_cubic.py

"""Interpolation algorithms using piecewise cubic polynomials."""

from __future__ import division, print_function, absolute_import

import numpy as np

from scipy._lib.six import string_types

from . import BPoly, PPoly
from .polyint import _isscalar
from scipy._lib._util import _asarray_validated
from scipy.linalg import solve_banded, solve


__all__ = ["PchipInterpolator", "pchip_interpolate", "pchip",
           "Akima1DInterpolator", "CubicSpline"]


class PchipInterpolator(BPoly):
    r"""PCHIP 1-d monotonic cubic interpolation.

    `x` and `y` are arrays of values used to approximate some function f,
    with ``y = f(x)``. The interpolant uses monotonic cubic splines
    to find the value of new points. (PCHIP stands for Piecewise Cubic
    Hermite Interpolating Polynomial).

    Parameters
    ----------
    x : ndarray
        A 1-D array of monotonically increasing real values.  `x` cannot
        include duplicate values (otherwise f is overspecified)
    y : ndarray
        A 1-D array of real values. `y`'s length along the interpolation
        axis must be equal to the length of `x`. If N-D array, use `axis`
        parameter to select correct axis.
    axis : int, optional
        Axis in the y array corresponding to the x-coordinate values.
    extrapolate : bool, optional
        Whether to extrapolate to out-of-bounds points based on first
        and last intervals, or to return NaNs.

    Methods
    -------
    __call__
    derivative
    antiderivative
    roots

    See Also
    --------
    Akima1DInterpolator
    CubicSpline
    BPoly

    Notes
    -----
    The interpolator preserves monotonicity in the interpolation data and does
    not overshoot if the data is not smooth.

    The first derivatives are guaranteed to be continuous, but the second
    derivatives may jump at :math:`x_k`.

    Determines the derivatives at the points :math:`x_k`, :math:`f'_k`,
    by using PCHIP algorithm [1]_.

    Let :math:`h_k = x_{k+1} - x_k`, and  :math:`d_k = (y_{k+1} - y_k) / h_k`
    are the slopes at internal points :math:`x_k`.
    If the signs of :math:`d_k` and :math:`d_{k-1}` are different or either of
    them equals zero, then :math:`f'_k = 0`. Otherwise, it is given by the
    weighted harmonic mean

    .. math::

        \frac{w_1 + w_2}{f'_k} = \frac{w_1}{d_{k-1}} + \frac{w_2}{d_k}

    where :math:`w_1 = 2 h_k + h_{k-1}` and :math:`w_2 = h_k + 2 h_{k-1}`.

    The end slopes are set using a one-sided scheme [2]_.


    References
    ----------
    .. [1] F. N. Fritsch and R. E. Carlson, Monotone Piecewise Cubic Interpolation,
           SIAM J. Numer. Anal., 17(2), 238 (1980).
           :doi:`10.1137/0717021`.
    .. [2] see, e.g., C. Moler, Numerical Computing with Matlab, 2004.
           :doi:`10.1137/1.9780898717952`


    """
    def __init__(self, x, y, axis=0, extrapolate=None):
        x = _asarray_validated(x, check_finite=False, as_inexact=True)
        y = _asarray_validated(y, check_finite=False, as_inexact=True)

        axis = axis % y.ndim

        xp = x.reshape((x.shape[0],) + (1,)*(y.ndim-1))
        yp = np.rollaxis(y, axis)

        dk = self._find_derivatives(xp, yp)
        data = np.hstack((yp[:, None, ...], dk[:, None, ...]))

        _b = BPoly.from_derivatives(x, data, orders=None)
        super(PchipInterpolator, self).__init__(_b.c, _b.x,
                                                extrapolate=extrapolate)
        self.axis = axis

    def roots(self):
        """
        Return the roots of the interpolated function.
        """
        return (PPoly.from_bernstein_basis(self)).roots()

    @staticmethod
    def _edge_case(h0, h1, m0, m1):
        # one-sided three-point estimate for the derivative
        d = ((2*h0 + h1)*m0 - h0*m1) / (h0 + h1)

