venv/lib/python3.11/site-packages/scipy/interpolate/_fitpack_py.py

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


import numpy as np

# These are in the API for fitpack even if not used in fitpack.py itself.
from ._fitpack_impl import bisplrep, bisplev, dblint  # noqa: F401
from . import _fitpack_impl as _impl
from ._bsplines import BSpline


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):
    """
    Find the B-spline representation of an N-D curve.

    .. legacy:: function

        Specifically, we recommend using `make_splprep` in new code.

    Given a list of N rank-1 arrays, `x`, which represent a curve in
    N-dimensional space parametrized by `u`, find a smooth approximating
    spline curve g(`u`). Uses the FORTRAN routine parcur from FITPACK.

    Parameters
    ----------
    x : array_like
        A list of sample vector arrays representing the curve.
    w : array_like, optional
        Strictly positive rank-1 array of weights the same length as `x[0]`.
        The weights are used in computing the weighted least-squares spline
        fit. If the errors in the `x` values have standard-deviation given by
        the vector d, then `w` should be 1/d. Default is ``ones(len(x[0]))``.
    u : array_like, optional
        An array of parameter values. If not given, these values are
        calculated automatically as ``M = len(x[0])``, where

            v[0] = 0

            v[i] = v[i-1] + distance(`x[i]`, `x[i-1]`)

            u[i] = v[i] / v[M-1]

    ub, ue : int, optional
        The end-points of the parameters interval.  Defaults to
        u[0] and u[-1].
    k : int, optional
        Degree of the spline. Cubic splines are recommended.
        Even values of `k` should be avoided especially with a small s-value.
        ``1 <= k <= 5``, default is 3.
    task : int, optional
        If task==0 (default), find t and c for a given smoothing factor, s.
        If task==1, find t and c for another value of the smoothing factor, s.
        There must have been a previous call with task=0 or task=1
        for the same set of data.
        If task=-1 find the weighted least square spline for a given set of
        knots, t.
    s : float, optional
        A smoothing condition.  The amount of smoothness is determined by
        satisfying the conditions: ``sum((w * (y - g))**2,axis=0) <= s``,
        where g(x) is the smoothed interpolation of (x,y).  The user can
        use `s` to control the trade-off between closeness and smoothness
        of fit.  Larger `s` means more smoothing while smaller values of `s`
        indicate less smoothing. Recommended values of `s` depend on the
        weights, w.  If the weights represent the inverse of the
        standard-deviation of y, then a good `s` value should be found in
        the range ``(m-sqrt(2*m),m+sqrt(2*m))``, where m is the number of
        data points in x, y, and w.
    t : array, optional
        The knots needed for ``task=-1``.
        There must be at least ``2*k+2`` knots.
    full_output : int, optional
        If non-zero, then return optional outputs.
    nest : int, optional
        An over-estimate of the total number of knots of the spline to
        help in determining the storage space.  By default nest=m/2.
        Always large enough is nest=m+k+1.
    per : int, optional
       If non-zero, data points are considered periodic with period
       ``x[m-1] - x[0]`` and a smooth periodic spline approximation is
       returned.  Values of ``y[m-1]`` and ``w[m-1]`` are not used.
    quiet : int, optional
         Non-zero to suppress messages.

    Returns
    -------
    tck : tuple
        A tuple, ``(t,c,k)`` containing the vector of knots, the B-spline
        coefficients, and the degree of the spline.
    u : array
        An array of the values of the parameter.
    fp : float
        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
    --------
    splrep, splev, sproot, spalde, splint,
    bisplrep, bisplev
    UnivariateSpline, BivariateSpline
    BSpline
    make_interp_spline

    Notes
    -----
    See `splev` for evaluation of the spline and its derivatives.
    The number of dimensions N must be smaller than 11.

    The number of coefficients in the `c` array is ``k+1`` less than the number
    of knots, ``len(t)``. This is in contrast with `splrep`, which zero-pads
    the array of coefficients to have the same length as the array of knots.
    These additional coefficients are ignored by evaluation routines, `splev`
    and `BSpline`.

    References
    ----------
    .. [1] P. Dierckx, "Algorithms for smoothing data with periodic and
        parametric splines, Computer Graphics and Image Processing",
        20 (1982) 171-184.
    .. [2] P. Dierckx, "Algorithms for smoothing data with periodic and
        parametric splines", report tw55, Dept. Computer Science,
        K.U.Leuven, 1981.
    .. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs on
        Numerical Analysis, Oxford University Press, 1993.

