def test_mgs_orthogonality_xxl():
"""Test if MGS orthogonalises the directions -- large vectors."""
np.random.seed(777)
v = np.random.randn(333, 33)
w = modified_gram_schmidt(v)
for i in range(v.shape[1]):
for j in range(i):
npt.assert_almost_equal(np.dot(w[:, i], w[:, j]), 0.0)
def test_mgs_vs_qr():
"""Test if MGS result spans the same subspace as the QR decomposition."""
np.random.seed(777)
v = np.random.randn(100, 20)
w = modified_gram_schmidt(v)
q, _ = np.linalg.qr(v)
npt.assert_almost_equal(
sp.linalg.subspace_angles(q, w),
np.zeros((v.shape[1], )),
)
def test_mgs_already_orthonormal():
"""Test MGS if vectors are already orthonormal."""
v = np.array([
[1.0, 0.0, 0.0],
[0.0, 1.0, 0.0],
[0.0, 0.0, 1.0],
[0.0, 0.0, 0.0],
])
w = modified_gram_schmidt(v)
assert np.all(v == w)
def test_mgs_copy():
"""Test if MGS preserves the original matrix."""
v = np.array([
[1.0, 1.0, 3.0],
[0.0, 1.0, 0.0],
[2.0, 0.0, 1.0],
[0.0, 3.0, 4.0],
])
v_copy = v.copy()
_ = modified_gram_schmidt(v)
assert np.all(v_copy == v)
def test_mgs_orthogonality():
"""Test if MGS orthogonalises the directions."""
v = np.array([
[1.0, 1.0, 3.0],
[0.0, 1.0, 0.0],
[2.0, 0.0, 1.0],
[0.0, 3.0, 4.0],
])
w = modified_gram_schmidt(v)
for i in range(v.shape[1]):
for j in range(i):
npt.assert_almost_equal(np.dot(w[:, i], w[:, j]), 0.0)
def test_mgs_normalisation():
"""Test if MGS normalises correctly."""
v = np.array([
[1.0, 1.0, 3.0],
[0.0, 1.0, 0.0],
[2.0, 0.0, 1.0],
[0.0, 3.0, 4.0],
])
w = modified_gram_schmidt(v)
npt.assert_almost_equal(
np.linalg.norm(w, axis=0),
np.ones((v.shape[1], )),
)