import warnings import numpy as np from scipy.special import factorial from scipy._lib._util import (_asarray_validated, float_factorial, check_random_state, _transition_to_rng) __all__ = ["KroghInterpolator", "krogh_interpolate", "BarycentricInterpolator", "barycentric_interpolate", "approximate_taylor_polynomial"] def _isscalar(x): """Check whether x is if a scalar type, or 0-dim""" return np.isscalar(x) or hasattr(x, 'shape') and x.shape == () class _Interpolator1D: """ Common features in univariate interpolation Deal with input data type and interpolation axis rolling. The actual interpolator can assume the y-data is of shape (n, r) where `n` is the number of x-points, and `r` the number of variables, and use self.dtype as the y-data type. Attributes ---------- _y_axis Axis along which the interpolation goes in the original array _y_extra_shape Additional trailing shape of the input arrays, excluding the interpolation axis. dtype Dtype of the y-data arrays. Can be set via _set_dtype, which forces it to be float or complex. Methods ------- __call__ _prepare_x _finish_y _reshape_yi _set_yi _set_dtype _evaluate """ __slots__ = ('_y_axis', '_y_extra_shape', 'dtype') def __init__(self, xi=None, yi=None, axis=None): self._y_axis = axis self._y_extra_shape = None self.dtype = None if yi is not None: self._set_yi(yi, xi=xi, axis=axis) def __call__(self, x): """ Evaluate the interpolant Parameters ---------- x : array_like Point or points at which to evaluate the interpolant. Returns ------- y : array_like Interpolated values. Shape is determined by replacing the interpolation axis in the original array with the shape of `x`. Notes ----- Input values `x` must be convertible to `float` values like `int` or `float`. """ x, x_shape = self._prepare_x(x) y = self._evaluate(x) return self._finish_y(y, x_shape) def _evaluate(self, x): """ Actually evaluate the value of the interpolator. """ raise NotImplementedError() def _prepare_x(self, x): """Reshape input x array to 1-D""" x = _asarray_validated(x, check_finite=False, as_inexact=True) x_shape = x.shape return x.ravel(), x_shape def _finish_y(self, y, x_shape): """Reshape interpolated y back to an N-D array similar to initial y""" y = y.reshape(x_shape + self._y_extra_shape) if self._y_axis != 0 and x_shape != (): nx = len(x_shape) ny = len(self._y_extra_shape) s = (list(range(nx, nx + self._y_axis)) + list(range(nx)) + list(range(nx+self._y_axis, nx+ny))) y = y.transpose(s) return y def _reshape_yi(self, yi, check=False): yi = np.moveaxis(np.asarray(yi), self._y_axis, 0) if check and yi.shape[1:] != self._y_extra_shape: ok_shape = (f"{self._y_extra_shape[-self._y_axis:]!r} + (N,) + " f"{self._y_extra_shape[:-self._y_axis]!r}") raise ValueError(f"Data must be of shape {ok_shape}") return yi.reshape((yi.shape[0], -1)) def _set_yi(self, yi, xi=None, axis=None): if axis is None: axis = self._y_axis if axis is None: raise ValueError("no interpolation axis specified") yi = np.asarray(yi) shape = yi.shape if shape == (): shape = (1,) if xi is not None and shape[axis] != len(xi): raise ValueError("x and y arrays must be equal in length along " "interpolation axis.") self._y_axis = (axis % yi.ndim) self._y_extra_shape = yi.shape[:self._y_axis] + yi.shape[self._y_axis+1:] self.dtype = None self._set_dtype(yi.dtype) def _set_dtype(self, dtype, union=False): if np.issubdtype(dtype, np.complexfloating) \ or np.issubdtype(self.dtype, np.complexfloating): self.dtype = np.complex128 else: if not union or self.dtype != np.complex128: self.dtype = np.float64 class _Interpolator1DWithDerivatives(_Interpolator1D): def derivatives(self, x, der=None): """ Evaluate several derivatives of the polynomial at the point `x` Produce an array of derivatives evaluated at the point `x`. Parameters ---------- x : array_like Point or points at which to evaluate the derivatives der : int or list or None, optional How many derivatives to evaluate, or None for all potentially nonzero derivatives (that is, a number equal to the number of points), or a list of derivatives to evaluate. This number includes the function value as the '0th' derivative. Returns ------- d : ndarray Array with derivatives; ``d[j]`` contains the jth