547 lines
20 KiB
Python
547 lines
20 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 scipy.linalg.matfuncs module
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"""
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from __future__ import division, print_function, absolute_import
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import math
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import numpy as np
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from numpy import array, eye, exp, random
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from numpy.linalg import matrix_power
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from numpy.testing import (
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assert_allclose, assert_, assert_array_almost_equal, assert_equal,
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assert_array_almost_equal_nulp)
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from scipy._lib._numpy_compat import suppress_warnings
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from scipy.sparse import csc_matrix, SparseEfficiencyWarning
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from scipy.sparse.construct import eye as speye
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from scipy.sparse.linalg.matfuncs import (expm, _expm,
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ProductOperator, MatrixPowerOperator,
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_onenorm_matrix_power_nnm)
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from scipy.linalg import logm
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from scipy.special import factorial, binom
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import scipy.sparse
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import scipy.sparse.linalg
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def _burkardt_13_power(n, p):
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"""
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A helper function for testing matrix functions.
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Parameters
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----------
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n : integer greater than 1
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Order of the square matrix to be returned.
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p : non-negative integer
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Power of the matrix.
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Returns
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-------
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out : ndarray representing a square matrix
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A Forsythe matrix of order n, raised to the power p.
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"""
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# Input validation.
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if n != int(n) or n < 2:
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raise ValueError('n must be an integer greater than 1')
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n = int(n)
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if p != int(p) or p < 0:
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raise ValueError('p must be a non-negative integer')
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p = int(p)
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# Construct the matrix explicitly.
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a, b = divmod(p, n)
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large = np.power(10.0, -n*a)
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small = large * np.power(10.0, -n)
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return np.diag([large]*(n-b), b) + np.diag([small]*b, b-n)
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def test_onenorm_matrix_power_nnm():
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np.random.seed(1234)
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for n in range(1, 5):
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for p in range(5):
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M = np.random.random((n, n))
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Mp = np.linalg.matrix_power(M, p)
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observed = _onenorm_matrix_power_nnm(M, p)
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expected = np.linalg.norm(Mp, 1)
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assert_allclose(observed, expected)
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class TestExpM(object):
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def test_zero_ndarray(self):
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a = array([[0.,0],[0,0]])
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assert_array_almost_equal(expm(a),[[1,0],[0,1]])
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def test_zero_sparse(self):
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a = csc_matrix([[0.,0],[0,0]])
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assert_array_almost_equal(expm(a).toarray(),[[1,0],[0,1]])
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def test_zero_matrix(self):
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a = np.matrix([[0.,0],[0,0]])
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assert_array_almost_equal(expm(a),[[1,0],[0,1]])
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def test_misc_types(self):
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A = expm(np.array([[1]]))
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assert_allclose(expm(((1,),)), A)
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assert_allclose(expm([[1]]), A)
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assert_allclose(expm(np.matrix([[1]])), A)
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assert_allclose(expm(np.array([[1]])), A)
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assert_allclose(expm(csc_matrix([[1]])).A, A)
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B = expm(np.array([[1j]]))
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assert_allclose(expm(((1j,),)), B)
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assert_allclose(expm([[1j]]), B)
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assert_allclose(expm(np.matrix([[1j]])), B)
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assert_allclose(expm(csc_matrix([[1j]])).A, B)
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def test_bidiagonal_sparse(self):
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A = csc_matrix([
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[1, 3, 0],
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[0, 1, 5],
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[0, 0, 2]], dtype=float)
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e1 = math.exp(1)
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e2 = math.exp(2)
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expected = np.array([
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[e1, 3*e1, 15*(e2 - 2*e1)],
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[0, e1, 5*(e2 - e1)],
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[0, 0, e2]], dtype=float)
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observed = expm(A).toarray()
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assert_array_almost_equal(observed, expected)
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def test_padecases_dtype_float(self):
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for dtype in [np.float32, np.float64]:
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for scale in [1e-2, 1e-1, 5e-1, 1, 10]:
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A = scale * eye(3, dtype=dtype)
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observed = expm(A)
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expected = exp(scale) * eye(3, dtype=dtype)
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assert_array_almost_equal_nulp(observed, expected, nulp=100)
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def test_padecases_dtype_complex(self):
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for dtype in [np.complex64, np.complex128]:
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for scale in [1e-2, 1e-1, 5e-1, 1, 10]:
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A = scale * eye(3, dtype=dtype)
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observed = expm(A)
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expected = exp(scale) * eye(3, dtype=dtype)
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assert_array_almost_equal_nulp(observed, expected, nulp=100)
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def test_padecases_dtype_sparse_float(self):
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# float32 and complex64 lead to errors in spsolve/UMFpack
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dtype = np.float64
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for scale in [1e-2, 1e-1, 5e-1, 1, 10]:
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a = scale * speye(3, 3, dtype=dtype, format='csc')
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e = exp(scale) * eye(3, dtype=dtype)
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with suppress_warnings() as sup:
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sup.filter(SparseEfficiencyWarning,
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"Changing the sparsity structure of a csc_matrix is expensive.")