    Examples
    --------
    Generate a discretization of a limacon curve in the polar coordinates:

    >>> import numpy as np
    >>> phi = np.linspace(0, 2.*np.pi, 40)
    >>> r = 0.5 + np.cos(phi)         # polar coords
    >>> x, y = r * np.cos(phi), r * np.sin(phi)    # convert to cartesian

    And interpolate:

    >>> from scipy.interpolate import splprep, splev
    >>> tck, u = splprep([x, y], s=0)
    >>> new_points = splev(u, tck)

    Notice that (i) we force interpolation by using ``s=0``,
    (ii) the parameterization, ``u``, is generated automatically.
    Now plot the result:

    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, y, 'ro')
    >>> ax.plot(new_points[0], new_points[1], 'r-')
    >>> plt.show()

    """

    res = _impl.splprep(x, w, u, ub, ue, k, task, s, t, full_output, nest, per,
                        quiet)
    return res


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):
    """
    Find the B-spline representation of a 1-D curve.

    .. legacy:: function

        Specifically, we recommend using `make_splrep` in new code.


    Given the set of data points ``(x[i], y[i])`` determine a smooth spline
    approximation of degree k on the interval ``xb <= x <= xe``.

    Parameters
    ----------
    x, y : array_like
        The data points defining a curve ``y = f(x)``.
    w : array_like, optional
        Strictly positive rank-1 array of weights the same length as `x` and `y`.
        The weights are used in computing the weighted least-squares spline
        fit. If the errors in the `y` values have standard-deviation given by the
        vector ``d``, then `w` should be ``1/d``. Default is ``ones(len(x))``.
    xb, xe : float, optional
        The interval to fit.  If None, these default to ``x[0]`` and ``x[-1]``
        respectively.
    k : int, optional
        The degree of the spline fit. It is recommended to use cubic splines.
        Even values of `k` should be avoided especially with small `s` values.
        ``1 <= k <= 5``.
    task : {1, 0, -1}, optional
        If ``task==0``, find ``t`` and ``c`` for a given smoothing factor, `s`.

        If ``task==1`` find ``t`` and ``c`` for another value of the smoothing factor,
        `s`. There must have been a previous call with ``task=0`` or ``task=1`` for
        the same set of data (``t`` will be stored an used internally)

        If ``task=-1`` find the weighted least square spline for a given set of
        knots, ``t``. These should be interior knots as knots on the ends will be
        added automatically.
    s : float, optional
        A smoothing condition. The amount of smoothness is determined by
        satisfying the conditions: ``sum((w * (y - g))**2,axis=0) <= s`` where ``g(x)``
        is the smoothed interpolation of ``(x,y)``. The user can use `s` to control
        the tradeoff between closeness and smoothness of fit. Larger `s` means
        more smoothing while smaller values of `s` indicate less smoothing.
        Recommended values of `s` depend on the weights, `w`. If the weights
        represent the inverse of the standard-deviation of `y`, then a good `s`
        value should be found in the range ``(m-sqrt(2*m),m+sqrt(2*m))`` where ``m`` is
        the number of datapoints in `x`, `y`, and `w`. default : ``s=m-sqrt(2*m)`` if
        weights are supplied. ``s = 0.0`` (interpolating) if no weights are
        supplied.
    t : array_like, optional
        The knots needed for ``task=-1``. If given then task is automatically set
        to ``-1``.
    full_output : bool, optional
        If non-zero, then return optional outputs.
    per : bool, optional
        If non-zero, data points are considered periodic with period ``x[m-1]`` -
        ``x[0]`` and a smooth periodic spline approximation is returned. Values of
        ``y[m-1]`` and ``w[m-1]`` are not used.
        The default is zero, corresponding to boundary condition 'not-a-knot'.
    quiet : bool, optional
        Non-zero to suppress messages.

    Returns
    -------
    tck : tuple
        A tuple ``(t,c,k)`` containing the vector of knots, the B-spline
        coefficients, and the degree of the spline.
    fp : array, optional
        The weighted sum of squared residuals of the spline approximation.
    ier : int, optional
        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, optional
        A message corresponding to the integer flag, `ier`.

