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| #!/usr/bin/python | |
| # -*- coding: utf-8 -*- | |
| from mpmath import mp | |
| from mpmath import libmp | |
| xrange = libmp.backend.xrange | |
| def run_eigsy(A, verbose = False): | |
| if verbose: | |
| print("original matrix:\n", str(A)) | |
| D, Q = mp.eigsy(A) | |
| B = Q * mp.diag(D) * Q.transpose() | |
| C = A - B | |
| E = Q * Q.transpose() - mp.eye(A.rows) | |
| if verbose: | |
| print("eigenvalues:\n", D) | |
| print("eigenvectors:\n", Q) | |
| NC = mp.mnorm(C) | |
| NE = mp.mnorm(E) | |
| if verbose: | |
| print("difference:", NC, "\n", C, "\n") | |
| print("difference:", NE, "\n", E, "\n") | |
| eps = mp.exp( 0.8 * mp.log(mp.eps)) | |
| assert NC < eps | |
| assert NE < eps | |
| return NC | |
| def run_eighe(A, verbose = False): | |
| if verbose: | |
| print("original matrix:\n", str(A)) | |
| D, Q = mp.eighe(A) | |
| B = Q * mp.diag(D) * Q.transpose_conj() | |
| C = A - B | |
| E = Q * Q.transpose_conj() - mp.eye(A.rows) | |
| if verbose: | |
| print("eigenvalues:\n", D) | |
| print("eigenvectors:\n", Q) | |
| NC = mp.mnorm(C) | |
| NE = mp.mnorm(E) | |
| if verbose: | |
| print("difference:", NC, "\n", C, "\n") | |
| print("difference:", NE, "\n", E, "\n") | |
| eps = mp.exp( 0.8 * mp.log(mp.eps)) | |
| assert NC < eps | |
| assert NE < eps | |
| return NC | |
| def run_svd_r(A, full_matrices = False, verbose = True): | |
| m, n = A.rows, A.cols | |
| eps = mp.exp(0.8 * mp.log(mp.eps)) | |
| if verbose: | |
| print("original matrix:\n", str(A)) | |
| print("full", full_matrices) | |
| U, S0, V = mp.svd_r(A, full_matrices = full_matrices) | |
| S = mp.zeros(U.cols, V.rows) | |
| for j in xrange(min(m, n)): | |
| S[j,j] = S0[j] | |
| if verbose: | |
| print("U:\n", str(U)) | |
| print("S:\n", str(S0)) | |
| print("V:\n", str(V)) | |
| C = U * S * V - A | |
| err = mp.mnorm(C) | |
| if verbose: | |
| print("C\n", str(C), "\n", err) | |
| assert err < eps | |
| D = V * V.transpose() - mp.eye(V.rows) | |
| err = mp.mnorm(D) | |
| if verbose: | |
| print("D:\n", str(D), "\n", err) | |
| assert err < eps | |
| E = U.transpose() * U - mp.eye(U.cols) | |
| err = mp.mnorm(E) | |
| if verbose: | |
| print("E:\n", str(E), "\n", err) | |
| assert err < eps | |
| def run_svd_c(A, full_matrices = False, verbose = True): | |
| m, n = A.rows, A.cols | |
| eps = mp.exp(0.8 * mp.log(mp.eps)) | |
| if verbose: | |
| print("original matrix:\n", str(A)) | |
| print("full", full_matrices) | |
| U, S0, V = mp.svd_c(A, full_matrices = full_matrices) | |
| S = mp.zeros(U.cols, V.rows) | |
| for j in xrange(min(m, n)): | |
| S[j,j] = S0[j] | |
| if verbose: | |
| print("U:\n", str(U)) | |
| print("S:\n", str(S0)) | |
| print("V:\n", str(V)) | |
| C = U * S * V - A | |
| err = mp.mnorm(C) | |
| if verbose: | |
| print("C\n", str(C), "\n", err) | |
| assert err < eps | |
| D = V * V.transpose_conj() - mp.eye(V.rows) | |