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exact_onenorm = _expm(a, use_exact_onenorm=True).toarray()
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inexact_onenorm = _expm(a, use_exact_onenorm=False).toarray()
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assert_array_almost_equal_nulp(exact_onenorm, e, nulp=100)
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assert_array_almost_equal_nulp(inexact_onenorm, e, nulp=100)
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def test_padecases_dtype_sparse_complex(self):
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# float32 and complex64 lead to errors in spsolve/UMFpack
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dtype = np.complex128
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for scale in [1e-2, 1e-1, 5e-1, 1, 10]:
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a = scale * speye(3, 3, dtype=dtype, format='csc')
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e = exp(scale) * eye(3, dtype=dtype)
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with suppress_warnings() as sup:
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sup.filter(SparseEfficiencyWarning,
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"Changing the sparsity structure of a csc_matrix is expensive.")
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assert_array_almost_equal_nulp(expm(a).toarray(), e, nulp=100)
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def test_logm_consistency(self):
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random.seed(1234)
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for dtype in [np.float64, np.complex128]:
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for n in range(1, 10):
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for scale in [1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1, 1e2]:
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# make logm(A) be of a given scale
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A = (eye(n) + random.rand(n, n) * scale).astype(dtype)
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if np.iscomplexobj(A):
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A = A + 1j * random.rand(n, n) * scale
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assert_array_almost_equal(expm(logm(A)), A)
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def test_integer_matrix(self):
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Q = np.array([
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[-3, 1, 1, 1],
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[1, -3, 1, 1],
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[1, 1, -3, 1],
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[1, 1, 1, -3]])
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assert_allclose(expm(Q), expm(1.0 * Q))
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def test_triangularity_perturbation(self):
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# Experiment (1) of
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# 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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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, 3.221e-1, 3e4],
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[0, 0, 0, 3.0744e-1]],
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dtype=float)
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A_logm = np.array([
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[-1.12867982029050462e+00, 9.61418377142025565e+04,
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-4.52485573953179264e+09, 2.92496941103871812e+14],
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[0.00000000000000000e+00, -1.20101052953082288e+00,
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9.63469687211303099e+04, -4.68104828911105442e+09],
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[0.00000000000000000e+00, 0.00000000000000000e+00,
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-1.13289322264498393e+00, 9.53249183094775653e+04],
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[0.00000000000000000e+00, 0.00000000000000000e+00,
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0.00000000000000000e+00, -1.17947533272554850e+00]],
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dtype=float)
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assert_allclose(expm(A_logm), A, rtol=1e-4)
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# Perturb the upper triangular matrix by tiny amounts,
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# so that it becomes technically not upper triangular.
