import os import numpy as np from numpy.testing import assert_array_almost_equal, assert_allclose import pytest from pytest import raises as assert_raises from scipy.linalg import solve_sylvester from scipy.linalg import solve_continuous_lyapunov, solve_discrete_lyapunov from scipy.linalg import solve_continuous_are, solve_discrete_are from scipy.linalg import block_diag, solve, LinAlgError from scipy.sparse._sputils import matrix def _load_data(name): """ Load npz data file under data/ Returns a copy of the data, rather than keeping the npz file open. """ filename = os.path.join(os.path.abspath(os.path.dirname(__file__)), 'data', name) with np.load(filename) as f: return dict(f.items()) class TestSolveLyapunov: cases = [ (np.array([[1, 2], [3, 4]]), np.array([[9, 10], [11, 12]])), # a, q all complex. (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]), np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])), # a real; q complex. (np.array([[1.0, 2.0], [3.0, 5.0]]), np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])), # a complex; q real. (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]), np.array([[2.0, 2.0], [-1.0, 2.0]])), # An example from Kitagawa, 1977 (np.array([[3, 9, 5, 1, 4], [1, 2, 3, 8, 4], [4, 6, 6, 6, 3], [1, 5, 2, 0, 7], [5, 3, 3, 1, 5]]), np.array([[2, 4, 1, 0, 1], [4, 1, 0, 2, 0], [1, 0, 3, 0, 3], [0, 2, 0, 1, 0], [1, 0, 3, 0, 4]])), # Companion matrix example. a complex; q real; a.shape[0] = 11 (np.array([[0.100+0.j, 0.091+0.j, 0.082+0.j, 0.073+0.j, 0.064+0.j, 0.055+0.j, 0.046+0.j, 0.037+0.j, 0.028+0.j, 0.019+0.j, 0.010+0.j], [1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j], [0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j], [0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j], [0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j], [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j], [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j], [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j], [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j], [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j], [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j]]), np.eye(11)), # https://github.com/scipy/scipy/issues/4176 (matrix([[0, 1], [-1/2, -1]]), (matrix([0, 3]).T @ matrix([0, 3]).T.T)), # https://github.com/scipy/scipy/issues/4176 (matrix([[0, 1], [-1/2, -1]]), (np.array(matrix([0, 3]).T @ matrix([0, 3]).T.T))), ] def test_continuous_squareness_and_shape(self): nsq = np.ones((3, 2)) sq = np.eye(3) assert_raises(ValueError, solve_continuous_lyapunov, nsq, sq) assert_raises(ValueError, solve_continuous_lyapunov, sq, nsq) assert_raises(ValueError, solve_continuous_lyapunov, sq, np.eye(2)) def check_continuous_case(self, a, q): x = solve_continuous_lyapunov(a, q) assert_array_almost_equal( np.dot(a, x) + np.dot(x, a.conj().transpose()), q) def check_discrete_case(self, a, q, method=None): x = solve_discrete_lyapunov(a, q, method=method) assert_array_almost_equal( np.dot(np.dot(a, x), a.conj().transpose()) - x, -1.0*q) def test_cases(self): for case in self.cases: self.check_continuous_case(case[0], case[1]) self.check_discrete_case(case[0], case[1]) self.check_discrete_case(case[0], case[1], method='direct') self.check_discrete_case(case[0], case[1], method='bilinear') class TestSolveContinuousAre: mat6 = _load_data('carex_6_data.npz') mat15 = _load_data('carex_15_data.npz') mat18 = _load_data('carex_18_data.npz') mat19 = _load_data('carex_19_data.npz') mat20 = _load_data('carex_20_data.npz') cases = [ # Carex examples taken from (with default parameters): # [1] P.BENNER, A.J. LAUB, V. MEHRMANN: 'A Collection of Benchmark # Examples for the Numerical Solution of Algebraic Riccati # Equations II: Continuous-Time Case', Tech. Report SPC 95_23, # Fak. f. Mathematik, TU Chemnitz-Zwickau (Germany), 1995. # # The format of the data is (a, b, q, r, knownfailure), where # knownfailure is None if the test passes or a string # indicating the reason for failure. # # Test Case 0: carex #1 (np.diag([1.], 1), np.array([[0], [1]]), block_diag(1., 2.), 1, None), # Test Case 1: carex #2 (np.array([[4, 3], [-4.5, -3.5]]), np.array([[1], [-1]]), np.array([[9, 6], [6, 4.]]), 1, None), # Test Case 2: carex #3 (np.array([[0, 1, 0, 0], [0, -1.89, 0.39, -5.53], [0, -0.034, -2.98, 2.43], [0.034, -0.0011, -0.99, -0.21]]), np.array([[0, 0], [0.36, -1.6], [-0.95, -0.032], [0.03, 0]]), np.array([[2.313, 2.727, 0.688, 0.023], [2.727, 4.271, 1.148, 0.323], [0.688, 1.148, 0.313, 0.102], [0.023, 0.323, 0.102, 0.083]]), np.eye(2), None), # Test Case 3: carex #4 (np.array([[-0.991, 0.529, 0, 0, 0, 0, 0, 0], [0.522, -1.051, 0.596, 0, 0, 0, 0, 0], [0, 0.522, -1.118, 0.596, 0, 0, 0, 0], [0, 0, 0.522, -1.548, 0.718, 0, 0, 0], [0, 0, 0, 0.922, -1.64, 0.799, 0, 0], [0, 0, 0, 0, 0.922, -1.721, 0.901, 0], [0, 0, 0, 0, 0, 0.922, -1.823, 1.021], [0, 0, 0, 0, 0, 0, 0.922, -1.943]]), np.array([[3.84, 4.00, 37.60, 3.08, 2.36, 2.88, 3.08, 3.00], [-2.88, -3.04, -2.80, -2.32, -3.32, -3.82, -4.12, -3.96]] ).T * 0.001, np.array([[1.0, 0.0, 0.0, 0.0, 0.5, 0.0, 0.0, 0.1], [0.0, 1.0, 0.0, 0.0, 0.1, 0.0, 0.0, 0.0], [0.0, 0.0, 1.0, 0.0, 0.0, 0.5, 0.0, 0.0], [0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0], [0.5, 0.1, 0.0, 0.0, 0.1, 0.0, 0.0, 0.0], [0.0, 0.0, 0.5, 0.0, 0.0, 0.1, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.1, 0.0], [0.1, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.1]]), np.eye(2), None), # Test Case 4: carex #5 (np.array( [[-4.019, 5.120, 0., 0., -2.082, 0., 0., 0., 0.870], [-0.346, 0.986, 0., 0., -2.340, 0., 0., 0., 0.970], [-7.909, 15.407, -4.069, 0., -6.450, 0., 0., 0., 2.680], [-21.816, 35.606, -0.339, -3.870, -17.800, 0., 0., 0., 7.390], [-60.196, 98.188, -7.907, 0.340, -53.008, 0., 0., 0., 20.400], [0, 0, 0, 0, 94.000, -147.200, 0., 53.200, 0.], [0, 0, 0, 0, 0, 94.000, -147.200, 0, 0], [0, 0, 0, 0, 0, 