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from __future__ import division, print_function, absolute_import
import numpy as np
import scipy.linalg
from scipy.sparse import csc_matrix
from scipy.optimize._trustregion_constr.projections \
import projections, orthogonality
from numpy.testing import (TestCase, assert_array_almost_equal,
assert_array_equal, assert_array_less,
assert_raises, assert_equal, assert_,
run_module_suite, assert_allclose, assert_warns,
dec)
import pytest
import sys
import platform
try:
from sksparse.cholmod import cholesky_AAt
sksparse_available = True
available_sparse_methods = ("NormalEquation", "AugmentedSystem")
except ImportError:
import warnings
sksparse_available = False
available_sparse_methods = ("AugmentedSystem",)
available_dense_methods = ('QRFactorization', 'SVDFactorization')
class TestProjections(TestCase):
def test_nullspace_and_least_squares_sparse(self):
A_dense = np.array([[1, 2, 3, 4, 0, 5, 0, 7],
[0, 8, 7, 0, 1, 5, 9, 0],
[1, 0, 0, 0, 0, 1, 2, 3]])
At_dense = A_dense.T
A = csc_matrix(A_dense)
test_points = ([1, 2, 3, 4, 5, 6, 7, 8],
[1, 10, 3, 0, 1, 6, 7, 8],
[1.12, 10, 0, 0, 100000, 6, 0.7, 8])
for method in available_sparse_methods:
Z, LS, _ = projections(A, method)
for z in test_points:
# Test if x is in the null_space
x = Z.matvec(z)
assert_array_almost_equal(A.dot(x), 0)
# Test orthogonality
assert_array_almost_equal(orthogonality(A, x), 0)
# Test if x is the least square solution
x = LS.matvec(z)
x2 = scipy.linalg.lstsq(At_dense, z)[0]
assert_array_almost_equal(x, x2)
def test_iterative_refinements_sparse(self):
A_dense = np.array([[1, 2, 3, 4, 0, 5, 0, 7],
[0, 8, 7, 0, 1, 5, 9, 0],
[1, 0, 0, 0, 0, 1, 2, 3]])
A = csc_matrix(A_dense)
test_points = ([1, 2, 3, 4, 5, 6, 7, 8],
[1, 10, 3, 0, 1, 6, 7, 8],
[1.12, 10, 0, 0, 100000, 6, 0.7, 8],
[1, 0, 0, 0, 0, 1, 2, 3+1e-10])
for method in available_sparse_methods:
Z, LS, _ = projections(A, method, orth_tol=1e-18, max_refin=100)
for z in test_points:
# Test if x is in the null_space
x = Z.matvec(z)
atol = 1e-13 * abs(x).max()
err = abs(A.dot(x)).max()
assert_allclose(A.dot(x), 0, atol=atol)
# Test orthogonality
assert_allclose(orthogonality(A, x), 0, atol=1e-13)
def test_rowspace_sparse(self):
A_dense = np.array([[1, 2, 3, 4, 0, 5, 0, 7],
[0, 8, 7, 0, 1, 5, 9, 0],
[1, 0, 0, 0, 0, 1, 2, 3]])
A = csc_matrix(A_dense)
test_points = ([1, 2, 3],
[1, 10, 3],
[1.12, 10, 0])
for method in available_sparse_methods:
_, _, Y = projections(A, method)
for z in test_points:
# Test if x is solution of A x = z
x = Y.matvec(z)
assert_array_almost_equal(A.dot(x), z)
# Test if x is in the return row space of A
A_ext = np.vstack((A_dense, x))
assert_equal(np.linalg.matrix_rank(A_dense),
np.linalg.matrix_rank(A_ext))
def test_nullspace_and_least_squares_dense(self):
A = np.array([[1, 2, 3, 4, 0, 5, 0, 7],
[0, 8, 7, 0, 1, 5, 9, 0],
[1, 0, 0, 0, 0, 1, 2, 3]])
At = A.T
test_points = ([1, 2, 3, 4, 5, 6, 7, 8],
[1, 10, 3, 0, 1, 6, 7, 8],
[1.12, 10, 0, 0, 100000, 6, 0.7, 8])
for method in available_dense_methods:
Z, LS, _ = projections(A, method)
for z in test_points:
# Test if x is in the null_space
x = Z.matvec(z)
assert_array_almost_equal(A.dot(x), 0)
# Test orthogonality
assert_array_almost_equal(orthogonality(A, x), 0)
# Test if x is the least square solution
x = LS.matvec(z)
