''' All the software contained in this library is protected by copyright. Permission to use, copy, modify, and distribute this software for any purpose without fee is hereby granted, provided that this entire notice is included in all copies of any software which is or includes a copy or modification of this software and in all copies of the supporting documentation for such software. THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED WARRANTY. IN NO EVENT, NEITHER THE AUTHORS, NOR THE PUBLISHER, NOR ANY MEMBER OF THE EDITORIAL BOARD OF THE JOURNAL "NUMERICAL ALGORITHMS", NOR ITS EDITOR-IN-CHIEF, BE LIABLE FOR ANY ERROR IN THE SOFTWARE, ANY MISUSE OF IT OR ANY DAMAGE ARISING OUT OF ITS USE. THE ENTIRE RISK OF USING THE SOFTWARE LIES WITH THE PARTY DOING SO. ANY USE OF THE SOFTWARE CONSTITUTES ACCEPTANCE OF THE TERMS OF THE ABOVE STATEMENT. AUTHORS: Per A Brodtkorb Python code Based on matlab code written by: Marco Caliari University of Verona, Italy E-mail: marco.caliari@univr.it Stefano de Marchi, Alvise Sommariva, Marco Vianello University of Padua, Italy E-mail: demarchi@math.unipd.it, alvise@math.unipd.it, marcov@math.unipd.it Reference --------- Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave NUMERICAL ALGORITHMS, 56 (2011), PP. 45-60 Padua module ------------ In polynomial interpolation of two variables, the Padua points are the first known example (and up to now the only one) of a unisolvent point set (that is, the interpolating polynomial is unique) with minimal growth of their Lebesgue constant, proven to be O(log2 n). This module provides all the functions needed to perform interpolation and cubature at the Padua points, together with the functions and the demos used in the paper. pdint.m : main function for interpolation at the Padua points pdcub.m : main function for cubature at the Padua points pdpts.m : function for the computation of the Padua points padua_fit.m : function for the computation of the interpolation coefficients by FFT (recommended) pdcfsMM.m : function for the computation of the interpolation coefficients by matrix multiplications pdval.m : function for the evaluation of the interpolation polynomial pdwtsFFT.m : function for the computation of the cubature weights by FFT pdwtsMM.m : function for the computation of the cubature weights by matrix multiplications (recommended) funct.m : function containing some test functions demo_pdint.m : demo script for pdint demo_cputime_pdint.m : demo script for the computation of CPU time for interpolation demo_errors_pdint.m : demo script for the comparison of interpolation with coefficients computed by FFT or by matrix multiplications demo_pdcub : demo script for pdcub demo_cputime_pdcub.m : demo script for the computation of CPU time for cubature demo_errors_pdcub.m : demo script for the comparison of cubature with weights computed by FFT or by matrix multiplications demo_errors_pdcub_gl.m : demo script for the comparison of different cubature formulas cubature_square.m : function for the computation of some cubature formulas for the square omelyan_solovyan_rule.m : function for the computation of Omelyan-Solovyan cubature points and weights Contents.m : Contents file for Matlab ''' from __future__ import absolute_import, division import numpy as np from numpy.fft import fft from wafo.dctpack import dct from wafo.polynomial import map_from_interval, map_to_interval # from scipy.fftpack.realtransforms import dct class _ExampleFunctions(object): ''' Computes test function in the points (x, y) Parameters ---------- x,y : array-like evaluate the function in the points (x,y) i : scalar int (default 0) defining which test function to use. Options are 0: franke 1: half_sphere 