12.800, 0.000, -31.600, 0], [0, 0, 0, 0, 12.800, 0.000, 0.000, 18.800, -31.600]]), np.array([[0.010, -0.011, -0.151], [0.003, -0.021, 0.000], [0.009, -0.059, 0.000], [0.024, -0.162, 0.000], [0.068, -0.445, 0.000], [0.000, 0.000, 0.000], [0.000, 0.000, 0.000], [0.000, 0.000, 0.000], [0.000, 0.000, 0.000]]), np.eye(9), np.eye(3), None), # Test Case 5: carex #6 (mat6['A'], mat6['B'], mat6['Q'], mat6['R'], None), # Test Case 6: carex #7 (np.array([[1, 0], [0, -2.]]), np.array([[1e-6], [0]]), np.ones((2, 2)), 1., 'Bad residual accuracy'), # Test Case 7: carex #8 (block_diag(-0.1, -0.02), np.array([[0.100, 0.000], [0.001, 0.010]]), np.array([[100, 1000], [1000, 10000]]), np.ones((2, 2)) + block_diag(1e-6, 0), None), # Test Case 8: carex #9 (np.array([[0, 1e6], [0, 0]]), np.array([[0], [1.]]), np.eye(2), 1., None), # Test Case 9: carex #10 (np.array([[1.0000001, 1], [1., 1.0000001]]), np.eye(2), np.eye(2), np.eye(2), None), # Test Case 10: carex #11 (np.array([[3, 1.], [4, 2]]), np.array([[1], [1]]), np.array([[-11, -5], [-5, -2.]]), 1., None), # Test Case 11: carex #12 (np.array([[7000000., 2000000., -0.], [2000000., 6000000., -2000000.], [0., -2000000., 5000000.]]) / 3, np.eye(3), np.array([[1., -2., -2.], [-2., 1., -2.], [-2., -2., 1.]]).dot( np.diag([1e-6, 1, 1e6])).dot( np.array([[1., -2., -2.], [-2., 1., -2.], [-2., -2., 1.]])) / 9, np.eye(3) * 1e6, 'Bad Residual Accuracy'), # Test Case 12: carex #13 (np.array([[0, 0.4, 0, 0], [0, 0, 0.345, 0], [0, -0.524e6, -0.465e6, 0.262e6], [0, 0, 0, -1e6]]), np.array([[0, 0, 0, 1e6]]).T, np.diag([1, 0, 1, 0]), 1., None), # Test Case 13: carex #14 (np.array([[-1e-6, 1, 0, 0], [-1, -1e-6, 0, 0], [0, 0, 1e-6, 1], [0, 0, -1, 1e-6]]), np.ones((4, 1)), np.ones((4, 4)), 1., None), # Test Case 14: carex #15 (mat15['A'], mat15['B'], mat15['Q'], mat15['R'], None), # Test Case 15: carex #16 (np.eye(64, 64, k=-1) + np.eye(64, 64)*(-2.) + np.rot90( block_diag(1, np.zeros((62, 62)), 1)) + np.eye(64, 64, k=1), np.eye(64), np.eye(64), np.eye(64), None), # Test Case 16: carex #17 (np.diag(np.ones((20, )), 1), np.flipud(np.eye(21, 1)), np.eye(21, 1) * np.eye(21, 1).T, 1, 'Bad Residual Accuracy'), # Test Case 17: carex #18 (mat18['A'], mat18['B'], mat18['Q'], mat18['R'], None), # Test Case 18: carex #19 (mat19['A'], mat19['B'], mat19['Q'], mat19['R'], 'Bad Residual Accuracy'), # Test Case 19: carex #20 (mat20['A'], mat20['B'], mat20['Q'], mat20['R'], 'Bad Residual Accuracy') ] # Makes the minimum precision requirements customized to the test. # Here numbers represent the number of decimals that agrees with zero # matrix when the solution x is plugged in to the equation. # # res = array([[8e-3,1e-16],[1e-16,1e-20]]) --> min_decimal[k] = 2 # # If the test is failing use "None" for that entry. # min_decimal = (14, 12, 13, 14, 11, 6, None, 5, 7, 14, 14, None, 9, 14, 13, 14, None, 12, None, None) @pytest.mark.parametrize("j, case", enumerate(cases)) def