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from numpy.testing import TestCase, assert_array_almost_equal
from numpy import array, transpose, dot, conjugate, zeros_like
from numpy.random import rand
from scipy.linalg import cholesky, cholesky_banded, cho_solve_banded, \
cho_factor, cho_solve
from scipy.linalg._testutils import assert_no_overwrite
def random(size):
return rand(*size)
class TestCholesky(TestCase):
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_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(TestCase):
"""Tests for cholesky_banded() and cho_solve_banded."""
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[range(4),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[range(4),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[range(4),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[range(4),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,)])
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