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random.seed(1234)
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tiny = 1e-17
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A_logm_perturbed = A_logm.copy()
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A_logm_perturbed[1, 0] = tiny
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with suppress_warnings() as sup:
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sup.filter(RuntimeWarning,
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"scipy.linalg.solve\nIll-conditioned.*")
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A_expm_logm_perturbed = expm(A_logm_perturbed)
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rtol = 1e-4
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atol = 100 * tiny
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assert_(not np.allclose(A_expm_logm_perturbed, A, rtol=rtol, atol=atol))
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def test_burkardt_1(self):
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# This matrix is diagonal.
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# The calculation of the matrix exponential is simple.
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#
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# This is the first of a series of matrix exponential tests
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# collected by John Burkardt from the following sources.
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#
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# Alan Laub,
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# Review of "Linear System Theory" by Joao Hespanha,
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# SIAM Review,
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# Volume 52, Number 4, December 2010, pages 779--781.
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#
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# Cleve Moler and Charles Van Loan,
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# Nineteen Dubious Ways to Compute the Exponential of a Matrix,
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# Twenty-Five Years Later,
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# SIAM Review,
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# Volume 45, Number 1, March 2003, pages 3--49.
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#
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# Cleve Moler,
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# Cleve's Corner: A Balancing Act for the Matrix Exponential,
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# 23 July 2012.
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#
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# Robert Ward,
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# Numerical computation of the matrix exponential
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# with accuracy estimate,
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# SIAM Journal on Numerical Analysis,
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# Volume 14, Number 4, September 1977, pages 600--610.
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exp1 = np.exp(1)
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exp2 = np.exp(2)
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A = np.array([
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[1, 0],
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[0, 2],
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], dtype=float)
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desired = np.array([
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[exp1, 0],
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[0, exp2],
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], dtype=float)
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actual = expm(A)
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assert_allclose(actual, desired)
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def test_burkardt_2(self):
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# This matrix is symmetric.
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# The calculation of the matrix exponential is straightforward.
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A = np.array([
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[1, 3],
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[3, 2],
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], dtype=float)
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desired = np.array([
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[39.322809708033859, 46.166301438885753],
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[46.166301438885768, 54.711576854329110],
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], dtype=float)
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actual = expm(A)
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assert_allclose(actual, desired)
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def test_burkardt_3(self):
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# This example is due to Laub.
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# This matrix is ill-suited for the Taylor series approach.
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# As powers of A are computed, the entries blow up too quickly.
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exp1 = np.exp(1)
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exp39 = np.exp(39)
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A = np.array([
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[0, 1],
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[-39, -40],
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], dtype=float)
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desired = np.array([
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[
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39/(38*exp1) - 1/(38*exp39),
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-np.expm1(-38) / (38*exp1)],
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[
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39*np.expm1(-38) / (38*exp1),
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-1/(38*exp1) + 39/(38*exp39)],
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], dtype=float)
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actual = expm(A)
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assert_allclose(actual, desired)
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def test_burkardt_4(self):
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# This example is due to Moler and Van Loan.
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# The example will cause problems for the series summation approach,
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# as well as for diagonal Pade approximations.
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A = np.array([
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[-49, 24],
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[-64, 31],
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], dtype=float)
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U = np.array([[3, 1], [4, 2]], dtype=float)
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V = np.array([[1, -1/2], [-2, 3/2]], dtype=float)
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w = np.array([-17, -1], dtype=float)
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desired = np.dot(U * np.exp(w), V)
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actual = expm(A)
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assert_allclose(actual, desired)
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def test_burkardt_5(self):
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# This example is due to Moler and Van Loan.
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# This matrix is strictly upper triangular
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# All powers of A are zero beyond some (low) limit.
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# This example will cause problems for Pade approximations.
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A = np.array([
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[0, 6, 0, 0],
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[0, 0, 6, 0],
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[0, 0, 0, 6],
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[0, 0, 0, 0],
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], dtype=float)
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desired = np.array([
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[1, 6, 18, 36],
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[0, 1, 6, 18],
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[0, 0, 1, 6],
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[0, 0, 0, 1],
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], dtype=float)
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actual = expm(A)
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assert_allclose(actual, desired)
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def test_burkardt_6(self):
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# This example is due to Moler and Van Loan.