test_solve_continuous_are(self, j, case): """Checks if 0 = XA + A'X - XB(R)^{-1} B'X + Q is true""" a, b, q, r, knownfailure = case if knownfailure: pytest.xfail(reason=knownfailure) dec = self.min_decimal[j] x = solve_continuous_are(a, b, q, r) res = x @ a + a.conj().T @ x + q out_fact = x @ b res -= out_fact @ solve(np.atleast_2d(r), out_fact.conj().T) assert_array_almost_equal(res, np.zeros_like(res), decimal=dec) class TestSolveDiscreteAre: cases = [ # Darex examples taken from (with default parameters): # [1] P.BENNER, A.J. LAUB, V. MEHRMANN: 'A Collection of Benchmark # Examples for the Numerical Solution of Algebraic Riccati # Equations II: Discrete-Time Case', Tech. Report SPC 95_23, # Fak. f. Mathematik, TU Chemnitz-Zwickau (Germany), 1995. # [2] T. GUDMUNDSSON, C. KENNEY, A.J. LAUB: 'Scaling of the # Discrete-Time Algebraic Riccati Equation to Enhance Stability # of the Schur Solution Method', IEEE Trans.Aut.Cont., vol.37(4) # # The format of the data is (a, b, q, r, knownfailure), where # knownfailure is None if the test passes or a string # indicating the reason for failure. # # TEST CASE 0 : Complex a; real b, q, r (np.array([[2, 1-2j], [0, -3j]]), np.array([[0], [1]]), np.array([[1, 0], [0, 2]]), np.array([[1]]), None), # TEST CASE 1 :Real a, q, r; complex b (np.array([[2, 1], [0, -1]]), np.array([[-2j], [1j]]), np.array([[1, 0], [0, 2]]), np.array([[1]]), None), # TEST CASE 2 : Real a, b; complex q, r (np.array([[3, 1], [0, -1]]), np.array([[1, 2], [1, 3]]), np.array([[1, 1+1j], [1-1j, 2]]), np.array([[2, -2j], [2j, 3]]), None), # TEST CASE 3 : User-reported gh-2251 (Trac #1732) (np.array([[0.63399379, 0.54906824, 0.76253406], [0.5404729, 0.53745766, 0.08731853], [0.27524045, 0.84922129, 0.4681622]]), np.array([[0.96861695], [0.05532739], [0.78934047]]), np.eye(3), np.eye(1), None), # TEST CASE 4 : darex #1 (np.array([[4, 3], [-4.5, -3.5]]), np.array([[1], [-1]]), np.array([[9, 6], [6, 4]]), np.array([[1]]), None), # TEST CASE 5 : darex #2 (np.array([[0.9512, 0], [0, 0.9048]]), np.array([[4.877, 4.877], [-1.1895, 3.569]]), np.array([[0.005, 0], [0, 0.02]]), np.array([[1/3, 0], [0, 3]]), None), # TEST CASE 6 : darex #3 (np.array([[2, -1], [1, 0]]), np.array([[1], [0]]), np.array([[0, 0], [0, 1]]), np.array([[0]]), None), # TEST CASE 7 : darex #4 (skipped the gen. Ric. term S) (np.array([[0, 1], [0, -1]]), np.array([[1, 0], [2, 1]]), np.array([[-4, -4], [-4, 7]]) * (1/11), np.array([[9, 3], [3, 1]]), None), # TEST CASE 8 : darex #5 (np.array([[0, 1], [0, 0]]), np.array([[0], [1]]), np.array([[1, 2], [2, 4]]), np.array([[1]]), None), # TEST CASE 9 : darex #6 (np.array([[0.998, 0.067, 0, 0], [-.067, 0.998, 0, 0], [0, 0, 0.998, 0.153], [0, 0, -.153, 0.998]]), np.array([[0.0033, 0.0200], [0.1000, -.0007], [0.0400, 0.0073], [-.0028, 0.1000]]), np.array([[1.87, 0, 0, -0.244], [0, 0.744, 0.205, 0], [0, 0.205, 0.589, 0], [-0.244, 0, 0, 1.048]]), np.eye(2), None), # TEST CASE 10 : darex #7 (np.array([[0.984750, -.079903, 