derivative. Shape of ``d[j]`` is determined by replacing the interpolation axis in the original array with the shape of `x`. Examples -------- >>> from scipy.interpolate import KroghInterpolator >>> KroghInterpolator([0,0,0],[1,2,3]).derivatives(0) array([1.0,2.0,3.0]) >>> KroghInterpolator([0,0,0],[1,2,3]).derivatives([0,0]) array([[1.0,1.0], [2.0,2.0], [3.0,3.0]]) """ x, x_shape = self._prepare_x(x) y = self._evaluate_derivatives(x, der) y = y.reshape((y.shape[0],) + x_shape + self._y_extra_shape) if self._y_axis != 0 and x_shape != (): nx = len(x_shape) ny = len(self._y_extra_shape) s = ([0] + list(range(nx+1, nx + self._y_axis+1)) + list(range(1, nx+1)) + list(range(nx+1+self._y_axis, nx+ny+1))) y = y.transpose(s) return y def derivative(self, x, der=1): """ Evaluate a single derivative of the polynomial at the point `x`. Parameters ---------- x : array_like Point or points at which to evaluate the derivatives der : integer, optional Which derivative to evaluate (default: first derivative). This number includes the function value as 0th derivative. Returns ------- d : ndarray Derivative interpolated at the x-points. Shape of `d` is determined by replacing the interpolation axis in the original array with the shape of `x`. Notes ----- This may be computed by evaluating all derivatives up to the desired one (using self.derivatives()) and then discarding the rest. """ x, x_shape = self._prepare_x(x) y = self._evaluate_derivatives(x, der+1) return self._finish_y(y[der], x_shape) def _evaluate_derivatives(self, x, der=None): """ Actually evaluate the derivatives. Parameters ---------- x : array_like 1D array of points at which to evaluate the derivatives der : integer, optional The number of derivatives to evaluate, from 'order 0' (der=1) to order der-1. If omitted, return all possibly-non-zero derivatives, ie 0 to order n-1. Returns ------- d : ndarray Array of shape ``(der, x.size, self.yi.shape[1])`` containing the derivatives from 0 to der-1 """ raise NotImplementedError() class KroghInterpolator(_Interpolator1DWithDerivatives): """ Interpolating polynomial for a set of points. The polynomial passes through all the pairs ``(xi, yi)``. One may additionally specify a number of derivatives at each point `xi`; this is done by repeating the value `xi` and specifying the derivatives as successive `yi` values. Allows evaluation of the polynomial and all its derivatives. For reasons of numerical stability, this function does not compute the coefficients of the polynomial, although they can be obtained by evaluating all the derivatives. Parameters ---------- xi : array_like, shape (npoints, ) Known x-coordinates. Must be sorted in increasing order. yi : array_like, shape (..., npoints, ...) Known y-coordinates. When an xi occurs two or more times in a row, the corresponding yi's represent derivative values. The length of `yi` along the interpolation axis must be equal to the length of `xi`. Use the `axis` parameter to select the correct axis. axis : int, optional Axis in the `yi` array corresponding to the x-coordinate values. Defaults to ``axis=0``. Notes ----- Be aware that the algorithms implemented here are not necessarily the most numerically stable known. Moreover, even in a world of exact computation, unless the x coordinates are chosen very carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon. In general, even with well-chosen x values, degrees higher than about thirty cause problems with numerical instability in this code. Based on [1]_. References ---------- .. [1] Krogh, "Efficient Algorithms for Polynomial Interpolation and Numerical Differentiation", 1970. Examples -------- To produce a polynomial that is zero at 0 and 1 and has derivative 2 at 0, call >>> from scipy.interpolate import KroghInterpolator >>> KroghInterpolator([0,0,1],[0,2,0]) This constructs the quadratic :math:`2x^2-2x`. The derivative condition is indicated by the repeated zero in the `xi` array; the corresponding yi values are 0, the function value, and 2, the derivative value. For another example, given `xi`, `yi`, and a derivative `ypi` for each point, appropriate arrays can be constructed as: >>> import numpy as np >>> rng = np.random.default_rng() >>> xi = np.linspace(0, 1, 5) >>> yi, ypi = rng.random((2, 5)) >>> xi_k, yi_k = np.repeat(xi, 2), np.ravel(np.dstack((yi,ypi))) >>> KroghInterpolator(xi_k, yi_k) To produce a vector-valued polynomial, supply a higher-dimensional array for `yi`: >>> KroghInterpolator([0,1],[[2,3],[4,5]]) This constructs a linear polynomial giving (2,3) at 0 and (4,5) at 1. """ def __init__(self, xi, yi, axis=0): super().