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# This matrix does not have a complete set of eigenvectors.
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# That means the eigenvector approach will fail.
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exp1 = np.exp(1)
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A = np.array([
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[1, 1],
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[0, 1],
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], dtype=float)
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desired = np.array([
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[exp1, exp1],
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[0, exp1],
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], dtype=float)
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actual = expm(A)
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assert_allclose(actual, desired)
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def test_burkardt_7(self):
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# This example is due to Moler and Van Loan.
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# This matrix is very close to example 5.
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|
# Mathematically, it has a complete set of eigenvectors.
|
||
|
|
# Numerically, however, the calculation will be suspect.
|
||
|
|
exp1 = np.exp(1)
|
||
|
|
eps = np.spacing(1)
|
||
|
|
A = np.array([
|
||
|
|
[1 + eps, 1],
|
||
|
|
[0, 1 - eps],
|
||
|
|
], dtype=float)
|
||
|
|
desired = np.array([
|
||
|
|
[exp1, exp1],
|
||
|
|
[0, exp1],
|
||
|
|
], dtype=float)
|
||
|
|
actual = expm(A)
|
||
|
|
assert_allclose(actual, desired)
|
||
|
|
|
||
|
|
def test_burkardt_8(self):
|
||
|
|
# This matrix was an example in Wikipedia.
|
||
|
|
exp4 = np.exp(4)
|
||
|
|
exp16 = np.exp(16)
|
||
|
|
A = np.array([
|
||
|
|
[21, 17, 6],
|
||
|
|
[-5, -1, -6],
|
||
|
|
[4, 4, 16],
|
||
|
|
], dtype=float)
|
||
|
|
desired = np.array([
|
||
|
|
[13*exp16 - exp4, 13*exp16 - 5*exp4, 2*exp16 - 2*exp4],
|
||
|
|
[-9*exp16 + exp4, -9*exp16 + 5*exp4, -2*exp16 + 2*exp4],
|
||
|
|
[16*exp16, 16*exp16, 4*exp16],
|
||
|
|
], dtype=float) * 0.25
|
||
|
|
actual = expm(A)
|
||
|
|
assert_allclose(actual, desired)
|
||
|
|
|
||
|
|
def test_burkardt_9(self):
|
||
|
|
# This matrix is due to the NAG Library.
|
||
|
|
# It is an example for function F01ECF.
|
||
|
|
A = np.array([
|
||
|
|
[1, 2, 2, 2],
|
||
|
|
[3, 1, 1, 2],
|
||
|
|
[3, 2, 1, 2],
|
||
|
|
[3, 3, 3, 1],
|
||
|
|
], dtype=float)
|
||
|
|
desired = np.array([
|
||
|
|
[740.7038, 610.8500, 542.2743, 549.1753],
|
||
|
|
[731.2510, 603.5524, 535.0884, 542.2743],
|
||
|
|
[823.7630, 679.4257, 603.5524, 610.8500],
|
||
|
|
[998.4355, 823.7630, 731.2510, 740.7038],
|
||
|
|
], dtype=float)
|
||
|
|
actual = expm(A)
|
||
|
|
assert_allclose(actual, desired)
|
||
|
|
|
||
|
|
def test_burkardt_10(self):
|
||
|
|
# This is Ward's example #1.
|
||
|
|
# It is defective and nonderogatory.
|
||
|
|
A = np.array([
|
||
|
|
[4, 2, 0],
|
||
|
|
[1, 4, 1],
|
||
|
|
[1, 1, 4],
|
||
|
|
], dtype=float)
|
||
|
|
assert_allclose(sorted(scipy.linalg.eigvals(A)), (3, 3, 6))
|
||
|
|
desired = np.array([
|
||
|
|
[147.8666224463699, 183.7651386463682, 71.79703239999647],
|
||
|
|
[127.7810855231823, 183.7651386463682, 91.88256932318415],
|
||
|
|
[127.7810855231824, 163.6796017231806, 111.9681062463718],
|
||
|
|
], dtype=float)
|
||
|
|
actual = expm(A)
|
||
|
|
assert_allclose(actual, desired)
|
||
|
|
|
||
|
|
def test_burkardt_11(self):
