896 lines
34 KiB
Python
896 lines
34 KiB
Python
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#
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# Created by: Pearu Peterson, March 2002
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#
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""" Test functions for linalg.matfuncs module
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"""
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import random
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import functools
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import numpy as np
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from numpy import array, identity, dot, sqrt
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from numpy.testing import (
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assert_array_equal, assert_array_less, assert_equal,
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assert_array_almost_equal,
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assert_allclose, assert_, assert_warns)
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import pytest
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import scipy.linalg
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from scipy.linalg import (funm, signm, logm, sqrtm, fractional_matrix_power,
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expm, expm_frechet, expm_cond, norm, khatri_rao)
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from scipy.linalg import _matfuncs_inv_ssq
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import scipy.linalg._expm_frechet
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from scipy.optimize import minimize
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def _get_al_mohy_higham_2012_experiment_1():
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"""
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Return the test matrix from Experiment (1) of [1]_.
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References
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----------
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.. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012)
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"Improved Inverse Scaling and Squaring Algorithms
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for the Matrix Logarithm."
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SIAM Journal on Scientific Computing, 34 (4). C152-C169.
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ISSN 1095-7197
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"""
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A = np.array([
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[3.2346e-1, 3e4, 3e4, 3e4],
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[0, 3.0089e-1, 3e4, 3e4],
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[0, 0, 3.2210e-1, 3e4],
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[0, 0, 0, 3.0744e-1]], dtype=float)
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return A
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class TestSignM(object):
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def test_nils(self):
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a = array([[29.2, -24.2, 69.5, 49.8, 7.],
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[-9.2, 5.2, -18., -16.8, -2.],
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[-10., 6., -20., -18., -2.],
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[-9.6, 9.6, -25.5, -15.4, -2.],
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[9.8, -4.8, 18., 18.2, 2.]])
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cr = array([[11.94933333,-2.24533333,15.31733333,21.65333333,-2.24533333],
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[-3.84266667,0.49866667,-4.59066667,-7.18666667,0.49866667],
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[-4.08,0.56,-4.92,-7.6,0.56],
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[-4.03466667,1.04266667,-5.59866667,-7.02666667,1.04266667],
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[4.15733333,-0.50133333,4.90933333,7.81333333,-0.50133333]])
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r = signm(a)
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assert_array_almost_equal(r,cr)
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def test_defective1(self):
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a = array([[0.0,1,0,0],[1,0,1,0],[0,0,0,1],[0,0,1,0]])
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signm(a, disp=False)
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#XXX: what would be the correct result?
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def test_defective2(self):
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a = array((
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[29.2,-24.2,69.5,49.8,7.0],
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[-9.2,5.2,-18.0,-16.8,-2.0],
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[-10.0,6.0,-20.0,-18.0,-2.0],
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[-9.6,9.6,-25.5,-15.4,-2.0],
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[9.8,-4.8,18.0,18.2,2.0]))
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signm(a, disp=False)
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#XXX: what would be the correct result?
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def test_defective3(self):
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a = array([[-2., 25., 0., 0., 0., 0., 0.],
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[0., -3., 10., 3., 3., 3., 0.],
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[0., 0., 2., 15., 3., 3., 0.],
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[0., 0., 0., 0., 15., 3., 0.],
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[0., 0., 0., 0., 3., 10., 0.],
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[0., 0., 0., 0., 0., -2., 25.],
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[0., 0., 0., 0., 0., 0., -3.]])
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signm(a, disp=False)
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#XXX: what would be the correct result?
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class TestLogM(object):
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def test_nils(self):
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a = array([[-2., 25., 0., 0., 0., 0., 0.],
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[0., -3., 10., 3., 3., 3., 0.],
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[0., 0., 2., 15., 3., 3., 0.],
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[0., 0., 0., 0., 15., 3., 0.],
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[0., 0., 0., 0., 3., 10., 0.],
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[0., 0., 0., 0., 0., -2., 25.],
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[0., 0., 0., 0., 0., 0., -3.]])
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m = (identity(7)*3.1+0j)-a
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logm(m, disp=False)
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#XXX: what would be the correct result?
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def test_al_mohy_higham_2012_experiment_1_logm(self):
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# The logm completes the round trip successfully.
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# Note that the expm leg of the round trip is badly conditioned.
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A = _get_al_mohy_higham_2012_experiment_1()
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A_logm, info = logm(A, disp=False)
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A_round_trip = expm(A_logm)
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assert_allclose(A_round_trip, A, rtol=1e-5, atol=1e-14)
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def test_al_mohy_higham_2012_experiment_1_funm_log(self):
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# The raw funm with np.log does not complete the round trip.
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# Note that the expm leg of the round trip is badly conditioned.
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A = _get_al_mohy_higham_2012_experiment_1()
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A_funm_log, info = funm(A, np.log, disp=False)
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A_round_trip = expm(A_funm_log)
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assert_(not np.allclose(A_round_trip, A, rtol=1e-5, atol=1e-14))
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def test_round_trip_random_float(self):
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np.random.seed(1234)
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for n in range(1, 6):
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M_unscaled = np.random.randn(n, n)
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for scale in np.logspace(-4, 4, 9):
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M = M_unscaled * scale
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# Eigenvalues are related to the branch cut.
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W = np.linalg.eigvals(M)
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err_msg = 'M:{0} eivals:{1}'.format(M, W)
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# Check sqrtm round trip because it is used within logm.
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M_sqrtm, info = sqrtm(M, disp=False)
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M_sqrtm_round_trip = M_sqrtm.dot(M_sqrtm)
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assert_allclose(M_sqrtm_round_trip, M)
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# Check logm round trip.