    See Also
    --------
    UnivariateSpline, BivariateSpline
    splprep, splev, sproot, spalde, splint
    bisplrep, bisplev
    BSpline
    make_interp_spline

    Notes
    -----
    See `splev` for evaluation of the spline and its derivatives. Uses the
    FORTRAN routine ``curfit`` from FITPACK.

    The user is responsible for assuring that the values of `x` are unique.
    Otherwise, `splrep` will not return sensible results.

    If provided, knots `t` must satisfy the Schoenberg-Whitney conditions,
    i.e., there must be a subset of data points ``x[j]`` such that
    ``t[j] < x[j] < t[j+k+1]``, for ``j=0, 1,...,n-k-2``.

    This routine zero-pads the coefficients array ``c`` to have the same length
    as the array of knots ``t`` (the trailing ``k + 1`` coefficients are ignored
    by the evaluation routines, `splev` and `BSpline`.) This is in contrast with
    `splprep`, which does not zero-pad the coefficients.

    The default boundary condition is 'not-a-knot', i.e. the first and second
    segment at a curve end are the same polynomial. More boundary conditions are
    available in `CubicSpline`.

    References
    ----------
    Based on algorithms described in [1]_, [2]_, [3]_, and [4]_:

    .. [1] P. Dierckx, "An algorithm for smoothing, differentiation and
       integration of experimental data using spline functions",
       J.Comp.Appl.Maths 1 (1975) 165-184.
    .. [2] P. Dierckx, "A fast algorithm for smoothing data on a rectangular
       grid while using spline functions", SIAM J.Numer.Anal. 19 (1982)
       1286-1304.
    .. [3] P. Dierckx, "An improved algorithm for curve fitting with spline
       functions", report tw54, Dept. Computer Science,K.U. Leuven, 1981.
    .. [4] P. Dierckx, "Curve and surface fitting with splines", Monographs on
       Numerical Analysis, Oxford University Press, 1993.

    Examples
    --------
    You can interpolate 1-D points with a B-spline curve.
    Further examples are given in
    :ref:`in the tutorial <tutorial-interpolate_splXXX>`.

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy.interpolate import splev, splrep
    >>> x = np.linspace(0, 10, 10)
    >>> y = np.sin(x)
    >>> spl = splrep(x, y)
    >>> x2 = np.linspace(0, 10, 200)
    >>> y2 = splev(x2, spl)
    >>> plt.plot(x, y, 'o', x2, y2)
    >>> plt.show()

    """
    res = _impl.splrep(x, y, w, xb, xe, k, task, s, t, full_output, per, quiet)
    return res


def splev(x, tck, der=0, ext=0):
    """
    Evaluate a B-spline or its derivatives.

    .. legacy:: function

        Specifically, we recommend constructing a `BSpline` object and using
        its ``__call__`` method.

    Given the knots and coefficients of a B-spline representation, evaluate
    the value of the smoothing polynomial and its derivatives. This is a
    wrapper around the FORTRAN routines splev and splder of FITPACK.

    Parameters
    ----------
    x : array_like
        An array of points at which to return the value of the smoothed
        spline or its derivatives. If `tck` was returned from `splprep`,
        then the parameter values, u should be given.
    tck : BSpline instance or tuple
        If a tuple, then it should be a sequence of length 3 returned by
        `splrep` or `splprep` containing the knots, coefficients, and degree
        of the spline. (Also see Notes.)
    der : int, optional
        The order of derivative of the spline to compute (must be less than
        or equal to k, the degree of the spline).
    ext : int, optional
        Controls the value returned for elements of ``x`` not in the
        interval defined by the knot sequence.

        * if ext=0, return the extrapolated value.
        * if ext=1, return 0
        * if ext=2, raise a ValueError
        * if ext=3, return the boundary value.

        The default value is 0.

    Returns
    -------
    y : ndarray or list of ndarrays
        An array of values representing the spline function evaluated at
        the points in `x`.  If `tck` was returned from `splprep`, then this
        is a list of arrays representing the curve in an N-D space.

    See Also
    --------
    splprep, splrep, sproot, spalde, splint
    bisplrep, bisplev
    BSpline

    Notes
    -----
    Manipulating the tck-tuples directly is not recommended. In new code,
    prefer using `BSpline` objects.

    References
    ----------
    .. [1] C. de Boor, "On calculating with b-splines", J. Approximation
        Theory, 6, p.50-62, 1972.
    .. [2] M. G. Cox, "The numerical evaluation of b-splines", J. Inst. Maths
        Applics, 10, p.134-149, 1972.
    .. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs
        on Numerical Analysis, Oxford University Press, 1993.