        # try to preserve shape
        mask = np.sign(d) != np.sign(m0)
        mask2 = (np.sign(m0) != np.sign(m1)) & (np.abs(d) > 3.*np.abs(m0))
        mmm = (~mask) & mask2

        d[mask] = 0.
        d[mmm] = 3.*m0[mmm]

        return d

    @staticmethod
    def _find_derivatives(x, y):
        # Determine the derivatives at the points y_k, d_k, by using
        #  PCHIP algorithm is:
        # We choose the derivatives at the point x_k by
        # Let m_k be the slope of the kth segment (between k and k+1)
        # If m_k=0 or m_{k-1}=0 or sgn(m_k) != sgn(m_{k-1}) then d_k == 0
        # else use weighted harmonic mean:
        #   w_1 = 2h_k + h_{k-1}, w_2 = h_k + 2h_{k-1}
        #   1/d_k = 1/(w_1 + w_2)*(w_1 / m_k + w_2 / m_{k-1})
        #   where h_k is the spacing between x_k and x_{k+1}
        y_shape = y.shape
        if y.ndim == 1:
            # So that _edge_case doesn't end up assigning to scalars
            x = x[:, None]
            y = y[:, None]

        hk = x[1:] - x[:-1]
        mk = (y[1:] - y[:-1]) / hk

        if y.shape[0] == 2:
            # edge case: only have two points, use linear interpolation
            dk = np.zeros_like(y)
            dk[0] = mk
            dk[1] = mk
            return dk.reshape(y_shape)

        smk = np.sign(mk)
        condition = (smk[1:] != smk[:-1]) | (mk[1:] == 0) | (mk[:-1] == 0)

        w1 = 2*hk[1:] + hk[:-1]
        w2 = hk[1:] + 2*hk[:-1]

        # values where division by zero occurs will be excluded
        # by 'condition' afterwards
        with np.errstate(divide='ignore'):
            whmean = (w1/mk[:-1] + w2/mk[1:]) / (w1 + w2)

        dk = np.zeros_like(y)
        dk[1:-1][condition] = 0.0
        dk[1:-1][~condition] = 1.0 / whmean[~condition]

        # special case endpoints, as suggested in
        # Cleve Moler, Numerical Computing with MATLAB, Chap 3.4
        dk[0] = PchipInterpolator._edge_case(hk[0], hk[1], mk[0], mk[1])
        dk[-1] = PchipInterpolator._edge_case(hk[-1], hk[-2], mk[-1], mk[-2])

        return dk.reshape(y_shape)


def pchip_interpolate(xi, yi, x, der=0, axis=0):
    """
    Convenience function for pchip interpolation.
    xi and yi are arrays of values used to approximate some function f,
    with ``yi = f(xi)``.  The interpolant uses monotonic cubic splines
    to find the value of new points x and the derivatives there.

    See `PchipInterpolator` for details.

    Parameters
    ----------
    xi : array_like
        A sorted list of x-coordinates, of length N.
    yi :  array_like
        A 1-D array of real values.  `yi`'s length along the interpolation
        axis must be equal to the length of `xi`. If N-D array, use axis
        parameter to select correct axis.
    x : scalar or array_like
        Of length M.
    der : int or list, optional
        Derivatives to extract.  The 0-th derivative can be included to
        return the function value.
    axis : int, optional
        Axis in the yi array corresponding to the x-coordinate values.

    See Also
    --------
    PchipInterpolator

    Returns
    -------
    y : scalar or array_like
        The result, of length R or length M or M by R,

    """
    P = PchipInterpolator(xi, yi, axis=axis)

    if der == 0:
        return P(x)
    elif _isscalar(der):
        return P.derivative(der)(x)
    else:
        return [P.derivative(nu)(x) for nu in der]


# Backwards compatibility
pchip = PchipInterpolator


class Akima1DInterpolator(PPoly):
    """
    Akima interpolator

    Fit piecewise cubic polynomials, given vectors x and y. The interpolation
    method by Akima uses a continuously differentiable sub-spline built from
    piecewise cubic polynomials. The resultant curve passes through the given
    data points and will appear smooth and natural.

    Parameters
    ----------
    x : ndarray, shape (m, )
        1-D array of monotonically increasing real values.
    y : ndarray, shape (m, ...)
        N-D array of real values. The length of `y` along the first axis must
        be equal to the length of `x`.
    axis : int, optional
        Specifies the axis of `y` along which to interpolate. Interpolation
        defaults to the first axis of `y`.

