur, surev
"""

__all__ = ['splrep', 'splprep', 'splev', 'splint', 'sproot', 'spalde',
           'bisplrep', 'bisplev', 'insert', 'splder', 'splantider']

import warnings
import numpy as np
from . import _fitpack
from numpy import (atleast_1d, array, ones, zeros, sqrt, ravel, transpose,
                   empty, iinfo, asarray)

# Try to replace _fitpack interface with
#  f2py-generated version
from . import dfitpack


dfitpack_int = dfitpack.types.intvar.dtype


def _int_overflow(x, exception, msg=None):
    """Cast the value to an dfitpack_int and raise an OverflowError if the value
    cannot fit.
    """
    if x > iinfo(dfitpack_int).max:
        if msg is None:
            msg = f'{x!r} cannot fit into an {dfitpack_int!r}'
        raise exception(msg)
    return dfitpack_int.type(x)


_iermess = {
    0: ["The spline has a residual sum of squares fp such that "
        "abs(fp-s)/s<=0.001", None],
    -1: ["The spline is an interpolating spline (fp=0)", None],
    -2: ["The spline is weighted least-squares polynomial of degree k.\n"
         "fp gives the upper bound fp0 for the smoothing factor s", None],
    1: ["The required storage space exceeds the available storage space.\n"
        "Probable causes: data (x,y) size is too small or smoothing parameter"
        "\ns is too small (fp>s).", ValueError],
    2: ["A theoretically impossible result when finding a smoothing spline\n"
        "with fp = s. Probable cause: s too small. (abs(fp-s)/s>0.001)",
        ValueError],
    3: ["The maximal number of iterations (20) allowed for finding smoothing\n"
        "spline with fp=s has been reached. Probable cause: s too small.\n"
        "(abs(fp-s)/s>0.001)", ValueError],
    10: ["Error on input data", ValueError],
    'unknown': ["An error occurred", TypeError]
}

_iermess2 = {
    0: ["The spline has a residual sum of squares fp such that "
        "abs(fp-s)/s<=0.001", None],
    -1: ["The spline is an interpolating spline (fp=0)", None],
    -2: ["The spline is weighted least-squares polynomial of degree kx and ky."
         "\nfp gives the upper bound fp0 for the smoothing factor s", None],
    -3: ["Warning. The coefficients of the spline have been computed as the\n"
         "minimal norm least-squares solution of a rank deficient system.",
         None],
    1: ["The required storage space exceeds the available storage space.\n"
        "Probable causes: nxest or nyest too small or s is too small. (fp>s)",
        ValueError],
    2: ["A theoretically impossible result when finding a smoothing spline\n"
        "with fp = s. Probable causes: s too small or badly chosen eps.\n"
        "(abs(fp-s)/s>0.001)", ValueError],
    3: ["The maximal number of iterations (20) allowed for finding smoothing\n"
        "spline with fp=s has been reached. Probable cause: s too small.\n"
        "(abs(fp-s)/s>0.001)", ValueError],
    4: ["No more knots can be added because the number of B-spline\n"
        "coefficients already exceeds the number of data points m.\n"
        "Probable causes: either s or m too small. (fp>s)", ValueError],
    5: ["No more knots can be added because the additional knot would\n"
        "coincide with an old one. Probable cause: s too small or too large\n"
        "a weight to an inaccurate data point. (fp>s)", ValueError],
    10: ["Error on input data", ValueError],
    11: ["rwrk2 too small, i.e., there is not enough workspace for computing\n"
         "the minimal least-squares solution of a rank deficient system of\n"
         "linear equations.", ValueError],
    'unknown': ["An error occurred", TypeError]
}

_parcur_cache = {'t': array([], float), 'wrk': array([], float),
                 'iwrk': array([], dfitpack_int), 'u': array([], float),
                 'ub': 0, 'ue': 1}


