1616 lines
59 KiB
Python
1616 lines
59 KiB
Python
import warnings
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import itertools
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import numpy as np
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from numpy import (arange, array, dot, zeros, identity, conjugate, transpose,
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float32)
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import numpy.linalg as linalg
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from numpy.random import random
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from numpy.testing import (assert_equal, assert_almost_equal, assert_,
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assert_array_almost_equal, assert_allclose,
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assert_array_equal, suppress_warnings)
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import pytest
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from pytest import raises as assert_raises
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from scipy.linalg import (solve, inv, det, lstsq, pinv, pinv2, pinvh, norm,
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solve_banded, solveh_banded, solve_triangular,
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solve_circulant, circulant, LinAlgError, block_diag,
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matrix_balance, LinAlgWarning)
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from scipy.linalg._testutils import assert_no_overwrite
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from scipy._lib._testutils import check_free_memory
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from scipy.linalg.blas import HAS_ILP64
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REAL_DTYPES = [np.float32, np.float64, np.longdouble]
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COMPLEX_DTYPES = [np.complex64, np.complex128, np.clongdouble]
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DTYPES = REAL_DTYPES + COMPLEX_DTYPES
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def _eps_cast(dtyp):
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"""Get the epsilon for dtype, possibly downcast to BLAS types."""
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dt = dtyp
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if dt == np.longdouble:
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dt = np.float64
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elif dt == np.clongdouble:
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dt = np.complex128
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return np.finfo(dt).eps
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class TestSolveBanded(object):
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def test_real(self):
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a = array([[1.0, 20, 0, 0],
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[-30, 4, 6, 0],
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[2, 1, 20, 2],
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[0, -1, 7, 14]])
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ab = array([[0.0, 20, 6, 2],
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[1, 4, 20, 14],
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[-30, 1, 7, 0],
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[2, -1, 0, 0]])
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l, u = 2, 1
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b4 = array([10.0, 0.0, 2.0, 14.0])
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b4by1 = b4.reshape(-1, 1)
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b4by2 = array([[2, 1],
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[-30, 4],
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[2, 3],
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[1, 3]])
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b4by4 = array([[1, 0, 0, 0],
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[0, 0, 0, 1],
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[0, 1, 0, 0],
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[0, 1, 0, 0]])
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for b in [b4, b4by1, b4by2, b4by4]:
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x = solve_banded((l, u), ab, b)
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assert_array_almost_equal(dot(a, x), b)
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def test_complex(self):
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a = array([[1.0, 20, 0, 0],
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[-30, 4, 6, 0],
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[2j, 1, 20, 2j],
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[0, -1, 7, 14]])
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ab = array([[0.0, 20, 6, 2j],
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[1, 4, 20, 14],
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[-30, 1, 7, 0],
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[2j, -1, 0, 0]])
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l, u = 2, 1
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b4 = array([10.0, 0.0, 2.0, 14.0j])
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b4by1 = b4.reshape(-1, 1)
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b4by2 = array([[2, 1],
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[-30, 4],
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[2, 3],
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[1, 3]])
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b4by4 = array([[1, 0, 0, 0],
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[0, 0, 0, 1j],
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[0, 1, 0, 0],
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[0, 1, 0, 0]])
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for b in [b4, b4by1, b4by2, b4by4]:
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x = solve_banded((l, u), ab, b)
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assert_array_almost_equal(dot(a, x), b)
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def test_tridiag_real(self):
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ab = array([[0.0, 20, 6, 2],
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[1, 4, 20, 14],
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[-30, 1, 7, 0]])
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a = np.diag(ab[0, 1:], 1) + np.diag(ab[1, :], 0) + np.diag(
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ab[2, :-1], -1)
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b4 = array([10.0, 0.0, 2.0, 14.0])
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b4by1 = b4.reshape(-1, 1)
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b4by2 = array([[2, 1],
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[-30, 4],
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[2, 3],
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[1, 3]])
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b4by4 = array([[1, 0, 0, 0],
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[0, 0, 0, 1],
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[0, 1, 0, 0],
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[0, 1, 0, 0]])
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for b in [b4, b4by1, b4by2, b4by4]:
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x = solve_banded((1, 1), ab, b)
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assert_array_almost_equal(dot(a, x), b)
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def test_tridiag_complex(self):
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ab = array([[0.0, 20, 6, 2j],
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[1, 4, 20, 14],
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[-30, 1, 7, 0]])
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a = np.diag(ab[0, 1:], 1) + np.diag(ab[1, :], 0) + np.diag(
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ab[2, :-1], -1)
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b4 = array([10.0, 0.0, 2.0, 14.0j])
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b4by1 = b4.reshape(-1, 1)
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b4by2 = array([[2, 1],
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[-30, 4],
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[2, 3],
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[1, 3]])
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b4by4 = array([[1, 0, 0, 0],
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[0, 0, 0, 1],
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[0, 1, 0, 0],
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[0, 1, 0, 0]])
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for b in [b4, b4by1, b4by2, b4by4]:
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x = solve_banded((1, 1), ab, b)
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assert_array_almost_equal(dot(a, x), b)
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def test_check_finite(self):
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a = array([[1.0, 20, 0, 0],
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[-30, 4, 6, 0],
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[2, 1, 20, 2],
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[0, -1, 7, 14]])
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ab = array([[0.0, 20, 6, 2],
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[1, 4, 20, 14],
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[-30, 1, 7, 0],
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[2, -1, 0, 0]])
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l, u = 2, 1
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b4 = array([10.0, 0.0, 2.0, 14.0])
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x = solve_banded((l, u), ab, b4, check_finite=False)
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assert_array_almost_equal(dot(a, x), b4)
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def test_bad_shape(self):
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ab = array([[0.0, 20, 6, 2],
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[1, 4, 20, 14],
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[-30, 1, 7, 0],
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[2, -1, 0, 0]])
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l, u = 2, 1
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bad = array([1.0, 2.0, 3.0, 4.0]).reshape(-1, 4)
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assert_raises(ValueError, solve_banded, (l, u), ab, bad)
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assert_raises(ValueError, solve_banded, (l, u), ab, [1.0, 2.0])
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# Values of (l,u) are not compatible with ab.
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assert_raises(ValueError, solve_banded, (1, 1), ab, [1.0, 2.0])
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def test_1x1(self):
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b = array([[1., 2., 3.]])
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x = solve_banded((1, 1), [[0], [2], [0]], b)
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assert_array_equal(x, [[0.5, 1.0, 1.5]])
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assert_equal(x.dtype, np.dtype('f8'))
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assert_array_equal(b, [[1.0, 2.0, 3.0]])
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def test_native_list_arguments(self):
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a = [[1.0, 20, 0, 0],
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[-30, 4, 6, 0],
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[2, 1, 20, 2],
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[0, -1, 7, 14]]
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ab = [[0.0, 20, 6, 2],
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[1, 4, 20, 14],
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[-30, 1, 7, 0],
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[2, -1, 0, 0]]
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l, u = 2, 1
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b = [10.0, 0.0, 2.0, 14.0]
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x = solve_banded((l, u), ab, b)
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assert_array_almost_equal(dot(a, x), b)
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class TestSolveHBanded(object):
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def test_01_upper(self):
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# Solve
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# [ 4 1 2 0] [1]
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# [ 1 4 1 2] X = [4]
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# [ 2 1 4 1] [1]
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# [ 0 2 1 4] [2]
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# with the RHS as a 1D array.
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ab = array([[0.0, 0.0, 2.0, 2.0],
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[-99, 1.0, 1.0, 1.0],
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[4.0, 4.0, 4.0, 4.0]])
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b = array([1.0, 4.0, 1.0, 2.0])
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x = solveh_banded(ab, b)
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assert_array_almost_equal(x, [0.0, 1.0, 0.0, 0.0])
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def test_02_upper(self):
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# Solve
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# [ 4 1 2 0] [1 6]
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# [ 1 4 1 2] X = [4 2]
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# [ 2 1 4 1] [1 6]
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# [ 0 2 1 4] [2 1]
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#
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ab = array([[0.0, 0.0, 2.0, 2.0],
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[-99, 1.0, 1.0, 1.0],
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[4.0, 4.0, 4.0, 4.0]])
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b = array([[1.0, 6.0],
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[4.0, 2.0],
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[1.0, 6.0],
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[2.0, 1.0]])
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x = solveh_banded(ab, b)
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expected = array([[0.0, 1.0],
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[1.0, 0.0],
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[0.0, 1.0],
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[0.0, 0.0]])
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assert_array_almost_equal(x, expected)
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def test_03_upper(self):
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# Solve
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# [ 4 1 2 0] [1]
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# [ 1 4 1 2] X = [4]
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# [ 2 1 4 1] [1]
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# [ 0 2 1 4] [2]
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# with the RHS as a 2D array with shape (3,1).