|
||
|
|
# This is Ward's example #2.
|
||
|
|
# It is a symmetric matrix.
|
||
|
|
A = np.array([
|
||
|
|
[29.87942128909879, 0.7815750847907159, -2.289519314033932],
|
||
|
|
[0.7815750847907159, 25.72656945571064, 8.680737820540137],
|
||
|
|
[-2.289519314033932, 8.680737820540137, 34.39400925519054],
|
||
|
|
], dtype=float)
|
||
|
|
assert_allclose(scipy.linalg.eigvalsh(A), (20, 30, 40))
|
||
|
|
desired = np.array([
|
||
|
|
[
|
||
|
|
5.496313853692378E+15,
|
||
|
|
-1.823188097200898E+16,
|
||
|
|
-3.047577080858001E+16],
|
||
|
|
[
|
||
|
|
-1.823188097200899E+16,
|
||
|
|
6.060522870222108E+16,
|
||
|
|
1.012918429302482E+17],
|
||
|
|
[
|
||
|
|
-3.047577080858001E+16,
|
||
|
|
1.012918429302482E+17,
|
||
|
|
1.692944112408493E+17],
|
||
|
|
], dtype=float)
|
||
|
|
actual = expm(A)
|
||
|
|
assert_allclose(actual, desired)
|
||
|
|
|
||
|
|
def test_burkardt_12(self):
|
||
|
|
# This is Ward's example #3.
|
||
|
|
# Ward's algorithm has difficulty estimating the accuracy
|
||
|
|
# of its results.
|
||
|
|
A = np.array([
|
||
|
|
[-131, 19, 18],
|
||
|
|
[-390, 56, 54],
|
||
|
|
[-387, 57, 52],
|
||
|
|
], dtype=float)
|
||
|
|
assert_allclose(sorted(scipy.linalg.eigvals(A)), (-20, -2, -1))
|
||
|
|
desired = np.array([
|
||
|
|
[-1.509644158793135, 0.3678794391096522, 0.1353352811751005],
|
||
|
|
[-5.632570799891469, 1.471517758499875, 0.4060058435250609],
|
||
|
|
[-4.934938326088363, 1.103638317328798, 0.5413411267617766],
|
||
|
|
], dtype=float)
|
||
|
|
actual = expm(A)
|
||
|
|
assert_allclose(actual, desired)
|
||
|
|
|
||
|
|
def test_burkardt_13(self):
|
||
|
|
# This is Ward's example #4.
|
||
|
|
# This is a version of the Forsythe matrix.
|
||
|
|
# The eigenvector problem is badly conditioned.
|
||
|
|
# Ward's algorithm has difficulty esimating the accuracy
|
||
|
|
# of its results for this problem.
|
||
|
|
#
|
||
|
|
# Check the construction of one instance of this family of matrices.
|
||
|
|
A4_actual = _burkardt_13_power(4, 1)
|
||
|
|
A4_desired = [[0, 1, 0, 0],
|
||
|
|
[0, 0, 1, 0],
|
||
|
|
[0, 0, 0, 1],
|
||
|
|
[1e-4, 0, 0, 0]]
|
||
|
|
assert_allclose(A4_actual, A4_desired)
|
||
|
|
# Check the expm for a few instances.
|
||
|
|
for n in (2, 3, 4, 10):