def splprep(x, w=None, u=None, ub=None, ue=None, k=3, task=0, s=None, t=None,
            full_output=0, nest=None, per=0, quiet=1):
    # see the docstring of `_fitpack_py/splprep`
    if task <= 0:
        _parcur_cache = {'t': array([], float), 'wrk': array([], float),
                         'iwrk': array([], dfitpack_int), 'u': array([], float),
                         'ub': 0, 'ue': 1}
    x = atleast_1d(x)
    idim, m = x.shape
    if per:
        for i in range(idim):
            if x[i][0] != x[i][-1]:
                if not quiet:
                    warnings.warn(RuntimeWarning('Setting x[%d][%d]=x[%d][0]' %
                                                 (i, m, i)),
                                  stacklevel=2)
                x[i][-1] = x[i][0]
    if not 0 < idim < 11:
        raise TypeError('0 < idim < 11 must hold')
    if w is None:
        w = ones(m, float)
    else:
        w = atleast_1d(w)
    ipar = (u is not None)
    if ipar:
        _parcur_cache['u'] = u
        if ub is None:
            _parcur_cache['ub'] = u[0]
        else:
            _parcur_cache['ub'] = ub
        if ue is None:
            _parcur_cache['ue'] = u[-1]
        else:
            _parcur_cache['ue'] = ue
    else:
        _parcur_cache['u'] = zeros(m, float)
    if not (1 <= k <= 5):
        raise TypeError('1 <= k= %d <=5 must hold' % k)
    if not (-1 <= task <= 1):
        raise TypeError('task must be -1, 0 or 1')
    if (not len(w) == m) or (ipar == 1 and (not len(u) == m)):
        raise TypeError('Mismatch of input dimensions')
    if s is None:
        s = m - sqrt(2*m)
    if t is None and task == -1:
        raise TypeError('Knots must be given for task=-1')
    if t is not None:
        _parcur_cache['t'] = atleast_1d(t)
    n = len(_parcur_cache['t'])
    if task == -1 and n < 2*k + 2:
        raise TypeError('There must be at least 2*k+2 knots for task=-1')
    if m <= k:
        raise TypeError('m > k must hold')
    if nest is None:
        nest = m + 2*k

    if (task >= 0 and s == 0) or (nest < 0):
        if per:
            nest = m + 2*k
        else:
            nest = m + k + 1
    nest = max(nest, 2*k + 3)
    u = _parcur_cache['u']
    ub = _parcur_cache['ub']
    ue = _parcur_cache['ue']
    t = _parcur_cache['t']
    wrk = _parcur_cache['wrk']
    iwrk = _parcur_cache['iwrk']
    t, c, o = _fitpack._parcur(ravel(transpose(x)), w, u, ub, ue, k,
                               task, ipar, s, t, nest, wrk, iwrk, per)
    _parcur_cache['u'] = o['u']
    _parcur_cache['ub'] = o['ub']
    _parcur_cache['ue'] = o['ue']
    _parcur_cache['t'] = t
    _parcur_cache['wrk'] = o['wrk']
    _parcur_cache['iwrk'] = o['iwrk']
    ier = o['ier']
    fp = o['fp']
    n = len(t)
    u = o['u']
    c.shape = idim, n - k - 1
    tcku = [t, list(c), k], u
    if ier <= 0 and not quiet:
        warnings.warn(RuntimeWarning(_iermess[ier][0] +
                                     "\tk=%d n=%d m=%d fp=%f s=%f" %
                                     (k, len(t), m, fp, s)),
                      stacklevel=2)
    if ier > 0 and not full_output:
        if ier in [1, 2, 3]:
            warnings.warn(RuntimeWarning(_iermess[ier][0]), stacklevel=2)
        else:
            try:
                raise _iermess[ier][1](_iermess[ier][0])
            except KeyError as e:
                raise _iermess['unknown'][1](_iermess['unknown'][0]) from e
    if full_output:
        try:
            return tcku, fp, ier, _iermess[ier][0]
        except KeyError:
            return tcku, fp, ier, _iermess['unknown'][0]
    else:
        return tcku


_curfit_cache = {'t': array([], float), 'wrk': array([], float),
                 'iwrk': array([], dfitpack_int)}