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ab = array([[0.0, 0.0, 2.0, 2.0],
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[-99, 1.0, 1.0, 1.0],
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[4.0, 4.0, 4.0, 4.0]])
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b = array([1.0, 4.0, 1.0, 2.0]).reshape(-1, 1)
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x = solveh_banded(ab, b)
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assert_array_almost_equal(x, array([0., 1., 0., 0.]).reshape(-1, 1))
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def test_01_lower(self):
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# Solve
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# [ 4 1 2 0] [1]
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# [ 1 4 1 2] X = [4]
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# [ 2 1 4 1] [1]
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# [ 0 2 1 4] [2]
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#
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ab = array([[4.0, 4.0, 4.0, 4.0],
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[1.0, 1.0, 1.0, -99],
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[2.0, 2.0, 0.0, 0.0]])
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b = array([1.0, 4.0, 1.0, 2.0])
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x = solveh_banded(ab, b, lower=True)
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assert_array_almost_equal(x, [0.0, 1.0, 0.0, 0.0])
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def test_02_lower(self):
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# Solve
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# [ 4 1 2 0] [1 6]
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# [ 1 4 1 2] X = [4 2]
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# [ 2 1 4 1] [1 6]
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# [ 0 2 1 4] [2 1]
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#
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ab = array([[4.0, 4.0, 4.0, 4.0],
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[1.0, 1.0, 1.0, -99],
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[2.0, 2.0, 0.0, 0.0]])
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b = array([[1.0, 6.0],
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[4.0, 2.0],
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[1.0, 6.0],
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[2.0, 1.0]])
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x = solveh_banded(ab, b, lower=True)
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expected = array([[0.0, 1.0],
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[1.0, 0.0],
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[0.0, 1.0],
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[0.0, 0.0]])
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assert_array_almost_equal(x, expected)
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def test_01_float32(self):
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# Solve
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# [ 4 1 2 0] [1]
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# [ 1 4 1 2] X = [4]
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# [ 2 1 4 1] [1]
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# [ 0 2 1 4] [2]
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#
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ab = array([[0.0, 0.0, 2.0, 2.0],
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[-99, 1.0, 1.0, 1.0],
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[4.0, 4.0, 4.0, 4.0]], dtype=float32)
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b = array([1.0, 4.0, 1.0, 2.0], dtype=float32)
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x = solveh_banded(ab, b)
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assert_array_almost_equal(x, [0.0, 1.0, 0.0, 0.0])
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def test_02_float32(self):
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# Solve
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# [ 4 1 2 0] [1 6]
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# [ 1 4 1 2] X = [4 2]
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# [ 2 1 4 1] [1 6]
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# [ 0 2 1 4] [2 1]
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#
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ab = array([[0.0, 0.0, 2.0, 2.0],
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[-99, 1.0, 1.0, 1.0],
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[4.0, 4.0, 4.0, 4.0]], dtype=float32)
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b = array([[1.0, 6.0],
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[4.0, 2.0],
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[1.0, 6.0],
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[2.0, 1.0]], dtype=float32)
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x = solveh_banded(ab, b)
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expected = array([[0.0, 1.0],
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[1.0, 0.0],
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[0.0, 1.0],
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[0.0, 0.0]])
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assert_array_almost_equal(x, expected)
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def test_01_complex(self):
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# Solve
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# [ 4 -j 2 0] [2-j]
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# [ j 4 -j 2] X = [4-j]
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# [ 2 j 4 -j] [4+j]
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# [ 0 2 j 4] [2+j]
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#
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ab = array([[0.0, 0.0, 2.0, 2.0],
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[-99, -1.0j, -1.0j, -1.0j],
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[4.0, 4.0, 4.0, 4.0]])
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b = array([2-1.0j, 4.0-1j, 4+1j, 2+1j])
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x = solveh_banded(ab, b)
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assert_array_almost_equal(x, [0.0, 1.0, 1.0, 0.0])
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def test_02_complex(self):
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# Solve
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# [ 4 -j 2 0] [2-j 2+4j]
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# [ j 4 -j 2] X = [4-j -1-j]
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# [ 2 j 4 -j] [4+j 4+2j]
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# [ 0 2 j 4] [2+j j]
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#
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ab = array([[0.0, 0.0, 2.0, 2.0],
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[-99, -1.0j, -1.0j, -1.0j],
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[4.0, 4.0, 4.0, 4.0]])
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b = array([[2-1j, 2+4j],
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[4.0-1j, -1-1j],
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[4.0+1j, 4+2j],
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[2+1j, 1j]])
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x = solveh_banded(ab, b)
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expected = array([[0.0, 1.0j],
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[1.0, 0.0],
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[1.0, 1.0],
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[0.0, 0.0]])
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assert_array_almost_equal(x, expected)
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def test_tridiag_01_upper(self):
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# Solve
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# [ 4 1 0] [1]
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# [ 1 4 1] X = [4]
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# [ 0 1 4] [1]
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# with the RHS as a 1D array.
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ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]])
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b = array([1.0, 4.0, 1.0])
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x = solveh_banded(ab, b)
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assert_array_almost_equal(x, [0.0, 1.0, 0.0])
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def test_tridiag_02_upper(self):
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# Solve
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# [ 4 1 0] [1 4]
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# [ 1 4 1] X = [4 2]
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# [ 0 1 4] [1 4]
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#
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ab = array([[-99, 1.0, 1.0],
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[4.0, 4.0, 4.0]])
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b = array([[1.0, 4.0],
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[4.0, 2.0],
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[1.0, 4.0]])
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x = solveh_banded(ab, b)
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expected = array([[0.0, 1.0],
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[1.0, 0.0],
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[0.0, 1.0]])
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assert_array_almost_equal(x, expected)
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def test_tridiag_03_upper(self):
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# Solve
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# [ 4 1 0] [1]
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# [ 1 4 1] X = [4]
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# [ 0 1 4] [1]
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# with the RHS as a 2D array with shape (3,1).
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ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]])
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b = array([1.0, 4.0, 1.0]).reshape(-1, 1)
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x = solveh_banded(ab, b)
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assert_array_almost_equal(x, array([0.0, 1.0, 0.0]).reshape(-1, 1))
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def test_tridiag_01_lower(self):
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# Solve
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# [ 4 1 0] [1]
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# [ 1 4 1] X = [4]
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# [ 0 1 4] [1]
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#
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ab = array([[4.0, 4.0, 4.0],
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[1.0, 1.0, -99]])
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b = array([1.0, 4.0, 1.0])
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x = solveh_banded(ab, b, lower=True)
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assert_array_almost_equal(x, [0.0, 1.0, 0.0])
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def test_tridiag_02_lower(self):
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# Solve
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# [ 4 1 0] [1 4]
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# [ 1 4 1] X = [4 2]
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# [ 0 1 4] [1 4]
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#
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ab = array([[4.0, 4.0, 4.0],
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[1.0, 1.0, -99]])
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b = array([[1.0, 4.0],
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[4.0, 2.0],
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[1.0, 4.0]])
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x = solveh_banded(ab, b, lower=True)
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expected = array([[0.0, 1.0],
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[1.0, 0.0],
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[0.0, 1.0]])
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assert_array_almost_equal(x, expected)
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def test_tridiag_01_float32(self):
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# Solve
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# [ 4 1 0] [1]
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# [ 1 4 1] X = [4]
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# [ 0 1 4] [1]
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#
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ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]], dtype=float32)
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b = array([1.0, 4.0, 1.0], dtype=float32)
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x = solveh_banded(ab, b)
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assert_array_almost_equal(x, [0.0, 1.0, 0.0])
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def test_tridiag_02_float32(self):
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# Solve
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# [ 4 1 0] [1 4]
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# [ 1 4 1] X = [4 2]
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# [ 0 1 4] [1 4]
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#
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ab = array([[-99, 1.0, 1.0],
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[4.0, 4.0, 4.0]], dtype=float32)
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b = array([[1.0, 4.0],
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[4.0, 2.0],
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|
[1.0, 4.0]], dtype=float32)
|
|
x = solveh_banded(ab, b)
|
|
expected = array([[0.0, 1.0],
|
|
[1.0, 0.0],
|
|
[0.0, 1.0]])
|
|
assert_array_almost_equal(x, expected)
|
|
|
|
def test_tridiag_01_complex(self):
|
|
# Solve
|
|
# [ 4 -j 0] [ -j]
|
|
# [ j 4 -j] X = [4-j]
|
|
# [ 0 j 4] [4+j]
|
|
#
|
|
ab = array([[-99, -1.0j, -1.0j], [4.0, 4.0, 4.0]])
|
|
b = array([-1.0j, 4.0-1j, 4+1j])
|
|
x = solveh_banded(ab, b)
|
|
assert_array_almost_equal(x, [0.0, 1.0, 1.0])
|
|
|
|
def test_tridiag_02_complex(self):
|
|
# Solve
|
|
# [ 4 -j 0] [ -j 4j]
|
|
# [ j 4 -j] X = [4-j -1-j]
|
|
# [ 0 j 4] [4+j 4 ]
|
|
#
|
|
ab = array([[-99, -1.0j, -1.0j],
|
|
[4.0, 4.0, 4.0]])
|
|
b = array([[-1j, 4.0j],
|
|
[4.0-1j, -1.0-1j],
|
|
[4.0+1j, 4.0]])
|
|
x = solveh_banded(ab, b)
|
|
expected = array([[0.0, 1.0j],
|
|
[1.0, 0.0],
|
|
[1.0, 1.0]])
|
|
assert_array_almost_equal(x, expected)
|
|
|
|
def test_check_finite(self):
