# # Created by: Pearu Peterson, March 2002 # """ Test functions for scipy.linalg.matfuncs module """ from __future__ import division, print_function, absolute_import import math import numpy as np from numpy import array, eye, exp, random from numpy.linalg import matrix_power from numpy.testing import ( assert_allclose, assert_, assert_array_almost_equal, assert_equal, assert_array_almost_equal_nulp) from scipy._lib._numpy_compat import suppress_warnings from scipy.sparse import csc_matrix, SparseEfficiencyWarning from scipy.sparse.construct import eye as speye from scipy.sparse.linalg.matfuncs import (expm, _expm, ProductOperator, MatrixPowerOperator, _onenorm_matrix_power_nnm) from scipy.linalg import logm from scipy.special import factorial, binom import scipy.sparse import scipy.sparse.linalg def _burkardt_13_power(n, p): """ A helper function for testing matrix functions. Parameters ---------- n : integer greater than 1 Order of the square matrix to be returned. p : non-negative integer Power of the matrix. Returns ------- out : ndarray representing a square matrix A Forsythe matrix of order n, raised to the power p. """ # Input validation. if n != int(n) or n < 2: raise ValueError('n must be an integer greater than 1') n = int(n) if p != int(p) or p < 0: raise ValueError('p must be a non-negative integer') p = int(p) # Construct the matrix explicitly. a, b = divmod(p, n) large = np.power(10.0, -n*a) small = large * np.power(10.0, -n) return np.diag([large]*(n-b), b) + np.diag([small]*b, b-n) def test_onenorm_matrix_power_nnm(): np.random.seed(1234) for n in range(1, 5): for p in range(5): M = np.random.random((n, n)) Mp = np.linalg.matrix_power(M, p) observed = _onenorm_matrix_power_nnm(M, p) expected = np.linalg.norm(Mp, 1) assert_allclose(observed, expected) class TestExpM(object): def test_zero_ndarray(self): a = array([[0.,0],[0,0]]) assert_array_almost_equal(expm(a),[[1,0],[0,1]]) def test_zero_sparse(self): a = csc_matrix([[0.,0],[0,0]]) assert_array_almost_equal(expm(a).toarray(),[[1,0],[0,1]]) def test_zero_matrix(self): a = np.matrix([[0.,0],[0,0]]) assert_array_almost_equal(expm(a),[[1,0],[0,1]]) def test_misc_types(self): A = expm(np.array([[1]])) assert_allclose(expm(((1,),)), A) assert_allclose(expm([[1]]), A) assert_allclose(expm(np.matrix([[1]])), A) assert_allclose(expm(np.array([[1]])), A) assert_allclose(expm(csc_matrix([[1]])).A, A) B = expm(np.array([[1j]])) assert_allclose(expm(((1j,),)), B) assert_allclose(expm([[1j]]), B) assert_allclose(expm(np.matrix([[1j]])), B) assert_allclose(expm(csc_matrix([[1j]])).A, B) def test_bidiagonal_sparse(self): A = csc_matrix([ [1, 3, 0], [0, 1, 5], [0, 0, 2]], dtype=float) e1 = math.exp(1) e2 = math.exp(2) expected = np.array([ [e1, 3*e1, 15*(e2 - 2*e1)], [0, e1, 5*(e2 - e1)], [0, 0, e2]], dtype=float) observed = expm(A).toarray() assert_array_almost_equal(observed, expected) def test_padecases_dtype_float(self): for dtype in [np.float32, np.float64]: for scale in [1e-2, 1e-1, 5e-1, 1, 10]: A = scale * eye(3, dtype=dtype) observed = expm(A) expected = exp(scale) * eye(3, dtype=dtype) assert_array_almost_equal_nulp(observed, expected, nulp=100) def test_padecases_dtype_complex(self): for dtype in [np.complex64, np.complex128]: for scale in [1e-2, 1e-1, 5e-1, 