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 ("QRFactorization", ):
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_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_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]])
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], [1, 0, 0, 0, 0, 1, 2, 3 + 1e-10])
for method in ("NormalEquation", "AugmentedSystem"):
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)
assert_array_almost_equal(A.dot(x), 0, decimal=14)
# Test orthogonality
assert_array_almost_equal(orthogonality(A, x), 0, decimal=16)
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 ("QRFactorization", ):
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 = np.linalg.lstsq(At, z)[0]
assert_array_almost_equal(x, x2)
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 ("NormalEquation", "AugmentedSystem"):
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 = np.linalg.lstsq(At_dense, z)[0]
assert_array_almost_equal(x, x2)