|
|
# Solve
|
|
# [ 4 1 0] [1]
|
|
# [ 1 4 1] X = [4]
|
|
# [ 0 1 4] [1]
|
|
# with the RHS as a 1D array.
|
|
ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]])
|
|
b = array([1.0, 4.0, 1.0])
|
|
x = solveh_banded(ab, b, check_finite=False)
|
|
assert_array_almost_equal(x, [0.0, 1.0, 0.0])
|
|
|
|
def test_bad_shapes(self):
|
|
ab = array([[-99, 1.0, 1.0],
|
|
[4.0, 4.0, 4.0]])
|
|
b = array([[1.0, 4.0],
|
|
[4.0, 2.0]])
|
|
assert_raises(ValueError, solveh_banded, ab, b)
|
|
assert_raises(ValueError, solveh_banded, ab, [1.0, 2.0])
|
|
assert_raises(ValueError, solveh_banded, ab, [1.0])
|
|
|
|
def test_1x1(self):
|
|
x = solveh_banded([[1]], [[1, 2, 3]])
|
|
assert_array_equal(x, [[1.0, 2.0, 3.0]])
|
|
assert_equal(x.dtype, np.dtype('f8'))
|
|
|
|
def test_native_list_arguments(self):
|
|
# Same as test_01_upper, using python's native list.
|
|
ab = [[0.0, 0.0, 2.0, 2.0],
|
|
[-99, 1.0, 1.0, 1.0],
|
|
[4.0, 4.0, 4.0, 4.0]]
|
|
b = [1.0, 4.0, 1.0, 2.0]
|
|
x = solveh_banded(ab, b)
|
|
assert_array_almost_equal(x, [0.0, 1.0, 0.0, 0.0])
|
|
|
|
|
|
class TestSolve(object):
|
|
def setup_method(self):
|
|
np.random.seed(1234)
|
|
|
|
def test_20Feb04_bug(self):
|
|
a = [[1, 1], [1.0, 0]] # ok
|
|
x0 = solve(a, [1, 0j])
|
|
assert_array_almost_equal(dot(a, x0), [1, 0])
|
|
|
|
# gives failure with clapack.zgesv(..,rowmajor=0)
|
|
a = [[1, 1], [1.2, 0]]
|
|
b = [1, 0j]
|
|
x0 = solve(a, b)
|
|
assert_array_almost_equal(dot(a, x0), [1, 0])
|
|
|
|
def test_simple(self):
|
|
a = [[1, 20], [-30, 4]]
|
|
for b in ([[1, 0], [0, 1]], [1, 0],
|
|
[[2, 1], [-30, 4]]):
|
|
x = solve(a, b)
|
|
assert_array_almost_equal(dot(a, x), b)
|
|
|
|
def test_simple_sym(self):
|
|
a = [[2, 3], [3, 5]]
|
|
for lower in [0, 1]:
|
|
for b in ([[1, 0], [0, 1]], [1, 0]):
|
|
x = solve(a, b, sym_pos=1, lower=lower)
|
|
assert_array_almost_equal(dot(a, x), b)
|
|
|
|
def test_simple_sym_complex(self):
|
|
a = [[5, 2], [2, 4]]
|
|
for b in [[1j, 0],
|
|
[[1j, 1j],
|
|
[0, 2]],
|
|
]:
|
|
x = solve(a, b, sym_pos=1)
|
|
assert_array_almost_equal(dot(a, x), b)
|
|
|
|
def test_simple_complex(self):
|
|
a = array([[5, 2], [2j, 4]], 'D')
|
|
for b in [[1j, 0],
|
|
[[1j, 1j],
|
|
[0, 2]],
|
|
[1, 0j],
|
|
array([1, 0], 'D'),
|
|
]:
|
|
x = solve(a, b)
|
|
assert_array_almost_equal(dot(a, x), b)
|
|
|
|
def test_nils_20Feb04(self):
|
|
n = 2
|
|
A = random([n, n])+random([n, n])*1j
|
|
X = zeros((n, n), 'D')
|
|
Ainv = inv(A)
|
|
R = identity(n)+identity(n)*0j
|
|
for i in arange(0, n):
|
|
r = R[:, i]
|
|
X[:, i] = solve(A, r)
|
|
assert_array_almost_equal(X, Ainv)
|
|
|
|
def test_random(self):
|
|
|
|
n = 20
|
|
a = random([n, n])
|
|
for i in range(n):
|
|
a[i, i] = 20*(.1+a[i, i])
|
|
for i in range(4):
|
|
b = random([n, 3])
|
|
x = solve(a, b)
|
|
assert_array_almost_equal(dot(a, x), b)
|
|
|
|
def test_random_complex(self):
|
|
n = 20
|
|
a = random([n, n]) + 1j * random([n, n])
|
|
for i in range(n):
|
|
a[i, i] = 20*(.1+a[i, i])
|
|
for i in range(2):
|
|
b = random([n, 3])
|
|
x = solve(a, b)
|
|
assert_array_almost_equal(dot(a, x), b)
|
|
|
|
def test_random_sym(self):
|
|
n = 20
|
|
a = random([n, n])
|
|
for i in range(n):
|
|
a[i, i] = abs(20*(.1+a[i, i]))
|
|
for j in range(i):
|
|
a[i, j] = a[j, i]
|
|
for i in range(4):
|
|
b = random([n])
|
|
x = solve(a, b, sym_pos=1)
|
|
assert_array_almost_equal(dot(a, x), b)
|
|
|
|
def test_random_sym_complex(self):
|
|
n = 20
|
|
a = random([n, n])
|
|
a = a + 1j*random([n, n])
|
|
for i in range(n):
|
|
a[i, i] = abs(20*(.1+a[i, i]))
|
|
for j in range(i):
|
|
a[i, j] = conjugate(a[j, i])
|
|
b = random([n])+2j*random([n])
|
|
for i in range(2):
|
|
x = solve(a, b, sym_pos=1)
|
|
assert_array_almost_equal(dot(a, x), b)
|
|
|
|
def test_check_finite(self):
|
|
a = [[1, 20], [-30, 4]]
|
|
for b in ([[1, 0], [0, 1]], [1, 0],
|
|
[[2, 1], [-30, 4]]):
|
|
x = solve(a, b, check_finite=False)
|
|
assert_array_almost_equal(dot(a, x), b)
|
|
|
|
def test_scalar_a_and_1D_b(self):
|
|
a = 1
|
|
b = [1, 2, 3]
|
|
x = solve(a, b)
|
|
assert_array_almost_equal(x.ravel(), b)
|
|
assert_(x.shape == (3,), 'Scalar_a_1D_b test returned wrong shape')
|
|
|
|
def test_simple2(self):
|
|
a = np.array([[1.80, 2.88, 2.05, -0.89],
|
|
[525.00, -295.00, -95.00, -380.00],
|
|
[1.58, -2.69, -2.90, -1.04],
|
|
[-1.11, -0.66, -0.59, 0.80]])
|
|
|
|
b = np.array([[9.52, 18.47],
|
|
[2435.00, 225.00],
|
|
[0.77, -13.28],
|
|
[-6.22, -6.21]])
|
|
|
|
x = solve(a, b)
|
|
assert_array_almost_equal(x, np.array([[1., -1, 3, -5],
|
|
[3, 2, 4, 1]]).T)
|
|
|
|
def test_simple_complex2(self):
|
|
a = np.array([[-1.34+2.55j, 0.28+3.17j, -6.39-2.20j, 0.72-0.92j],
|
|
[-1.70-14.10j, 33.10-1.50j, -1.50+13.40j, 12.90+13.80j],
|
|
[-3.29-2.39j, -1.91+4.42j, -0.14-1.35j, 1.72+1.35j],
|
|
[2.41+0.39j, -0.56+1.47j, -0.83-0.69j, -1.96+0.67j]])
|
|
|
|
b = np.array([[26.26+51.78j, 31.32-6.70j],
|
|
[64.30-86.80j, 158.60-14.20j],
|
|
[-5.75+25.31j, -2.15+30.19j],
|
|
[1.16+2.57j, -2.56+7.55j]])
|
|
|
|
x = solve(a, b)
|
|
assert_array_almost_equal(x, np. array([[1+1.j, -1-2.j],
|
|
[2-3.j, 5+1.j],
|
|
[-4-5.j, -3+4.j],
|
|
[6.j, 2-3.j]]))
|
|
|
|
def test_hermitian(self):
|
|
# An upper triangular matrix will be used for hermitian matrix a
|
|
a = np.array([[-1.84, 0.11-0.11j, -1.78-1.18j, 3.91-1.50j],
|
|
[0, -4.63, -1.84+0.03j, 2.21+0.21j],
|
|
[0, 0, -8.87, 1.58-0.90j],
|
|
[0, 0, 0, -1.36]])
|
|
b = np.array([[2.98-10.18j, 28.68-39.89j],
|
|
[-9.58+3.88j, -24.79-8.40j],
|
|
[-0.77-16.05j, 4.23-70.02j],
|
|
[7.79+5.48j, -35.39+18.01j]])
|
|
res = np.array([[2.+1j, -8+6j],
|
|
[3.-2j, 7-2j],
|
|
[-1+2j, -1+5j],
|
|
[1.-1j, 3-4j]])
|
|
x = solve(a, b, assume_a='her')
|
|
assert_array_almost_equal(x, res)
|
|
# Also conjugate a and test for lower triangular data
|
|
x = solve(a.conj().T, b, assume_a='her', lower=True)
|
|
assert_array_almost_equal(x, res)
|
|
|
|
def test_pos_and_sym(self):
|
|
A = np.arange(1, 10).reshape(3, 3)
|
|
x = solve(np.tril(A)/9, np.ones(3), assume_a='pos')
|
|
assert_array_almost_equal(x, [9., 1.8, 1.])