0.0009054, -.0010765], [0.041588, 0.998990, -.0358550, 0.0126840], [-.546620, 0.044916, -.3299100, 0.1931800], [2.662400, -.100450, -.9245500, -.2632500]]), np.array([[0.0037112, 0.0007361], [-.0870510, 9.3411e-6], [-1.198440, -4.1378e-4], [-3.192700, 9.2535e-4]]), np.eye(4)*1e-2, np.eye(2), None), # TEST CASE 11 : darex #8 (np.array([[-0.6000000, -2.2000000, -3.6000000, -5.4000180], [1.0000000, 0.6000000, 0.8000000, 3.3999820], [0.0000000, 1.0000000, 1.8000000, 3.7999820], [0.0000000, 0.0000000, 0.0000000, -0.9999820]]), np.array([[1.0, -1.0, -1.0, -1.0], [0.0, 1.0, -1.0, -1.0], [0.0, 0.0, 1.0, -1.0], [0.0, 0.0, 0.0, 1.0]]), np.array([[2, 1, 3, 6], [1, 2, 2, 5], [3, 2, 6, 11], [6, 5, 11, 22]]), np.eye(4), None), # TEST CASE 12 : darex #9 (np.array([[95.4070, 1.9643, 0.3597, 0.0673, 0.0190], [40.8490, 41.3170, 16.0840, 4.4679, 1.1971], [12.2170, 26.3260, 36.1490, 15.9300, 12.3830], [4.1118, 12.8580, 27.2090, 21.4420, 40.9760], [0.1305, 0.5808, 1.8750, 3.6162, 94.2800]]) * 0.01, np.array([[0.0434, -0.0122], [2.6606, -1.0453], [3.7530, -5.5100], [3.6076, -6.6000], [0.4617, -0.9148]]) * 0.01, np.eye(5), np.eye(2), None), # TEST CASE 13 : darex #10 (np.kron(np.eye(2), np.diag([1, 1], k=1)), np.kron(np.eye(2), np.array([[0], [0], [1]])), np.array([[1, 1, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -1, 0], [0, 0, 0, -1, 1, 0], [0, 0, 0, 0, 0, 0]]), np.array([[3, 0], [0, 1]]), None), # TEST CASE 14 : darex #11 (0.001 * np.array( [[870.1, 135.0, 11.59, .5014, -37.22, .3484, 0, 4.242, 7.249], [76.55, 897.4, 12.72, 0.5504, -40.16, .3743, 0, 4.53, 7.499], [-127.2, 357.5, 817, 1.455, -102.8, .987, 0, 11.85, 18.72], [-363.5, 633.9, 74.91, 796.6, -273.5, 2.653, 0, 31.72, 48.82], [-960, 1645.9, -128.9, -5.597, 71.42, 7.108, 0, 84.52, 125.9], [-664.4, 112.96, -88.89, -3.854, 84.47, 13.6, 0, 144.3, 101.6], [-410.2, 693, -54.71, -2.371, 66.49, 12.49, .1063, 99.97, 69.67], [-179.9, 301.7, -23.93, -1.035, 60.59, 22.16, 0, 213.9, 35.54], [-345.1, 580.4, -45.96, -1.989, 105.6, 19.86, 0, 219.1, 215.2]]), np.array([[4.7600, -0.5701, -83.6800], [0.8790, -4.7730, -2.7300], [1.4820, -13.1200, 8.8760], [3.8920, -35.1300, 24.8000], [10.3400, -92.7500, 66.8000], [7.2030, -61.5900, 38.3400], [4.4540, -36.8300, 20.2900], [1.9710, -15.5400, 6.9370], [3.7730, -30.2800, 14.6900]]) * 0.001, np.diag([50, 0, 0, 0, 50, 0, 0, 0, 0]), np.eye(3), None), # TEST CASE 15 : darex #12 - numerically least accurate example (np.array([[0, 1e6], [0, 0]]), np.array([[0], [1]]), np.eye(2), np.array([[1]]), "Presumed issue with OpenBLAS, see gh-16926"), # TEST CASE 16 : darex #13 (np.array([[16, 10, -2], [10, 13, -8], [-2, -8, 7]]) * (1/9), np.eye(3), 1e6 * np.eye(3), 1e6 * np.eye(3), "Issue with OpenBLAS, see gh-16926"), # TEST CASE 17 : darex #14 (np.array([[1 - 1/1e8, 0, 0, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]]), np.array([[1e-08], [0], [0], [0]]), np.diag([0, 0, 0, 1]), np.array([[0.25]]), None), # TEST