    Methods
    -------
    __call__
    derivative
    antiderivative
    roots

    See Also
    --------
    PchipInterpolator
    CubicSpline
    PPoly

    Notes
    -----
    .. versionadded:: 0.14

    Use only for precise data, as the fitted curve passes through the given
    points exactly. This routine is useful for plotting a pleasingly smooth
    curve through a few given points for purposes of plotting.

    References
    ----------
    [1] A new method of interpolation and smooth curve fitting based
        on local procedures. Hiroshi Akima, J. ACM, October 1970, 17(4),
        589-602.

    """

    def __init__(self, x, y, axis=0):
        # Original implementation in MATLAB by N. Shamsundar (BSD licensed), see
        # http://www.mathworks.de/matlabcentral/fileexchange/1814-akima-interpolation
        x, y = map(np.asarray, (x, y))
        axis = axis % y.ndim

        if np.any(np.diff(x) < 0.):
            raise ValueError("x must be strictly ascending")
        if x.ndim != 1:
            raise ValueError("x must be 1-dimensional")
        if x.size < 2:
            raise ValueError("at least 2 breakpoints are needed")
        if x.size != y.shape[axis]:
            raise ValueError("x.shape must equal y.shape[%s]" % axis)

        # move interpolation axis to front
        y = np.rollaxis(y, axis)

        # determine slopes between breakpoints
        m = np.empty((x.size + 3, ) + y.shape[1:])
        dx = np.diff(x)
        dx = dx[(slice(None), ) + (None, ) * (y.ndim - 1)]
        m[2:-2] = np.diff(y, axis=0) / dx

        # add two additional points on the left ...
        m[1] = 2. * m[2] - m[3]
        m[0] = 2. * m[1] - m[2]
        # ... and on the right
        m[-2] = 2. * m[-3] - m[-4]
        m[-1] = 2. * m[-2] - m[-3]

        # if m1 == m2 != m3 == m4, the slope at the breakpoint is not defined.
        # This is the fill value:
        t = .5 * (m[3:] + m[:-3])
        # get the denominator of the slope t
        dm = np.abs(np.diff(m, axis=0))
        f1 = dm[2:]
        f2 = dm[:-2]
        f12 = f1 + f2
        # These are the mask of where the the slope at breakpoint is defined:
        ind = np.nonzero(f12 > 1e-9 * np.max(f12))
        x_ind, y_ind = ind[0], ind[1:]
        # Set the slope at breakpoint
        t[ind] = (f1[ind] * m[(x_ind + 1,) + y_ind] +
                  f2[ind] * m[(x_ind + 2,) + y_ind]) / f12[ind]
        # calculate the higher order coefficients
        c = (3. * m[2:-2] - 2. * t[:-1] - t[1:]) / dx
        d = (t[:-1] + t[1:] - 2. * m[2:-2]) / dx ** 2

        coeff = np.zeros((4, x.size - 1) + y.shape[1:])
        coeff[3] = y[:-1]
        coeff[2] = t[:-1]
        coeff[1] = c
        coeff[0] = d

        super(Akima1DInterpolator, self).__init__(coeff, x, extrapolate=False)
        self.axis = axis

    def extend(self, c, x, right=True):
        raise NotImplementedError("Extending a 1D Akima interpolator is not "
                                  "yet implemented")

    # These are inherited from PPoly, but they do not produce an Akima
    # interpolator. Hence stub them out.
    @classmethod
    def from_spline(cls, tck, extrapolate=None):
        raise NotImplementedError("This method does not make sense for "
                                  "an Akima interpolator.")

    @classmethod
    def from_bernstein_basis(cls, bp, extrapolate=None):
        raise NotImplementedError("This method does not make sense for "
                                  "an Akima interpolator.")


class CubicSpline(PPoly):
    """Cubic spline data interpolator.

    Interpolate data with a piecewise cubic polynomial which is twice
    continuously differentiable [1]_. The result is represented as a `PPoly`
    instance with breakpoints matching the given data.

    Parameters
    ----------
    x : array_like, shape (n,)
        1-d array containing values of the independent variable.
        Values must be real, finite and in strictly increasing order.
    y : array_like
        Array containing values of the dependent variable. It can have
        arbitrary number of dimensions, but the length along `axis` (see below)
        must match the length of `x`. Values must be finite.
    axis : int, optional
        Axis along which `y` is assumed to be varying. Meaning that for
        ``x[i]`` the corresponding values are ``np.take(y, i, axis=axis)``.
        Default is 0.
    bc_type : string or 2-tuple, optional
        Boundary condition type. Two additional equations, given by the
        boundary conditions, are required to determine all coefficients of
        polynomials on each segment [2]_.