|
|
x = solve(np.tril(A)/9, np.ones(3), assume_a='sym')
|
|
assert_array_almost_equal(x, [9., 1.8, 1.])
|
|
|
|
def test_singularity(self):
|
|
a = np.array([[1, 0, 0, 0, 0, 0, 1, 0, 1],
|
|
[1, 1, 1, 0, 0, 0, 1, 0, 1],
|
|
[0, 1, 1, 0, 0, 0, 1, 0, 1],
|
|
[1, 0, 1, 1, 1, 1, 0, 0, 0],
|
|
[1, 0, 1, 1, 1, 1, 0, 0, 0],
|
|
[1, 0, 1, 1, 1, 1, 0, 0, 0],
|
|
[1, 0, 1, 1, 1, 1, 0, 0, 0],
|
|
[1, 1, 1, 1, 1, 1, 1, 1, 1],
|
|
[1, 1, 1, 1, 1, 1, 1, 1, 1]])
|
|
b = np.arange(9)[:, None]
|
|
assert_raises(LinAlgError, solve, a, b)
|
|
|
|
def test_ill_condition_warning(self):
|
|
a = np.array([[1, 1], [1+1e-16, 1-1e-16]])
|
|
b = np.ones(2)
|
|
with warnings.catch_warnings():
|
|
warnings.simplefilter('error')
|
|
assert_raises(LinAlgWarning, solve, a, b)
|
|
|
|
def test_empty_rhs(self):
|
|
a = np.eye(2)
|
|
b = [[], []]
|
|
x = solve(a, b)
|
|
assert_(x.size == 0, 'Returned array is not empty')
|
|
assert_(x.shape == (2, 0), 'Returned empty array shape is wrong')
|
|
|
|
def test_multiple_rhs(self):
|
|
a = np.eye(2)
|
|
b = np.random.rand(2, 3, 4)
|
|
x = solve(a, b)
|
|
assert_array_almost_equal(x, b)
|
|
|
|
def test_transposed_keyword(self):
|
|
A = np.arange(9).reshape(3, 3) + 1
|
|
x = solve(np.tril(A)/9, np.ones(3), transposed=True)
|
|
assert_array_almost_equal(x, [1.2, 0.2, 1])
|
|
x = solve(np.tril(A)/9, np.ones(3), transposed=False)
|
|
assert_array_almost_equal(x, [9, -5.4, -1.2])
|
|
|
|
def test_transposed_notimplemented(self):
|
|
a = np.eye(3).astype(complex)
|
|
with assert_raises(NotImplementedError):
|
|
solve(a, a, transposed=True)
|
|
|
|
def test_nonsquare_a(self):
|
|
assert_raises(ValueError, solve, [1, 2], 1)
|
|
|
|
def test_size_mismatch_with_1D_b(self):
|
|
assert_array_almost_equal(solve(np.eye(3), np.ones(3)), np.ones(3))
|
|
assert_raises(ValueError, solve, np.eye(3), np.ones(4))
|
|
|
|
def test_assume_a_keyword(self):
|
|
assert_raises(ValueError, solve, 1, 1, assume_a='zxcv')
|
|
|
|
@pytest.mark.skip(reason="Failure on OS X (gh-7500), "
|
|
"crash on Windows (gh-8064)")
|
|
def test_all_type_size_routine_combinations(self):
|
|
sizes = [10, 100]
|
|
assume_as = ['gen', 'sym', 'pos', 'her']
|
|
dtypes = [np.float32, np.float64, np.complex64, np.complex128]
|
|
for size, assume_a, dtype in itertools.product(sizes, assume_as,
|
|
dtypes):
|
|
is_complex = dtype in (np.complex64, np.complex128)
|
|
if assume_a == 'her' and not is_complex:
|
|
continue
|
|
|
|
err_msg = ("Failed for size: {}, assume_a: {},"
|
|
"dtype: {}".format(size, assume_a, dtype))
|
|
|
|
a = np.random.randn(size, size).astype(dtype)
|
|
b = np.random.randn(size).astype(dtype)
|
|
if is_complex:
|
|
a = a + (1j*np.random.randn(size, size)).astype(dtype)
|
|
|
|
if assume_a == 'sym': # Can still be complex but only symmetric
|
|
a = a + a.T
|
|
elif assume_a == 'her': # Handle hermitian matrices here instead
|
|
a = a + a.T.conj()
|
|
elif assume_a == 'pos':
|
|
a = a.conj().T.dot(a) + 0.1*np.eye(size)
|
|
|
|
tol = 1e-12 if dtype in (np.float64, np.complex128) else 1e-6
|
|
|
|
if assume_a in ['gen', 'sym', 'her']:
|
|
# We revert the tolerance from before
|
|
# 4b4a6e7c34fa4060533db38f9a819b98fa81476c
|
|
if dtype in (np.float32, np.complex64):
|
|
tol *= 10
|
|
|
|
x = solve(a, b, assume_a=assume_a)
|
|
assert_allclose(a.dot(x), b,
|
|
atol=tol * size,
|
|
rtol=tol * size,
|
|
err_msg=err_msg)
|
|
|
|
if assume_a == 'sym' and dtype not in (np.complex64,
|
|
np.complex128):
|
|
x = solve(a, b, assume_a=assume_a, transposed=True)
|
|
assert_allclose(a.dot(x), b,
|
|
atol=tol * size,
|
|
rtol=tol * size,
|
|
err_msg=err_msg)
|
|
|
|
|
|
class TestSolveTriangular(object):
|
|
|
|
def test_simple(self):
|
|
"""
|
|
solve_triangular on a simple 2x2 matrix.
|
|
"""
|
|
A = array([[1, 0], [1, 2]])
|
|
b = [1, 1]
|
|
sol = solve_triangular(A, b, lower=True)
|
|
assert_array_almost_equal(sol, [1, 0])
|
|
|
|
# check that it works also for non-contiguous matrices
|
|
sol = solve_triangular(A.T, b, lower=False)
|
|
assert_array_almost_equal(sol, [.5, .5])
|
|
|
|
# and that it gives the same result as trans=1
|
|
sol = solve_triangular(A, b, lower=True, trans=1)
|
|
assert_array_almost_equal(sol, [.5, .5])
|
|
|
|
b = identity(2)
|
|
sol = solve_triangular(A, b, lower=True, trans=1)
|
|
assert_array_almost_equal(sol, [[1., -.5], [0, 0.5]])
|
|
|
|
def test_simple_complex(self):
|
|
"""
|
|
solve_triangular on a simple 2x2 complex matrix
|
|
"""
|
|
A = array([[1+1j, 0], [1j, 2]])
|
|
b = identity(2)
|
|
sol = solve_triangular(A, b, lower=True, trans=1)
|
|
assert_array_almost_equal(sol, [[.5-.5j, -.25-.25j], [0, 0.5]])
|
|
|
|
# check other option combinations with complex rhs
|
|
b = np.diag([1+1j, 1+2j])
|
|
sol = solve_triangular(A, b, lower=True, trans=0)
|
|
assert_array_almost_equal(sol, [[1, 0], [-0.5j, 0.5+1j]])
|
|
|
|
sol = solve_triangular(A, b, lower=True, trans=1)
|
|
assert_array_almost_equal(sol, [[1, 0.25-0.75j], [0, 0.5+1j]])
|
|
|
|
sol = solve_triangular(A, b, lower=True, trans=2)
|
|
assert_array_almost_equal(sol, [[1j, -0.75-0.25j], [0, 0.5+1j]])
|
|
|
|
sol = solve_triangular(A.T, b, lower=False, trans=0)
|
|
assert_array_almost_equal(sol, [[1, 0.25-0.75j], [0, 0.5+1j]])
|
|
|
|
sol = solve_triangular(A.T, b, lower=False, trans=1)
|
|
assert_array_almost_equal(sol, [[1, 0], [-0.5j, 0.5+1j]])
|
|
|
|
sol = solve_triangular(A.T, b, lower=False, trans=2)
|
|
assert_array_almost_equal(sol, [[1j, 0], [-0.5, 0.5+1j]])
|
|
|
|
def test_check_finite(self):
|
|
"""
|
|
solve_triangular on a simple 2x2 matrix.