CASE 18 : darex #15 (np.eye(100, k=1), np.flipud(np.eye(100, 1)), np.eye(100), np.array([[1]]), None) ] # Makes the minimum precision requirements customized to the test. # Here numbers represent the number of decimals that agrees with zero # matrix when the solution x is plugged in to the equation. # # res = array([[8e-3,1e-16],[1e-16,1e-20]]) --> min_decimal[k] = 2 # # If the test is failing use "None" for that entry. # min_decimal = (12, 14, 13, 14, 13, 16, 18, 14, 14, 13, 14, 13, 13, 14, 12, 2, 5, 6, 10) max_tol = [1.5 * 10**-ind for ind in min_decimal] # relaxed tolerance in gh-18012 after bump to OpenBLAS max_tol[11] = 2.5e-13 @pytest.mark.parametrize("j, case", enumerate(cases)) def test_solve_discrete_are(self, j, case): """Checks if X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q) is true""" a, b, q, r, knownfailure = case if knownfailure: pytest.xfail(reason=knownfailure) atol = self.max_tol[j] x = solve_discrete_are(a, b, q, r) bH = b.conj().T xa, xb = x @ a, x @ b res = a.conj().T @ xa - x + q res -= a.conj().T @ xb @ (solve(r + bH @ xb, bH) @ xa) # changed from # assert_array_almost_equal(res, np.zeros_like(res), decimal=dec) # in gh-18012 as it's easier to relax a tolerance and allclose is # preferred assert_allclose(res, np.zeros_like(res), atol=atol) def test_infeasible(self): # An infeasible example taken from https://arxiv.org/abs/1505.04861v1 A = np.triu(np.ones((3, 3))) A[0, 1] = -1 B = np.array([[1, 1, 0], [0, 0, 1]]).T Q = np.full_like(A, -2) + np.diag([8, -1, -1.9]) R = np.diag([-10, 0.1]) assert_raises(LinAlgError, solve_continuous_are, A, B, Q, R) def test_solve_generalized_continuous_are(): cases = [ # Two random examples differ by s term # in the absence of any literature for demanding examples. (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01], [4.617139e-02, 6.948286e-01, 3.444608e-02], [9.713178e-02, 3.170995e-01, 4.387444e-01]]), np.array([[3.815585e-01, 1.868726e-01], [7.655168e-01, 4.897644e-01], [7.951999e-01, 4.455862e-01]]), np.eye(3), np.eye(2), np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01], [7.093648e-01, 6.797027e-01, 1.189977e-01], [7.546867e-01, 6.550980e-01, 4.983641e-01]]), np.zeros((3, 2)), None), (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01], [4.617139e-02, 6.948286e-01, 3.444608e-02], [9.713178e-02, 3.170995e-01, 4.387444e-01]]), np.array([[3.815585e-01, 1.868726e-01], [7.655168e-01, 4.897644e-01], [7.951999e-01, 4.455862e-01]]), np.eye(3), np.eye(2), np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01], [7.093648e-01, 6.797027e-01, 1.189977e-01], [7.546867e-01, 6.550980e-01, 4.983641e-01]]), np.ones((3, 2)), None) ] min_decimal = (10, 10) def _test_factory(case, dec): """Checks if X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q) is true""" a, b, q, r, e, s, knownfailure = case if knownfailure: pytest.xfail(reason=knownfailure) x = solve_continuous_are(a, b, q, r, e, s) res = a.conj().T.dot(x.dot(e)) + e.conj().T.dot(x.dot(a)) + q out_fact = e.conj().T.dot(x).dot(b) + s res -= out_fact.dot(solve(np.atleast_2d(r), out_fact.conj().T)) assert_array_almost_equal(res, np.zeros_like(res), decimal=dec) for ind, case in enumerate(cases): _test_factory(case, min_decimal[ind]) def test_solve_generalized_discrete_are(): mat20170120 = _load_data('gendare_20170120_data.npz') cases = [ # Two random examples differ by s term # in the absence of any literature for demanding examples. (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01], [4.617139e-02, 6.948286e-01, 3.444608e-02], [9.713178e-02, 3.170995e-01, 4.387444e-01]]), np.array([[3.815585e-01, 1.868726e-01], [7.655168e-01, 4.897644e-01], [7.951999e-01, 4.455862e-01]]), np.eye(3), np.eye(2), np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01], [7.093648e-01, 6.797027e-01, 1.189977e-01], [7.546867e-01, 6.550980e-01, 4.983641e-01]]), np.zeros((3, 2)), None), (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01], [4.617139e-02, 6.948286e-01, 3.444608e-02], [9.713178e-02, 3.170995e-01, 4.387444e-01]]), np.array([[3.815585e-01, 1.868726e-01], [7.655168e-01, 4.897644e-01], [7.951999e-01, 4.455862e-01]]), np.eye(3), np.eye(2), np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01], [7.093648e-01, 6.797027e-01, 1.189977e-01], [7.546867e-01, 6.550980e-01, 4.983641e-01]]), np.ones((3, 2)), "Presumed issue with OpenBLAS, see gh-16926"), # user-reported (under PR-6616) 20-Jan-2017 # tests against the case where E is None but S is provided (mat20170120['A'], mat20170120['B'], mat20170120['Q'], mat20170120['R'], None, mat20170120['S'], None), ] max_atol = (1.5e-11, 1.5e-11, 3.5e-16) def _test_factory(case, atol): """Checks if X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q) is true""" a, b, q, r, e, s, knownfailure = case if knownfailure: pytest.xfail(reason=knownfailure) x = solve_discrete_are(a, b, q, r, e, s) if e is None: e = np.eye(a.shape[0]) if s is None: s = np.zeros_like(b) res = a.conj().T.dot(x.dot(a)) - e.conj().T.dot(x.dot(e)) + q res -= (a.conj().T.dot(x.dot(b)) + s).dot( solve(r+b.conj().T.dot(x.dot(b)), (b.conj().T.dot(x.dot(a)) + s.conj().T) ) ) # changed from: # assert_array_almost_equal(res, np.zeros_like(res), decimal=dec) # in gh-17950 because of a Linux 32 bit fail. assert_allclose(res, np.zeros_like(res), atol=atol) for ind, case in enumerate(cases): _test_factory(case, max_atol[ind]) def test_are_validate_args(): def test_square_shape(): nsq = np.ones((3, 2)) sq = np.eye(3) for x in (solve_continuous_are, solve_discrete_are): assert_raises(ValueError, x, nsq, 1, 1, 1) assert_raises(ValueError, x, sq, sq, nsq, 1) assert_raises(ValueError, x, sq, sq, sq, nsq) assert_raises(ValueError, x, sq, sq, sq, sq, nsq) def test_compatible_sizes(): nsq = np.ones((3, 2)) sq = np.eye(4) for x in (solve_continuous_are, solve_discrete_are): assert_raises(ValueError, x, sq, nsq, 1, 