def test_pod(vector_array, method):
A = vector_array
# TODO assumption here masks a potential issue with the algorithm
# where it fails in internal lapack instead of a proper error
assume(len(A) > 1 or A.dim > 1)
assume(not contains_zero_vector(A, rtol=1e-13, atol=1e-13))
B = A.copy()
orth_tol = 1e-10
U, s = pod(A, method=method, orth_tol=orth_tol)
assert np.all(almost_equal(A, B))
assert len(U) == len(s)
assert np.allclose(U.gramian(), np.eye(len(s)), atol=orth_tol)
def test_gram_schmidt(vector_array):
U = vector_array
# TODO assumption here masks a potential issue with the algorithm
# where it fails in del instead of a proper error
assume(len(U) > 1 or not contains_zero_vector(U))
V = U.copy()
onb = gram_schmidt(U, copy=True)
assert np.all(almost_equal(U, V))
assert np.allclose(onb.dot(onb), np.eye(len(onb)))
# TODO maybe raise tolerances again
assert np.all(
almost_equal(U, onb.lincomb(onb.dot(U).T), atol=1e-13, rtol=1e-13))
onb2 = gram_schmidt(U, copy=False)
assert np.all(almost_equal(onb, onb2))
assert np.all(almost_equal(onb, U))
def test_method_of_snapshots(vector_array, method):
A = vector_array[0]
# TODO assumption here masks a potential issue with the algorithm
# where it fails in internal lapack instead of a proper error
assume(len(A) > 1 or A.dim > 1)
assume(not contains_zero_vector(A, rtol=1e-13, atol=1e-13))
B = A.copy()
with log_levels({"pymor.algorithms": "ERROR"}):
U, s, Vh = method(A)
assert np.all(almost_equal(A, B))
assert len(U) == len(s) == Vh.shape[0]
assert Vh.shape[1] == len(A)
assert np.allclose(Vh @ Vh.T.conj(), np.eye(len(s)))
U.scal(s)
UsVh = U.lincomb(Vh.T)
assert np.all(almost_equal(A, UsVh, rtol=4e-8))
def test_gram_schmidt_with_R(vector_array):
U = vector_array
# TODO assumption here masks a potential issue with the algorithm
# where it fails in del instead of a proper error
assume(len(U) > 1 or not contains_zero_vector(U))
V = U.copy()
onb, R = gram_schmidt(U, return_R=True, copy=True)
assert np.all(almost_equal(U, V))
assert np.allclose(onb.dot(onb), np.eye(len(onb)))
lc = onb.lincomb(onb.dot(U).T)
rtol = atol = 1e-13
assert np.all(almost_equal(U, lc, rtol=rtol, atol=atol))
assert np.all(almost_equal(V, onb.lincomb(R.T), rtol=rtol, atol=atol))
onb2, R2 = gram_schmidt(U, return_R=True, copy=False)
assert np.all(almost_equal(onb, onb2))
assert np.all(R == R2)
assert np.all(almost_equal(onb, U))