|
|
"""
|
|
A = array([[1, 0], [1, 2]])
|
|
b = [1, 1]
|
|
sol = solve_triangular(A, b, lower=True, check_finite=False)
|
|
assert_array_almost_equal(sol, [1, 0])
|
|
|
|
|
|
class TestInv(object):
|
|
def setup_method(self):
|
|
np.random.seed(1234)
|
|
|
|
def test_simple(self):
|
|
a = [[1, 2], [3, 4]]
|
|
a_inv = inv(a)
|
|
assert_array_almost_equal(dot(a, a_inv), np.eye(2))
|
|
a = [[1, 2, 3], [4, 5, 6], [7, 8, 10]]
|
|
a_inv = inv(a)
|
|
assert_array_almost_equal(dot(a, a_inv), np.eye(3))
|
|
|
|
def test_random(self):
|
|
n = 20
|
|
for i in range(4):
|
|
a = random([n, n])
|
|
for i in range(n):
|
|
a[i, i] = 20*(.1+a[i, i])
|
|
a_inv = inv(a)
|
|
assert_array_almost_equal(dot(a, a_inv),
|
|
identity(n))
|
|
|
|
def test_simple_complex(self):
|
|
a = [[1, 2], [3, 4j]]
|
|
a_inv = inv(a)
|
|
assert_array_almost_equal(dot(a, a_inv), [[1, 0], [0, 1]])
|
|
|
|
def test_random_complex(self):
|
|
n = 20
|
|
for i in range(4):
|
|
a = random([n, n])+2j*random([n, n])
|
|
for i in range(n):
|
|
a[i, i] = 20*(.1+a[i, i])
|
|
a_inv = inv(a)
|
|
assert_array_almost_equal(dot(a, a_inv),
|
|
identity(n))
|
|
|
|
def test_check_finite(self):
|
|
a = [[1, 2], [3, 4]]
|
|
a_inv = inv(a, check_finite=False)
|
|
assert_array_almost_equal(dot(a, a_inv), [[1, 0], [0, 1]])
|
|
|
|
|
|
class TestDet(object):
|
|
def setup_method(self):
|
|
np.random.seed(1234)
|
|
|
|
def test_simple(self):
|
|
a = [[1, 2], [3, 4]]
|
|
a_det = det(a)
|
|
assert_almost_equal(a_det, -2.0)
|
|
|
|
def test_simple_complex(self):
|
|
a = [[1, 2], [3, 4j]]
|
|
a_det = det(a)
|
|
assert_almost_equal(a_det, -6+4j)
|
|
|
|
def test_random(self):
|
|
basic_det = linalg.det
|
|
n = 20
|
|
for i in range(4):
|
|
a = random([n, n])
|
|
d1 = det(a)
|
|
d2 = basic_det(a)
|
|
assert_almost_equal(d1, d2)
|
|
|
|
def test_random_complex(self):
|
|
basic_det = linalg.det
|
|
n = 20
|
|
for i in range(4):
|
|
a = random([n, n]) + 2j*random([n, n])
|
|
d1 = det(a)
|
|
d2 = basic_det(a)
|
|
assert_allclose(d1, d2, rtol=1e-13)
|
|
|
|
def test_check_finite(self):
|
|
a = [[1, 2], [3, 4]]
|
|
a_det = det(a, check_finite=False)
|
|
assert_almost_equal(a_det, -2.0)
|
|
|
|
|
|
def direct_lstsq(a, b, cmplx=0):
|
|
at = transpose(a)
|
|
if cmplx:
|
|
at = conjugate(at)
|
|
a1 = dot(at, a)
|
|
b1 = dot(at, b)
|
|
return solve(a1, b1)
|
|
|
|
|
|
class TestLstsq(object):
|
|
|
|
lapack_drivers = ('gelsd', 'gelss', 'gelsy', None)
|
|
|
|
def setup_method(self):
|
|
np.random.seed(1234)
|
|
|
|
def test_simple_exact(self):
|
|
for dtype in REAL_DTYPES:
|
|
a = np.array([[1, 20], [-30, 4]], dtype=dtype)
|
|
for lapack_driver in TestLstsq.lapack_drivers:
|
|
for overwrite in (True, False):
|
|
for bt in (((1, 0), (0, 1)), (1, 0),
|
|
((2, 1), (-30, 4))):
|
|
# Store values in case they are overwritten
|
|
# later
|
|
a1 = a.copy()
|
|
b = np.array(bt, dtype=dtype)
|
|
b1 = b.copy()
|
|
out = lstsq(a1, b1,
|
|
lapack_driver=lapack_driver,
|
|
overwrite_a=overwrite,
|
|
overwrite_b=overwrite)
|
|
x = out[0]
|
|
r = out[2]
|
|
assert_(r == 2,
|
|
'expected efficient rank 2, got %s' % r)
|
|
assert_allclose(dot(a, x), b,
|
|
atol=25 * _eps_cast(a1.dtype),
|
|
rtol=25 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
|
|
def test_simple_overdet(self):
|
|
for dtype in REAL_DTYPES:
|
|
a = np.array([[1, 2], [4, 5], [3, 4]], dtype=dtype)
|
|
b = np.array([1, 2, 3], dtype=dtype)
|
|
for lapack_driver in TestLstsq.lapack_drivers:
|
|
for overwrite in (True, False):
|
|
# Store values in case they are overwritten later
|
|
a1 = a.copy()
|
|
b1 = b.copy()
|
|
out = lstsq(a1, b1, lapack_driver=lapack_driver,
|
|
overwrite_a=overwrite,
|
|
overwrite_b=overwrite)
|
|
x = out[0]
|
|
if lapack_driver == 'gelsy':
|
|
residuals = np.sum((b - a.dot(x))**2)
|
|
else:
|
|
residuals = out[1]
|
|
r = out[2]
|
|
assert_(r == 2, 'expected efficient rank 2, got %s' % r)
|
|
assert_allclose(abs((dot(a, x) - b)**2).sum(axis=0),
|
|
residuals,
|
|
rtol=25 * _eps_cast(a1.dtype),
|
|
atol=25 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
assert_allclose(x, (-0.428571428571429, 0.85714285714285),
|
|
rtol=25 * _eps_cast(a1.dtype),
|
|
atol=25 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
|
|
def test_simple_overdet_complex(self):
|
|
for dtype in COMPLEX_DTYPES:
|
|
a = np.array([[1+2j, 2], [4, 5], [3, 4]], dtype=dtype)
|
|
b = np.array([1, 2+4j, 3], dtype=dtype)
|
|
for lapack_driver in TestLstsq.lapack_drivers:
|
|
for overwrite in (True, False):
|
|
# Store values in case they are overwritten later
|
|
a1 = a.copy()
|
|
b1 = b.copy()
|
|
out = lstsq(a1, b1, lapack_driver=lapack_driver,
|
|
overwrite_a=overwrite,
|
|
overwrite_b=overwrite)
|
|
|
|
x = out[0]
|
|
if lapack_driver == 'gelsy':
|
|
res = b - a.dot(x)
|
|
residuals = np.sum(res * res.conj())
|
|
else:
|
|
residuals = out[1]
|
|
r = out[2]
|
|
assert_(r == 2, 'expected efficient rank 2, got %s' % r)
|
|
assert_allclose(abs((dot(a, x) - b)**2).sum(axis=0),
|
|
residuals,
|
|
rtol=25 * _eps_cast(a1.dtype),
|
|
atol=25 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
assert_allclose(
|
|
x, (-0.4831460674157303 + 0.258426966292135j,
|
|
0.921348314606741 + 0.292134831460674j),
|
|
rtol=25 * _eps_cast(a1.dtype),
|
|
atol=25 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
|
|
def test_simple_underdet(self):
|
|
for dtype in REAL_DTYPES:
|
|
a = np.array([[1, 2, 3], [4, 5, 6]], dtype=dtype)
|
|
b = np.array([1, 2], dtype=dtype)
|
|
for lapack_driver in TestLstsq.lapack_drivers:
|
|
for overwrite in (True, False):
|
|
# Store values in case they are overwritten later
|
|
a1 = a.copy()
|
|
b1 = b.copy()
|
|
out = lstsq(a1, b1, lapack_driver=lapack_driver,
|
|
overwrite_a=overwrite,
|
|
overwrite_b=overwrite)
|
|
|
|
x = out[0]
|
|
r = out[2]
|
|
assert_(r == 2, 'expected efficient rank 2, got %s' % r)
|
|
assert_allclose(x, (-0.055555555555555, 0.111111111111111,
|
|
0.277777777777777),
|
|
rtol=25 * _eps_cast(a1.dtype),
|
|
atol=25 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