def splrep(x, y, w=None, xb=None, xe=None, k=3, task=0, s=None, t=None,
           full_output=0, per=0, quiet=1):
    # see the docstring of `_fitpack_py/splrep`
    if task <= 0:
        _curfit_cache = {}
    x, y = map(atleast_1d, [x, y])
    m = len(x)
    if w is None:
        w = ones(m, float)
        if s is None:
            s = 0.0
    else:
        w = atleast_1d(w)
        if s is None:
            s = m - sqrt(2*m)
    if not len(w) == m:
        raise TypeError('len(w)=%d is not equal to m=%d' % (len(w), m))
    if (m != len(y)) or (m != len(w)):
        raise TypeError('Lengths of the first three arguments (x,y,w) must '
                        'be equal')
    if not (1 <= k <= 5):
        raise TypeError('Given degree of the spline (k=%d) is not supported. '
                        '(1<=k<=5)' % k)
    if m <= k:
        raise TypeError('m > k must hold')
    if xb is None:
        xb = x[0]
    if xe is None:
        xe = x[-1]
    if not (-1 <= task <= 1):
        raise TypeError('task must be -1, 0 or 1')
    if t is not None:
        task = -1
    if task == -1:
        if t is None:
            raise TypeError('Knots must be given for task=-1')
        numknots = len(t)
        _curfit_cache['t'] = empty((numknots + 2*k + 2,), float)
        _curfit_cache['t'][k+1:-k-1] = t
        nest = len(_curfit_cache['t'])
    elif task == 0:
        if per:
            nest = max(m + 2*k, 2*k + 3)
        else:
            nest = max(m + k + 1, 2*k + 3)
        t = empty((nest,), float)
        _curfit_cache['t'] = t
    if task <= 0:
        if per:
            _curfit_cache['wrk'] = empty((m*(k + 1) + nest*(8 + 5*k),), float)
        else:
            _curfit_cache['wrk'] = empty((m*(k + 1) + nest*(7 + 3*k),), float)
        _curfit_cache['iwrk'] = empty((nest,), dfitpack_int)
    try:
        t = _curfit_cache['t']
        wrk = _curfit_cache['wrk']
        iwrk = _curfit_cache['iwrk']
    except KeyError as e:
        raise TypeError("must call with task=1 only after"
                        " call with task=0,-1") from e
    if not per:
        n, c, fp, ier = dfitpack.curfit(task, x, y, w, t, wrk, iwrk,
                                        xb, xe, k, s)
    else:
        n, c, fp, ier = dfitpack.percur(task, x, y, w, t, wrk, iwrk, k, s)
    tck = (t[:n], c[:n], k)
    if ier <= 0 and not quiet:
        _mess = (_iermess[ier][0] + "\tk=%d n=%d m=%d fp=%f s=%f" %
                 (k, len(t), m, fp, s))
        warnings.warn(RuntimeWarning(_mess), stacklevel=2)
    if ier > 0 and not full_output:
        if ier in [1, 2, 3]:
            warnings.warn(RuntimeWarning(_iermess[ier][0]), stacklevel=2)
        else:
            try:
                raise _iermess[ier][1](_iermess[ier][0])
            except KeyError as e:
                raise _iermess['unknown'][1](_iermess['unknown'][0]) from e
    if full_output:
        try:
            return tck, fp, ier, _iermess[ier][0]
        except KeyError:
            return tck, fp, ier, _iermess['unknown'][0]
    else:
        return tck


def splev(x, tck, der=0, ext=0):
    # see the docstring of `_fitpack_py/splev`
    t, c, k = tck
    try:
        c[0][0]
        parametric = True
    except Exception:
        parametric = False
    if parametric:
        return list(map(lambda c, x=x, t=t, k=k, der=der:
                        splev(x, [t, c, k], der, ext), c))
    else:
        if not (0 <= der <= k):
            raise ValueError("0<=der=%d<=k=%d must hold" % (der, k))
        if ext not in (0, 1, 2, 3):
            raise ValueError("ext = %s not in (0, 1, 2, 3) " % ext)

        x = asarray(x)
        shape = x.shape
        x = atleast_1d(x).ravel()
        if der == 0:
            y, ier = dfitpack.splev(t, c, k, x, ext)
        else:
            y, ier = dfitpack.splder(t, c, k, x, der, ext)

        if ier == 10:
            raise ValueError("Invalid input data")
        if ier == 1:
            raise ValueError("Found x value not in the domain")
        if ier:
            raise TypeError("An error occurred")

        return y.reshape(shape)


def splint(a, b, tck, full_output=0):
    # see the docstring of `_fitpack_py/splint`
    t, c, k = tck
    try:
        c[0][0]
        parametric = True
    except Exception:
        parametric = False
    if parametric:
        return list(map(lambda c, a=a, b=b, t=t, k=k:
                        splint(a, b, [t, c, k]), c))
    else:
        aint, wrk = dfitpack.splint(t, c, k, a, b)
        if full_output:
            return aint, wrk
        else:
            return aint