| err = mp.mnorm(D) | |
| if verbose: | |
| print("D:\n", str(D), "\n", err) | |
| assert err < eps | |
| E = U.transpose_conj() * U - mp.eye(U.cols) | |
| err = mp.mnorm(E) | |
| if verbose: | |
| print("E:\n", str(E), "\n", err) | |
| assert err < eps | |
| def run_gauss(qtype, a, b): | |
| eps = 1e-5 | |
| d, e = mp.gauss_quadrature(len(a), qtype) | |
| d -= mp.matrix(a) | |
| e -= mp.matrix(b) | |
| assert mp.mnorm(d) < eps | |
| assert mp.mnorm(e) < eps | |
| def irandmatrix(n, range = 10): | |
| """ | |
| random matrix with integer entries | |
| """ | |
| A = mp.matrix(n, n) | |
| for i in xrange(n): | |
| for j in xrange(n): | |
| A[i,j]=int( (2 * mp.rand() - 1) * range) | |
| return A | |
| ####################### | |
| def test_eighe_fixed_matrix(): | |
| A = mp.matrix([[2, 3], [3, 5]]) | |
| run_eigsy(A) | |
| run_eighe(A) | |
| A = mp.matrix([[7, -11], [-11, 13]]) | |
| run_eigsy(A) | |
| run_eighe(A) | |
| A = mp.matrix([[2, 11, 7], [11, 3, 13], [7, 13, 5]]) | |
| run_eigsy(A) | |
| run_eighe(A) | |
| A = mp.matrix([[2, 0, 7], [0, 3, 1], [7, 1, 5]]) | |
| run_eigsy(A) | |
| run_eighe(A) | |
| # | |
| A = mp.matrix([[2, 3+7j], [3-7j, 5]]) | |
| run_eighe(A) | |
| A = mp.matrix([[2, -11j, 0], [+11j, 3, 29j], [0, -29j, 5]]) | |
| run_eighe(A) | |
| A = mp.matrix([[2, 11 + 17j, 7 + 19j], [11 - 17j, 3, -13 + 23j], [7 - 19j, -13 - 23j, 5]]) | |
| run_eighe(A) | |
| def test_eigsy_randmatrix(): | |
| N = 5 | |
| for a in xrange(10): | |
| A = 2 * mp.randmatrix(N, N) - 1 | |
| for i in xrange(0, N): | |
| for j in xrange(i + 1, N): | |
| A[j,i] = A[i,j] | |
| run_eigsy(A) | |
| def test_eighe_randmatrix(): | |
| N = 5 | |
| for a in xrange(10): | |
| A = (2 * mp.randmatrix(N, N) - 1) + 1j * (2 * mp.randmatrix(N, N) - 1) | |
| for i in xrange(0, N): | |
| A[i,i] = mp.re(A[i,i]) | |
| for j in xrange(i + 1, N): | |
| A[j,i] = mp.conj(A[i,j]) | |
| run_eighe(A) | |
| def test_eigsy_irandmatrix(): | |
| N = 4 | |
| R = 4 | |
| for a in xrange(10): | |
| A=irandmatrix(N, R) | |
| for i in xrange(0, N): | |
| for j in xrange(i + 1, N): | |
| A[j,i] = A[i,j] | |
| run_eigsy(A) | |
| def test_eighe_irandmatrix(): | |
| N = 4 | |
| R = 4 | |
| for a in xrange(10): | |
| A=irandmatrix(N, R) + 1j * irandmatrix(N, R) | |
| for i in xrange(0, N): | |
| A[i,i] = mp.re(A[i,i]) | |
| for j in xrange(i + 1, N): | |
| A[j,i] = mp.conj(A[i,j]) | |
| run_eighe(A) | |
| def test_svd_r_rand(): | |
| for i in xrange(5): | |
| full = mp.rand() > 0.5 | |
| m = 1 + int(mp.rand() * 10) | |
| n = 1 + int(mp.rand() * 10) | |
| A = 2 * mp.randmatrix(m, n) - 1 | |
| if mp.rand() > 0.5: | |
| A *= 10 | |
| for x in xrange(m): | |
| for y in xrange(n): | |
| A[x,y]=int(A[x,y]) | |
| run_svd_r(A, full_matrices = full, verbose = False) | |
| def test_svd_c_rand(): | |
| for i in xrange(5): | |
| full = mp.rand() > 0.5 | |
| m = 1 + int(mp.rand() * 10) | |
| n = 1 + int(mp.rand() * 10) | |