2: poly_degree20 3: exp_fun1 4: exp_fun100 5: cos30 6: constant 7: exp_xy 8: runge 9: abs_cubed 10: gauss 11: exp_inv Returns ------- z : array-like value of the function in the points (x,y) ''' @staticmethod def franke(x, y): '''Franke function. The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 500, is 2.1547794245591083e+000 with an estimated absolute error of 8.88e-016. The value of the definite integral on the square [0,1] x [0,1], obtained using a Padua Points cubature formula of degree 500, is 4.06969589491556e-01 with an estimated absolute error of 8.88e-016. Maple: 0.40696958949155611906 ''' def _exp(x, y, loc, scale, p2=2): return np.exp(- (x - loc[0])**2 / scale[0] - (y - loc[1])**p2 / scale[1]) # exp = np.exp x9, y9 = 9. * x, 9. * y return (3. / 4 * _exp(x9, y9, [2, 2], [4, 4]) + 3. / 4 * _exp(x9, y9, [-1, -1], [49, 10], p2=1) + 1. / 2 * _exp(x9, y9, [7, 3], [4, 4]) - 1. / 5 * _exp(x9, y9, [4, 7], [1, 1])) @staticmethod def half_sphere(x, y): '''The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 2000, is 3.9129044444568244e+000 with an estimated absolute error of 3.22e-010. ''' return ((x - 0.5)**2 + (y - 0.5)**2)**(1. / 2) @staticmethod def poly_degree20(x, y): ''''Bivariate polynomial having moderate degree. The value of the definite integral on the square [-1,1] x [-1,1], up to machine precision, is 18157.16017316017 (see ref. 6). The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 500, is 1.8157160173160162e+004. 2D modification of an example by L.N.Trefethen (see ref. 7), f(x)=x^20. ''' return (x + y)**20 @staticmethod def exp_fun1(x, y): ''' The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 2000, is 2.1234596326670683e+001 with an estimated absolute error of 7.11e-015. ''' return np.exp((x - 0.5)**2 + (y - 0.5)**2) @staticmethod def exp_fun100(x, y): '''The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 2000, is 3.1415926535849605e-002 with an estimated absolute error of 3.47e-017. ''' return np.exp(-100 * ((x - 0.5)**2 + (y - 0.5)**2)) @staticmethod def cos30(x, y): ''' The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 500, is 4.3386955120336568e-003 with an estimated absolute error of 2.95e-017. ''' return np.cos(30 * (x + y)) @staticmethod def constant(x, y): '''Constant. To test interpolation and cubature at degree 0. The value of the definite integral on the square [-1,1] x [-1,1] is 4. ''' return np.ones(np.shape(x + y)) @staticmethod def exp_xy(x, y): '''The value of the definite integral on the square [-1,1] x [-1,1] is up to machine precision is 5.524391382167263 (see ref. 6). 2. The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 500, is 5.5243913821672628e+000 with an estimated absolute error of 0.00e+000. 2D modification of an example by L.N.Trefethen (see ref. 7), f(x)=exp(x). ''' return np.exp(x + y) @staticmethod def runge(x, y): ''' Bivariate Runge function: as 1D complex function is analytic in a neighborhood of [-1; 1] but not throughout the complex plane. The value of the definite integral on the square [-1,1] x [-1,1], up to machine precision, is 0.597388947274307 (see ref. 6). The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 500, is 5.9738894727430725e-001 with an estimated absolute error of 0.00e+000. 2D modification of an example by L.N.Trefethen (see ref. 7), f(x)=1/(1+16*x^2). ''' return 1. / (1 + 16 * (x**2 + y**2)) @staticmethod def abs_cubed(x, y): '''Low regular function. The value of the definite integral on the square [-1,1] x [-1,1], up to machine precision, is 2.508723139534059 (see ref. 6). The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 500, is 2.5087231395340579e+000 with an estimated absolute error of 0.00e+000. 