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M_logm, info = logm(M, disp=False)
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M_logm_round_trip = expm(M_logm)
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assert_allclose(M_logm_round_trip, M, err_msg=err_msg)
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def test_round_trip_random_complex(self):
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np.random.seed(1234)
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for n in range(1, 6):
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M_unscaled = np.random.randn(n, n) + 1j * np.random.randn(n, n)
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for scale in np.logspace(-4, 4, 9):
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M = M_unscaled * scale
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M_logm, info = logm(M, disp=False)
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M_round_trip = expm(M_logm)
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assert_allclose(M_round_trip, M)
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def test_logm_type_preservation_and_conversion(self):
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# The logm matrix function should preserve the type of a matrix
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# whose eigenvalues are positive with zero imaginary part.
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# Test this preservation for variously structured matrices.
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complex_dtype_chars = ('F', 'D', 'G')
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for matrix_as_list in (
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[[1, 0], [0, 1]],
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[[1, 0], [1, 1]],
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[[2, 1], [1, 1]],
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[[2, 3], [1, 2]]):
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# check that the spectrum has the expected properties
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W = scipy.linalg.eigvals(matrix_as_list)
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assert_(not any(w.imag or w.real < 0 for w in W))
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# check float type preservation
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A = np.array(matrix_as_list, dtype=float)
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A_logm, info = logm(A, disp=False)
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assert_(A_logm.dtype.char not in complex_dtype_chars)
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# check complex type preservation
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A = np.array(matrix_as_list, dtype=complex)
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A_logm, info = logm(A, disp=False)
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assert_(A_logm.dtype.char in complex_dtype_chars)
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# check float->complex type conversion for the matrix negation
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A = -np.array(matrix_as_list, dtype=float)
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A_logm, info = logm(A, disp=False)
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assert_(A_logm.dtype.char in complex_dtype_chars)
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def test_complex_spectrum_real_logm(self):
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# This matrix has complex eigenvalues and real logm.
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# Its output dtype depends on its input dtype.
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M = [[1, 1, 2], [2, 1, 1], [1, 2, 1]]
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for dt in float, complex:
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X = np.array(M, dtype=dt)
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w = scipy.linalg.eigvals(X)
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assert_(1e-2 < np.absolute(w.imag).sum())
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Y, info = logm(X, disp=False)
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assert_(np.issubdtype(Y.dtype, np.inexact))
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assert_allclose(expm(Y), X)
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def test_real_mixed_sign_spectrum(self):
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# These matrices have real eigenvalues with mixed signs.
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# The output logm dtype is complex, regardless of input dtype.
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for M in (
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[[1, 0], [0, -1]],
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[[0, 1], [1, 0]]):
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for dt in float, complex:
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A = np.array(M, dtype=dt)
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A_logm, info = logm(A, disp=False)
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assert_(np.issubdtype(A_logm.dtype, np.complexfloating))
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def test_exactly_singular(self):
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A = np.array([[0, 0], [1j, 1j]])
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B = np.asarray([[1, 1], [0, 0]])
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for M in A, A.T, B, B.T:
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expected_warning = _matfuncs_inv_ssq.LogmExactlySingularWarning
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L, info = assert_warns(expected_warning, logm, M, disp=False)
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E = expm(L)
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assert_allclose(E, M, atol=1e-14)
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def test_nearly_singular(self):
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M = np.array([[1e-100]])
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expected_warning = _matfuncs_inv_ssq.LogmNearlySingularWarning
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L, info = assert_warns(expected_warning, logm, M, disp=False)
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E = expm(L)
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assert_allclose(E, M, atol=1e-14)
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def test_opposite_sign_complex_eigenvalues(self):
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# See gh-6113
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E = [[0, 1], [-1, 0]]
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L = [[0, np.pi*0.5], [-np.pi*0.5, 0]]
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assert_allclose(expm(L), E, atol=1e-14)
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assert_allclose(logm(E), L, atol=1e-14)
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E = [[1j, 4], [0, -1j]]
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L = [[1j*np.pi*0.5, 2*np.pi], [0, -1j*np.pi*0.5]]
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assert_allclose(expm(L), E, atol=1e-14)
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assert_allclose(logm(E), L, atol=1e-14)
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E = [[1j, 0], [0, -1j]]
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L = [[1j*np.pi*0.5, 0], [0, -1j*np.pi*0.5]]
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assert_allclose(expm(L), E, atol=1e-14)
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assert_allclose(logm(E), L, atol=1e-14)
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class TestSqrtM(object):
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def test_round_trip_random_float(self):
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np.random.seed(1234)
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for n in range(1, 6):
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M_unscaled = np.random.randn(n, n)
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for scale in np.logspace(-4, 4, 9):
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M = M_unscaled * scale
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M_sqrtm, info = sqrtm(M, disp=False)
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M_sqrtm_round_trip = M_sqrtm.dot(M_sqrtm)
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assert_allclose(M_sqrtm_round_trip, M)
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def test_round_trip_random_complex(self):
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np.random.seed(1234)
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for n in range(1, 6):
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M_unscaled = np.random.randn(n, n) + 1j * np.random.randn(n, n)
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for scale in np.logspace(-4, 4, 9):
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M = M_unscaled * scale
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M_sqrtm, info = sqrtm(M, disp=False)
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M_sqrtm_round_trip = M_sqrtm.dot(M_sqrtm)
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assert_allclose(M_sqrtm_round_trip, M)
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def test_bad(self):
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# See https://web.archive.org/web/20051220232650/http://www.maths.man.ac.uk/~nareports/narep336.ps.gz
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e = 2**-5
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se = sqrt(e)
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a = array([[1.0,0,0,1],
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[0,e,0,0],
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[0,0,e,0],
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[0,0,0,1]])
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sa = array([[1,0,0,0.5],
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[0,se,0,0],
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[0,0,se,0],
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[0,0,0,1]])
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n = a.shape[0]
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assert_array_almost_equal(dot(sa,sa),a)
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# Check default sqrtm.