|
||
|
|
# Approximate expm using Taylor series.
|
||
|
|
# This works well for this matrix family
|
||
|
|
# because each matrix in the summation,
|
||
|
|
# even before dividing by the factorial,
|
||
|
|
# is entrywise positive with max entry 10**(-floor(p/n)*n).
|
||
|
|
k = max(1, int(np.ceil(16/n)))
|
||
|
|
desired = np.zeros((n, n), dtype=float)
|
||
|
|
for p in range(n*k):
|
||
|
|
Ap = _burkardt_13_power(n, p)
|
||
|
|
assert_equal(np.min(Ap), 0)
|
||
|
|
assert_allclose(np.max(Ap), np.power(10, -np.floor(p/n)*n))
|
||
|
|
desired += Ap / factorial(p)
|
||
|
|
actual = expm(_burkardt_13_power(n, 1))
|
||
|
|
assert_allclose(actual, desired)
|
||
|
|
|
||
|
|
def test_burkardt_14(self):
|
||
|
|
# This is Moler's example.
|
||
|
|
# This badly scaled matrix caused problems for MATLAB's expm().
|
||
|
|
A = np.array([
|
||
|
|
[0, 1e-8, 0],
|
||
|
|
[-(2e10 + 4e8/6.), -3, 2e10],
|
||
|
|
[200./3., 0, -200./3.],
|
||
|
|
], dtype=float)
|
||
|
|
desired = np.array([
|
||
|
|
[0.446849468283175, 1.54044157383952e-09, 0.462811453558774],
|
||
|
|
[-5743067.77947947, -0.0152830038686819, -4526542.71278401],
|
||
|
|
[0.447722977849494, 1.54270484519591e-09, 0.463480648837651],
|
||
|
|
], dtype=float)
|
||
|
|
actual = expm(A)
|
||
|
|
assert_allclose(actual, desired)
|
||
|
|
|
||
|
|
def test_pascal(self):
|
||
|
|
# Test pascal triangle.
|
||
|
|
# Nilpotent exponential, used to trigger a failure (gh-8029)
|
||
|
|
|
||
|
|
for scale in [1.0, 1e-3, 1e-6]:
|
||
|
|
for n in range(120):
|
||
|
|
A = np.diag(np.arange(1, n + 1), -1) * scale
|
||
|
|
B = expm(A)
|
||
|
|
|
||
|
|
sc = scale**np.arange(n, -1, -1)
|
||
|
|
if np.any(sc < 1e-300):
|
||
|
|
continue
|
||
|
|
|
||
|
|
got = B
|
||
|
|
expected = binom(np.arange(n + 1)[:,None],
|
||
|
|
np.arange(n + 1)[None,:]) * sc[None,:] / sc[:,None]
|
||
|
|
err = abs(expected - got).max()
|
||
|
|
atol = 1e-13 * abs(expected).max()
|
||
|
|
assert_allclose(got, expected, atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
class TestOperators(object):
|
||
|
|
|
||
|
|
def test_product_operator(self):
|
||
|
|
random.seed(1234)
|
||
|
|
n = 5
|
||
|
|
k = 2
|
||
|
|
nsamples = 10
|
||
|
|
for i in range(nsamples):
|
||
|
|
A = np.random.randn(n, n)
|
||
|
|
B = np.random.randn(n, n)
|
||
|
|
C = np.random.randn(n, n)
|
||
|
|
D = np.random.randn(n, k)
|
||
|
|
op = ProductOperator(A, B, C)
|
||
|
|
assert_allclose(op.matmat(D), A.dot(B).dot(C).dot(D))
|
||
|
|
assert_allclose(op.T.matmat(D), (A.dot(B).dot(C)).T.dot(D))
|
||
|
|
|
||
|
|
def test_matrix_power_operator(self):
|
||
|
|
random.seed(1234)
|
||
|
|
n = 5
|
||
|
|
k = 2
|
||
|
|
p = 3
|
||
|
|
nsamples = 10
|
||
|
|
for i in range(nsamples):
|
||
|
|
A = np.random.randn(n, n)
|
||
|
|
B = np.random.randn(n, k)
|
||
|
|
op = MatrixPowerOperator(A, p)
|
||
|
|
assert_allclose(op.matmat(B), matrix_power(A, p).dot(B))
|
||
|
|
assert_allclose(op.T.matmat(B), matrix_power(A, p).T.dot(B))
|
||
|
|
|