__init__(xi, yi, axis) self.xi = np.asarray(xi) self.yi = self._reshape_yi(yi) self.n, self.r = self.yi.shape if (deg := self.xi.size) > 30: warnings.warn(f"{deg} degrees provided, degrees higher than about" " thirty cause problems with numerical instability " "with 'KroghInterpolator'", stacklevel=2) c = np.zeros((self.n+1, self.r), dtype=self.dtype) c[0] = self.yi[0] Vk = np.zeros((self.n, self.r), dtype=self.dtype) for k in range(1, self.n): s = 0 while s <= k and xi[k-s] == xi[k]: s += 1 s -= 1 Vk[0] = self.yi[k]/float_factorial(s) for i in range(k-s): if xi[i] == xi[k]: raise ValueError("Elements of `xi` can't be equal.") if s == 0: Vk[i+1] = (c[i]-Vk[i])/(xi[i]-xi[k]) else: Vk[i+1] = (Vk[i+1]-Vk[i])/(xi[i]-xi[k]) c[k] = Vk[k-s] self.c = c def _evaluate(self, x): pi = 1 p = np.zeros((len(x), self.r), dtype=self.dtype) p += self.c[0,np.newaxis,:] for k in range(1, self.n): w = x - self.xi[k-1] pi = w*pi p += pi[:,np.newaxis] * self.c[k] return p def _evaluate_derivatives(self, x, der=None): n = self.n r = self.r if der is None: der = self.n pi = np.zeros((n, len(x))) w = np.zeros((n, len(x))) pi[0] = 1 p = np.zeros((len(x), self.r), dtype=self.dtype) p += self.c[0, np.newaxis, :] for k in range(1, n): w[k-1] = x - self.xi[k-1] pi[k] = w[k-1] * pi[k-1] p += pi[k, :, np.newaxis] * self.c[k] cn = np.zeros((max(der, n+1), len(x), r), dtype=self.dtype) cn[:n+1, :, :] += self.c[:n+1, np.newaxis, :] cn[0] = p for k in range(1, n): for i in range(1, n-k+1): pi[i] = w[k+i-1]*pi[i-1] + pi[i] cn[k] = cn[k] + pi[i, :, np.newaxis]*cn[k+i] cn[k] *= float_factorial(k) cn[n, :, :] = 0 return cn[:der] def krogh_interpolate(xi, yi, x, der=0, axis=0): """ Convenience function for polynomial interpolation. See `KroghInterpolator` for more details. Parameters ---------- xi : array_like Interpolation points (known x-coordinates). yi : array_like Known y-coordinates, of shape ``(xi.size, R)``. Interpreted as vectors of length R, or scalars if R=1. x : array_like Point or points at which to evaluate the derivatives. der : int or list or None, optional How many derivatives to evaluate, or None for all potentially nonzero derivatives (that is, a number equal to the number of points), or a list of derivatives to evaluate. This number includes the function value as the '0th' derivative. axis : int, optional Axis in the `yi` array corresponding to the x-coordinate values. Returns ------- d : ndarray If the interpolator's values are R-D then the returned array will be the number of derivatives by N by R. If `x` is a scalar, the middle dimension will be dropped; if the `yi` are scalars then the last dimension will be dropped. See Also -------- KroghInterpolator : Krogh interpolator Notes ----- Construction of the interpolating polynomial is a relatively expensive process. If you want to evaluate it repeatedly consider using the class KroghInterpolator (which is what this function uses). Examples -------- We can interpolate 2D observed data using Krogh interpolation: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.interpolate import krogh_interpolate >>> x_observed = np.linspace(0.0, 10.0, 11) >>> y_observed = np.sin(x_observed) >>> x = np.linspace(min(x_observed), max(x_observed), num=100) >>> y = krogh_interpolate(x_observed, y_observed, x) >>> plt.plot(x_observed, y_observed, "o", label="observation") >>> plt.plot(x, y, label="krogh interpolation") >>> plt.legend() >>> plt.show() """ P = KroghInterpolator(xi, yi, axis=axis) if der == 0: return P(x) elif _isscalar(der): return P.derivative(x, der=der) else: return P.derivatives(x, der=np.amax(der)+1)[der] def approximate_taylor_polynomial(f,x,degree,scale,order=None): """ Estimate the Taylor polynomial of f at x by polynomial fitting. Parameters ---------- f : callable The function whose Taylor