|
|
def test_random_exact(self):
|
|
for dtype in REAL_DTYPES:
|
|
for n in (20, 200):
|
|
for lapack_driver in TestLstsq.lapack_drivers:
|
|
for overwrite in (True, False):
|
|
a = np.asarray(random([n, n]), dtype=dtype)
|
|
for i in range(n):
|
|
a[i, i] = 20 * (0.1 + a[i, i])
|
|
for i in range(4):
|
|
b = np.asarray(random([n, 3]), dtype=dtype)
|
|
# Store values in case they are overwritten later
|
|
a1 = a.copy()
|
|
b1 = b.copy()
|
|
out = lstsq(a1, b1,
|
|
lapack_driver=lapack_driver,
|
|
overwrite_a=overwrite,
|
|
overwrite_b=overwrite)
|
|
x = out[0]
|
|
r = out[2]
|
|
assert_(r == n, 'expected efficient rank %s, '
|
|
'got %s' % (n, r))
|
|
if dtype is np.float32:
|
|
assert_allclose(
|
|
dot(a, x), b,
|
|
rtol=500 * _eps_cast(a1.dtype),
|
|
atol=500 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
else:
|
|
assert_allclose(
|
|
dot(a, x), b,
|
|
rtol=1000 * _eps_cast(a1.dtype),
|
|
atol=1000 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
|
|
def test_random_complex_exact(self):
|
|
for dtype in COMPLEX_DTYPES:
|
|
for n in (20, 200):
|
|
for lapack_driver in TestLstsq.lapack_drivers:
|
|
for overwrite in (True, False):
|
|
a = np.asarray(random([n, n]) + 1j*random([n, n]),
|
|
dtype=dtype)
|
|
for i in range(n):
|
|
a[i, i] = 20 * (0.1 + a[i, i])
|
|
for i in range(2):
|
|
b = np.asarray(random([n, 3]), dtype=dtype)
|
|
# Store values in case they are overwritten later
|
|
a1 = a.copy()
|
|
b1 = b.copy()
|
|
out = lstsq(a1, b1, lapack_driver=lapack_driver,
|
|
overwrite_a=overwrite,
|
|
overwrite_b=overwrite)
|
|
x = out[0]
|
|
r = out[2]
|
|
assert_(r == n, 'expected efficient rank %s, '
|
|
'got %s' % (n, r))
|
|
if dtype is np.complex64:
|
|
assert_allclose(
|
|
dot(a, x), b,
|
|
rtol=400 * _eps_cast(a1.dtype),
|
|
atol=400 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
else:
|
|
assert_allclose(
|
|
dot(a, x), b,
|
|
rtol=1000 * _eps_cast(a1.dtype),
|
|
atol=1000 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
|
|
def test_random_overdet(self):
|
|
for dtype in REAL_DTYPES:
|
|
for (n, m) in ((20, 15), (200, 2)):
|
|
for lapack_driver in TestLstsq.lapack_drivers:
|
|
for overwrite in (True, False):
|
|
a = np.asarray(random([n, m]), dtype=dtype)
|
|
for i in range(m):
|
|
a[i, i] = 20 * (0.1 + a[i, i])
|
|
for i in range(4):
|
|
b = np.asarray(random([n, 3]), dtype=dtype)
|
|
# Store values in case they are overwritten later
|
|
a1 = a.copy()
|
|
b1 = b.copy()
|
|
out = lstsq(a1, b1,
|
|
lapack_driver=lapack_driver,
|
|
overwrite_a=overwrite,
|
|
overwrite_b=overwrite)
|
|
x = out[0]
|
|
r = out[2]
|
|
assert_(r == m, 'expected efficient rank %s, '
|
|
'got %s' % (m, r))
|
|
assert_allclose(
|
|
x, direct_lstsq(a, b, cmplx=0),
|
|
rtol=25 * _eps_cast(a1.dtype),
|
|
atol=25 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
|
|
def test_random_complex_overdet(self):
|
|
for dtype in COMPLEX_DTYPES:
|
|
for (n, m) in ((20, 15), (200, 2)):
|
|
for lapack_driver in TestLstsq.lapack_drivers:
|
|
for overwrite in (True, False):
|
|
a = np.asarray(random([n, m]) + 1j*random([n, m]),
|
|
dtype=dtype)
|
|
for i in range(m):
|
|
a[i, i] = 20 * (0.1 + a[i, i])
|
|
for i in range(2):
|
|
b = np.asarray(random([n, 3]), dtype=dtype)
|
|
# Store values in case they are overwritten
|
|
# later
|
|
a1 = a.copy()
|
|
b1 = b.copy()
|
|
out = lstsq(a1, b1,
|
|
lapack_driver=lapack_driver,
|
|
overwrite_a=overwrite,
|
|
overwrite_b=overwrite)
|
|
x = out[0]
|
|
r = out[2]
|
|
assert_(r == m, 'expected efficient rank %s, '
|
|
'got %s' % (m, r))
|
|
assert_allclose(
|
|
x, direct_lstsq(a, b, cmplx=1),
|
|
rtol=25 * _eps_cast(a1.dtype),
|
|
atol=25 * _eps_cast(a1.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
|
|
def test_check_finite(self):
|
|
with suppress_warnings() as sup:
|
|
# On (some) OSX this tests triggers a warning (gh-7538)
|
|
sup.filter(RuntimeWarning,
|
|
"internal gelsd driver lwork query error,.*"
|
|
"Falling back to 'gelss' driver.")
|
|
|
|
at = np.array(((1, 20), (-30, 4)))
|
|
for dtype, bt, lapack_driver, overwrite, check_finite in \
|
|
itertools.product(REAL_DTYPES,
|
|
(((1, 0), (0, 1)), (1, 0), ((2, 1), (-30, 4))),
|
|
TestLstsq.lapack_drivers,
|
|
(True, False),
|
|
(True, False)):
|
|
|
|
a = at.astype(dtype)
|
|
b = np.array(bt, dtype=dtype)
|
|
# Store values in case they are overwritten
|
|
# later
|
|
a1 = a.copy()
|
|
b1 = b.copy()
|
|
out = lstsq(a1, b1, lapack_driver=lapack_driver,
|
|
check_finite=check_finite, overwrite_a=overwrite,
|
|
overwrite_b=overwrite)
|
|
x = out[0]
|
|
r = out[2]
|
|
assert_(r == 2, 'expected efficient rank 2, got %s' % r)
|
|
assert_allclose(dot(a, x), b,
|
|
rtol=25 * _eps_cast(a.dtype),
|
|
atol=25 * _eps_cast(a.dtype),
|
|
err_msg="driver: %s" % lapack_driver)
|
|
|
|
def test_zero_size(self):
|
|
for a_shape, b_shape in (((0, 2), (0,)),
|
|
((0, 4), (0, 2)),
|
|
((4, 0), (4,)),
|
|
((4, 0), (4, 2))):
|
|
b = np.ones(b_shape)
|
|
x, residues, rank, s = lstsq(np.zeros(a_shape), b)
|
|
assert_equal(x, np.zeros((a_shape[1],) + b_shape[1:]))
|
|
residues_should_be = (np.empty((0,)) if a_shape[1]
|
|
else np.linalg.norm(b, axis=0)**2)
|
|
assert_equal(residues, residues_should_be)
|
|
assert_(rank == 0, 'expected rank 0')
|
|
assert_equal(s, np.empty((0,)))
|
|
|
|
|
|
class TestPinv(object):
|
|
|
|
def test_simple_real(self):
|
|
a = array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], dtype=float)
|
|
a_pinv = pinv(a)
|
|
assert_array_almost_equal(dot(a, a_pinv), np.eye(3))
|
|
a_pinv = pinv2(a)
|
|
assert_array_almost_equal(dot(a, a_pinv), np.eye(3))
|
|
|
|
def test_simple_complex(self):
|
|
a = (array([[1, 2, 3], [4, 5, 6], [7, 8, 10]],
|
|
dtype=float) + 1j * array([[10, 8, 7], [6, 5, 4], [3, 2, 1]],
|
|
dtype=float))
|
|
a_pinv = pinv(a)
|
|
assert_array_almost_equal(dot(a, a_pinv), np.eye(3))
|
|
a_pinv = pinv2(a)
|
|
assert_array_almost_equal(dot(a, a_pinv), np.eye(3))
|
|
|
|
def test_simple_singular(self):
