from __future__ import division, print_function, absolute_import import numpy as np from numpy.linalg import norm from numpy.testing import (assert_, assert_allclose, assert_equal) from scipy.linalg import polar, eigh diag2 = np.array([[2, 0], [0, 3]]) a13 = np.array([[1, 2, 2]]) precomputed_cases = [ [[[0]], 'right', [[1]], [[0]]], [[[0]], 'left', [[1]], [[0]]], [[[9]], 'right', [[1]], [[9]]], [[[9]], 'left', [[1]], [[9]]], [diag2, 'right', np.eye(2), diag2], [diag2, 'left', np.eye(2), diag2], [a13, 'right', a13/norm(a13[0]), a13.T.dot(a13)/norm(a13[0])], ] verify_cases = [ [[1, 2], [3, 4]], [[1, 2, 3]], [[1], [2], [3]], [[1, 2, 3], [3, 4, 0]], [[1, 2], [3, 4], [5, 5]], [[1, 2], [3, 4+5j]], [[1, 2, 3j]], [[1], [2], [3j]], [[1, 2, 3+2j], [3, 4-1j, -4j]], [[1, 2], [3-2j, 4+0.5j], [5, 5]], [[10000, 10, 1], [-1, 2, 3j], [0, 1, 2]], ] def check_precomputed_polar(a, side, expected_u, expected_p): # Compare the result of the polar decomposition to a # precomputed result. u, p = polar(a, side=side) assert_allclose(u, expected_u, atol=1e-15) assert_allclose(p, expected_p, atol=1e-15) def verify_polar(a): # Compute the polar decomposition, and then verify that # the result has all the expected properties. product_atol = np.sqrt(np.finfo(float).eps) aa = np.asarray(a) m, n = aa.shape u, p = polar(a, side='right') assert_equal(u.shape, (m, n)) assert_equal(p.shape, (n, n)) # a = up assert_allclose(u.dot(p), a, atol=product_atol) if m >= n: assert_allclose(u.conj().T.dot(u), np.eye(n), atol=1e-15) else: assert_allclose(u.dot(u.conj().T), np.eye(m), atol=1e-15) # p is Hermitian positive semidefinite. assert_allclose(p.conj().T, p) evals = eigh(p, eigvals_only=True) nonzero_evals = evals[abs(evals) > 1e-14] assert_((nonzero_evals >= 0).all()) u, p = polar(a, side='left') assert_equal(u.shape, (m, n)) assert_equal(p.shape, (m, m)) # a = pu assert_allclose(p.dot(u), a, atol=product_atol) if m >= n: assert_allclose(u.conj().T.dot(u), np.eye(n), atol=1e-15) else: assert_allclose(u.dot(u.conj().T), np.eye(m), atol=1e-15) # p is Hermitian positive semidefinite. assert_allclose(p.conj().T, p) evals = eigh(p, eigvals_only=True) nonzero_evals = evals[abs(evals) > 1e-14] assert_((nonzero_evals >= 0).all()) def test_precomputed_cases(): for a, side, expected_u, expected_p in precomputed_cases: check_precomputed_polar(a, side, expected_u, expected_p) def test_verify_cases(): for a in verify_cases: verify_polar(a)