1, 10]: A = scale * eye(3, dtype=dtype) observed = expm(A) expected = exp(scale) * eye(3, dtype=dtype) assert_array_almost_equal_nulp(observed, expected, nulp=100) def test_padecases_dtype_sparse_float(self): # float32 and complex64 lead to errors in spsolve/UMFpack dtype = np.float64 for scale in [1e-2, 1e-1, 5e-1, 1, 10]: a = scale * speye(3, 3, dtype=dtype, format='csc') e = exp(scale) * eye(3, dtype=dtype) with suppress_warnings() as sup: sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure of a csc_matrix is expensive.") exact_onenorm = _expm(a, use_exact_onenorm=True).toarray() inexact_onenorm = _expm(a, use_exact_onenorm=False).toarray() assert_array_almost_equal_nulp(exact_onenorm, e, nulp=100) assert_array_almost_equal_nulp(inexact_onenorm, e, nulp=100) def test_padecases_dtype_sparse_complex(self): # float32 and complex64 lead to errors in spsolve/UMFpack dtype = np.complex128 for scale in [1e-2, 1e-1, 5e-1, 1, 10]: a = scale * speye(3, 3, dtype=dtype, format='csc') e = exp(scale) * eye(3, dtype=dtype) with suppress_warnings() as sup: sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure of a csc_matrix is expensive.") assert_array_almost_equal_nulp(expm(a).toarray(), e, nulp=100) def test_logm_consistency(self): random.seed(1234) for dtype in [np.float64, np.complex128]: for n in range(1, 10): for scale in [1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1, 1e2]: # make logm(A) be of a given scale A = (eye(n) + random.rand(n, n) * scale).astype(dtype) if np.iscomplexobj(A): A = A + 1j * random.rand(n, n) * scale assert_array_almost_equal(expm(logm(A)), A) def test_integer_matrix(self): Q = np.array([ [-3, 1, 1, 1], [1, -3, 1, 1], [1, 1, -3, 1], [1, 1, 1, -3]]) assert_allclose(expm(Q), expm(1.0 * Q)) def test_integer_matrix_2(self): # Check for integer overflows Q = np.array([[-500, 500, 0, 0], [0, -550, 360, 190], [0, 630, -630, 0], [0, 0, 0, 0]], dtype=np.int16) assert_allclose(expm(Q), expm(1.0 * Q)) Q = csc_matrix(Q) assert_allclose(expm(Q).A, expm(1.0 * Q).A) def test_triangularity_perturbation(self): # Experiment (1) of # Awad H. Al-Mohy and Nicholas J. Higham (2012) # Improved Inverse Scaling and Squaring Algorithms # for the Matrix Logarithm. A = np.array([ [3.2346e-1, 3e4, 3e4, 3e4], [0, 3.0089e-1, 3e4, 3e4], [0, 0, 3.221e-1, 3e4], [0, 0, 0, 3.0744e-1]], dtype=float) A_logm = np.array([ [-1.12867982029050462e+00, 9.61418377142025565e+04, -4.52485573953179264e+09, 2.92496941103871812e+14], [0.00000000000000000e+00, -1.20101052953082288e+00, 9.63469687211303099e+04, -4.68104828911105442e+09], [0.00000000000000000e+00, 0.00000000000000000e+00, -1.13289322264498393e+00, 9.53249183094775653e+04], [0.00000000000000000e+00, 0.00000000000000000e+00, 0.00000000000000000e+00, -1.17947533272554850e+00]], dtype=float) assert_allclose(expm(A_logm), A, rtol=1e-4) # Perturb the upper triangular matrix by tiny amounts, # so that it becomes technically not upper triangular. random.seed(1234) tiny = 1e-17 A_logm_perturbed = A_logm.copy() A_logm_perturbed[1, 0] = tiny with suppress_warnings() as sup: sup.filter(RuntimeWarning, "Ill-conditioned.