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esa = sqrtm(a, disp=False, blocksize=n)[0]
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assert_array_almost_equal(dot(esa,esa),a)
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# Check sqrtm with 2x2 blocks.
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esa = sqrtm(a, disp=False, blocksize=2)[0]
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assert_array_almost_equal(dot(esa,esa),a)
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def test_sqrtm_type_preservation_and_conversion(self):
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# The sqrtm matrix function should preserve the type of a matrix
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# whose eigenvalues are nonnegative with zero imaginary part.
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# Test this preservation for variously structured matrices.
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complex_dtype_chars = ('F', 'D', 'G')
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for matrix_as_list in (
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[[1, 0], [0, 1]],
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[[1, 0], [1, 1]],
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[[2, 1], [1, 1]],
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[[2, 3], [1, 2]],
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[[1, 1], [1, 1]]):
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# check that the spectrum has the expected properties
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W = scipy.linalg.eigvals(matrix_as_list)
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assert_(not any(w.imag or w.real < 0 for w in W))
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# check float type preservation
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A = np.array(matrix_as_list, dtype=float)
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A_sqrtm, info = sqrtm(A, disp=False)
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assert_(A_sqrtm.dtype.char not in complex_dtype_chars)
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# check complex type preservation
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A = np.array(matrix_as_list, dtype=complex)
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A_sqrtm, info = sqrtm(A, disp=False)
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assert_(A_sqrtm.dtype.char in complex_dtype_chars)
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# check float->complex type conversion for the matrix negation
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A = -np.array(matrix_as_list, dtype=float)
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A_sqrtm, info = sqrtm(A, disp=False)
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assert_(A_sqrtm.dtype.char in complex_dtype_chars)
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def test_sqrtm_type_conversion_mixed_sign_or_complex_spectrum(self):
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complex_dtype_chars = ('F', 'D', 'G')
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for matrix_as_list in (
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[[1, 0], [0, -1]],
|
||
|
|
[[0, 1], [1, 0]],
|
||
|
|
[[0, 1, 0], [0, 0, 1], [1, 0, 0]]):
|
||
|
|
|
||
|
|
# check that the spectrum has the expected properties
|
||
|
|
W = scipy.linalg.eigvals(matrix_as_list)
|
||
|
|
assert_(any(w.imag or w.real < 0 for w in W))
|
||
|
|
|
||
|
|
# check complex->complex
|
||
|
|
A = np.array(matrix_as_list, dtype=complex)
|
||
|
|
A_sqrtm, info = sqrtm(A, disp=False)
|
||
|
|
assert_(A_sqrtm.dtype.char in complex_dtype_chars)
|
||
|
|
|
||
|
|
# check float->complex
|
||
|
|
A = np.array(matrix_as_list, dtype=float)
|
||
|
|
A_sqrtm, info = sqrtm(A, disp=False)
|
||
|
|
assert_(A_sqrtm.dtype.char in complex_dtype_chars)
|
||
|
|
|
||
|
|
def test_blocksizes(self):
|
||
|
|
# Make sure I do not goof up the blocksizes when they do not divide n.
|
||
|
|
np.random.seed(1234)
|
||
|
|
for n in range(1, 8):
|
||
|
|
A = np.random.rand(n, n) + 1j*np.random.randn(n, n)
|
||
|
|
A_sqrtm_default, info = sqrtm(A, disp=False, blocksize=n)
|
||
|
|
assert_allclose(A, np.linalg.matrix_power(A_sqrtm_default, 2))
|
||
|
|
for blocksize in range(1, 10):
|
||
|
|
A_sqrtm_new, info = sqrtm(A, disp=False, blocksize=blocksize)
|
||
|
|
assert_allclose(A_sqrtm_default, A_sqrtm_new)
|
||
|
|
|
||
|
|
def test_al_mohy_higham_2012_experiment_1(self):
|
||
|
|
# Matrix square root of a tricky upper triangular matrix.
|
||
|
|
A = _get_al_mohy_higham_2012_experiment_1()
|
||
|
|
A_sqrtm, info = sqrtm(A, disp=False)
|
||
|
|
A_round_trip = A_sqrtm.dot(A_sqrtm)
|
||
|
|
assert_allclose(A_round_trip, A, rtol=1e-5)
|
||
|
|
assert_allclose(np.tril(A_round_trip), np.tril(A))
|
||
|
|
|
||
|
|
def test_strict_upper_triangular(self):
|
||
|
|
# This matrix has no square root.
|
||
|
|
for dt in int, float:
|
||
|
|
A = np.array([
|
||
|
|
[0, 3, 0, 0],
|
||
|
|
[0, 0, 3, 0],
|
||
|
|
[0, 0, 0, 3],
|
||
|
|
[0, 0, 0, 0]], dtype=dt)
|
||
|
|
A_sqrtm, info = sqrtm(A, disp=False)
|
||
|
|
assert_(np.isnan(A_sqrtm).all())
|
||
|
|
|
||
|
|
def test_weird_matrix(self):
|
||
|
|
# The square root of matrix B exists.
|
||
|
|
for dt in int, float:
|
||
|
|
A = np.array([
|
||
|
|
[0, 0, 1],
|
||
|
|
[0, 0, 0],
|
||
|
|
[0, 1, 0]], dtype=dt)
|
||
|
|
B = np.array([
|
||
|
|
[0, 1, 0],
|
||
|
|
[0, 0, 0],
|
||
|
|
[0, 0, 0]], dtype=dt)
|
||
|
|
assert_array_equal(B, A.dot(A))