|
|
a = array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=float)
|
|
a_pinv = pinv(a)
|
|
a_pinv2 = pinv2(a)
|
|
assert_array_almost_equal(a_pinv, a_pinv2)
|
|
|
|
def test_simple_cols(self):
|
|
a = array([[1, 2, 3], [4, 5, 6]], dtype=float)
|
|
a_pinv = pinv(a)
|
|
a_pinv2 = pinv2(a)
|
|
assert_array_almost_equal(a_pinv, a_pinv2)
|
|
|
|
def test_simple_rows(self):
|
|
a = array([[1, 2], [3, 4], [5, 6]], dtype=float)
|
|
a_pinv = pinv(a)
|
|
a_pinv2 = pinv2(a)
|
|
assert_array_almost_equal(a_pinv, a_pinv2)
|
|
|
|
def test_check_finite(self):
|
|
a = array([[1, 2, 3], [4, 5, 6.], [7, 8, 10]])
|
|
a_pinv = pinv(a, check_finite=False)
|
|
assert_array_almost_equal(dot(a, a_pinv), np.eye(3))
|
|
a_pinv = pinv2(a, check_finite=False)
|
|
assert_array_almost_equal(dot(a, a_pinv), np.eye(3))
|
|
|
|
def test_native_list_argument(self):
|
|
a = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
|
|
a_pinv = pinv(a)
|
|
a_pinv2 = pinv2(a)
|
|
assert_array_almost_equal(a_pinv, a_pinv2)
|
|
|
|
def test_tall_transposed(self):
|
|
a = random([10, 2])
|
|
a_pinv = pinv(a)
|
|
# The result will be transposed internally hence will be a C-layout
|
|
# instead of the typical LAPACK output with Fortran-layout
|
|
assert a_pinv.flags['C_CONTIGUOUS']
|
|
|
|
|
|
class TestPinvSymmetric(object):
|
|
|
|
def test_simple_real(self):
|
|
a = array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], dtype=float)
|
|
a = np.dot(a, a.T)
|
|
a_pinv = pinvh(a)
|
|
assert_array_almost_equal(np.dot(a, a_pinv), np.eye(3))
|
|
|
|
def test_nonpositive(self):
|
|
a = array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=float)
|
|
a = np.dot(a, a.T)
|
|
u, s, vt = np.linalg.svd(a)
|
|
s[0] *= -1
|
|
a = np.dot(u * s, vt) # a is now symmetric non-positive and singular
|
|
a_pinv = pinv2(a)
|
|
a_pinvh = pinvh(a)
|
|
assert_array_almost_equal(a_pinv, a_pinvh)
|
|
|
|
def test_simple_complex(self):
|
|
a = (array([[1, 2, 3], [4, 5, 6], [7, 8, 10]],
|
|
dtype=float) + 1j * array([[10, 8, 7], [6, 5, 4], [3, 2, 1]],
|
|
dtype=float))
|
|
a = np.dot(a, a.conj().T)
|
|
a_pinv = pinvh(a)
|
|
assert_array_almost_equal(np.dot(a, a_pinv), np.eye(3))
|
|
|
|
def test_native_list_argument(self):
|
|
a = array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], dtype=float)
|
|
a = np.dot(a, a.T)
|
|
a_pinv = pinvh(a.tolist())
|
|
assert_array_almost_equal(np.dot(a, a_pinv), np.eye(3))
|
|
|
|
|
|
def test_pinv_pinv2_comparison(): # As reported in gh-8861
|
|
I_6 = np.eye(6)
|
|
Ts = np.diag([-1] * 4 + [-2], k=-1) + np.diag([-2] + [-1] * 4, k=1)
|
|
T = I_6 + Ts
|
|
A = 25 * (np.kron(I_6, T) + np.kron(Ts, I_6))
|
|
|
|
Ap, Ap2 = pinv(A), pinv2(A)
|
|
|
|
tol = 1e-11
|
|
assert_allclose(A @ Ap @ A - A, A @ Ap2 @ A - A, rtol=0., atol=tol)
|
|
assert_allclose(Ap @ A @ Ap - Ap, Ap2 @ A @ Ap2 - Ap2, rtol=0., atol=tol)
|
|
|
|
|
|
@pytest.mark.parametrize('scale', (1e-20, 1., 1e20))
|
|
@pytest.mark.parametrize('pinv_', (pinv, pinvh, pinv2))
|
|
def test_auto_rcond(scale, pinv_):
|
|
x = np.array([[1, 0], [0, 1e-10]]) * scale
|
|
expected = np.diag(1. / np.diag(x))
|
|
x_inv = pinv_(x)
|
|
assert_allclose(x_inv, expected)
|
|
|
|
|
|
class TestVectorNorms(object):
|
|
|
|
def test_types(self):
|
|
for dtype in np.typecodes['AllFloat']:
|
|
x = np.array([1, 2, 3], dtype=dtype)
|
|
tol = max(1e-15, np.finfo(dtype).eps.real * 20)
|
|
assert_allclose(norm(x), np.sqrt(14), rtol=tol)
|
|
assert_allclose(norm(x, 2), np.sqrt(14), rtol=tol)
|
|
|
|
for dtype in np.typecodes['Complex']:
|
|
x = np.array([1j, 2j, 3j], dtype=dtype)
|
|
tol = max(1e-15, np.finfo(dtype).eps.real * 20)
|
|
assert_allclose(norm(x), np.sqrt(14), rtol=tol)
|
|
assert_allclose(norm(x, 2), np.sqrt(14), rtol=tol)
|
|
|
|
def test_overflow(self):
|
|
# unlike numpy's norm, this one is
|
|
# safer on overflow
|
|
a = array([1e20], dtype=float32)
|
|
assert_almost_equal(norm(a), a)
|
|
|
|
def test_stable(self):
|
|
# more stable than numpy's norm
|
|
a = array([1e4] + [1]*10000, dtype=float32)
|
|
try:
|
|
# snrm in double precision; we obtain the same as for float64
|
|
# -- large atol needed due to varying blas implementations
|
|
assert_allclose(norm(a) - 1e4, 0.5, atol=1e-2)
|
|
except AssertionError:
|
|
# snrm implemented in single precision, == np.linalg.norm result
|
|
msg = ": Result should equal either 0.0 or 0.5 (depending on " \
|
|
"implementation of snrm2)."
|
|
assert_almost_equal(norm(a) - 1e4, 0.0, err_msg=msg)
|
|
|
|
def test_zero_norm(self):
|
|
assert_equal(norm([1, 0, 3], 0), 2)
|
|
assert_equal(norm([1, 2, 3], 0), 3)
|
|
|
|
def test_axis_kwd(self):
|
|
a = np.array([[[2, 1], [3, 4]]] * 2, 'd')
|
|
assert_allclose(norm(a, axis=1), [[3.60555128, 4.12310563]] * 2)
|
|
assert_allclose(norm(a, 1, axis=1), [[5.] * 2] * 2)
|
|
|
|
def test_keepdims_kwd(self):
|
|
a = np.array([[[2, 1], [3, 4]]] * 2, 'd')
|
|
b = norm(a, axis=1, keepdims=True)
|
|
assert_allclose(b, [[[3.60555128, 4.12310563]]] * 2)
|
|
assert_(b.shape == (2, 1, 2))
|
|
assert_allclose(norm(a, 1, axis=2, keepdims=True), [[[3.], [7.]]] * 2)
|
|
|
|
@pytest.mark.skipif(not HAS_ILP64, reason="64-bit BLAS required")
|
|
def test_large_vector(self):
|
|
check_free_memory(free_mb=17000)
|
|
x = np.zeros([2**31], dtype=np.float64)
|
|
x[-1] = 1
|
|
res = norm(x)
|
|
del x
|
|
assert_allclose(res, 1.0)
|
|
|
|
|
|
class TestMatrixNorms(object):
|
|
|
|
def test_matrix_norms(self):
|
|
# Not all of these are matrix norms in the most technical sense.
|
|
np.random.seed(1234)
|
|
for n, m in (1, 1), (1, 3), (3, 1), (4, 4), (4, 5), (5, 4):
|
|
for t in np.single, np.double, np.csingle, np.cdouble, np.int64:
|
|
A = 10 * np.random.randn(n, m).astype(t)
|
|
if np.issubdtype(A.dtype, np.complexfloating):
|
|
A = (A + 10j * np.random.randn(n, m)).astype(t)
|
|
t_high = np.cdouble
|
|
else:
|
|
t_high = np.double
|
|
for order in (None, 'fro', 1, -1, 2, -2, np.inf, -np.inf):
|
|
actual = norm(A, ord=order)
|
|
desired = np.linalg.norm(A, ord=order)
