import cupy from cupyx.scipy.interpolate._interpolate import PPoly def _isscalar(x): """Check whether x is if a scalar type, or 0-dim""" return cupy.isscalar(x) or hasattr(x, 'shape') and x.shape == () def prepare_input(x, y, axis, dydx=None): """Prepare input for cubic spline interpolators. All data are converted to numpy arrays and checked for correctness. Axes equal to `axis` of arrays `y` and `dydx` are moved to be the 0th axis. The value of `axis` is converted to lie in [0, number of dimensions of `y`). """ x, y = map(cupy.asarray, (x, y)) if cupy.issubdtype(x.dtype, cupy.complexfloating): raise ValueError("`x` must contain real values.") x = x.astype(float) if cupy.issubdtype(y.dtype, cupy.complexfloating): dtype = complex else: dtype = float if dydx is not None: dydx = cupy.asarray(dydx) if y.shape != dydx.shape: raise ValueError("The shapes of `y` and `dydx` must be identical.") if cupy.issubdtype(dydx.dtype, cupy.complexfloating): dtype = complex dydx = dydx.astype(dtype, copy=False) 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 cupy.all(cupy.isfinite(x)): raise ValueError("`x` must contain only finite values.") if not cupy.all(cupy.isfinite(y)): raise ValueError("`y` must contain only finite values.") if dydx is not None and not cupy.all(cupy.isfinite(dydx)): raise ValueError("`dydx` must contain only finite values.") dx = cupy.diff(x) if cupy.any(dx <= 0): raise ValueError("`x` must be strictly increasing sequence.") y = cupy.moveaxis(y, axis, 0) if dydx is not None: dydx = cupy.moveaxis(dydx, axis, 0) return x, dx, y, axis, dydx class CubicHermiteSpline(PPoly): """Piecewise-cubic interpolator matching values and first derivatives. The result is represented as a `PPoly` instance. [1]_ 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. dydx : array_like Array containing derivatives 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 ``cupy.take(y, i, axis=axis)``. Default is 0. 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), it is set to True. 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. See Also -------- Akima1DInterpolator : Akima 1D interpolator. PchipInterpolator : PCHIP 1-D monotonic cubic interpolator. PPoly : Piecewise polynomial in terms of coefficients and breakpoints Notes ----- If you want to create a higher-order spline matching higher-order derivatives, use `BPoly.from_derivatives`. References ---------- .. [1] `Cubic Hermite spline `_ on Wikipedia. """ def __init__(self, x, y, dydx, axis=0, extrapolate=None): if extrapolate is None: extrapolate = True x, dx, y, axis, dydx = prepare_input(x, y, axis, dydx) dxr = dx.reshape([dx.shape[0]] + [1] * (y.ndim - 1)) slope = cupy.diff(y, axis=0) / dxr t = (dydx[:-1] + dydx[1:] - 2 * slope) / dxr c = cupy.empty((4, len(x) - 1) + y.shape[1:], dtype=t.dtype) c[0] = t / dxr c[1] = (slope - dydx[:-1]) / dxr - t c[2] = dydx[:-1] c[3] = y[:-1] super().__init__(c, x, extrapolate=extrapolate) self.axis = axis class PchipInterpolator(CubicHermiteSpline): 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. See Also -------- CubicHermiteSpline : Piecewise-cubic interpolator. Akima1DInterpolator : Akima 1D interpolator. PPoly : Piecewise polynomial in terms of coefficients and breakpoints. 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 J. Butland, A method for constructing local monotone piecewise cubic interpolants, SIAM J. Sci. Comput., 5(2), 300-304 (1984). `10.1137/0905021 `_. .. [2] see, e.g., C. Moler, Numerical Computing with Matlab, 2004. `10.1137/1.9780898717952 `_ """ def __init__(self, x, y, axis=0, extrapolate=None): x, _, y, axis, _ = prepare_input(x, y, axis) xp = x.reshape((x.shape[0],) + (1,)*(y.ndim-1)) dk = self._find_derivatives(xp, y) super().__init__(x, y, dk, axis=0, extrapolate=extrapolate) self.axis = axis @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 = cupy.sign(d) != cupy.sign(m0) mask2 = (cupy.sign(m0) != cupy.sign(m1)) & ( cupy.abs(d) > 3.*cupy.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 = cupy.zeros_like(y) dk[0] = mk dk[1] = mk return dk.reshape(y_shape) smk = cupy.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 whmean = (w1 / mk[:-1] + w2 / mk[1:]) / (w1 + w2) dk = cupy.zeros_like(y) dk[1:-1] = cupy.where(condition, 0.0, 1.0 / whmean) # special case endpoints, as suggested in # Cleve Moler, Numerical Computing with MATLAB, Chap 3.6 (pchiptx.m) 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 `scipy.interpolate.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 0th 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 : PCHIP 1-D monotonic cubic interpolator. 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] class Akima1DInterpolator(CubicHermiteSpline): """ 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 [1]_. 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``. See Also -------- CubicHermiteSpline : Piecewise-cubic interpolator. PchipInterpolator : PCHIP 1-D monotonic cubic interpolator. PPoly : Piecewise polynomial in terms of coefficients and breakpoints Notes ----- 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) # https://www.mathworks.com/matlabcentral/fileexchange/1814-akima-interpolation # noqa: E501 x, dx, y, axis, _ = prepare_input(x, y, axis) # determine slopes between breakpoints m = cupy.empty((x.size + 3, ) + y.shape[1:]) dx = dx[(slice(None), ) + (None, ) * (y.ndim - 1)] m[2:-2] = cupy.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 = cupy.abs(cupy.diff(m, axis=0)) f1 = dm[2:] f2 = dm[:-2] f12 = f1 + f2 # These are the mask of where the slope at breakpoint is defined: max_value = -cupy.inf if y.size == 0 else cupy.max(f12) ind = cupy.nonzero(f12 > 1e-9 * max_value) 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] super().__init__(x, y, t, axis=0, extrapolate=False) self.axis = axis def extend(self, c, x, right=True): raise NotImplementedError("Extending a 1-D 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.")