*") A_expm_logm_perturbed = expm(A_logm_perturbed) rtol = 1e-4 atol = 100 * tiny assert_(not np.allclose(A_expm_logm_perturbed, A, rtol=rtol, atol=atol)) def test_burkardt_1(self): # This matrix is diagonal. # The calculation of the matrix exponential is simple. # # This is the first of a series of matrix exponential tests # collected by John Burkardt from the following sources. # # Alan Laub, # Review of "Linear System Theory" by Joao Hespanha, # SIAM Review, # Volume 52, Number 4, December 2010, pages 779--781. # # Cleve Moler and Charles Van Loan, # Nineteen Dubious Ways to Compute the Exponential of a Matrix, # Twenty-Five Years Later, # SIAM Review, # Volume 45, Number 1, March 2003, pages 3--49. # # Cleve Moler, # Cleve's Corner: A Balancing Act for the Matrix Exponential, # 23 July 2012. # # Robert Ward, # Numerical computation of the matrix exponential # with accuracy estimate, # SIAM Journal on Numerical Analysis, # Volume 14, Number 4, September 1977, pages 600--610. exp1 = np.exp(1) exp2 = np.exp(2) A = np.array([ [1, 0], [0, 2], ], dtype=float) desired = np.array([ [exp1, 0], [0, exp2], ], dtype=float) actual = expm(A) assert_allclose(actual, desired) def test_burkardt_2(self): # This matrix is symmetric. # The calculation of the matrix exponential is straightforward. A = np.array([ [1, 3], [3, 2], ], dtype=float) desired = np.array([ [39.322809708033859, 46.166301438885753], [46.166301438885768, 54.711576854329110], ], dtype=float) actual = expm(A) assert_allclose(actual, desired) def test_burkardt_3(self): # This example is due to Laub. # This matrix is ill-suited for the Taylor series approach. # As powers of A are computed, the entries blow up too quickly. exp1 = np.exp(1) exp39 = np.exp(39) A = np.array([ [0, 1], [-39, -40], ], dtype=float) desired = np.array([ [ 39/(38*exp1) - 1/(38*exp39), -np.expm1(-38) / (38*exp1)], [ 39*np.expm1(-38) / (38*exp1), -1/(38*exp1) + 39/(38*exp39)], ], dtype=float) actual = expm(A) assert_allclose(actual, desired) def test_burkardt_4(self): # This example is due to Moler and Van Loan. # The example will cause problems for the series summation approach, # as well as for diagonal Pade approximations. A = np.array([ [-49, 24], [-64, 31], ], dtype=float) U = np.array([[3, 1], [4, 2]], dtype=float) V = np.array([[1, -1/2], [-2, 3/2]], dtype=float) w = np.array([-17, -1], dtype=float) desired = np.dot(U * np.exp(w), V) actual = expm(A) assert_allclose(actual, desired) def test_burkardt_5(self): # This example is due to Moler and Van Loan. # This matrix is strictly upper triangular # All powers of A are zero beyond some (low) limit. # This example will cause problems for Pade approximations. A = np.array([ [0, 6, 0, 0], [0, 0, 6, 0], [0, 0, 0, 6], [0, 0, 0, 0], ], dtype=float) desired = np.array([ [1, 6, 18, 36], [0, 1, 6, 18], [0, 0, 1, 6], [0, 0, 0, 1], ], dtype=float) actual = expm(A) assert_allclose(actual, desired) def test_burkardt_6(self): # This example is due to Moler and Van Loan. # This matrix does not have a complete set of eigenvectors. # That means the eigenvector approach will fail. exp1 = np.exp(1) A = np.array([ [1, 1], [0, 1], ], dtype=float) desired = np.array([ [exp1, exp1], [0, exp1], ], dtype=float) actual = expm(A) assert_allclose(actual, desired) def test_burkardt_7(self): # This example is due to Moler and Van Loan. # This matrix is very close to example 5. # 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))