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

    A comparison between `splev`, `splder` and `spalde` to compute the derivatives of a 
    B-spline can be found in the `spalde` examples section.

    """
    if isinstance(tck, BSpline):
        if tck.c.ndim > 1:
            mesg = ("Calling splev() with BSpline objects with c.ndim > 1 is "
                    "not allowed. Use BSpline.__call__(x) instead.")
            raise ValueError(mesg)

        # remap the out-of-bounds behavior
        try:
            extrapolate = {0: True, }[ext]
        except KeyError as e:
            raise ValueError(f"Extrapolation mode {ext} is not supported "
                             "by BSpline.") from e

        return tck(x, der, extrapolate=extrapolate)
    else:
        return _impl.splev(x, tck, der, ext)


def splint(a, b, tck, full_output=0):
    """
    Evaluate the definite integral of a B-spline between two given points.

    .. legacy:: function

        Specifically, we recommend constructing a `BSpline` object and using its
        ``integrate`` method.

    Parameters
    ----------
    a, b : float
        The end-points of the integration interval.
    tck : tuple or a BSpline instance
        If a tuple, then it should be a sequence of length 3, containing the
        vector of knots, the B-spline coefficients, and the degree of the
        spline (see `splev`).
    full_output : int, optional
        Non-zero to return optional output.

    Returns
    -------
    integral : float
        The resulting integral.
    wrk : ndarray
        An array containing the integrals of the normalized B-splines
        defined on the set of knots.
        (Only returned if `full_output` is non-zero)

    See Also
    --------
    splprep, splrep, sproot, spalde, splev
    bisplrep, bisplev
    BSpline

    Notes
    -----
    `splint` silently assumes that the spline function is zero outside the data
    interval (`a`, `b`).

    Manipulating the tck-tuples directly is not recommended. In new code,
    prefer using the `BSpline` objects.

    References
    ----------
    .. [1] P.W. Gaffney, The calculation of indefinite integrals of b-splines",
        J. Inst. Maths Applics, 17, p.37-41, 1976.
    .. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs
        on Numerical Analysis, Oxford University Press, 1993.

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

    """
    if isinstance(tck, BSpline):
        if tck.c.ndim > 1:
            mesg = ("Calling splint() with BSpline objects with c.ndim > 1 is "
                    "not allowed. Use BSpline.integrate() instead.")
            raise ValueError(mesg)

        if full_output != 0:
            mesg = (f"full_output = {full_output} is not supported. Proceeding as if "
                    "full_output = 0")

        return tck.integrate(a, b, extrapolate=False)
    else:
        return _impl.splint(a, b, tck, full_output)


def sproot(tck, mest=10):
    """
    Find the roots of a cubic B-spline.

    .. legacy:: function

        Specifically, we recommend constructing a `BSpline` object and using the
        following pattern: `PPoly.from_spline(spl).roots()`.

    Given the knots (>=8) and coefficients of a cubic B-spline return the
    roots of the spline.

    Parameters
    ----------
    tck : tuple or a BSpline object
        If a tuple, then it should be a sequence of length 3, containing the
        vector of knots, the B-spline coefficients, and the degree of the
        spline.
        The number of knots must be >= 8, and the degree must be 3.
        The knots must be a montonically increasing sequence.
    mest : int, optional
        An estimate of the number of zeros (Default is 10).

    Returns
    -------
    zeros : ndarray
        An array giving the roots of the spline.

    See Also
    --------
    splprep, splrep, splint, spalde, splev
    bisplrep, bisplev
    BSpline

    Notes
    -----
    Manipulating the tck-tuples directly is not recommended. In new code,
    prefer using the `BSpline` objects.

    References
    ----------
    .. [1] C. de Boor, "On calculating with b-splines", J. Approximation
        Theory, 6, p.50-62, 1972.
    .. [2] M. G. Cox, "The numerical evaluation of b-splines", J. Inst. Maths
        Applics, 10, p.134-149, 1972.
    .. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs
        on Numerical Analysis, Oxford University Press, 1993.

    Examples
    --------

    For some data, this method may miss a root. This happens when one of
    the spline knots (which FITPACK places automatically) happens to
    coincide with the true root. A workaround is to convert to `PPoly`,
    which uses a different root-finding algorithm.