        If `bc_type` is a string, then the specified condition will be applied
        at both ends of a spline. Available conditions are:

        * 'not-a-knot' (default): The first and second segment at a curve end
          are the same polynomial. It is a good default when there is no
          information on boundary conditions.
        * 'periodic': The interpolated functions is assumed to be periodic
          of period ``x[-1] - x[0]``. The first and last value of `y` must be
          identical: ``y[0] == y[-1]``. This boundary condition will result in
          ``y'[0] == y'[-1]`` and ``y''[0] == y''[-1]``.
        * 'clamped': The first derivative at curves ends are zero. Assuming
          a 1D `y`, ``bc_type=((1, 0.0), (1, 0.0))`` is the same condition.
        * 'natural': The second derivative at curve ends are zero. Assuming
          a 1D `y`, ``bc_type=((2, 0.0), (2, 0.0))`` is the same condition.

        If `bc_type` is a 2-tuple, the first and the second value will be
        applied at the curve start and end respectively. The tuple values can
        be one of the previously mentioned strings (except 'periodic') or a
        tuple `(order, deriv_values)` allowing to specify arbitrary
        derivatives at curve ends:

        * `order`: the derivative order, 1 or 2.
        * `deriv_value`: array_like containing derivative values, shape must
          be the same as `y`, excluding `axis` dimension. For example, if `y`
          is 1D, then `deriv_value` must be a scalar. If `y` is 3D with the
          shape (n0, n1, n2) and axis=2, then `deriv_value` must be 2D
          and have the shape (n0, n1).
    extrapolate : {bool, 'periodic', None}, optional
        If bool, determines whether to extrapolate to out-of-bounds points
        based on first and last intervals, or to return NaNs. If 'periodic',
        periodic extrapolation is used. If None (default), `extrapolate` is
        set to 'periodic' for ``bc_type='periodic'`` and to True otherwise.

    Attributes
    ----------
    x : ndarray, shape (n,)
        Breakpoints. The same `x` which was passed to the constructor.
    c : ndarray, shape (4, n-1, ...)
        Coefficients of the polynomials on each segment. The trailing
        dimensions match the dimensions of `y`, excluding `axis`. For example,
        if `y` is 1-d, then ``c[k, i]`` is a coefficient for
        ``(x-x[i])**(3-k)`` on the segment between ``x[i]`` and ``x[i+1]``.
    axis : int
        Interpolation axis. The same `axis` which was passed to the
        constructor.

    Methods
    -------
    __call__
    derivative
    antiderivative
    integrate
    roots

    See Also
    --------
    Akima1DInterpolator
    PchipInterpolator
    PPoly

    Notes
    -----
    Parameters `bc_type` and `interpolate` work independently, i.e. the former
    controls only construction of a spline, and the latter only evaluation.

    When a boundary condition is 'not-a-knot' and n = 2, it is replaced by
    a condition that the first derivative is equal to the linear interpolant
    slope. When both boundary conditions are 'not-a-knot' and n = 3, the
    solution is sought as a parabola passing through given points.

    When 'not-a-knot' boundary conditions is applied to both ends, the
    resulting spline will be the same as returned by `splrep` (with ``s=0``)
    and `InterpolatedUnivariateSpline`, but these two methods use a
    representation in B-spline basis.

    .. versionadded:: 0.18.0

    Examples
    --------
    In this example the cubic spline is used to interpolate a sampled sinusoid.
    You can see that the spline continuity property holds for the first and
    second derivatives and violates only for the third derivative.

    >>> from scipy.interpolate import CubicSpline
    >>> import matplotlib.pyplot as plt
    >>> x = np.arange(10)
    >>> y = np.sin(x)
    >>> cs = CubicSpline(x, y)
    >>> xs = np.arange(-0.5, 9.6, 0.1)
    >>> plt.figure(figsize=(6.5, 4))
    >>> plt.plot(x, y, 'o', label='data')
    >>> plt.plot(xs, np.sin(xs), label='true')
    >>> plt.plot(xs, cs(xs), label="S")
    >>> plt.plot(xs, cs(xs, 1), label="S'")
    >>> plt.plot(xs, cs(xs, 2), label="S''")
    >>> plt.plot(xs, cs(xs, 3), label="S'''")
    >>> plt.xlim(-0.5, 9.5)
    >>> plt.legend(loc='lower left', ncol=2)
    >>> plt.show()

    In the second example, the unit circle is interpolated with a spline. A
    periodic boundary condition is used. You can see that the first derivative
    values, ds/dx=0, ds/dy=1 at the periodic point (1, 0) are correctly
    computed. Note that a circle cannot be exactly represented by a cubic
    spline. To increase precision, more breakpoints would be required.