polynomial is sought. Should accept a vector of `x` values. x : scalar The point at which the polynomial is to be evaluated. degree : int The degree of the Taylor polynomial scale : scalar The width of the interval to use to evaluate the Taylor polynomial. Function values spread over a range this wide are used to fit the polynomial. Must be chosen carefully. order : int or None, optional The order of the polynomial to be used in the fitting; `f` will be evaluated ``order+1`` times. If None, use `degree`. Returns ------- p : poly1d instance The Taylor polynomial (translated to the origin, so that for example p(0)=f(x)). Notes ----- The appropriate choice of "scale" is a trade-off; too large and the function differs from its Taylor polynomial too much to get a good answer, too small and round-off errors overwhelm the higher-order terms. The algorithm used becomes numerically unstable around order 30 even under ideal circumstances. Choosing order somewhat larger than degree may improve the higher-order terms. Examples -------- We can calculate Taylor approximation polynomials of sin function with various degrees: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.interpolate import approximate_taylor_polynomial >>> x = np.linspace(-10.0, 10.0, num=100) >>> plt.plot(x, np.sin(x), label="sin curve") >>> for degree in np.arange(1, 15, step=2): ... sin_taylor = approximate_taylor_polynomial(np.sin, 0, degree, 1, ... order=degree + 2) ... plt.plot(x, sin_taylor(x), label=f"degree={degree}") >>> plt.legend(bbox_to_anchor=(1.05, 1), loc='upper left', ... borderaxespad=0.0, shadow=True) >>> plt.tight_layout() >>> plt.axis([-10, 10, -10, 10]) >>> plt.show() """ if order is None: order = degree n = order+1 # Choose n points that cluster near the endpoints of the interval in # a way that avoids the Runge phenomenon. Ensure, by including the # endpoint or not as appropriate, that one point always falls at x # exactly. xs = scale*np.cos(np.linspace(0,np.pi,n,endpoint=n % 1)) + x P = KroghInterpolator(xs, f(xs)) d = P.derivatives(x,der=degree+1) return np.poly1d((d/factorial(np.arange(degree+1)))[::-1]) class BarycentricInterpolator(_Interpolator1DWithDerivatives): r"""Interpolating polynomial for a set of points. Constructs a polynomial that passes through a given set of points. Allows evaluation of the polynomial and all its derivatives, efficient changing of the y-values to be interpolated, and updating by adding more x- and y-values. For reasons of numerical stability, this function does not compute the coefficients of the polynomial. The values `yi` need to be provided before the function is evaluated, but none of the preprocessing depends on them, so rapid updates are possible. Parameters ---------- xi : array_like, shape (npoints, ) 1-D array of x coordinates of the points the polynomial should pass through yi : array_like, shape (..., npoints, ...), optional N-D array of y coordinates of the points the polynomial should pass through. If None, the y values will be supplied later via the `set_y` method. The length of `yi` along the interpolation axis must be equal to the length of `xi`. Use the ``axis`` parameter to select correct axis. axis : int, optional Axis in the yi array corresponding to the x-coordinate values. Defaults to ``axis=0``. wi : array_like, optional The barycentric weights for the chosen interpolation points `xi`. If absent or None, the weights will be computed from `xi` (default). This allows for the reuse of the weights `wi` if several interpolants are being calculated using the same nodes `xi`, without re-computation. rng : {None, int, `numpy.random.Generator`}, optional If `rng` is passed by keyword, types other than `numpy.random.Generator` are passed to `numpy.random.default_rng` to instantiate a ``Generator``. If `rng` is already a ``Generator`` instance, then the provided instance is used. Specify `rng` for repeatable interpolation. If this argument `random_state` is passed by keyword, legacy behavior for the argument `random_state` applies: - If `random_state` is None (or `numpy.random`), the `numpy.random.RandomState` singleton is used. - If `random_state` is an int, a new ``RandomState`` instance is used, seeded