|
||
|
|
|
||
|
|
# But scipy sqrtm is not clever enough to find it.
|
||
|
|
B_sqrtm, info = sqrtm(B, disp=False)
|
||
|
|
assert_(np.isnan(B_sqrtm).all())
|
||
|
|
|
||
|
|
def test_disp(self):
|
||
|
|
np.random.seed(1234)
|
||
|
|
|
||
|
|
A = np.random.rand(3, 3)
|
||
|
|
B = sqrtm(A, disp=True)
|
||
|
|
assert_allclose(B.dot(B), A)
|
||
|
|
|
||
|
|
def test_opposite_sign_complex_eigenvalues(self):
|
||
|
|
M = [[2j, 4], [0, -2j]]
|
||
|
|
R = [[1+1j, 2], [0, 1-1j]]
|
||
|
|
assert_allclose(np.dot(R, R), M, atol=1e-14)
|
||
|
|
assert_allclose(sqrtm(M), R, atol=1e-14)
|
||
|
|
|
||
|
|
def test_gh4866(self):
|
||
|
|
M = np.array([[1, 0, 0, 1],
|
||
|
|
[0, 0, 0, 0],
|
||
|
|
[0, 0, 0, 0],
|
||
|
|
[1, 0, 0, 1]])
|
||
|
|
R = np.array([[sqrt(0.5), 0, 0, sqrt(0.5)],
|
||
|
|
[0, 0, 0, 0],
|
||
|
|
[0, 0, 0, 0],
|
||
|
|
[sqrt(0.5), 0, 0, sqrt(0.5)]])
|
||
|
|
assert_allclose(np.dot(R, R), M, atol=1e-14)
|
||
|
|
assert_allclose(sqrtm(M), R, atol=1e-14)
|
||
|
|
|
||
|
|
def test_gh5336(self):
|
||
|
|
M = np.diag([2, 1, 0])
|
||
|
|
R = np.diag([sqrt(2), 1, 0])
|
||
|
|
assert_allclose(np.dot(R, R), M, atol=1e-14)
|
||
|
|
assert_allclose(sqrtm(M), R, atol=1e-14)
|
||
|
|
|
||
|
|
def test_gh7839(self):
|
||
|
|
M = np.zeros((2, 2))
|
||
|
|
R = np.zeros((2, 2))
|
||
|
|
assert_allclose(np.dot(R, R), M, atol=1e-14)
|
||
|
|
assert_allclose(sqrtm(M), R, atol=1e-14)
|
||
|
|
|
||
|
|
|
||
|
|
class TestFractionalMatrixPower(object):
|
||
|
|
def test_round_trip_random_complex(self):
|
||
|
|
np.random.seed(1234)
|
||
|
|
for p in range(1, 5):
|
||
|
|
for n in range(1, 5):
|
||
|
|
M_unscaled = np.random.randn(n, n) + 1j * np.random.randn(n, n)
|
||
|
|
for scale in np.logspace(-4, 4, 9):
|
||
|
|
M = M_unscaled * scale
|
||
|
|
M_root = fractional_matrix_power(M, 1/p)
|
||
|
|
M_round_trip = np.linalg.matrix_power(M_root, p)
|
||
|
|
assert_allclose(M_round_trip, M)
|
||
|
|
|
||
|
|
def test_round_trip_random_float(self):
|
||
|
|
# This test is more annoying because it can hit the branch cut;
|
||
|
|
# this happens when the matrix has an eigenvalue
|
||
|
|
# with no imaginary component and with a real negative component,
|
||
|
|
# and it means that the principal branch does not exist.
|
||
|
|
np.random.seed(1234)
|
||
|
|
for p in range(1, 5):
|
||
|
|
for n in range(1, 5):
|
||
|
|
M_unscaled = np.random.randn(n, n)
|
||
|
|
for scale in np.logspace(-4, 4, 9):
|
||
|
|
M = M_unscaled * scale
|
||
|
|
M_root = fractional_matrix_power(M, 1/p)
|
||
|
|
M_round_trip = np.linalg.matrix_power(M_root, p)
|
||
|
|
assert_allclose(M_round_trip, M)
|
||
|
|
|
||
|
|
def test_larger_abs_fractional_matrix_powers(self):
|
||
|
|
np.random.seed(1234)
|
||
|
|
for n in (2, 3, 5):
|
||
|
|
for i in range(10):
|
||
|
|
M = np.random.randn(n, n) + 1j * np.random.randn(n, n)
|
||
|
|
M_one_fifth = fractional_matrix_power(M, 0.2)
|
||
|
|
# Test the round trip.
|
||
|
|
M_round_trip = np.linalg.matrix_power(M_one_fifth, 5)
|
||
|
|
assert_allclose(M, M_round_trip)
|
||
|
|
# Test a large abs fractional power.
|
||
|
|
X = fractional_matrix_power(M, -5.4)
|
||
|
|
Y = np.linalg.matrix_power(M_one_fifth, -27)
|
||
|
|
assert_allclose(X, Y)
|
||
|
|
# Test another large abs fractional power.
|
||
|
|
X = fractional_matrix_power(M, 3.8)
|
||
|
|
Y = np.linalg.matrix_power(M_one_fifth, 19)
|
||
|
|
assert_allclose(X, Y)
|
||
|
|
|
||
|
|
def test_random_matrices_and_powers(self):
|
||
|
|
# Each independent iteration of this fuzz test picks random parameters.
|
||
|
|
# It tries to hit some edge cases.
|
||
|
|
np.random.seed(1234)
|
||
|
|
nsamples = 20
|
||
|
|
for i in range(nsamples):
|
||
|
|
# Sample a matrix size and a random real power.
|
||
|
|
n = random.randrange(1, 5)
|
||
|
|
p = np.random.randn()
|
||
|
|
|
||
|
|
# Sample a random real or complex matrix.
|
||
|
|
matrix_scale = np.exp(random.randrange(-4, 5))
|
||
|
|
A = np.random.randn(n, n)
|
||
|
|
if random.choice((True, False)):
|
||
|
|
A = A + 1j * np.random.randn(n, n)