    >>> theta = 2 * np.pi * np.linspace(0, 1, 5)
    >>> y = np.c_[np.cos(theta), np.sin(theta)]
    >>> cs = CubicSpline(theta, y, bc_type='periodic')
    >>> print("ds/dx={:.1f} ds/dy={:.1f}".format(cs(0, 1)[0], cs(0, 1)[1]))
    ds/dx=0.0 ds/dy=1.0
    >>> xs = 2 * np.pi * np.linspace(0, 1, 100)
    >>> plt.figure(figsize=(6.5, 4))
    >>> plt.plot(y[:, 0], y[:, 1], 'o', label='data')
    >>> plt.plot(np.cos(xs), np.sin(xs), label='true')
    >>> plt.plot(cs(xs)[:, 0], cs(xs)[:, 1], label='spline')
    >>> plt.axes().set_aspect('equal')
    >>> plt.legend(loc='center')
    >>> plt.show()

    The third example is the interpolation of a polynomial y = x**3 on the
    interval 0 <= x<= 1. A cubic spline can represent this function exactly.
    To achieve that we need to specify values and first derivatives at
    endpoints of the interval. Note that y' = 3 * x**2 and thus y'(0) = 0 and
    y'(1) = 3.

    >>> cs = CubicSpline([0, 1], [0, 1], bc_type=((1, 0), (1, 3)))
    >>> x = np.linspace(0, 1)
    >>> np.allclose(x**3, cs(x))
    True

    References
    ----------
    .. [1] `Cubic Spline Interpolation
            <https://en.wikiversity.org/wiki/Cubic_Spline_Interpolation>`_
            on Wikiversity.
    .. [2] Carl de Boor, "A Practical Guide to Splines", Springer-Verlag, 1978.
    """
    def __init__(self, x, y, axis=0, bc_type='not-a-knot', extrapolate=None):
        x, y = map(np.asarray, (x, y))

        if np.issubdtype(x.dtype, np.complexfloating):
            raise ValueError("`x` must contain real values.")

        if np.issubdtype(y.dtype, np.complexfloating):
            dtype = complex
        else:
            dtype = float
        y = y.astype(dtype, copy=False)

        axis = axis % y.ndim
        if x.ndim != 1:
            raise ValueError("`x` must be 1-dimensional.")
        if x.shape[0] < 2:
            raise ValueError("`x` must contain at least 2 elements.")
        if x.shape[0] != y.shape[axis]:
            raise ValueError("The length of `y` along `axis`={0} doesn't "
                             "match the length of `x`".format(axis))

        if not np.all(np.isfinite(x)):
            raise ValueError("`x` must contain only finite values.")
        if not np.all(np.isfinite(y)):
            raise ValueError("`y` must contain only finite values.")

        dx = np.diff(x)
        if np.any(dx <= 0):
            raise ValueError("`x` must be strictly increasing sequence.")

        n = x.shape[0]
        y = np.rollaxis(y, axis)

        bc, y = self._validate_bc(bc_type, y, y.shape[1:], axis)

        if extrapolate is None:
            if bc[0] == 'periodic':
                extrapolate = 'periodic'
            else:
                extrapolate = True

        dxr = dx.reshape([dx.shape[0]] + [1] * (y.ndim - 1))
        slope = np.diff(y, axis=0) / dxr

        # If bc is 'not-a-knot' this change is just a convention.
        # If bc is 'periodic' then we already checked that y[0] == y[-1],
        # and the spline is just a constant, we handle this case in the same
        # way by setting the first derivatives to slope, which is 0.
        if n == 2:
            if bc[0] in ['not-a-knot', 'periodic']:
                bc[0] = (1, slope[0])
            if bc[1] in ['not-a-knot', 'periodic']:
                bc[1] = (1, slope[0])