2D modification of an example by L.N.Trefethen (see ref. 7), f(x)=abs(x)^3. ''' return (x**2 + y**2)**(3 / 2) @staticmethod def gauss(x, y): '''Bivariate gaussian: smooth function. The value of the definite integral on the square [-1,1] x [-1,1], up to machine precision, is 2.230985141404135 (see ref. 6). The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 500, is 2.2309851414041333e+000 with an estimated absolute error of 2.66e-015. 2D modification of an example by L.N.Trefethen (see ref. 7), f(x)=exp(-x^2). ''' return np.exp(-x**2 - y**2) @staticmethod def exp_inv(x, y): '''Bivariate example stemming from a 1D C-infinity function. The value of the definite integral on the square [-1,1] x [-1,1], up to machine precision, is 0.853358758654305 (see ref. 6). The value of the definite integral on the square [-1,1] x [-1,1], obtained using a Padua Points cubature formula of degree 2000, is 8.5335875865430544e-001 with an estimated absolute error of 3.11e-015. 2D modification of an example by L.N.Trefethen (see ref. 7), f(x)=exp(-1/x^2). ''' arg_z = (x**2 + y**2) # Avoid cases in which "arg_z=0", setting only in those instances # "arg_z=eps". arg_z = arg_z + (1 - np.abs(np.sign(arg_z))) * 1.e-100 arg_z = 1. / arg_z return np.exp(-arg_z) def __call__(self, x, y, i=0): s = self test_function = [s.franke, s.half_sphere, s.poly_degree20, s.exp_fun1, s.exp_fun100, s.cos30, s.constant, s.exp_xy, s.runge, s.abs_cubed, s.gauss, s.exp_inv] return test_function[i](x, y) example_functions = _ExampleFunctions() def _find_m(n): ix = np.r_[1:(n + 1) * (n + 2):2] if np.mod(n, 2) == 0: n2 = n // 2 offset = np.array([[0, 1] * n2 + [0, ]] * (n2 + 1)) ix = ix - offset.ravel(order='F') return ix def padua_points(n, domain=(-1, 1, -1, 1)): ''' Return Padua points Parameters ---------- n : scalar integer interpolation degree domain : vector [a,b,c,d] defining the rectangle [a,b] x [c,d]. (default domain = (-1,1,-1,1)) Returns ------- pad : array of shape (2 x (n+1)*(n+2)/2) such that (pad[0,:], pad[1,: ]) defines the Padua points in the domain rectangle [a,b] x [c,d]. or X1,Y1,X2,Y2 : arrays Two subgrids X1,Y1 and X2,Y2 defining the Padua points ------------------------------------------------------------------------------- ''' a, b, c, d = domain t0 = [np.pi] if n == 0 else np.linspace(0, np.pi, n + 1) t1 = np.linspace(0, np.pi, n + 2) zn = map_to_interval(np.cos(t0), a, b) zn1 = map_to_interval(np.cos(t1), c, d) Pad1, Pad2 = np.meshgrid(zn, zn1) ix = _find_m(n) return np.vstack((Pad1.ravel(order='F')[ix], Pad2.ravel(order='F')[ix])) def error_estimate(C0f): ''' Return interpolation error estimate from Padua coefficients ''' n = C0f.shape[1] C0f2 = np.fliplr(C0f) errest = sum(np.abs(np.diag(C0f2))) if (n >= 1): errest = errest + sum(np.abs(np.diag(C0f2, -1))) if (n >= 2): errest = errest + sum(np.abs(np.diag(C0f2, -2))) return 2 * errest def padua_fit(Pad, fun, *args): ''' Computes the Chebyshevs coefficients so that f(x, y) can be approximated by: f(x, y) = sum cjk*Tjk(x, y) Parameters ---------- Pad : array-like Padua points, as computed with padua_points function. fun : function to be interpolated in the form fun(x, y, *args), where *args are optional arguments for fun. Returns ------- coefficents: coefficient matrix abs_err : interpolation error estimate Example ------ >>> import numpy as np >>> import wafo.padua as wp >>> domain = [0, 1, 0, 1] >>> a, b, c, d = domain >>> points = wp.padua_points(21, domain) >>> C0f, abs_error = wp.padua_fit(points, wp.example_functions.franke) >>> x1 = np.linspace(a, b, 100) >>> x2 = np.linspace(c, d, 101) >>> val = wp.padua_val(x1, x2, C0f, domain, use_meshgrid=True) >>> X1, X2 = np.meshgrid(x1, x2) >>> true_val = wp.example_functions.franke(X1, X2) >>> np.allclose(val, true_val, atol=10*abs_error) True >>> np.allclose(np.abs(val-true_val).max(), 0.0073174614275738296) True >>> np.allclose(abs_error, 0.0022486904061664046) True import matplotlib.pyplot as plt plt.contour(x1, x2, val) ''' N = np.shape(Pad)[1] # recover the degree n from N = (n+1)(n+2)/2 n = int(round(-3 + np.sqrt(1 + 8 * N)) / 2) C0f = fun(Pad[0], Pad[1], *args) if (n > 0): ix = _find_m(n) GfT = np.zeros((n + 2) * (n + 1)) GfT[ix] = C0f * 2 / (n * (n + 1)) GfT = GfT.reshape(n + 1, n + 2) GfT = GfT.T GfT[0] = GfT[0] / 2 GfT[n + 1] = GfT[n + 1] / 2 GfT[:, 0] = GfT[:, 0] / 2 GfT[:, n] = GfT[:, n] / 2 Gf = GfT.T # compute the interpolation coefficient matrix C0f by FFT Gfhat = np.real(fft(Gf, 2 * n, axis=0)) Gfhathat = np.real(fft(Gfhat[:n + 1, :], 2 * (n + 1), axis=1)) C0f = 2 * Gfhathat[:, 0:n + 1] C0f[0] = C0f[0, :] / np.sqrt(2) C0f[:, 0] = C0f[:, 0] / np.sqrt(2) C0f = np.fliplr(np.triu(np.fliplr(C0f))) C0f[n] = C0f[n] / 2 return C0f, error_estimate(C0f) def paduavals2coefs(f): m = len(f) n = int(round(-1.5 + np.sqrt(.25 + 2 * m))) x = padua_points(n) idx = _find_m(n) w = 0 * x[0] + 1. / (n * (n + 1)) idx1 = np.all(np.abs(x) == 1, axis=0) w[idx1] = .5 * w[idx1] idx2 = np.all(np.abs(x) != 1, axis=0) w[idx2] = 2 * w[idx2] G = np.zeros(idx.max() + 1) G[idx] = 4 * w * f use_dct = 100 < n if use_dct: C = np.rot90(dct(dct(G.T).T)) # , axis=1) else: t1 = np.r_[0:n + 1].reshape(-1, 1) Tn1 = np.cos(t1 * t1.T * np.pi / n) t2 = np.r_[0:n + 2].reshape(-1, 1) Tn2 = np.cos(t2 * t2.T * np.pi / (n + 1)) C = np.dot(Tn2, np.dot(G, Tn1)) C[0] = .5 * C[0] C[:, 1] = .5 * C[:, 1] C[0, -1] = .5 * C[0, -1] del C[-1] # Take upper-left triangular part: return np.fliplr(np.triu(np.fliplr(C))) # C = triu(C(:,end:-1:1)); # C = C(:,end:-1:1); # TODO: padua_fit2 does not work correctly yet. def padua_fit2(Pad, fun, *args): # N = np.shape(Pad)[1] # recover the degree n from N = (n+1)(n+2)/2 # _n = int(round(-3 + np.sqrt(1 + 8 * N)) / 2) C0f = fun(Pad[0], Pad[1], *args) return paduavals2coefs(C0f) def _compute_moments(n): k = np.r_[0:n:2] mom = 2 * np.sqrt(2) / (1 - k ** 2) mom[0] = 2 return mom def padua_cubature(coefficients, domain=(-1, 1, -1, 1)): ''' Compute the integral through the coefficient matrix. ''' n = coefficients.shape[1] mom = _compute_moments(n) M1, M2 = np.meshgrid(mom, mom) M = M1 * M2 C0fM = coefficients[0:n:2, 0:n:2] * M a, b, c, d = domain integral = (b - a) * (d - c) * C0fM.sum() / 4 return integral def padua_val(X, Y, coefficients, domain=(-1, 1, -1, 1), use_meshgrid=False): ''' Evaluate polynomial in padua form at X, Y. Evaluate the interpolation polynomial defined through its coefficient matrix coefficients at the target points X(:,1),X(:,2) or at the meshgrid(X1,X2) Parameters ---------- X, Y: array-like evaluation points. coefficients : array-like coefficient matrix domain : a vector [a,b,c,d] defining the rectangle [a,b] x [c,d] use_meshgrid: bool If True interpolate at the points meshgrid(X, Y) Returns ------- fxy : array-like evaluation of the interpolation polynomial at the target points ''' def transform(tn, x, a, b): xn = map_from_interval(x, a, b).clip(min=-1, max=1).reshape(1, -1) tx = np.cos(tn * np.arccos(xn)) * np.sqrt(2) tx[0] = 1 return tx X, Y = np.atleast_1d(X, Y) original_shape = X.shape a, b, c, d = domain n = np.shape(coefficients)[1] tn = np.r_[0:n][:, None] tx1 = transform(tn, X.ravel(), a, b) tx2 = transform(tn, Y.ravel(), c, d) if use_meshgrid: # eval on meshgrid points return np.dot(tx1.T, np.dot(coefficients, tx2)).T # scattered points val = np.sum(np.dot(tx1.T, coefficients) * tx2.T, axis=1) return val.reshape(original_shape) if __name__ == '__main__': from wafo.testing import test_docstrings test_docstrings(__file__)