x2 = scipy.linalg.lstsq(At, z)[0]
assert_array_almost_equal(x, x2)
def test_compare_dense_and_sparse(self):
D = np.diag(range(1, 101))
A = np.hstack([D, D, D, D])
A_sparse = csc_matrix(A)
np.random.seed(0)
Z, LS, Y = projections(A)
Z_sparse, LS_sparse, Y_sparse = projections(A_sparse)
for k in range(20):
z = np.random.normal(size=(400,))
assert_array_almost_equal(Z.dot(z), Z_sparse.dot(z))
assert_array_almost_equal(LS.dot(z), LS_sparse.dot(z))
x = np.random.normal(size=(100,))
assert_array_almost_equal(Y.dot(x), Y_sparse.dot(x))
def test_compare_dense_and_sparse2(self):
D1 = np.diag([-1.7, 1, 0.5])
D2 = np.diag([1, -0.6, -0.3])
D3 = np.diag([-0.3, -1.5, 2])
A = np.hstack([D1, D2, D3])
A_sparse = csc_matrix(A)
np.random.seed(0)
Z, LS, Y = projections(A)
Z_sparse, LS_sparse, Y_sparse = projections(A_sparse)
for k in range(1):
z = np.random.normal(size=(9,))
assert_array_almost_equal(Z.dot(z), Z_sparse.dot(z))
assert_array_almost_equal(LS.dot(z), LS_sparse.dot(z))
x = np.random.normal(size=(3,))
assert_array_almost_equal(Y.dot(x), Y_sparse.dot(x))
def test_iterative_refinements_dense(self):
A = np.array([[1, 2, 3, 4, 0, 5, 0, 7],
[0, 8, 7, 0, 1, 5, 9, 0],
[1, 0, 0, 0, 0, 1, 2, 3]])
test_points = ([1, 2, 3, 4, 5, 6, 7, 8],
[1, 10, 3, 0, 1, 6, 7, 8],
[1, 0, 0, 0, 0, 1, 2, 3+1e-10])
for method in available_dense_methods:
Z, LS, _ = projections(A, method, orth_tol=1e-18, max_refin=10)
for z in test_points:
# Test if x is in the null_space
x = Z.matvec(z)
assert_array_almost_equal(A.dot(x), 0, decimal=14)
# Test orthogonality
assert_array_almost_equal(orthogonality(A, x), 0, decimal=16)
def test_rowspace_dense(self):
A = np.array([[1, 2, 3, 4, 0, 5, 0, 7],
[0, 8, 7, 0, 1, 5, 9, 0],
[1, 0, 0, 0, 0, 1, 2, 3]])
test_points = ([1, 2, 3],
[1, 10, 3],
[1.12, 10, 0])
for method in available_dense_methods:
_, _, Y = projections(A, method)
for z in test_points:
# Test if x is solution of A x = z
x = Y.matvec(z)
assert_array_almost_equal(A.dot(x), z)
# Test if x is in the return row space of A
A_ext = np.vstack((A, x))
assert_equal(np.linalg.matrix_rank(A),
np.linalg.matrix_rank(A_ext))
class TestOrthogonality(TestCase):
def test_dense_matrix(self):
A = np.array([[1, 2, 3, 4, 0, 5, 0, 7],
[0, 8, 7, 0, 1, 5, 9, 0],
[1, 0, 0, 0, 0, 1, 2, 3]])
test_vectors = ([-1.98931144, -1.56363389,
-0.84115584, 2.2864762,
5.599141, 0.09286976,
1.37040802, -0.28145812],
[697.92794044, -4091.65114008,
-3327.42316335, 836.86906951,
99434.98929065, -1285.37653682,
-4109.21503806, 2935.29289083])
test_expected_orth = (0, 0)
for i in range(len(test_vectors)):
x = test_vectors[i]
orth = test_expected_orth[i]
assert_array_almost_equal(orthogonality(A, x), orth)
def test_sparse_matrix(self):
A = np.array([[1, 2, 3, 4, 0, 5, 0, 7],
[0, 8, 7, 0, 1, 5, 9, 0],
[1, 0, 0, 0, 0, 1, 2, 3]])
A = csc_matrix(A)
test_vectors = ([-1.98931144, -1.56363389,
-0.84115584, 2.2864762,
5.599141, 0.09286976,
1.37040802, -0.28145812],
[697.92794044, -4091.65114008,
-3327.42316335, 836.86906951,
99434.98929065, -1285.37653682,
-4109.21503806, 2935.29289083])
test_expected_orth = (0, 0)
for i in range(len(test_vectors)):
x = test_vectors[i]
orth = test_expected_orth[i]
assert_array_almost_equal(orthogonality(A, x), orth)
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