|
||
|
|
A = A * matrix_scale
|
||
|
|
|
||
|
|
# Check a couple of analytically equivalent ways
|
||
|
|
# to compute the fractional matrix power.
|
||
|
|
# These can be compared because they both use the principal branch.
|
||
|
|
A_power = fractional_matrix_power(A, p)
|
||
|
|
A_logm, info = logm(A, disp=False)
|
||
|
|
A_power_expm_logm = expm(A_logm * p)
|
||
|
|
assert_allclose(A_power, A_power_expm_logm)
|
||
|
|
|
||
|
|
def test_al_mohy_higham_2012_experiment_1(self):
|
||
|
|
# Fractional powers of a tricky upper triangular matrix.
|
||
|
|
A = _get_al_mohy_higham_2012_experiment_1()
|
||
|
|
|
||
|
|
# Test remainder matrix power.
|
||
|
|
A_funm_sqrt, info = funm(A, np.sqrt, disp=False)
|
||
|
|
A_sqrtm, info = sqrtm(A, disp=False)
|
||
|
|
A_rem_power = _matfuncs_inv_ssq._remainder_matrix_power(A, 0.5)
|
||
|
|
A_power = fractional_matrix_power(A, 0.5)
|
||
|
|
assert_array_equal(A_rem_power, A_power)
|
||
|
|
assert_allclose(A_sqrtm, A_power)
|
||
|
|
assert_allclose(A_sqrtm, A_funm_sqrt)
|
||
|
|
|
||
|
|
# Test more fractional powers.
|
||
|
|
for p in (1/2, 5/3):
|
||
|
|
A_power = fractional_matrix_power(A, p)
|
||
|
|
A_round_trip = fractional_matrix_power(A_power, 1/p)
|
||
|
|
assert_allclose(A_round_trip, A, rtol=1e-2)
|
||
|
|
assert_allclose(np.tril(A_round_trip, 1), np.tril(A, 1))
|
||
|
|
|
||
|
|
def test_briggs_helper_function(self):
|
||
|
|
np.random.seed(1234)
|
||
|
|
for a in np.random.randn(10) + 1j * np.random.randn(10):
|
||
|
|
for k in range(5):
|
||
|
|
x_observed = _matfuncs_inv_ssq._briggs_helper_function(a, k)
|
||
|
|
x_expected = a ** np.exp2(-k) - 1
|
||
|
|
assert_allclose(x_observed, x_expected)
|
||
|
|
|
||
|
|
def test_type_preservation_and_conversion(self):
|
||
|
|
# The fractional_matrix_power matrix function should preserve
|
||
|
|
# the type of a matrix whose eigenvalues
|
||
|
|
# are positive with zero imaginary part.
|
||
|
|
# Test this preservation for variously structured matrices.
|
||
|
|
complex_dtype_chars = ('F', 'D', 'G')
|
||
|
|
for matrix_as_list in (
|
||
|
|
[[1, 0], [0, 1]],
|
||
|
|
[[1, 0], [1, 1]],
|
||
|
|
[[2, 1], [1, 1]],
|
||
|
|
[[2, 3], [1, 2]]):
|
||
|
|
|
||
|
|
# check that the spectrum has the expected properties
|
||
|
|
W = scipy.linalg.eigvals(matrix_as_list)
|
||
|
|
assert_(not any(w.imag or w.real < 0 for w in W))
|
||
|
|
|
||
|
|
# Check various positive and negative powers
|
||
|
|
# with absolute values bigger and smaller than 1.
|
||
|
|
for p in (-2.4, -0.9, 0.2, 3.3):
|
||
|
|
|
||
|
|
# check float type preservation
|
||
|
|
A = np.array(matrix_as_list, dtype=float)
|
||
|
|
A_power = fractional_matrix_power(A, p)
|
||
|
|
assert_(A_power.dtype.char not in complex_dtype_chars)
|
||
|
|
|
||
|
|
# check complex type preservation
|
||
|
|
A = np.array(matrix_as_list, dtype=complex)
|
||
|
|
A_power = fractional_matrix_power(A, p)
|
||
|
|
assert_(A_power.dtype.char in complex_dtype_chars)
|
||
|
|
|
||
|
|
# check float->complex for the matrix negation
|
||
|
|
A = -np.array(matrix_as_list, dtype=float)
|
||
|
|
A_power = fractional_matrix_power(A, p)
|
||
|
|
assert_(A_power.dtype.char in complex_dtype_chars)
|
||
|
|
|
||
|
|
def test_type_conversion_mixed_sign_or_complex_spectrum(self):
|
||
|
|
complex_dtype_chars = ('F', 'D', 'G')
|
||
|
|
for matrix_as_list in (
|
||
|
|
[[1, 0], [0, -1]],
|
||
|
|
[[0, 1], [1, 0]],
|
||
|
|
[[0, 1, 0], [0, 0, 1], [1, 0, 0]]):
|
||
|
|
|
||
|
|
# check that the spectrum has the expected properties
|
||
|
|
W = scipy.linalg.eigvals(matrix_as_list)
|
||
|
|
assert_(any(w.imag or w.real < 0 for w in W))
|
||
|
|
|
||
|
|
# Check various positive and negative powers
|
||
|
|
# with absolute values bigger and smaller than 1.
|
||
|
|
for p in (-2.4, -0.9, 0.2, 3.3):
|
||
|
|
|
||
|
|
# check complex->complex
|
||
|
|
A = np.array(matrix_as_list, dtype=complex)
|
||
|
|
A_power = fractional_matrix_power(A, p)
|
||
|
|
assert_(A_power.dtype.char in complex_dtype_chars)
|
||
|
|
|
||
|
|
# check float->complex
|
||
|
|
A = np.array(matrix_as_list, dtype=float)
|
||
|
|
A_power = fractional_matrix_power(A, p)
|
||
|
|
assert_(A_power.dtype.char in complex_dtype_chars)
|
||
|
|
|
||
|
|
@pytest.mark.xfail(reason='Too unstable across LAPACKs.')