def sproot(tck, mest=10):
    # see the docstring of `_fitpack_py/sproot`
    t, c, k = tck
    if k != 3:
        raise ValueError("sproot works only for cubic (k=3) splines")
    try:
        c[0][0]
        parametric = True
    except Exception:
        parametric = False
    if parametric:
        return list(map(lambda c, t=t, k=k, mest=mest:
                        sproot([t, c, k], mest), c))
    else:
        if len(t) < 8:
            raise TypeError("The number of knots %d>=8" % len(t))
        z, m, ier = dfitpack.sproot(t, c, mest)
        if ier == 10:
            raise TypeError("Invalid input data. "
                            "t1<=..<=t4<t5<..<tn-3<=..<=tn must hold.")
        if ier == 0:
            return z[:m]
        if ier == 1:
            warnings.warn(RuntimeWarning("The number of zeros exceeds mest"),
                          stacklevel=2)
            return z[:m]
        raise TypeError("Unknown error")


def spalde(x, tck):
    # see the docstring of `_fitpack_py/spalde`
    t, c, k = tck
    try:
        c[0][0]
        parametric = True
    except Exception:
        parametric = False
    if parametric:
        return list(map(lambda c, x=x, t=t, k=k:
                        spalde(x, [t, c, k]), c))
    else:
        x = atleast_1d(x)
        if len(x) > 1:
            return list(map(lambda x, tck=tck: spalde(x, tck), x))
        d, ier = dfitpack.spalde(t, c, k+1, x[0])
        if ier == 0:
            return d
        if ier == 10:
            raise TypeError("Invalid input data. t(k)<=x<=t(n-k+1) must hold.")
        raise TypeError("Unknown error")

# def _curfit(x,y,w=None,xb=None,xe=None,k=3,task=0,s=None,t=None,
#           full_output=0,nest=None,per=0,quiet=1):


_surfit_cache = {'tx': array([], float), 'ty': array([], float),
                 'wrk': array([], float), 'iwrk': array([], dfitpack_int)}


def bisplrep(x, y, z, w=None, xb=None, xe=None, yb=None, ye=None,
             kx=3, ky=3, task=0, s=None, eps=1e-16, tx=None, ty=None,
             full_output=0, nxest=None, nyest=None, quiet=1):
    """
    Find a bivariate B-spline representation of a surface.

    Given a set of data points (x[i], y[i], z[i]) representing a surface
    z=f(x,y), compute a B-spline representation of the surface. Based on
    the routine SURFIT from FITPACK.

    Parameters
    ----------
    x, y, z : ndarray
        Rank-1 arrays of data points.
    w : ndarray, optional
        Rank-1 array of weights. By default ``w=np.ones(len(x))``.
    xb, xe : float, optional
        End points of approximation interval in `x`.
        By default ``xb = x.min(), xe=x.max()``.
    yb, ye : float, optional
        End points of approximation interval in `y`.
        By default ``yb=y.min(), ye = y.max()``.
    kx, ky : int, optional
        The degrees of the spline (1 <= kx, ky <= 5).
        Third order (kx=ky=3) is recommended.
    task : int, optional
        If task=0, find knots in x and y and coefficients for a given
        smoothing factor, s.
        If task=1, find knots and coefficients for another value of the
        smoothing factor, s.  bisplrep must have been previously called
        with task=0 or task=1.
        If task=-1, find coefficients for a given set of knots tx, ty.
    s : float, optional
        A non-negative smoothing factor. If weights correspond
        to the inverse of the standard-deviation of the errors in z,
        then a good s-value should be found in the range
        ``(m-sqrt(2*m),m+sqrt(2*m))`` where m=len(x).
    eps : float, optional
        A threshold for determining the effective rank of an
        over-determined linear system of equations (0 < eps < 1).
        `eps` is not likely to need changing.
    tx, ty : ndarray, optional
        Rank-1 arrays of the knots of the spline for task=-1
    full_output : int, optional
        Non-zero to return optional outputs.
    nxest, nyest : int, optional
        Over-estimates of the total number of knots. If None then
        ``nxest = max(kx+sqrt(m/2),2*kx+3)``,
        ``nyest = max(ky+sqrt(m/2),2*ky+3)``.
    quiet : int, optional
        Non-zero to suppress printing of messages.