    For example,

    >>> x = [1.96, 1.97, 1.98, 1.99, 2.00, 2.01, 2.02, 2.03, 2.04, 2.05]
    >>> y = [-6.365470e-03, -4.790580e-03, -3.204320e-03, -1.607270e-03,
    ...      4.440892e-16,  1.616930e-03,  3.243000e-03,  4.877670e-03,
    ...      6.520430e-03,  8.170770e-03]
    >>> from scipy.interpolate import splrep, sproot, PPoly
    >>> tck = splrep(x, y, s=0)
    >>> sproot(tck)
    array([], dtype=float64)

    Converting to a PPoly object does find the roots at ``x=2``:

    >>> ppoly = PPoly.from_spline(tck)
    >>> ppoly.roots(extrapolate=False)
    array([2.])


    Further examples are given :ref:`in the tutorial
    <tutorial-interpolate_splXXX>`.

    """
    if isinstance(tck, BSpline):
        if tck.c.ndim > 1:
            mesg = ("Calling sproot() with BSpline objects with c.ndim > 1 is "
                    "not allowed.")
            raise ValueError(mesg)

        t, c, k = tck.tck

        # _impl.sproot expects the interpolation axis to be last, so roll it.
        # NB: This transpose is a no-op if c is 1D.
        sh = tuple(range(c.ndim))
        c = c.transpose(sh[1:] + (0,))
        return _impl.sproot((t, c, k), mest)
    else:
        return _impl.sproot(tck, mest)


def spalde(x, tck):
    """
    Evaluate a B-spline and all its derivatives at one point (or set of points) up
    to order k (the degree of the spline), being 0 the spline itself.

    .. legacy:: function

        Specifically, we recommend constructing a `BSpline` object and evaluate
        its derivative in a loop or a list comprehension.

    Parameters
    ----------
    x : array_like
        A point or a set of points at which to evaluate the derivatives.
        Note that ``t(k) <= x <= t(n-k+1)`` must hold for each `x`.
    tck : tuple
        A tuple (t,c,k) containing the vector of knots,
        the B-spline coefficients, and the degree of the spline whose 
        derivatives to compute.

    Returns
    -------
    results : {ndarray, list of ndarrays}
        An array (or a list of arrays) containing all derivatives
        up to order k inclusive for each point `x`, being the first element the 
        spline itself.

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

    References
    ----------
    .. [1] de Boor C : On calculating with b-splines, J. Approximation Theory
       6 (1972) 50-62.
    .. [2] Cox M.G. : The numerical evaluation of b-splines, J. Inst. Maths
       applics 10 (1972) 134-149.
    .. [3] Dierckx P. : Curve and surface fitting with splines, Monographs on
       Numerical Analysis, Oxford University Press, 1993.

    Examples
    --------
    To calculate the derivatives of a B-spline there are several aproaches. 
    In this example, we will demonstrate that `spalde` is equivalent to
    calling `splev` and `splder`.
    
    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy.interpolate import BSpline, spalde, splder, splev
    
    >>> # Store characteristic parameters of a B-spline
    >>> tck = ((-2, -2, -2, -2, -1, 0, 1, 2, 2, 2, 2),  # knots
    ...        (0, 0, 0, 6, 0, 0, 0),  # coefficients
    ...        3)  # degree (cubic)
    >>> # Instance a B-spline object
    >>> # `BSpline` objects are preferred, except for spalde()
    >>> bspl = BSpline(tck[0], tck[1], tck[2])
    >>> # Generate extra points to get a smooth curve
    >>> x = np.linspace(min(tck[0]), max(tck[0]), 100)
    
    Evaluate the curve and all derivatives
    
    >>> # The order of derivative must be less or equal to k, the degree of the spline
    >>> # Method 1: spalde()
    >>> f1_y_bsplin = [spalde(i, tck)[0] for i in x ]  # The B-spline itself
    >>> f1_y_deriv1 = [spalde(i, tck)[1] for i in x ]  # 1st derivative
    >>> f1_y_deriv2 = [spalde(i, tck)[2] for i in x ]  # 2nd derivative
    >>> f1_y_deriv3 = [spalde(i, tck)[3] for i in x ]  # 3rd derivative
    >>> # You can reach the same result by using `splev`and `splder`
    >>> f2_y_deriv3 = splev(x, bspl, der=3)
    >>> f3_y_deriv3 = splder(bspl, n=3)(x)
    