|
||
|
|
def test_singular(self):
|
||
|
|
# Negative fractional powers do not work with singular matrices.
|
||
|
|
for matrix_as_list in (
|
||
|
|
[[0, 0], [0, 0]],
|
||
|
|
[[1, 1], [1, 1]],
|
||
|
|
[[1, 2], [3, 6]],
|
||
|
|
[[0, 0, 0], [0, 1, 1], [0, -1, 1]]):
|
||
|
|
|
||
|
|
# Check fractional powers both for float and for complex types.
|
||
|
|
for newtype in (float, complex):
|
||
|
|
A = np.array(matrix_as_list, dtype=newtype)
|
||
|
|
for p in (-0.7, -0.9, -2.4, -1.3):
|
||
|
|
A_power = fractional_matrix_power(A, p)
|
||
|
|
assert_(np.isnan(A_power).all())
|
||
|
|
for p in (0.2, 1.43):
|
||
|
|
A_power = fractional_matrix_power(A, p)
|
||
|
|
A_round_trip = fractional_matrix_power(A_power, 1/p)
|
||
|
|
assert_allclose(A_round_trip, A)
|
||
|
|
|
||
|
|
def test_opposite_sign_complex_eigenvalues(self):
|
||
|
|
M = [[2j, 4], [0, -2j]]
|
||
|
|
R = [[1+1j, 2], [0, 1-1j]]
|
||
|
|
assert_allclose(np.dot(R, R), M, atol=1e-14)
|
||
|
|
assert_allclose(fractional_matrix_power(M, 0.5), R, atol=1e-14)
|
||
|
|
|
||
|
|
|
||
|
|
class TestExpM(object):
|
||
|
|
def test_zero(self):
|
||
|
|
a = array([[0.,0],[0,0]])
|
||
|
|
assert_array_almost_equal(expm(a),[[1,0],[0,1]])
|
||
|
|
|
||
|
|
def test_single_elt(self):
|
||
|
|
# See gh-5853
|
||
|
|
from scipy.sparse import csc_matrix
|
||
|
|
|
||
|
|
vOne = -2.02683397006j
|
||
|
|
vTwo = -2.12817566856j
|
||
|
|
|
||
|
|
mOne = csc_matrix([[vOne]], dtype='complex')
|
||
|
|
mTwo = csc_matrix([[vTwo]], dtype='complex')
|
||
|
|
|
||
|
|
outOne = expm(mOne)
|
||
|
|
outTwo = expm(mTwo)
|
||
|
|
|
||
|
|
assert_equal(type(outOne), type(mOne))
|
||
|
|
assert_equal(type(outTwo), type(mTwo))
|
||
|
|
|
||
|
|
assert_allclose(outOne[0, 0], complex(-0.44039415155949196,
|
||
|
|
-0.8978045395698304))
|
||
|
|
assert_allclose(outTwo[0, 0], complex(-0.52896401032626006,
|
||
|
|
-0.84864425749518878))
|
||
|
|
|
||
|
|
def test_empty_matrix_input(self):
|
||
|
|
# handle gh-11082
|
||
|
|
A = np.zeros((0, 0))
|
||
|
|
result = expm(A)
|
||
|
|
assert result.size == 0
|
||
|
|
|
||
|
|
|
||
|
|
class TestExpmFrechet(object):
|
||
|
|
|
||
|
|
def test_expm_frechet(self):
|
||
|
|
# a test of the basic functionality
|
||
|
|
M = np.array([
|
||
|
|
[1, 2, 3, 4],
|
||
|
|
[5, 6, 7, 8],
|
||
|
|
[0, 0, 1, 2],
|
||
|
|
[0, 0, 5, 6],
|
||
|
|
], dtype=float)
|
||
|
|
A = np.array([
|
||
|
|
[1, 2],
|
||
|
|
[5, 6],
|
||
|
|
], dtype=float)
|
||
|
|
E = np.array([
|
||
|
|
[3, 4],
|
||
|
|
[7, 8],
|
||
|
|
], dtype=float)
|
||
|
|
expected_expm = scipy.linalg.expm(A)
|
||
|
|
expected_frechet = scipy.linalg.expm(M)[:2, 2:]
|
||
|
|
for kwargs in ({}, {'method':'SPS'}, {'method':'blockEnlarge'}):
|
||
|
|
observed_expm, observed_frechet = expm_frechet(A, E, **kwargs)
|
||
|
|
assert_allclose(expected_expm, observed_expm)
|
||
|
|
assert_allclose(expected_frechet, observed_frechet)
|
||
|
|
|
||
|
|
def test_small_norm_expm_frechet(self):
|
||
|
|
# methodically test matrices with a range of norms, for better coverage
|
||
|
|
M_original = np.array([
|
||
|
|
[1, 2, 3, 4],
|
||
|
|
[5, 6, 7, 8],
|
||
|
|
[0, 0, 1, 2],
|
||
|
|
[0, 0, 5, 6],
|
||
|
|
], dtype=float)
|
||
|
|
A_original = np.array([
|
||
|
|
[1, 2],
|
||
|
|
[5, 6],
|
||
|
|
], dtype=float)
|
||
|
|
E_original = np.array([
|
||
|
|
[3, 4],
|
||
|
|
[7, 8],
|
||
|
|
], dtype=float)
|
||
|
|
A_original_norm_1 = scipy.linalg.norm(A_original, 1)
|
||
|
|
selected_m_list = [1, 3, 5, 7, 9, 11, 13, 15]
|
||
|
|
m_neighbor_pairs = zip(selected_m_list[:-1], selected_m_list[1:])
|
||
|
|
for ma, mb in m_neighbor_pairs:
|
||
|
|
ell_a = scipy.linalg._expm_frechet.ell_table_61[ma]
|
||
|
|
ell_b = scipy.linalg._expm_frechet.ell_table_61[mb]
|
||
|
|
target_norm_1 = 0.5 * (ell_a + ell_b)
|
||
|
|
scale = target_norm_1 / A_original_norm_1
|
||
|
|
M = scale * M_original
|
||
|
|
A = scale * A_original
|
||
|
|
E = scale * E_original
|
||
|
|
expected_expm = scipy.linalg.expm(A)
|
||
|
|
expected_frechet = scipy.linalg.expm(M)[:2, 2:]
|
||
|
|
observed_expm, observed_frechet = expm_frechet(A, E)
|
||
|
|
assert_allclose(expected_expm, observed_expm)
|
||
|
|
assert_allclose(expected_frechet, observed_frechet)
|
||
|
|
|
||
|
|
def test_fuzz(self):
|
||
|
|
# try a bunch of crazy inputs
|
||
|
|
rfuncs = (
|
||
|
|
np.random.uniform,
|