    Returns
    -------
    tck : array_like
        A list [tx, ty, c, kx, ky] containing the knots (tx, ty) and
        coefficients (c) of the bivariate B-spline representation of the
        surface along with the degree of the spline.
    fp : ndarray
        The weighted sum of squared residuals of the spline approximation.
    ier : int
        An integer flag about splrep success. Success is indicated if
        ier<=0. If ier in [1,2,3] an error occurred but was not raised.
        Otherwise an error is raised.
    msg : str
        A message corresponding to the integer flag, ier.

    See Also
    --------
    splprep, splrep, splint, sproot, splev
    UnivariateSpline, BivariateSpline

    Notes
    -----
    See `bisplev` to evaluate the value of the B-spline given its tck
    representation.

    If the input data is such that input dimensions have incommensurate
    units and differ by many orders of magnitude, the interpolant may have
    numerical artifacts. Consider rescaling the data before interpolation.

    References
    ----------
    .. [1] Dierckx P.:An algorithm for surface fitting with spline functions
       Ima J. Numer. Anal. 1 (1981) 267-283.
    .. [2] Dierckx P.:An algorithm for surface fitting with spline functions
       report tw50, Dept. Computer Science,K.U.Leuven, 1980.
    .. [3] Dierckx P.:Curve and surface fitting with splines, Monographs on
       Numerical Analysis, Oxford University Press, 1993.

    Examples
    --------
    Examples are given :ref:`in the tutorial <tutorial-interpolate_2d_spline>`.

    """
    x, y, z = map(ravel, [x, y, z])  # ensure 1-d arrays.
    m = len(x)
    if not (m == len(y) == len(z)):
        raise TypeError('len(x)==len(y)==len(z) must hold.')
    if w is None:
        w = ones(m, float)
    else:
        w = atleast_1d(w)
    if not len(w) == m:
        raise TypeError('len(w)=%d is not equal to m=%d' % (len(w), m))
    if xb is None:
        xb = x.min()
    if xe is None:
        xe = x.max()
    if yb is None:
        yb = y.min()
    if ye is None:
        ye = y.max()
    if not (-1 <= task <= 1):
        raise TypeError('task must be -1, 0 or 1')
    if s is None:
        s = m - sqrt(2*m)
    if tx is None and task == -1:
        raise TypeError('Knots_x must be given for task=-1')
    if tx is not None:
        _surfit_cache['tx'] = atleast_1d(tx)
    nx = len(_surfit_cache['tx'])
    if ty is None and task == -1:
        raise TypeError('Knots_y must be given for task=-1')
    if ty is not None:
        _surfit_cache['ty'] = atleast_1d(ty)
    ny = len(_surfit_cache['ty'])
    if task == -1 and nx < 2*kx+2:
        raise TypeError('There must be at least 2*kx+2 knots_x for task=-1')
    if task == -1 and ny < 2*ky+2:
        raise TypeError('There must be at least 2*ky+2 knots_x for task=-1')
    if not ((1 <= kx <= 5) and (1 <= ky <= 5)):
        raise TypeError('Given degree of the spline (kx,ky=%d,%d) is not '
                        'supported. (1<=k<=5)' % (kx, ky))
    if m < (kx + 1)*(ky + 1):
        raise TypeError('m >= (kx+1)(ky+1) must hold')
    if nxest is None:
        nxest = int(kx + sqrt(m/2))
    if nyest is None:
        nyest = int(ky + sqrt(m/2))
    nxest, nyest = max(nxest, 2*kx + 3), max(nyest, 2*ky + 3)
    if task >= 0 and s == 0:
        nxest = int(kx + sqrt(3*m))
        nyest = int(ky + sqrt(3*m))
    if task == -1:
        _surfit_cache['tx'] = atleast_1d(tx)
        _surfit_cache['ty'] = atleast_1d(ty)
    tx, ty = _surfit_cache['tx'], _surfit_cache['ty']
    wrk = _surfit_cache['wrk']
    u = nxest - kx - 1
    v = nyest - ky - 1
    km = max(kx, ky) + 1
    ne = max(nxest, nyest)
    bx, by = kx*v + ky + 1, ky*u + kx + 1
    b1, b2 = bx, bx + v - ky
    if bx > by:
        b1, b2 = by, by + u - kx
    msg = "Too many data points to interpolate"
    lwrk1 = _int_overflow(u*v*(2 + b1 + b2) +
                          2*(u + v + km*(m + ne) + ne - kx - ky) + b2 + 1,
                          OverflowError,
                          msg=msg)
    lwrk2 = _int_overflow(u*v*(b2 + 1) + b2, OverflowError, msg=msg)
    tx, ty, c, o = _fitpack._surfit(x, y, z, w, xb, xe, yb, ye, kx, ky,
                                    task, s, eps, tx, ty, nxest, nyest,
                                    wrk, lwrk1, lwrk2)
    _curfit_cache['tx'] = tx
    _curfit_cache['ty'] = ty
    _curfit_cache['wrk'] = o['wrk']
    ier, fp = o['ier'], o['fp']
    tck = [tx, ty, c, kx, ky]

