EVOLUTION-MANAGER

Edit File: polyint.py

from __future__ import division, print_function, absolute_import import warnings import numpy as np from scipy.special import factorial from scipy._lib.six import xrange from scipy._lib._util import _asarray_validated __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(object): """ 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 Points to evaluate the interpolant at. Returns ------- y : array_like Interpolated values. Shape 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(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 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.rollaxis(np.asarray(yi), self._y_axis) if check and yi.shape[1:] != self._y_extra_shape: ok_shape = "%r + (N,) + %r" % (self._y_extra_shape[-self._y_axis:], self._y_extra_shape[:-self._y_axis]) raise ValueError("Data must be of shape %s" % 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.complex_ else: if not union or self.dtype != np.complex_: self.dtype = np.float_ class _Interpolator1DWithDerivatives(_Interpolator1D): def derivatives(self, x, der=None): """ Evaluate many derivatives of the polynomial at the point x Produce an array of all derivative values at the point x. Parameters ---------- x : array_like Point or points at which to evaluate the derivatives der : int or None, optional How many derivatives to extract; None for all potentially nonzero derivatives (that is a number equal to the number of points). This number includes the function value as 0th derivative. Returns ------- d : ndarray Array with derivatives; d[j] contains the j-th 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 one 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 extract. 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 is 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) 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, length N Known x-coordinates. Must be sorted in increasing order. yi : array_like Known y-coordinates. When an xi occurs two or more times in a row, the corresponding yi's represent derivative values. axis : int, optional Axis in the yi array corresponding to the x-coordinate values. 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 2*X**2-2*X. 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: >>> xi = np.linspace(0, 1, 5) >>> yi, ypi = np.random.rand(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): _Interpolator1DWithDerivatives.__init__(self, xi, yi, axis) self.xi = np.asarray(xi) self.yi = self._reshape_yi(yi) self.n, self.r = self.yi.shape 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 xrange(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 xrange(k-s): if xi[i] == xi[k]: raise ValueError("Elements if `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 xrange(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 xrange(1, n): for i in xrange(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] *= 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 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, optional How many derivatives to extract; None for all potentially nonzero derivatives (that is a number equal to the number of points), or a list of derivatives to extract. This number includes the function value as 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-dimensional 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 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). """ 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. """ 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(_Interpolator1D): """The interpolating polynomial for a set of points Constructs a polynomial that passes through a given set of points. Allows evaluation of the polynomial, efficient changing of the y values to be interpolated, and updating by adding more x 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 1-d array of x coordinates of the points the polynomial should pass through yi : array_like, optional The y coordinates of the points the polynomial should pass through. If None, the y values will be supplied later via the `set_y` method. axis : int, optional Axis in the yi array corresponding to the x-coordinate values. 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". """ def __init__(self, xi, yi=None, axis=0): _Interpolator1D.__init__(self, xi, yi, axis) self.xi = np.asarray(xi) self.set_yi(yi) self.n = len(self.xi) self.wi = np.zeros(self.n) self.wi[0] = 1 for j in xrange(1,self.n): self.wi[:j] *= (self.xi[j]-self.xi[:j]) self.wi[j] = np.multiply.reduce(self.xi[:j]-self.xi[j]) self.wi **= -1 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 should pass through. If None, the y values will 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 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. """ 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 xrange(old_n,self.n): self.wi[:j] *= (self.xi[j]-self.xi[:j]) self.wi[j] = np.multiply.reduce(self.xi[:j]-self.xi[j]) self.wi **= -1 def __call__(self, x): """Evaluate the interpolating polynomial at the points x Parameters ---------- x : array_like Points to evaluate the interpolant at. 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 by 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 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 barycentric_interpolate(xi, yi, x, axis=0): """ 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 Points to evaluate the interpolator at. axis : int, optional Axis in the yi array corresponding to the x-coordinate values. 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 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. """ return BarycentricInterpolator(xi, yi, axis=axis)(x)