||
|
|
np.random.normal,
|
||
|
|
np.random.standard_cauchy,
|
||
|
|
np.random.exponential)
|
||
|
|
ntests = 100
|
||
|
|
for i in range(ntests):
|
||
|
|
rfunc = random.choice(rfuncs)
|
||
|
|
target_norm_1 = random.expovariate(1.0)
|
||
|
|
n = random.randrange(2, 16)
|
||
|
|
A_original = rfunc(size=(n,n))
|
||
|
|
E_original = rfunc(size=(n,n))
|
||
|
|
A_original_norm_1 = scipy.linalg.norm(A_original, 1)
|
||
|
|
scale = target_norm_1 / A_original_norm_1
|
||
|
|
A = scale * A_original
|
||
|
|
E = scale * E_original
|
||
|
|
M = np.vstack([
|
||
|
|
np.hstack([A, E]),
|
||
|
|
np.hstack([np.zeros_like(A), A])])
|
||
|
|
expected_expm = scipy.linalg.expm(A)
|
||
|
|
expected_frechet = scipy.linalg.expm(M)[:n, n:]
|
||
|
|
observed_expm, observed_frechet = expm_frechet(A, E)
|
||
|
|
assert_allclose(expected_expm, observed_expm)
|
||
|
|
assert_allclose(expected_frechet, observed_frechet)
|
||
|
|
|
||
|
|
def test_problematic_matrix(self):
|
||
|
|
# this test case uncovered a bug which has since been fixed
|
||
|
|
A = np.array([
|
||
|
|
[1.50591997, 1.93537998],
|
||
|
|
[0.41203263, 0.23443516],
|
||
|
|
], dtype=float)
|
||
|
|
E = np.array([
|
||
|
|
[1.87864034, 2.07055038],
|
||
|
|
[1.34102727, 0.67341123],
|
||
|
|
], dtype=float)
|
||
|
|
scipy.linalg.norm(A, 1)
|
||
|
|
sps_expm, sps_frechet = expm_frechet(
|
||
|
|
A, E, method='SPS')
|
||
|
|
blockEnlarge_expm, blockEnlarge_frechet = expm_frechet(
|
||
|
|
A, E, method='blockEnlarge')
|
||
|
|
assert_allclose(sps_expm, blockEnlarge_expm)
|
||
|
|
assert_allclose(sps_frechet, blockEnlarge_frechet)
|
||
|
|
|
||
|
|
@pytest.mark.slow
|
||
|
|
@pytest.mark.skip(reason='this test is deliberately slow')
|
||
|
|
def test_medium_matrix(self):
|
||
|
|
# profile this to see the speed difference
|
||
|
|
n = 1000
|
||
|
|
A = np.random.exponential(size=(n, n))
|
||
|
|
E = np.random.exponential(size=(n, n))
|
||
|
|
sps_expm, sps_frechet = expm_frechet(
|
||
|
|
A, E, method='SPS')
|
||
|
|
blockEnlarge_expm, blockEnlarge_frechet = expm_frechet(
|
||
|
|
A, E, method='blockEnlarge')
|
||
|
|
assert_allclose(sps_expm, blockEnlarge_expm)
|
||
|
|
assert_allclose(sps_frechet, blockEnlarge_frechet)
|
||
|
|
|
||
|
|
|
||
|
|
def _help_expm_cond_search(A, A_norm, X, X_norm, eps, p):
|
||
|
|
p = np.reshape(p, A.shape)
|
||
|
|
p_norm = norm(p)
|
||
|
|
perturbation = eps * p * (A_norm / p_norm)
|
||
|
|
X_prime = expm(A + perturbation)
|
||
|
|
scaled_relative_error = norm(X_prime - X) / (X_norm * eps)
|
||
|
|
return -scaled_relative_error
|
||
|
|
|
||
|
|
|
||
|
|
def _normalized_like(A, B):
|
||
|
|
return A * (scipy.linalg.norm(B) / scipy.linalg.norm(A))
|
||
|
|
|
||
|
|
|
||
|
|
def _relative_error(f, A, perturbation):
|
||
|
|
X = f(A)
|
||
|
|
X_prime = f(A + perturbation)
|
||
|
|
return norm(X_prime - X) / norm(X)
|
||
|
|
|
||
|
|
|
||
|
|
class TestExpmConditionNumber(object):
|
||
|
|
def test_expm_cond_smoke(self):
|
||
|
|
np.random.seed(1234)
|
||
|
|
for n in range(1, 4):
|
||
|
|
A = np.random.randn(n, n)
|
||
|
|
kappa = expm_cond(A)
|
||
|
|
assert_array_less(0, kappa)
|
||
|
|
|
||
|
|
def test_expm_bad_condition_number(self):
|
||
|
|
A = np.array([
|
||
|
|
[-1.128679820, 9.614183771e4, -4.524855739e9, 2.924969411e14],
|
||
|
|
[0, -1.201010529, 9.634696872e4, -4.681048289e9],
|
||
|
|
[0, 0, -1.132893222, 9.532491830e4],
|
||
|
|
[0, 0, 0, -1.179475332],
|
||
|
|
])
|
||
|
|
kappa = expm_cond(A)
|
||
|
|
assert_array_less(1e36, kappa)
|
||
|
|
|
||
|
|
def test_univariate(self):
|
||
|
|
np.random.seed(12345)
|
||
|
|
for x in np.linspace(-5, 5, num=11):
|
||
|
|
A = np.array([[x]])
|
||
|
|
assert_allclose(expm_cond(A), abs(x))
|
||
|
|
for x in np.logspace(-2, 2, num=11):
|
||
|
|
A = np.array([[x]])
|
||
|
|
assert_allclose(expm_cond(A), abs(x))
|
||
|
|
for i in range(10):
|
||
|
|
A = np.random.randn(1, 1)
|
||
|
|
assert_allclose(expm_cond(A), np.absolute(A)[0, 0])
|
||
|
|
|
||
|
|
@pytest.mark.slow
|
||
|
|
def test_expm_cond_fuzz(self):
|
||
|
|
np.random.seed(12345)
|
||
|
|
eps = 1e-5
|
||
|
|
nsamples = 10
|
||
|
|
for i in range(nsamples):
|
||
|
|
n = np.random.randint(2, 5)
|
||
|
|
A = np.random.randn(n, n)
|
||
|
|
A_norm = scipy.linalg.norm(A)
|
||
|
|
X = expm(A)
|
||
|
|
X_norm = scipy.linalg.norm(X)
|
||
|
|
kappa = expm_cond(A)