    ierm = min(11, max(-3, ier))
    if ierm <= 0 and not quiet:
        _mess = (_iermess2[ierm][0] +
                 "\tkx,ky=%d,%d nx,ny=%d,%d m=%d fp=%f s=%f" %
                 (kx, ky, len(tx), len(ty), m, fp, s))
        warnings.warn(RuntimeWarning(_mess), stacklevel=2)
    if ierm > 0 and not full_output:
        if ier in [1, 2, 3, 4, 5]:
            _mess = ("\n\tkx,ky=%d,%d nx,ny=%d,%d m=%d fp=%f s=%f" %
                     (kx, ky, len(tx), len(ty), m, fp, s))
            warnings.warn(RuntimeWarning(_iermess2[ierm][0] + _mess), stacklevel=2)
        else:
            try:
                raise _iermess2[ierm][1](_iermess2[ierm][0])
            except KeyError as e:
                raise _iermess2['unknown'][1](_iermess2['unknown'][0]) from e
    if full_output:
        try:
            return tck, fp, ier, _iermess2[ierm][0]
        except KeyError:
            return tck, fp, ier, _iermess2['unknown'][0]
    else:
        return tck


def bisplev(x, y, tck, dx=0, dy=0):
    """
    Evaluate a bivariate B-spline and its derivatives.

    Return a rank-2 array of spline function values (or spline derivative
    values) at points given by the cross-product of the rank-1 arrays `x` and
    `y`.  In special cases, return an array or just a float if either `x` or
    `y` or both are floats.  Based on BISPEV and PARDER from FITPACK.

    Parameters
    ----------
    x, y : ndarray
        Rank-1 arrays specifying the domain over which to evaluate the
        spline or its derivative.
    tck : tuple
        A sequence of length 5 returned by `bisplrep` containing the knot
        locations, the coefficients, and the degree of the spline:
        [tx, ty, c, kx, ky].
    dx, dy : int, optional
        The orders of the partial derivatives in `x` and `y` respectively.

    Returns
    -------
    vals : ndarray
        The B-spline or its derivative evaluated over the set formed by
        the cross-product of `x` and `y`.

    See Also
    --------
    splprep, splrep, splint, sproot, splev
    UnivariateSpline, BivariateSpline

    Notes
    -----
        See `bisplrep` to generate the `tck` representation.

    References
    ----------
    .. [1] Dierckx P. : An algorithm for surface fitting
       with spline functions
       Ima J. Numer. Anal. 1 (1981) 267-283.
    .. [2] Dierckx P. : An algorithm for surface fitting
       with spline functions
       report tw50, Dept. Computer Science,K.U.Leuven, 1980.
    .. [3] Dierckx P. : Curve and surface fitting with splines,
       Monographs on Numerical Analysis, Oxford University Press, 1993.

    Examples
    --------
    Examples are given :ref:`in the tutorial <tutorial-interpolate_2d_spline>`.

    """
    tx, ty, c, kx, ky = tck
    if not (0 <= dx < kx):
        raise ValueError("0 <= dx = %d < kx = %d must hold" % (dx, kx))
    if not (0 <= dy < ky):
        raise ValueError("0 <= dy = %d < ky = %d must hold" % (dy, ky))
    x, y = map(atleast_1d, [x, y])
    if (len(x.shape) != 1) or (len(y.shape) != 1):
        raise ValueError("First two entries should be rank-1 arrays.")

    msg = "Too many data points to interpolate."

