def assert_bidiagonalized_by(m,
p,
q,
rtol: float = DEFAULT_RTOL,
atol: float = DEFAULT_ATOL):
d = p.dot(m).dot(q)
try:
assert predicates.is_orthogonal(p)
assert predicates.is_orthogonal(q)
assert predicates.is_diagonal(d, rtol=rtol, atol=atol)
except AssertionError:
# coverage: ignore
print("m.round(3)")
print(np.round(m, 3))
print("p.round(3)")
print(np.round(p, 3))
print("q.round(3)")
print(np.round(q, 3))
print("np.log10(np.abs(p @ m @ q)).round(2)")
print(np.log10(np.abs(d)).round(2))
raise
def test_is_orthogonal_tolerance():
tol = Tolerance(atol=0.5)
# Pays attention to specified tolerance.
assert predicates.is_orthogonal(np.array([[1, 0], [-0.5, 1]]), tol)
assert not predicates.is_orthogonal(np.array([[1, 0], [-0.6, 1]]), tol)
# Error isn't accumulated across entries.
assert predicates.is_orthogonal(
np.array([[1.2, 0, 0], [0, 1.2, 0], [0, 0, 1.2]]), tol)
assert not predicates.is_orthogonal(
np.array([[1.2, 0, 0], [0, 1.3, 0], [0, 0, 1.2]]), tol)
def diagonalize_real_symmetric_matrix(
matrix: np.ndarray,
tolerance: Tolerance = Tolerance.DEFAULT) -> np.ndarray:
"""Returns an orthogonal matrix that diagonalizes the given matrix.
Args:
matrix: A real symmetric matrix to diagonalize.
tolerance: Numeric error thresholds.
Returns:
An orthogonal matrix P such that P.T @ matrix @ P is diagonal.
Raises:
ValueError: Matrix isn't real symmetric.
ArithmeticError: Failed to meet specified tolerance.
"""
if np.any(np.imag(matrix) != 0) or not predicates.is_hermitian(matrix):
raise ValueError('Input must be real and symmetric.')
_, result = np.linalg.eigh(matrix)
# Check acceptability vs tolerances.
if (not predicates.is_orthogonal(result, tolerance)
or not predicates.is_diagonal(
result.T.dot(matrix).dot(result), tolerance)):
raise ArithmeticError('Failed to diagonalize to specified tolerance.')
return result
def diagonalize_real_symmetric_matrix(
matrix: np.ndarray,
tolerance: Tolerance = Tolerance.DEFAULT
) -> np.ndarray:
"""Returns an orthogonal matrix that diagonalizes the given matrix.
Args:
matrix: A real symmetric matrix to diagonalize.
tolerance: Numeric error thresholds.
Returns:
An orthogonal matrix P such that P.T @ matrix @ P is diagonal.
Raises:
ValueError: Matrix isn't real symmetric.
ArithmeticError: Failed to meet specified tolerance.
"""
if np.any(np.imag(matrix) != 0) or not predicates.is_hermitian(matrix):
raise ValueError('Input must be real and symmetric.')
_, result = np.linalg.eigh(matrix)
# Check acceptability vs tolerances.
if (not predicates.is_orthogonal(result, tolerance) or
not predicates.is_diagonal(result.T.dot(matrix).dot(result),
tolerance)):
raise ArithmeticError('Failed to diagonalize to specified tolerance.')
return result
def assertdiagonalized_by(m, p, tol=Tolerance.DEFAULT):
d = p.T.dot(m).dot(p)
try:
assert predicates.is_orthogonal(p)
assert predicates.is_diagonal(d, tol)
except AssertionError:
# coverage: ignore
print("m.round(3)")
print(np.round(m, 3))
print("p.round(3)")
print(np.round(p, 3))
print("np.log10(np.abs(p.T @ m @ p)).round(2)")
print(np.log10(np.abs(d)).round(2))
raise
def test_is_orthogonal():
assert predicates.is_orthogonal(np.empty((0, 0)))
assert not predicates.is_orthogonal(np.empty((1, 0)))
assert not predicates.is_orthogonal(np.empty((0, 1)))
assert predicates.is_orthogonal(np.array([[1]]))
assert predicates.is_orthogonal(np.array([[-1]]))
assert not predicates.is_orthogonal(np.array([[1j]]))
assert not predicates.is_orthogonal(np.array([[5]]))
assert not predicates.is_orthogonal(np.array([[3j]]))
assert not predicates.is_orthogonal(np.array([[1, 0]]))
assert not predicates.is_orthogonal(np.array([[1], [0]]))
assert not predicates.is_orthogonal(np.array([[1, 0], [0, -2]]))
assert predicates.is_orthogonal(np.array([[1, 0], [0, -1]]))
assert not predicates.is_orthogonal(np.array([[1j, 0], [0, 1]]))
assert not predicates.is_orthogonal(np.array([[1, 0], [1, 1]]))
assert not predicates.is_orthogonal(np.array([[1, 1], [0, 1]]))
assert not predicates.is_orthogonal(np.array([[1, 1], [1, 1]]))
assert not predicates.is_orthogonal(np.array([[1, -1], [1, 1]]))
assert predicates.is_orthogonal\
(np.array([[1, -1], [1, 1]]) * np.sqrt(0.5))
assert not predicates.is_orthogonal(
np.array([[1, 1j], [1j, 1]]) * np.sqrt(0.5))
assert not predicates.is_orthogonal(
np.array([[1, -1j], [1j, 1]]) * np.sqrt(0.5))
assert predicates.is_orthogonal(np.array([[1, 1e-11], [0, 1 + 1e-11]]))