|
||
|
|
|
||
|
|
# Look for the small perturbation that gives the greatest
|
||
|
|
# relative error.
|
||
|
|
f = functools.partial(_help_expm_cond_search,
|
||
|
|
A, A_norm, X, X_norm, eps)
|
||
|
|
guess = np.ones(n*n)
|
||
|
|
out = minimize(f, guess, method='L-BFGS-B')
|
||
|
|
xopt = out.x
|
||
|
|
yopt = f(xopt)
|
||
|
|
p_best = eps * _normalized_like(np.reshape(xopt, A.shape), A)
|
||
|
|
p_best_relerr = _relative_error(expm, A, p_best)
|
||
|
|
assert_allclose(p_best_relerr, -yopt * eps)
|
||
|
|
|
||
|
|
# Check that the identified perturbation indeed gives greater
|
||
|
|
# relative error than random perturbations with similar norms.
|
||
|
|
for j in range(5):
|
||
|
|
p_rand = eps * _normalized_like(np.random.randn(*A.shape), A)
|
||
|
|
assert_allclose(norm(p_best), norm(p_rand))
|
||
|
|
p_rand_relerr = _relative_error(expm, A, p_rand)
|
||
|
|
assert_array_less(p_rand_relerr, p_best_relerr)
|
||
|
|
|
||
|
|
# The greatest relative error should not be much greater than
|
||
|
|
# eps times the condition number kappa.
|
||
|
|
# In the limit as eps approaches zero it should never be greater.
|
||
|
|
assert_array_less(p_best_relerr, (1 + 2*eps) * eps * kappa)
|
||
|
|
|
||
|
|
|
||
|
|
class TestKhatriRao(object):
|
||
|
|
|
||
|
|
def test_basic(self):
|
||
|
|
a = khatri_rao(array([[1, 2], [3, 4]]),
|
||
|
|
array([[5, 6], [7, 8]]))
|
||
|
|
|
||
|
|
assert_array_equal(a, array([[5, 12],
|
||
|
|
[7, 16],
|
||
|
|
[15, 24],
|
||
|
|
[21, 32]]))
|
||
|
|
|
||
|
|
b = khatri_rao(np.empty([2, 2]), np.empty([2, 2]))
|
||
|
|
assert_array_equal(b.shape, (4, 2))
|
||
|
|
|
||
|
|
def test_number_of_columns_equality(self):
|
||
|
|
with pytest.raises(ValueError):
|
||
|
|
a = array([[1, 2, 3],
|
||
|
|
[4, 5, 6]])
|
||
|
|
b = array([[1, 2],
|
||
|
|
[3, 4]])
|
||
|
|
khatri_rao(a, b)
|
||
|
|
|
||
|
|
def test_to_assure_2d_array(self):
|
||
|
|
with pytest.raises(ValueError):
|
||
|
|
# both arrays are 1-D
|
||
|
|
a = array([1, 2, 3])
|
||
|
|
b = array([4, 5, 6])
|
||
|
|
khatri_rao(a, b)
|
||
|
|
|
||
|
|
with pytest.raises(ValueError):
|
||
|
|
# first array is 1-D
|
||
|
|
a = array([1, 2, 3])
|
||
|
|
b = array([
|
||
|
|
[1, 2, 3],
|
||
|
|
[4, 5, 6]
|
||
|
|
])
|
||
|
|
khatri_rao(a, b)
|
||
|
|
|
||
|
|
with pytest.raises(ValueError):
|
||
|
|
# second array is 1-D
|
||
|
|
a = array([
|
||
|
|
[1, 2, 3],
|
||
|
|
[7, 8, 9]
|
||
|
|
])
|
||
|
|
b = array([4, 5, 6])
|
||
|
|
khatri_rao(a, b)
|
||
|
|
|
||
|
|
def test_equality_of_two_equations(self):
|
||
|
|
a = array([[1, 2], [3, 4]])
|
||
|
|
b = array([[5, 6], [7, 8]])
|
||
|
|
|
||
|
|
res1 = khatri_rao(a, b)
|
||
|
|
res2 = np.vstack([np.kron(a[:, k], b[:, k])
|
||
|
|
for k in range(b.shape[1])]).T
|
||
|
|
|
||
|
|
assert_array_equal(res1, res2)
|