    _int_overflow(x.size * y.size, MemoryError, msg=msg)

    if dx != 0 or dy != 0:
        _int_overflow((tx.size - kx - 1)*(ty.size - ky - 1),
                      MemoryError, msg=msg)
        z, ier = dfitpack.parder(tx, ty, c, kx, ky, dx, dy, x, y)
    else:
        z, ier = dfitpack.bispev(tx, ty, c, kx, ky, x, y)

    if ier == 10:
        raise ValueError("Invalid input data")
    if ier:
        raise TypeError("An error occurred")
    z.shape = len(x), len(y)
    if len(z) > 1:
        return z
    if len(z[0]) > 1:
        return z[0]
    return z[0][0]


def dblint(xa, xb, ya, yb, tck):
    """Evaluate the integral of a spline over area [xa,xb] x [ya,yb].

    Parameters
    ----------
    xa, xb : float
        The end-points of the x integration interval.
    ya, yb : float
        The end-points of the y integration interval.
    tck : list [tx, ty, c, kx, ky]
        A sequence of length 5 returned by bisplrep containing the knot
        locations tx, ty, the coefficients c, and the degrees kx, ky
        of the spline.

    Returns
    -------
    integ : float
        The value of the resulting integral.
    """
    tx, ty, c, kx, ky = tck
    return dfitpack.dblint(tx, ty, c, kx, ky, xa, xb, ya, yb)


def insert(x, tck, m=1, per=0):
    # see the docstring of `_fitpack_py/insert`
    t, c, k = tck
    try:
        c[0][0]
        parametric = True
    except Exception:
        parametric = False
    if parametric:
        cc = []
        for c_vals in c:
            tt, cc_val, kk = insert(x, [t, c_vals, k], m)
            cc.append(cc_val)
        return (tt, cc, kk)
    else:
        tt, cc, ier = _fitpack._insert(per, t, c, k, x, m)
        if ier == 10:
            raise ValueError("Invalid input data")
        if ier:
            raise TypeError("An error occurred")
        return (tt, cc, k)


def splder(tck, n=1):
    # see the docstring of `_fitpack_py/splder`
    if n < 0:
        return splantider(tck, -n)

    t, c, k = tck

    if n > k:
        raise ValueError(f"Order of derivative (n = {n!r}) must be <= "
                         f"order of spline (k = {tck[2]!r})")

    # Extra axes for the trailing dims of the `c` array:
    sh = (slice(None),) + ((None,)*len(c.shape[1:]))

    with np.errstate(invalid='raise', divide='raise'):
        try:
            for j in range(n):
                # See e.g. Schumaker, Spline Functions: Basic Theory, Chapter 5

                # Compute the denominator in the differentiation formula.
                # (and append trailing dims, if necessary)
                dt = t[k+1:-1] - t[1:-k-1]
                dt = dt[sh]
                # Compute the new coefficients
                c = (c[1:-1-k] - c[:-2-k]) * k / dt
                # Pad coefficient array to same size as knots (FITPACK
                # convention)
                c = np.r_[c, np.zeros((k,) + c.shape[1:])]
                # Adjust knots
                t = t[1:-1]
                k -= 1
        except FloatingPointError as e:
            raise ValueError(("The spline has internal repeated knots "
                              "and is not differentiable %d times") % n) from e

    return t, c, k


def splantider(tck, n=1):
    # see the docstring of `_fitpack_py/splantider`
    if n < 0:
        return splder(tck, -n)

    t, c, k = tck

    # Extra axes for the trailing dims of the `c` array:
    sh = (slice(None),) + (None,)*len(c.shape[1:])

    for j in range(n):
        # This is the inverse set of operations to splder.

        # Compute the multiplier in the antiderivative formula.
        dt = t[k+1:] - t[:-k-1]
        dt = dt[sh]
        # Compute the new coefficients
        c = np.cumsum(c[:-k-1] * dt, axis=0) / (k + 1)
        c = np.r_[np.zeros((1,) + c.shape[1:]),
                  c,
                  [c[-1]] * (k+2)]
        # New knots
        t = np.r_[t[0], t, t[-1]]
        k += 1

    return t, c, k