        # This is a very special case, when both conditions are 'not-a-knot'
        # and n == 3. In this case 'not-a-knot' can't be handled regularly
        # as the both conditions are identical. We handle this case by
        # constructing a parabola passing through given points.
        if n == 3 and bc[0] == 'not-a-knot' and bc[1] == 'not-a-knot':
            A = np.zeros((3, 3))  # This is a standard matrix.
            b = np.empty((3,) + y.shape[1:], dtype=y.dtype)

            A[0, 0] = 1
            A[0, 1] = 1
            A[1, 0] = dx[1]
            A[1, 1] = 2 * (dx[0] + dx[1])
            A[1, 2] = dx[0]
            A[2, 1] = 1
            A[2, 2] = 1

            b[0] = 2 * slope[0]
            b[1] = 3 * (dxr[0] * slope[1] + dxr[1] * slope[0])
            b[2] = 2 * slope[1]

            s = solve(A, b, overwrite_a=True, overwrite_b=True,
                      check_finite=False)
        else:
            # Find derivative values at each x[i] by solving a tridiagonal
            # system.
            A = np.zeros((3, n))  # This is a banded matrix representation.
            b = np.empty((n,) + y.shape[1:], dtype=y.dtype)

            # Filling the system for i=1..n-2
            #                         (x[i-1] - x[i]) * s[i-1] +\
            # 2 * ((x[i] - x[i-1]) + (x[i+1] - x[i])) * s[i]   +\
            #                         (x[i] - x[i-1]) * s[i+1] =\
            #       3 * ((x[i+1] - x[i])*(y[i] - y[i-1])/(x[i] - x[i-1]) +\
            #           (x[i] - x[i-1])*(y[i+1] - y[i])/(x[i+1] - x[i]))

            A[1, 1:-1] = 2 * (dx[:-1] + dx[1:])  # The diagonal
            A[0, 2:] = dx[:-1]                   # The upper diagonal
            A[-1, :-2] = dx[1:]                  # The lower diagonal

            b[1:-1] = 3 * (dxr[1:] * slope[:-1] + dxr[:-1] * slope[1:])

            bc_start, bc_end = bc

            if bc_start == 'periodic':
                # Due to the periodicity, and because y[-1] = y[0], the linear
                # system has (n-1) unknowns/equations instead of n:
                A = A[:, 0:-1]
                A[1, 0] = 2 * (dx[-1] + dx[0])
                A[0, 1] = dx[-1]

                b = b[:-1]

                # Also, due to the periodicity, the system is not tri-diagonal.
                # We need to compute a "condensed" matrix of shape (n-2, n-2).
                # See http://www.cfm.brown.edu/people/gk/chap6/node14.html for
                # more explanations.
                # The condensed matrix is obtained by removing the last column
                # and last row of the (n-1, n-1) system matrix. The removed
                # values are saved in scalar variables with the (n-1, n-1)
                # system matrix indices forming their names:
                a_m1_0 = dx[-2]  # lower left corner value: A[-1, 0]
                a_m1_m2 = dx[-1]
                a_m1_m1 = 2 * (dx[-1] + dx[-2])
                a_m2_m1 = dx[-2]
                a_0_m1 = dx[0]

                b[0] = 3 * (dxr[0] * slope[-1] + dxr[-1] * slope[0])
                b[-1] = 3 * (dxr[-1] * slope[-2] + dxr[-2] * slope[-1])

                Ac = A[:, :-1]
                b1 = b[:-1]
                b2 = np.zeros_like(b1)
                b2[0] = -a_0_m1
                b2[-1] = -a_m2_m1

                # s1 and s2 are the solutions of (n-2, n-2) system
                s1 = solve_banded((1, 1), Ac, b1, overwrite_ab=False,
                                  overwrite_b=False, check_finite=False)

                s2 = solve_banded((1, 1), Ac, b2, overwrite_ab=False,
                                  overwrite_b=False, check_finite=False)

                # computing the s[n-2] solution:
                s_m1 = ((b[-1] - a_m1_0 * s1[0] - a_m1_m2 * s1[-1]) /
                        (a_m1_m1 + a_m1_0 * s2[0] + a_m1_m2 * s2[-1]))