1) assert_raises(ValueError, x, sq, sq, sq, sq, sq, nsq) assert_raises(ValueError, x, sq, sq, np.eye(3), sq) assert_raises(ValueError, x, sq, sq, sq, np.eye(3)) assert_raises(ValueError, x, sq, sq, sq, sq, np.eye(3)) def test_symmetry(): nsym = np.arange(9).reshape(3, 3) sym = np.eye(3) for x in (solve_continuous_are, solve_discrete_are): assert_raises(ValueError, x, sym, sym, nsym, sym) assert_raises(ValueError, x, sym, sym, sym, nsym) def test_singularity(): sing = np.full((3, 3), 1e12) sing[2, 2] -= 1 sq = np.eye(3) for x in (solve_continuous_are, solve_discrete_are): assert_raises(ValueError, x, sq, sq, sq, sq, sing) assert_raises(ValueError, solve_continuous_are, sq, sq, sq, sing) def test_finiteness(): nm = np.full((2, 2), np.nan) sq = np.eye(2) for x in (solve_continuous_are, solve_discrete_are): assert_raises(ValueError, x, nm, sq, sq, sq) assert_raises(ValueError, x, sq, nm, sq, sq) assert_raises(ValueError, x, sq, sq, nm, sq) assert_raises(ValueError, x, sq, sq, sq, nm) assert_raises(ValueError, x, sq, sq, sq, sq, nm) assert_raises(ValueError, x, sq, sq, sq, sq, sq, nm) class TestSolveSylvester: cases = [ # a, b, c all real. (np.array([[1, 2], [0, 4]]), np.array([[5, 6], [0, 8]]), np.array([[9, 10], [11, 12]])), # a, b, c all real, 4x4. a and b have non-trival 2x2 blocks in their # quasi-triangular form. (np.array([[1.0, 0, 0, 0], [0, 1.0, 2.0, 0.0], [0, 0, 3.0, -4], [0, 0, 2, 5]]), np.array([[2.0, 0, 0, 1.0], [0, 1.0, 0.0, 0.0], [0, 0, 1.0, -1], [0, 0, 1, 1]]), np.array([[1.0, 0, 0, 0], [0, 1.0, 0, 0], [0, 0, 1.0, 0], [0, 0, 0, 1.0]])), # a, b, c all complex. (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]), np.array([[-1.0, 2j], [3.0, 4.0]]), np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])), # a and b real; c complex. (np.array([[1.0, 2.0], [3.0, 5.0]]), np.array([[-1.0, 0], [3.0, 4.0]]), np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])), # a and c complex; b real. (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]), np.array([[-1.0, 0], [3.0, 4.0]]), np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])), # a complex; b and c real. (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]), np.array([[-1.0, 0], [3.0, 4.0]]), np.array([[2.0, 2.0], [-1.0, 2.0]])), # not square matrices, real (np.array([[8, 1, 6], [3, 5, 7], [4, 9, 2]]), np.array([[2, 3], [4, 5]]), np.array([[1, 2], [3, 4], [5, 6]])), # not square matrices, complex (np.array([[8, 1j, 6+2j], [3, 5, 7], [4, 9, 2]]), np.array([[2, 3], [4, 5-1j]]), np.array([[1, 2j], [3, 4j], [5j, 6+7j]])), ] def check_case(self, a, b, c): x = solve_sylvester(a, b, c) assert_array_almost_equal(np.dot(a, x) + np.dot(x, b), c) def test_cases(self): for case in self.cases: self.check_case(case[0], case[1], case[2]) def test_trivial(self): a = np.array([[1.0, 0.0], [0.0, 1.0]]) b = np.array([[1.0]]) c = np.array([2.0, 2.0]).reshape(-1, 1) x = solve_sylvester(a, b, c) assert_array_almost_equal(x, np.array([1.0, 1.0]).reshape(-1, 1))