with `random_state`. - If `random_state` is already a ``Generator`` or ``RandomState`` instance then that instance is used. .. versionchanged:: 1.15.0 As part of the `SPEC-007 `_ transition from use of `numpy.random.RandomState` to `numpy.random.Generator` this keyword was changed from `random_state` to `rng`. For an interim period, both keywords will continue to work (only specify one of them). After the interim period using the `random_state` keyword will emit warnings. The behavior of the `random_state` and `rng` keywords is outlined above. Notes ----- This class uses a "barycentric interpolation" method that treats the problem as a special case of rational function interpolation. This algorithm is quite stable, numerically, but even in a world of exact computation, unless the x coordinates are chosen very carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon. Based on Berrut and Trefethen 2004, "Barycentric Lagrange Interpolation". Examples -------- To produce a quintic barycentric interpolant approximating the function :math:`\sin x`, and its first four derivatives, using six randomly-spaced nodes in :math:`(0, \frac{\pi}{2})`: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.interpolate import BarycentricInterpolator >>> rng = np.random.default_rng() >>> xi = rng.random(6) * np.pi/2 >>> f, f_d1, f_d2, f_d3, f_d4 = np.sin, np.cos, lambda x: -np.sin(x), lambda x: -np.cos(x), np.sin >>> P = BarycentricInterpolator(xi, f(xi), random_state=rng) >>> fig, axs = plt.subplots(5, 1, sharex=True, layout='constrained', figsize=(7,10)) >>> x = np.linspace(0, np.pi, 100) >>> axs[0].plot(x, P(x), 'r:', x, f(x), 'k--', xi, f(xi), 'xk') >>> axs[1].plot(x, P.derivative(x), 'r:', x, f_d1(x), 'k--', xi, f_d1(xi), 'xk') >>> axs[2].plot(x, P.derivative(x, 2), 'r:', x, f_d2(x), 'k--', xi, f_d2(xi), 'xk') >>> axs[3].plot(x, P.derivative(x, 3), 'r:', x, f_d3(x), 'k--', xi, f_d3(xi), 'xk') >>> axs[4].plot(x, P.derivative(x, 4), 'r:', x, f_d4(x), 'k--', xi, f_d4(xi), 'xk') >>> axs[0].set_xlim(0, np.pi) >>> axs[4].set_xlabel(r"$x$") >>> axs[4].set_xticks([i * np.pi / 4 for i in range(5)], ... ["0", r"$\frac{\pi}{4}$", r"$\frac{\pi}{2}$", r"$\frac{3\pi}{4}$", r"$\pi$"]) >>> axs[0].set_ylabel("$f(x)$") >>> axs[1].set_ylabel("$f'(x)$") >>> axs[2].set_ylabel("$f''(x)$") >>> axs[3].set_ylabel("$f^{(3)}(x)$") >>> axs[4].set_ylabel("$f^{(4)}(x)$") >>> labels = ['Interpolation nodes', 'True function $f$', 'Barycentric interpolation'] >>> axs[0].legend(axs[0].get_lines()[::-1], labels, bbox_to_anchor=(0., 1.02, 1., .102), ... loc='lower left', ncols=3, mode="expand", borderaxespad=0., frameon=False) >>> plt.show() """ # numpy/numpydoc#87 # noqa: E501 @_transition_to_rng("random_state", replace_doc=False) def __init__(self, xi, yi=None, axis=0, *, wi=None, rng=None): super().__init__(xi, yi, axis) rng = check_random_state(rng) self.xi = np.asarray(xi, dtype=np.float64) self.set_yi(yi) self.n = len(self.xi) # cache derivative object to avoid re-computing the weights with every call. self._diff_cij = None if wi is not None: self.wi = wi else: # See page 510 of Berrut and Trefethen 2004 for an explanation of the # capacity scaling and the suggestion of using a random permutation of # the input factors. # At the moment, the permutation is not performed for xi that are # appended later through the add_xi interface. It's not clear to me how # to implement that and it seems that most situations that require # these numerical stability improvements will be able to provide all # the points to the constructor. self._inv_capacity = 4.0 / (np.max(self.xi) - np.min(self.xi)) permute = rng.permutation(self.n, ) inv_permute = np.zeros(self.n, dtype=np.int32) inv_permute[permute] = np.arange(self.n) self.wi = np.zeros(self.n) for i in range(self.n): dist = self._inv_capacity * (self.xi[i] - self.xi[permute]) dist[inv_permute[i]] = 1.0 prod = np.prod(dist) if prod == 0.0: raise ValueError("Interpolation points xi must be" " distinct.") self.wi[i] = 1.0 / prod def set_yi(self, yi, axis=None): """ Update the y values to be interpolated The barycentric interpolation algorithm requires the calculation of weights, but these depend only on the `xi`. The `yi` can be