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from __future__ import division, print_function, absolute_import
from numpy.testing import assert_array_almost_equal, assert_array_equal
from pytest import raises as assert_raises
from numpy import array, transpose, dot, conjugate, zeros_like, empty
from numpy.random import random
from scipy.linalg import cholesky, cholesky_banded, cho_solve_banded, \
cho_factor, cho_solve
from scipy.linalg._testutils import assert_no_overwrite
class TestCholesky(object):
def test_simple(self):
a = [[8, 2, 3], [2, 9, 3], [3, 3, 6]]
c = cholesky(a)
assert_array_almost_equal(dot(transpose(c), c), a)
c = transpose(c)
a = dot(c, transpose(c))
assert_array_almost_equal(cholesky(a, lower=1), c)
def test_check_finite(self):
a = [[8, 2, 3], [2, 9, 3], [3, 3, 6]]
c = cholesky(a, check_finite=False)
assert_array_almost_equal(dot(transpose(c), c), a)
c = transpose(c)
a = dot(c, transpose(c))
assert_array_almost_equal(cholesky(a, lower=1, check_finite=False), c)
def test_simple_complex(self):
m = array([[3+1j, 3+4j, 5], [0, 2+2j, 2+7j], [0, 0, 7+4j]])
a = dot(transpose(conjugate(m)), m)
c = cholesky(a)
a1 = dot(transpose(conjugate(c)), c)
assert_array_almost_equal(a, a1)
c = transpose(c)
a = dot(c, transpose(conjugate(c)))
assert_array_almost_equal(cholesky(a, lower=1), c)
def test_random(self):
n = 20
for k in range(2):
m = random([n, n])
for i in range(n):
m[i, i] = 20*(.1+m[i, i])
a = dot(transpose(m), m)
c = cholesky(a)
a1 = dot(transpose(c), c)
assert_array_almost_equal(a, a1)
c = transpose(c)
a = dot(c, transpose(c))
assert_array_almost_equal(cholesky(a, lower=1), c)
def test_random_complex(self):
n = 20
for k in range(2):
m = random([n, n])+1j*random([n, n])
for i in range(n):
m[i, i] = 20*(.1+abs(m[i, i]))
a = dot(transpose(conjugate(m)), m)
c = cholesky(a)
a1 = dot(transpose(conjugate(c)), c)
assert_array_almost_equal(a, a1)
c = transpose(c)
a = dot(c, transpose(conjugate(c)))
assert_array_almost_equal(cholesky(a, lower=1), c)
class TestCholeskyBanded(object):
"""Tests for cholesky_banded() and cho_solve_banded."""
def test_check_finite(self):
# Symmetric positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0],
[1.0, 4.0, 0.5, 0.0],
[0.0, 0.5, 4.0, 0.2],
[0.0, 0.0, 0.2, 4.0]])
# Banded storage form of `a`.
ab = array([[-1.0, 1.0, 0.5, 0.2],
[4.0, 4.0, 4.0, 4.0]])
c = cholesky_banded(ab, lower=False, check_finite=False)
ufac = zeros_like(a)
ufac[list(range(4)), list(range(4))] = c[-1]
ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:]
assert_array_almost_equal(a, dot(ufac.T, ufac))
b = array([0.0, 0.5, 4.2, 4.2])
x = cho_solve_banded((c, False), b, check_finite=False)
assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0])
def test_upper_real(self):
# Symmetric positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0],
[1.0, 4.0, 0.5, 0.0],
[0.0, 0.5, 4.0, 0.2],
[0.0, 0.0, 0.2, 4.0]])
# Banded storage form of `a`.
ab = array([[-1.0, 1.0, 0.5, 0.2],
[4.0, 4.0, 4.0, 4.0]])
c = cholesky_banded(ab, lower=False)
ufac = zeros_like(a)
ufac[list(range(4)), list(range(4))] = c[-1]
ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:]
assert_array_almost_equal(a, dot(ufac.T, ufac))
b = array([0.0, 0.5, 4.2, 4.2])
x = cho_solve_banded((c, False), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0])
def test_upper_complex(self):
# Hermitian positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0],
[1.0, 4.0, 0.5, 0.0],
[0.0, 0.5, 4.0, -0.2j],
[0.0, 0.0, 0.2j, 4.0]])
# Banded storage form of `a`.
ab = array([[-1.0, 1.0, 0.5, -0.2j],
[4.0, 4.0, 4.0, 4.0]])
c = cholesky_banded(ab, lower=False)
ufac = zeros_like(a)
ufac[list(range(4)), list(range(4))] = c[-1]
ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:]
assert_array_almost_equal(a, dot(ufac.conj().T, ufac))
b = array([0.0, 0.5, 4.0-0.2j, 0.2j + 4.0])
x = cho_solve_banded((c, False), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0])
def test_lower_real(self):
# Symmetric positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0],
[1.0, 4.0, 0.5, 0.0],
[0.0, 0.5, 4.0, 0.2],
[0.0, 0.0, 0.2, 4.0]])
# Banded storage form of `a`.
ab = array([[4.0, 4.0, 4.0, 4.0],
[1.0, 0.5, 0.2, -1.0]])
c = cholesky_banded(ab, lower=True)
lfac = zeros_like(a)
lfac[list(range(4)), list(range(4))] = c[0]
lfac[(1, 2, 3), (0, 1, 2)] = c[1, :3]
assert_array_almost_equal(a, dot(lfac, lfac.T))
b = array([0.0, 0.5, 4.2, 4.2])
x = cho_solve_banded((c, True), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0])
def test_lower_complex(self):
# Hermitian positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0],
[1.0, 4.0, 0.5, 0.0],
[0.0, 0.5, 4.0, -0.2j],
[0.0, 0.0, 0.2j, 4.0]])
# Banded storage form of `a`.
ab = array([[4.0, 4.0, 4.0, 4.0],
[1.0, 0.5, 0.2j, -1.0]])
c = cholesky_banded(ab, lower=True)
lfac = zeros_like(a)
lfac[list(range(4)), list(range(4))] = c[0]
lfac[(1, 2, 3), (0, 1, 2)] = c[1, :3]
assert_array_almost_equal(a, dot(lfac, lfac.conj().T))
b = array([0.0, 0.5j, 3.8j, 3.8])
x = cho_solve_banded((c, True), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0j, 1.0])
class TestOverwrite(object):
def test_cholesky(self):
assert_no_overwrite(cholesky, [(3, 3)])
def test_cho_factor(self):
assert_no_overwrite(cho_factor, [(3, 3)])
def test_cho_solve(self):
x = array([[2, -1, 0], [-1, 2, -1], [0, -1, 2]])
xcho = cho_factor(x)
assert_no_overwrite(lambda b: cho_solve(xcho, b), [(3,)])
def test_cholesky_banded(self):
assert_no_overwrite(cholesky_banded, [(2, 3)])
def test_cho_solve_banded(self):
x = array([[0, -1, -1], [2, 2, 2]])
xcho = cholesky_banded(x)
assert_no_overwrite(lambda b: cho_solve_banded((xcho, False), b),
[(3,)])
class TestEmptyArray(object):
def test_cho_factor_empty_square(self):
a = empty((0, 0))
b = array([])
c = array([[]])
d = []
e = [[]]
x, _ = cho_factor(a)
assert_array_equal(x, a)
for x in ([b, c, d, e]):
assert_raises(ValueError, cho_factor, x)
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