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