| A = (2 * mp.randmatrix(m, n) - 1) + 1j * (2 * mp.randmatrix(m, n) - 1) | |
| if mp.rand() > 0.5: | |
| A *= 10 | |
| for x in xrange(m): | |
| for y in xrange(n): | |
| A[x,y]=int(mp.re(A[x,y])) + 1j * int(mp.im(A[x,y])) | |
| run_svd_c(A, full_matrices=full, verbose=False) | |
| def test_svd_test_case(): | |
| # a test case from Golub and Reinsch | |
| # (see wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).) | |
| eps = mp.exp(0.8 * mp.log(mp.eps)) | |
| a = [[22, 10, 2, 3, 7], | |
| [14, 7, 10, 0, 8], | |
| [-1, 13, -1, -11, 3], | |
| [-3, -2, 13, -2, 4], | |
| [ 9, 8, 1, -2, 4], | |
| [ 9, 1, -7, 5, -1], | |
| [ 2, -6, 6, 5, 1], | |
| [ 4, 5, 0, -2, 2]] | |
| a = mp.matrix(a) | |
| b = mp.matrix([mp.sqrt(1248), 20, mp.sqrt(384), 0, 0]) | |
| S = mp.svd_r(a, compute_uv = False) | |
| S -= b | |
| assert mp.mnorm(S) < eps | |
| S = mp.svd_c(a, compute_uv = False) | |
| S -= b | |
| assert mp.mnorm(S) < eps | |
| def test_gauss_quadrature_static(): | |
| a = [-0.57735027, 0.57735027] | |
| b = [ 1, 1] | |
| run_gauss("legendre", a , b) | |
| a = [ -0.906179846, -0.538469310, 0, 0.538469310, 0.906179846] | |
| b = [ 0.23692689, 0.47862867, 0.56888889, 0.47862867, 0.23692689] | |
| run_gauss("legendre", a , b) | |
| a = [ 0.06943184, 0.33000948, 0.66999052, 0.93056816] | |
| b = [ 0.17392742, 0.32607258, 0.32607258, 0.17392742] | |
| run_gauss("legendre01", a , b) | |
| a = [-0.70710678, 0.70710678] | |
| b = [ 0.88622693, 0.88622693] | |
| run_gauss("hermite", a , b) | |
| a = [ -2.02018287, -0.958572465, 0, 0.958572465, 2.02018287] | |
| b = [ 0.01995324, 0.39361932, 0.94530872, 0.39361932, 0.01995324] | |
| run_gauss("hermite", a , b) | |
| a = [ 0.41577456, 2.29428036, 6.28994508] | |
| b = [ 0.71109301, 0.27851773, 0.01038926] | |
| run_gauss("laguerre", a , b) | |
| def test_gauss_quadrature_dynamic(verbose = False): | |
| n = 5 | |
| A = mp.randmatrix(2 * n, 1) | |
| def F(x): | |
| r = 0 | |
| for i in xrange(len(A) - 1, -1, -1): | |
| r = r * x + A[i] | |
| return r | |
| def run(qtype, FW, R, alpha = 0, beta = 0): | |
| X, W = mp.gauss_quadrature(n, qtype, alpha = alpha, beta = beta) | |
| a = 0 | |
| for i in xrange(len(X)): | |
| a += W[i] * F(X[i]) | |
| b = mp.quad(lambda x: FW(x) * F(x), R) | |
| c = mp.fabs(a - b) | |
| if verbose: | |
| print(qtype, c, a, b) | |
| assert c < 1e-5 | |
| run("legendre", lambda x: 1, [-1, 1]) | |
| run("legendre01", lambda x: 1, [0, 1]) | |
| run("hermite", lambda x: mp.exp(-x*x), [-mp.inf, mp.inf]) | |
| run("laguerre", lambda x: mp.exp(-x), [0, mp.inf]) | |
| run("glaguerre", lambda x: mp.sqrt(x)*mp.exp(-x), [0, mp.inf], alpha = 1 / mp.mpf(2)) | |
| run("chebyshev1", lambda x: 1/mp.sqrt(1-x*x), [-1, 1]) | |
| run("chebyshev2", lambda x: mp.sqrt(1-x*x), [-1, 1]) | |
| run("jacobi", lambda x: (1-x)**(1/mp.mpf(3)) * (1+x)**(1/mp.mpf(5)), [-1, 1], alpha = 1 / mp.mpf(3), beta = 1 / mp.mpf(5) ) | |