changed at any time. Parameters ---------- yi : array_like The y-coordinates of the points the polynomial will pass through. If None, the y values must be supplied later. axis : int, optional Axis in the `yi` array corresponding to the x-coordinate values. """ if yi is None: self.yi = None return self._set_yi(yi, xi=self.xi, axis=axis) self.yi = self._reshape_yi(yi) self.n, self.r = self.yi.shape self._diff_baryint = None def add_xi(self, xi, yi=None): """ Add more x values to the set to be interpolated The barycentric interpolation algorithm allows easy updating by adding more points for the polynomial to pass through. Parameters ---------- xi : array_like The x coordinates of the points that the polynomial should pass through. yi : array_like, optional The y coordinates of the points the polynomial should pass through. Should have shape ``(xi.size, R)``; if R > 1 then the polynomial is vector-valued. If `yi` is not given, the y values will be supplied later. `yi` should be given if and only if the interpolator has y values specified. Notes ----- The new points added by `add_xi` are not randomly permuted so there is potential for numerical instability, especially for a large number of points. If this happens, please reconstruct interpolation from scratch instead. """ if yi is not None: if self.yi is None: raise ValueError("No previous yi value to update!") yi = self._reshape_yi(yi, check=True) self.yi = np.vstack((self.yi,yi)) else: if self.yi is not None: raise ValueError("No update to yi provided!") old_n = self.n self.xi = np.concatenate((self.xi,xi)) self.n = len(self.xi) self.wi **= -1 old_wi = self.wi self.wi = np.zeros(self.n) self.wi[:old_n] = old_wi for j in range(old_n, self.n): self.wi[:j] *= self._inv_capacity * (self.xi[j]-self.xi[:j]) self.wi[j] = np.multiply.reduce( self._inv_capacity * (self.xi[:j]-self.xi[j]) ) self.wi **= -1 self._diff_cij = None self._diff_baryint = None def __call__(self, x): """Evaluate the interpolating polynomial at the points x Parameters ---------- x : array_like Point or points at which to evaluate the interpolant. Returns ------- y : array_like Interpolated values. Shape is determined by replacing the interpolation axis in the original array with the shape of `x`. Notes ----- Currently the code computes an outer product between `x` and the weights, that is, it constructs an intermediate array of size ``(N, len(x))``, where N is the degree of the polynomial. """ return _Interpolator1D.__call__(self, x) def _evaluate(self, x): if x.size == 0: p = np.zeros((0, self.r), dtype=self.dtype) else: c = x[..., np.newaxis] - self.xi z = c == 0 c[z] = 1 c = self.wi / c with np.errstate(divide='ignore'): p = np.dot(c, self.yi) / np.sum(c, axis=-1)[..., np.newaxis] # Now fix where x==some xi r = np.nonzero(z) if len(r) == 1: # evaluation at a scalar if len(r[0]) > 0: # equals one of the points p = self.yi[r[0][0]] else: p[r[:-1]] = self.yi[r[-1]] return p def derivative(self, x, der=1): """ Evaluate a single derivative of the polynomial at the point x. Parameters ---------- x : array_like Point or points at which to evaluate the derivatives der : integer, optional Which derivative to evaluate (default: first derivative). This number includes the function value as 0th derivative. Returns ------- d : ndarray Derivative interpolated at the x-points. Shape of `d` is determined by replacing the interpolation axis in the original array with the shape of `x`. """ x, x_shape = self._prepare_x(x) y = self._evaluate_derivatives(x, der+1, all_lower=False) return self._finish_y(y, x_shape) def _evaluate_derivatives(self, x, der=None, all_lower=True): # NB: der here is not the order of the highest derivative; # instead, it is the size of the derivatives matrix that # would be returned with all_lower=True, including the # '0th' derivative (the undifferentiated function). # E.g. to evaluate the 5th derivative alone, call # _evaluate_derivatives(x, der=6, all_lower=False). if (not all_lower) and (x.size == 0 or self.r == 0): return np.zeros((0, self.r), dtype=self.dtype) if (not all_lower) and der == 1: return self._evaluate(x) if (not all_lower) and (der > self.n): return np.zeros((len(x), self.r), dtype=self.dtype) if