    >>> # Generate a figure with three axes for graphic comparison
    >>> fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(16, 5))
    >>> suptitle = fig.suptitle(f'Evaluate a B-spline and all derivatives')
    >>> # Plot B-spline and all derivatives using the three methods
    >>> orders = range(4)
    >>> linetypes = ['-', '--', '-.', ':']
    >>> labels = ['B-Spline', '1st deriv.', '2nd deriv.', '3rd deriv.']
    >>> functions = ['splev()', 'splder()', 'spalde()']
    >>> for order, linetype, label in zip(orders, linetypes, labels):
    ...     ax1.plot(x, splev(x, bspl, der=order), linetype, label=label)
    ...     ax2.plot(x, splder(bspl, n=order)(x), linetype, label=label)
    ...     ax3.plot(x, [spalde(i, tck)[order] for i in x], linetype, label=label)
    >>> for ax, function in zip((ax1, ax2, ax3), functions):
    ...     ax.set_title(function)
    ...     ax.legend()
    >>> plt.tight_layout()
    >>> plt.show()

    """
    if isinstance(tck, BSpline):
        raise TypeError("spalde does not accept BSpline instances.")
    else:
        return _impl.spalde(x, tck)


def insert(x, tck, m=1, per=0):
    """
    Insert knots into a B-spline.

    .. legacy:: function

        Specifically, we recommend constructing a `BSpline` object and using
        its ``insert_knot`` method.

    Given the knots and coefficients of a B-spline representation, create a
    new B-spline with a knot inserted `m` times at point `x`.
    This is a wrapper around the FORTRAN routine insert of FITPACK.

    Parameters
    ----------
    x (u) : float
        A knot value at which to insert a new knot.  If `tck` was returned
        from ``splprep``, then the parameter values, u should be given.
    tck : a `BSpline` instance or a tuple
        If tuple, then it is expected to be a tuple (t,c,k) containing
        the vector of knots, the B-spline coefficients, and the degree of
        the spline.
    m : int, optional
        The number of times to insert the given knot (its multiplicity).
        Default is 1.
    per : int, optional
        If non-zero, the input spline is considered periodic.

    Returns
    -------
    BSpline instance or a tuple
        A new B-spline with knots t, coefficients c, and degree k.
        ``t(k+1) <= x <= t(n-k)``, where k is the degree of the spline.
        In case of a periodic spline (``per != 0``) there must be
        either at least k interior knots t(j) satisfying ``t(k+1)<t(j)<=x``
        or at least k interior knots t(j) satisfying ``x<=t(j)<t(n-k)``.
        A tuple is returned iff the input argument `tck` is a tuple, otherwise
        a BSpline object is constructed and returned.

    Notes
    -----
    Based on algorithms from [1]_ and [2]_.

    Manipulating the tck-tuples directly is not recommended. In new code,
    prefer using the `BSpline` objects, in particular `BSpline.insert_knot`
    method.

    See Also
    --------
    BSpline.insert_knot

    References
    ----------
    .. [1] W. Boehm, "Inserting new knots into b-spline curves.",
        Computer Aided Design, 12, p.199-201, 1980.
    .. [2] P. Dierckx, "Curve and surface fitting with splines, Monographs on
        Numerical Analysis", Oxford University Press, 1993.

    Examples
    --------
    You can insert knots into a B-spline.

    >>> from scipy.interpolate import splrep, insert
    >>> import numpy as np
    >>> x = np.linspace(0, 10, 5)
    >>> y = np.sin(x)
    >>> tck = splrep(x, y)
    >>> tck[0]
    array([ 0.,  0.,  0.,  0.,  5., 10., 10., 10., 10.])

    A knot is inserted:

    >>> tck_inserted = insert(3, tck)
    >>> tck_inserted[0]
    array([ 0.,  0.,  0.,  0.,  3.,  5., 10., 10., 10., 10.])

    Some knots are inserted:

    >>> tck_inserted2 = insert(8, tck, m=3)
    >>> tck_inserted2[0]
    array([ 0.,  0.,  0.,  0.,  5.,  8.,  8.,  8., 10., 10., 10., 10.])