                # s is the solution of the (n, n) system:
                s = np.empty((n,) + y.shape[1:], dtype=y.dtype)
                s[:-2] = s1 + s_m1 * s2
                s[-2] = s_m1
                s[-1] = s[0]
            else:
                if bc_start == 'not-a-knot':
                    A[1, 0] = dx[1]
                    A[0, 1] = x[2] - x[0]
                    d = x[2] - x[0]
                    b[0] = ((dxr[0] + 2*d) * dxr[1] * slope[0] +
                            dxr[0]**2 * slope[1]) / d
                elif bc_start[0] == 1:
                    A[1, 0] = 1
                    A[0, 1] = 0
                    b[0] = bc_start[1]
                elif bc_start[0] == 2:
                    A[1, 0] = 2 * dx[0]
                    A[0, 1] = dx[0]
                    b[0] = -0.5 * bc_start[1] * dx[0]**2 + 3 * (y[1] - y[0])

                if bc_end == 'not-a-knot':
                    A[1, -1] = dx[-2]
                    A[-1, -2] = x[-1] - x[-3]
                    d = x[-1] - x[-3]
                    b[-1] = ((dxr[-1]**2*slope[-2] +
                             (2*d + dxr[-1])*dxr[-2]*slope[-1]) / d)
                elif bc_end[0] == 1:
                    A[1, -1] = 1
                    A[-1, -2] = 0
                    b[-1] = bc_end[1]
                elif bc_end[0] == 2:
                    A[1, -1] = 2 * dx[-1]
                    A[-1, -2] = dx[-1]
                    b[-1] = 0.5 * bc_end[1] * dx[-1]**2 + 3 * (y[-1] - y[-2])

                s = solve_banded((1, 1), A, b, overwrite_ab=True,
                                 overwrite_b=True, check_finite=False)

        # Compute coefficients in PPoly form.
        t = (s[:-1] + s[1:] - 2 * slope) / dxr
        c = np.empty((4, n - 1) + y.shape[1:], dtype=t.dtype)
        c[0] = t / dxr
        c[1] = (slope - s[:-1]) / dxr - t
        c[2] = s[:-1]
        c[3] = y[:-1]

        super(CubicSpline, self).__init__(c, x, extrapolate=extrapolate)
        self.axis = axis

    @staticmethod
    def _validate_bc(bc_type, y, expected_deriv_shape, axis):
        """Validate and prepare boundary conditions.

        Returns
        -------
        validated_bc : 2-tuple
            Boundary conditions for a curve start and end.
        y : ndarray
            y casted to complex dtype if one of the boundary conditions has
            complex dtype.
        """
        if isinstance(bc_type, string_types):
            if bc_type == 'periodic':
                if not np.allclose(y[0], y[-1], rtol=1e-15, atol=1e-15):
                    raise ValueError(
                        "The first and last `y` point along axis {} must "
                        "be identical (within machine precision) when "
                        "bc_type='periodic'.".format(axis))

            bc_type = (bc_type, bc_type)

        else:
            if len(bc_type) != 2:
                raise ValueError("`bc_type` must contain 2 elements to "
                                 "specify start and end conditions.")

            if 'periodic' in bc_type:
                raise ValueError("'periodic' `bc_type` is defined for both "
                                 "curve ends and cannot be used with other "
                                 "boundary conditions.")

        validated_bc = []
        for bc in bc_type:
            if isinstance(bc, string_types):
                if bc == 'clamped':
                    validated_bc.append((1, np.zeros(expected_deriv_shape)))
                elif bc == 'natural':
                    validated_bc.append((2, np.zeros(expected_deriv_shape)))
                elif bc in ['not-a-knot', 'periodic']:
                    validated_bc.append(bc)
                else:
                    raise ValueError("bc_type={} is not allowed.".format(bc))
            else:
                try:
                    deriv_order, deriv_value = bc
                except Exception:
                    raise ValueError("A specified derivative value must be "
                                     "given in the form (order, value).")

                if deriv_order not in [1, 2]:
                    raise ValueError("The specified derivative order must "
                                     "be 1 or 2.")

                deriv_value = np.asarray(deriv_value)
                if deriv_value.shape != expected_deriv_shape:
                    raise ValueError(
                        "`deriv_value` shape {} is not the expected one {}."
                        .format(deriv_value.shape, expected_deriv_shape))

                if np.issubdtype(deriv_value.dtype, np.complexfloating):
                    y = y.astype(complex, copy=False)

                validated_bc.append((deriv_order, deriv_value))

        return validated_bc, y