def test_all_zero():
atol = 5
assert all_near_zero(0, atol=atol)
assert all_near_zero(4.5, atol=atol)
assert not all_near_zero(5.5, atol=atol)
assert all_near_zero([-4.5, 0, 1, 4.5, 3], atol=atol)
assert not all_near_zero([-4.5, 0, 1, 4.5, 30], atol=atol)
def _perp_eigendecompose(matrix: np.ndarray,
rtol: float = DEFAULT_RTOL,
atol: float = DEFAULT_ATOL
) -> Tuple[np.array, List[np.ndarray]]:
"""An eigendecomposition that ensures eigenvectors are perpendicular.
numpy.linalg.eig doesn't guarantee that eigenvectors from the same
eigenspace will be perpendicular. This method uses Gram-Schmidt to recover
a perpendicular set. It further checks that all eigenvectors are
perpendicular and raises an ArithmeticError otherwise.
Args:
matrix: The matrix to decompose.
rtol: Relative threshold for determining whether eigenvalues are from the
same eigenspace and whether eigenvectors are perpendicular.
atol: Absolute threshold for determining whether eigenvalues are from the
same eigenspace and whether eigenvectors are perpendicular.
Returns:
The eigenvalues and column eigenvectors. The i'th eigenvalue is
associated with the i'th column eigenvector.
Raises:
ArithmeticError: Failed to find perpendicular eigenvectors.
"""
vals, cols = np.linalg.eig(matrix)
vecs = [cols[:, i] for i in range(len(cols))]
# Convert list of row arrays to list of column arrays.
for i in range(len(vecs)):
vecs[i] = np.reshape(vecs[i], (len(vecs[i]), vecs[i].ndim))
# Group by similar eigenvalue.
n = len(vecs)
groups = _group_similar(
list(range(n)),
lambda k1, k2: all_close(vals[k1], vals[k2], rtol=rtol, atol=atol))
# Remove overlap between eigenvectors with the same eigenvalue.
for g in groups:
q, _ = np.linalg.qr(np.hstack([vecs[i] for i in g]))
for i in range(len(g)):
vecs[g[i]] = q[:, i]
# Ensure no eigenvectors overlap.
for i in range(len(vecs)):
for j in range(i + 1, len(vecs)):
if not all_near_zero(np.dot(np.conj(vecs[i].T), vecs[j]), atol=atol, rtol=rtol):
raise ArithmeticError('Eigenvectors overlap.')
return vals, vecs
def is_diagonal(matrix: np.ndarray, *, atol: float = 1e-8) -> bool:
"""Determines if a matrix is a approximately diagonal.
A matrix is diagonal if i!=j implies m[i,j]==0.
Args:
matrix: The matrix to check.
atol: The per-matrix-entry absolute tolerance on equality.
Returns:
Whether the matrix is diagonal within the given tolerance.
"""
matrix = np.copy(matrix)
for i in range(min(matrix.shape)):
matrix[i, i] = 0
return tolerance.all_near_zero(matrix, atol=atol)
def bidiagonalize_real_matrix_pair_with_symmetric_products(
mat1: np.ndarray,
mat2: np.ndarray,
*,
rtol: float = 1e-5,
atol: float = 1e-8,
check_preconditions: bool = True) -> Tuple[np.ndarray, np.ndarray]:
"""Finds orthogonal matrices that diagonalize both mat1 and mat2.
Requires mat1 and mat2 to be real.
Requires mat1.T @ mat2 to be symmetric.
Requires mat1 @ mat2.T to be symmetric.
Args:
mat1: One of the real matrices.
mat2: The other real matrix.
rtol: Relative numeric error threshold.
atol: Absolute numeric error threshold.
check_preconditions: If set, verifies that the inputs are real, and that
mat1.T @ mat2 and mat1 @ mat2.T are both symmetric. Defaults to set.
Returns:
A tuple (L, R) of two orthogonal matrices, such that both L @ mat1 @ R
and L @ mat2 @ R are diagonal matrices.
Raises:
ValueError: Matrices don't meet preconditions (e.g. not real).
"""
if check_preconditions:
if np.any(np.imag(mat1) != 0):
raise ValueError('mat1 must be real.')
if np.any(np.imag(mat2) != 0):
raise ValueError('mat2 must be real.')
if not predicates.is_hermitian(mat1.dot(mat2.T), rtol=rtol, atol=atol):
raise ValueError('mat1 @ mat2.T must be symmetric.')
if not predicates.is_hermitian(mat1.T.dot(mat2), rtol=rtol, atol=atol):
raise ValueError('mat1.T @ mat2 must be symmetric.')
# Use SVD to bi-diagonalize the first matrix.
base_left, base_diag, base_right = _svd_handling_empty(np.real(mat1))
base_diag = np.diag(base_diag)
# Determine where we switch between diagonalization-fixup strategies.
dim = base_diag.shape[0]
rank = dim
while rank > 0 and tolerance.all_near_zero(base_diag[rank - 1, rank - 1],
atol=atol):
rank -= 1
base_diag = base_diag[:rank, :rank]
# Try diagonalizing the second matrix with the same factors as the first.
semi_corrected = base_left.T.dot(np.real(mat2)).dot(base_right.T)
# Fix up the part of the second matrix's diagonalization that's matched
# against non-zero diagonal entries in the first matrix's diagonalization
# by performing simultaneous diagonalization.
overlap = semi_corrected[:rank, :rank]
overlap_adjust = diagonalize_real_symmetric_and_sorted_diagonal_matrices(
overlap,
base_diag,
rtol=rtol,
atol=atol,
check_preconditions=check_preconditions)
# Fix up the part of the second matrix's diagonalization that's matched
# against zeros in the first matrix's diagonalization by performing an SVD.
extra = semi_corrected[rank:, rank:]
extra_left_adjust, _, extra_right_adjust = _svd_handling_empty(extra)
# Merge the fixup factors into the initial diagonalization.
left_adjust = combinators.block_diag(overlap_adjust, extra_left_adjust)
right_adjust = combinators.block_diag(overlap_adjust.T, extra_right_adjust)
left = left_adjust.T.dot(base_left.T)
right = base_right.T.dot(right_adjust.T)
return left, right