124 lines
4 KiB
Python
124 lines
4 KiB
Python
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"""Test functions for linalg._solve_toeplitz module
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"""
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from __future__ import division, print_function, absolute_import
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import numpy as np
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from scipy.linalg._solve_toeplitz import levinson
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from scipy.linalg import solve, toeplitz, solve_toeplitz
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from numpy.testing import assert_equal, assert_allclose
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import pytest
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from pytest import raises as assert_raises
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def test_solve_equivalence():
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# For toeplitz matrices, solve_toeplitz() should be equivalent to solve().
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random = np.random.RandomState(1234)
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for n in (1, 2, 3, 10):
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c = random.randn(n)
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if random.rand() < 0.5:
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c = c + 1j * random.randn(n)
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r = random.randn(n)
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if random.rand() < 0.5:
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r = r + 1j * random.randn(n)
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y = random.randn(n)
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if random.rand() < 0.5:
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y = y + 1j * random.randn(n)
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# Check equivalence when both the column and row are provided.
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actual = solve_toeplitz((c,r), y)
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desired = solve(toeplitz(c, r=r), y)
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assert_allclose(actual, desired)
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# Check equivalence when the column is provided but not the row.
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actual = solve_toeplitz(c, b=y)
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desired = solve(toeplitz(c), y)
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assert_allclose(actual, desired)
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def test_multiple_rhs():
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random = np.random.RandomState(1234)
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c = random.randn(4)
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r = random.randn(4)
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for offset in [0, 1j]:
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for yshape in ((4,), (4, 3), (4, 3, 2)):
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y = random.randn(*yshape) + offset
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actual = solve_toeplitz((c,r), b=y)
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desired = solve(toeplitz(c, r=r), y)
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assert_equal(actual.shape, yshape)
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assert_equal(desired.shape, yshape)
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assert_allclose(actual, desired)
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def test_native_list_arguments():
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c = [1,2,4,7]
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r = [1,3,9,12]
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y = [5,1,4,2]
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actual = solve_toeplitz((c,r), y)
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desired = solve(toeplitz(c, r=r), y)
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assert_allclose(actual, desired)
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def test_zero_diag_error():
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# The Levinson-Durbin implementation fails when the diagonal is zero.
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random = np.random.RandomState(1234)
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n = 4
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c = random.randn(n)
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r = random.randn(n)
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y = random.randn(n)
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c[0] = 0
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assert_raises(np.linalg.LinAlgError,
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solve_toeplitz, (c, r), b=y)
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def test_wikipedia_counterexample():
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# The Levinson-Durbin implementation also fails in other cases.
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# This example is from the talk page of the wikipedia article.
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random = np.random.RandomState(1234)
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c = [2, 2, 1]
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y = random.randn(3)
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assert_raises(np.linalg.LinAlgError, solve_toeplitz, c, b=y)
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def test_reflection_coeffs():
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# check that that the partial solutions are given by the reflection
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# coefficients
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random = np.random.RandomState(1234)
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y_d = random.randn(10)
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y_z = random.randn(10) + 1j
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reflection_coeffs_d = [1]
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reflection_coeffs_z = [1]
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for i in range(2, 10):
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reflection_coeffs_d.append(solve_toeplitz(y_d[:(i-1)], b=y_d[1:i])[-1])
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reflection_coeffs_z.append(solve_toeplitz(y_z[:(i-1)], b=y_z[1:i])[-1])
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y_d_concat = np.concatenate((y_d[-2:0:-1], y_d[:-1]))
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y_z_concat = np.concatenate((y_z[-2:0:-1].conj(), y_z[:-1]))
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_, ref_d = levinson(y_d_concat, b=y_d[1:])
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_, ref_z = levinson(y_z_concat, b=y_z[1:])
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assert_allclose(reflection_coeffs_d, ref_d[:-1])
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assert_allclose(reflection_coeffs_z, ref_z[:-1])
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@pytest.mark.xfail(reason='Instability of Levinson iteration')
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def test_unstable():
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# this is a "Gaussian Toeplitz matrix", as mentioned in Example 2 of
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# I. Gohbert, T. Kailath and V. Olshevsky "Fast Gaussian Elimination with
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# Partial Pivoting for Matrices with Displacement Structure"
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# Mathematics of Computation, 64, 212 (1995), pp 1557-1576
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# which can be unstable for levinson recursion.
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# other fast toeplitz solvers such as GKO or Burg should be better.
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random = np.random.RandomState(1234)
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n = 100
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c = 0.9 ** (np.arange(n)**2)
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y = random.randn(n)
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solution1 = solve_toeplitz(c, b=y)
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solution2 = solve(toeplitz(c), y)
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assert_allclose(solution1, solution2)
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