|
|
# SciPy may return higher precision matrix norms.
|
|
# This is a consequence of using LAPACK.
|
|
if not np.allclose(actual, desired):
|
|
desired = np.linalg.norm(A.astype(t_high), ord=order)
|
|
assert_allclose(actual, desired)
|
|
|
|
def test_axis_kwd(self):
|
|
a = np.array([[[2, 1], [3, 4]]] * 2, 'd')
|
|
b = norm(a, ord=np.inf, axis=(1, 0))
|
|
c = norm(np.swapaxes(a, 0, 1), ord=np.inf, axis=(0, 1))
|
|
d = norm(a, ord=1, axis=(0, 1))
|
|
assert_allclose(b, c)
|
|
assert_allclose(c, d)
|
|
assert_allclose(b, d)
|
|
assert_(b.shape == c.shape == d.shape)
|
|
b = norm(a, ord=1, axis=(1, 0))
|
|
c = norm(np.swapaxes(a, 0, 1), ord=1, axis=(0, 1))
|
|
d = norm(a, ord=np.inf, axis=(0, 1))
|
|
assert_allclose(b, c)
|
|
assert_allclose(c, d)
|
|
assert_allclose(b, d)
|
|
assert_(b.shape == c.shape == d.shape)
|
|
|
|
def test_keepdims_kwd(self):
|
|
a = np.arange(120, dtype='d').reshape(2, 3, 4, 5)
|
|
b = norm(a, ord=np.inf, axis=(1, 0), keepdims=True)
|
|
c = norm(a, ord=1, axis=(0, 1), keepdims=True)
|
|
assert_allclose(b, c)
|
|
assert_(b.shape == c.shape)
|
|
|
|
|
|
class TestOverwrite(object):
|
|
def test_solve(self):
|
|
assert_no_overwrite(solve, [(3, 3), (3,)])
|
|
|
|
def test_solve_triangular(self):
|
|
assert_no_overwrite(solve_triangular, [(3, 3), (3,)])
|
|
|
|
def test_solve_banded(self):
|
|
assert_no_overwrite(lambda ab, b: solve_banded((2, 1), ab, b),
|
|
[(4, 6), (6,)])
|
|
|
|
def test_solveh_banded(self):
|
|
assert_no_overwrite(solveh_banded, [(2, 6), (6,)])
|
|
|
|
def test_inv(self):
|
|
assert_no_overwrite(inv, [(3, 3)])
|
|
|
|
def test_det(self):
|
|
assert_no_overwrite(det, [(3, 3)])
|
|
|
|
def test_lstsq(self):
|
|
assert_no_overwrite(lstsq, [(3, 2), (3,)])
|
|
|
|
def test_pinv(self):
|
|
assert_no_overwrite(pinv, [(3, 3)])
|
|
|
|
def test_pinv2(self):
|
|
assert_no_overwrite(pinv2, [(3, 3)])
|
|
|
|
def test_pinvh(self):
|
|
assert_no_overwrite(pinvh, [(3, 3)])
|
|
|
|
|
|
class TestSolveCirculant(object):
|
|
|
|
def test_basic1(self):
|
|
c = np.array([1, 2, 3, 5])
|
|
b = np.array([1, -1, 1, 0])
|
|
x = solve_circulant(c, b)
|
|
y = solve(circulant(c), b)
|
|
assert_allclose(x, y)
|
|
|
|
def test_basic2(self):
|
|
# b is a 2-d matrix.
|
|
c = np.array([1, 2, -3, -5])
|
|
b = np.arange(12).reshape(4, 3)
|
|
x = solve_circulant(c, b)
|
|
y = solve(circulant(c), b)
|
|
assert_allclose(x, y)
|
|
|
|
def test_basic3(self):
|
|
# b is a 3-d matrix.
|
|
c = np.array([1, 2, -3, -5])
|
|
b = np.arange(24).reshape(4, 3, 2)
|
|
x = solve_circulant(c, b)
|
|
y = solve(circulant(c), b)
|
|
assert_allclose(x, y)
|
|
|
|
def test_complex(self):
|
|
# Complex b and c
|
|
c = np.array([1+2j, -3, 4j, 5])
|
|
b = np.arange(8).reshape(4, 2) + 0.5j
|
|
x = solve_circulant(c, b)
|
|
y = solve(circulant(c), b)
|
|
assert_allclose(x, y)
|
|
|
|
def test_random_b_and_c(self):
|
|
# Random b and c
|
|
np.random.seed(54321)
|
|
c = np.random.randn(50)
|
|
b = np.random.randn(50)
|
|
x = solve_circulant(c, b)
|
|
y = solve(circulant(c), b)
|
|
assert_allclose(x, y)
|
|
|
|
def test_singular(self):
|
|
# c gives a singular circulant matrix.
|
|
c = np.array([1, 1, 0, 0])
|
|
b = np.array([1, 2, 3, 4])
|
|
x = solve_circulant(c, b, singular='lstsq')
|
|
y, res, rnk, s = lstsq(circulant(c), b)
|
|
assert_allclose(x, y)
|
|
assert_raises(LinAlgError, solve_circulant, x, y)
|
|
|
|
def test_axis_args(self):
|
|
# Test use of caxis, baxis and outaxis.
|
|
|
|
# c has shape (2, 1, 4)
|
|
c = np.array([[[-1, 2.5, 3, 3.5]], [[1, 6, 6, 6.5]]])
|
|
|
|
# b has shape (3, 4)
|
|
b = np.array([[0, 0, 1, 1], [1, 1, 0, 0], [1, -1, 0, 0]])
|
|
|
|
x = solve_circulant(c, b, baxis=1)
|
|
assert_equal(x.shape, (4, 2, 3))
|
|
expected = np.empty_like(x)
|
|
expected[:, 0, :] = solve(circulant(c[0]), b.T)
|
|
expected[:, 1, :] = solve(circulant(c[1]), b.T)
|
|
assert_allclose(x, expected)
|
|
|
|
x = solve_circulant(c, b, baxis=1, outaxis=-1)
|
|
assert_equal(x.shape, (2, 3, 4))
|
|
assert_allclose(np.rollaxis(x, -1), expected)
|
|
|
|
# np.swapaxes(c, 1, 2) has shape (2, 4, 1); b.T has shape (4, 3).
|
|
x = solve_circulant(np.swapaxes(c, 1, 2), b.T, caxis=1)
|
|
assert_equal(x.shape, (4, 2, 3))
|
|
assert_allclose(x, expected)
|
|
|
|
def test_native_list_arguments(self):
|
|
# Same as test_basic1 using python's native list.
|
|
c = [1, 2, 3, 5]
|
|
b = [1, -1, 1, 0]
|
|
x = solve_circulant(c, b)
|
|
y = solve(circulant(c), b)
|
|
assert_allclose(x, y)
|
|
|
|
|
|
class TestMatrix_Balance(object):
|
|
|
|
def test_string_arg(self):
|
|
assert_raises(ValueError, matrix_balance, 'Some string for fail')
|
|
|
|
def test_infnan_arg(self):
|
|
assert_raises(ValueError, matrix_balance,
|
|
np.array([[1, 2], [3, np.inf]]))
|
|
assert_raises(ValueError, matrix_balance,
|
|
np.array([[1, 2], [3, np.nan]]))
|
|
|
|
def test_scaling(self):
|
|
_, y = matrix_balance(np.array([[1000, 1], [1000, 0]]))
|
|
# Pre/post LAPACK 3.5.0 gives the same result up to an offset
|
|
# since in each case col norm is x1000 greater and
|
|
# 1000 / 32 ~= 1 * 32 hence balanced with 2 ** 5.
|
|
assert_allclose(int(np.diff(np.log2(np.diag(y)))), 5)
|
|
|
|
def test_scaling_order(self):
|
|
A = np.array([[1, 0, 1e-4], [1, 1, 1e-2], [1e4, 1e2, 1]])
|
|
x, y = matrix_balance(A)
|
|
assert_allclose(solve(y, A).dot(y), x)
|
|
|
|
def test_separate(self):
|
|
_, (y, z) = matrix_balance(np.array([[1000, 1], [1000, 0]]),
|
|
separate=1)
|
|
assert_equal(int(np.diff(np.log2(y))), 5)
|
|
assert_allclose(z, np.arange(2))
|
|
|
|
def test_permutation(self):
|
|
A = block_diag(np.ones((2, 2)), np.tril(np.ones((2, 2))),
|
|
np.ones((3, 3)))
|
|
x, (y, z) = matrix_balance(A, separate=1)
|
|
assert_allclose(y, np.ones_like(y))
|
|
assert_allclose(z, np.array([0, 1, 6, 5, 4, 3, 2]))
|
|
|
|
def test_perm_and_scaling(self):
|
|
# Matrix with its diagonal removed
|
|
cases = ( # Case 0
|
|
np.array([[0., 0., 0., 0., 0.000002],
|
|
[0., 0., 0., 0., 0.],
|
|
[2., 2., 0., 0., 0.],
|
|
[2., 2., 0., 0., 0.],
|
|
[0., 0., 0.000002, 0., 0.]]),
|
|
# Case 1 user reported GH-7258
|
|
np.array([[-0.5, 0., 0., 0.],
|
|
[0., -1., 0., 0.],
|
|
[1., 0., -0.5, 0.],
|
|
[0., 1., 0., -1.]]),
|
|
# Case 2 user reported GH-7258
|
|
np.array([[-3., 0., 1., 0.],
|
|
[-1., -1., -0., 1.],
|
|
[-3., -0., -0., 0.],
|
|
[-1., -0., 1., -1.]])
|
|
)
|
|
|
|
for A in cases:
|
|
x, y = matrix_balance(A)
|
|
x, (s, p) = matrix_balance(A, separate=1)
|
|
ip = np.empty_like(p)
|
|
ip[p] = np.arange(A.shape[0])
|
|
assert_allclose(y, np.diag(s)[ip, :])
|
|
assert_allclose(solve(y, A).dot(y), x)
|