der is None: der = self.n if all_lower and (x.size == 0 or self.r == 0): return np.zeros((der, len(x), self.r), dtype=self.dtype) if self._diff_cij is None: # c[i,j] = xi[i] - xi[j] c = self.xi[:, np.newaxis] - self.xi # avoid division by 0 (diagonal entries are so far zero by construction) np.fill_diagonal(c, 1) # c[i,j] = (w[j] / w[i]) / (xi[i] - xi[j]) (equation 9.4) c = self.wi/ (c * self.wi[..., np.newaxis]) # fill in correct diagonal entries: each column sums to 0 np.fill_diagonal(c, 0) # calculate diagonal # c[j,j] = -sum_{i != j} c[i,j] (equation 9.5) d = -c.sum(axis=1) # c[i,j] = l_j(x_i) np.fill_diagonal(c, d) self._diff_cij = c if self._diff_baryint is None: # initialise and cache derivative interpolator and cijs; # reuse weights wi (which depend only on interpolation points xi), # to avoid unnecessary re-computation self._diff_baryint = BarycentricInterpolator(xi=self.xi, yi=self._diff_cij @ self.yi, wi=self.wi) self._diff_baryint._diff_cij = self._diff_cij if all_lower: # assemble matrix of derivatives from order 0 to order der-1, # in the format required by _Interpolator1DWithDerivatives. cn = np.zeros((der, len(x), self.r), dtype=self.dtype) for d in range(der): cn[d, :, :] = self._evaluate_derivatives(x, d+1, all_lower=False) return cn # recursively evaluate only the derivative requested return self._diff_baryint._evaluate_derivatives(x, der-1, all_lower=False) def barycentric_interpolate(xi, yi, x, axis=0, *, der=0, rng=None): """ Convenience function for polynomial interpolation. Constructs a polynomial that passes through a given set of points, then evaluates the polynomial. For reasons of numerical stability, this function does not compute the coefficients of the polynomial. This function uses a "barycentric interpolation" method that treats the problem as a special case of rational function interpolation. This algorithm is quite stable, numerically, but even in a world of exact computation, unless the `x` coordinates are chosen very carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon. Parameters ---------- xi : array_like 1-D array of x coordinates of the points the polynomial should pass through yi : array_like The y coordinates of the points the polynomial should pass through. x : scalar or array_like Point or points at which to evaluate the interpolant. axis : int, optional Axis in the `yi` array corresponding to the x-coordinate values. der : int or list or None, optional How many derivatives to evaluate, or None for all potentially nonzero derivatives (that is, a number equal to the number of points), or a list of derivatives to evaluate. This number includes the function value as the '0th' derivative. rng : `numpy.random.Generator`, optional Pseudorandom number generator state. When `rng` is None, a new `numpy.random.Generator` is created using entropy from the operating system. Types other than `numpy.random.Generator` are passed to `numpy.random.default_rng` to instantiate a ``Generator``. Returns ------- y : scalar or array_like Interpolated values. Shape is determined by replacing the interpolation axis in the original array with the shape of `x`. See Also -------- BarycentricInterpolator : Barycentric interpolator Notes ----- Construction of the interpolation weights is a relatively slow process. If you want to call this many times with the same xi (but possibly varying yi or x) you should use the class `BarycentricInterpolator`. This is what this function uses internally. Examples -------- We can interpolate 2D observed data using barycentric interpolation: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.interpolate import barycentric_interpolate >>> x_observed = np.linspace(0.0, 10.0, 11) >>> y_observed = np.sin(x_observed) >>> x = np.linspace(min(x_observed), max(x_observed), num=100) >>> y = barycentric_interpolate(x_observed, y_observed, x) >>> plt.plot(x_observed, y_observed, "o", label="observation") >>> plt.plot(x, y, label="barycentric interpolation") >>> plt.legend() >>> plt.show() """ P = BarycentricInterpolator(xi, yi, axis=axis, rng=rng) if der == 0: return P(x) elif _isscalar(der): return P.derivative(x, der=der) else: return P.derivatives(x, der=np.amax(der)+1)[der]