    """
    if isinstance(tck, BSpline):

        t, c, k = tck.tck

        # FITPACK expects the interpolation axis to be last, so roll it over
        # NB: if c array is 1D, transposes are no-ops
        sh = tuple(range(c.ndim))
        c = c.transpose(sh[1:] + (0,))
        t_, c_, k_ = _impl.insert(x, (t, c, k), m, per)

        # and roll the last axis back
        c_ = np.asarray(c_)
        c_ = c_.transpose((sh[-1],) + sh[:-1])
        return BSpline(t_, c_, k_)
    else:
        return _impl.insert(x, tck, m, per)


def splder(tck, n=1):
    """
    Compute the spline representation of the derivative of a given spline

    .. legacy:: function

        Specifically, we recommend constructing a `BSpline` object and using its
        ``derivative`` method.

    Parameters
    ----------
    tck : BSpline instance or tuple
        BSpline instance or a tuple (t,c,k) containing the vector of knots,
        the B-spline coefficients, and the degree of the spline whose 
        derivative to compute
    n : int, optional
        Order of derivative to evaluate. Default: 1

    Returns
    -------
    `BSpline` instance or tuple
        Spline of order k2=k-n representing the derivative
        of the input spline.
        A tuple is returned if the input argument `tck` is a tuple, otherwise
        a BSpline object is constructed and returned.

    See Also
    --------
    splantider, splev, spalde
    BSpline

    Notes
    -----

    .. versionadded:: 0.13.0

    Examples
    --------
    This can be used for finding maxima of a curve:

    >>> from scipy.interpolate import splrep, splder, sproot
    >>> import numpy as np
    >>> x = np.linspace(0, 10, 70)
    >>> y = np.sin(x)
    >>> spl = splrep(x, y, k=4)

    Now, differentiate the spline and find the zeros of the
    derivative. (NB: `sproot` only works for order 3 splines, so we
    fit an order 4 spline):

    >>> dspl = splder(spl)
    >>> sproot(dspl) / np.pi
    array([ 0.50000001,  1.5       ,  2.49999998])

    This agrees well with roots :math:`\\pi/2 + n\\pi` of
    :math:`\\cos(x) = \\sin'(x)`.

    A comparison between `splev`, `splder` and `spalde` to compute the derivatives of a 
    B-spline can be found in the `spalde` examples section.

    """
    if isinstance(tck, BSpline):
        return tck.derivative(n)
    else:
        return _impl.splder(tck, n)


def splantider(tck, n=1):
    """
    Compute the spline for the antiderivative (integral) of a given spline.

    .. legacy:: function

        Specifically, we recommend constructing a `BSpline` object and using its
        ``antiderivative`` method.

    Parameters
    ----------
    tck : BSpline instance or a tuple of (t, c, k)
        Spline whose antiderivative to compute
    n : int, optional
        Order of antiderivative to evaluate. Default: 1

    Returns
    -------
    BSpline instance or a tuple of (t2, c2, k2)
        Spline of order k2=k+n representing the antiderivative of the input
        spline.
        A tuple is returned iff the input argument `tck` is a tuple, otherwise
        a BSpline object is constructed and returned.

    See Also
    --------
    splder, splev, spalde
    BSpline

    Notes
    -----
    The `splder` function is the inverse operation of this function.
    Namely, ``splder(splantider(tck))`` is identical to `tck`, modulo
    rounding error.

    .. versionadded:: 0.13.0

    Examples
    --------
    >>> from scipy.interpolate import splrep, splder, splantider, splev
    >>> import numpy as np
    >>> x = np.linspace(0, np.pi/2, 70)
    >>> y = 1 / np.sqrt(1 - 0.8*np.sin(x)**2)
    >>> spl = splrep(x, y)

    The derivative is the inverse operation of the antiderivative,
    although some floating point error accumulates:

    >>> splev(1.7, spl), splev(1.7, splder(splantider(spl)))
    (array(2.1565429877197317), array(2.1565429877201865))

    Antiderivative can be used to evaluate definite integrals:

    >>> ispl = splantider(spl)
    >>> splev(np.pi/2, ispl) - splev(0, ispl)
    2.2572053588768486

    This is indeed an approximation to the complete elliptic integral
    :math:`K(m) = \\int_0^{\\pi/2} [1 - m\\sin^2 x]^{-1/2} dx`:

    >>> from scipy.special import ellipk
    >>> ellipk(0.8)
    2.2572053268208538

    """
    if isinstance(tck, BSpline):
        return tck.antiderivative(n)
    else:
        return _impl.splantider(tck, n)