def _solve_P_Q(U, V, structure=None):
"""
A helper function for expm_2009.
Parameters
----------
U : ndarray
Pade numerator.
V : ndarray
Pade denominator.
structure : str, optional
A string describing the structure of both matrices `U` and `V`.
Only `upper_triangular` is currently supported.
Notes
-----
The `structure` argument is inspired by similar args
for theano and cvxopt functions.
"""
P = U + V
Q = -U + V
if isspmatrix(U) or is_pydata_spmatrix(U):
return spsolve(Q, P)
elif structure is None:
return solve(Q, P)
elif structure == UPPER_TRIANGULAR:
return solve_triangular(Q, P)
else:
raise ValueError('unsupported matrix structure: ' + str(structure))
def _smart_matrix_product(A, B, alpha=None, structure=None):
"""
A matrix product that knows about sparse and structured matrices.
Parameters
----------
A : 2d ndarray
First matrix.
B : 2d ndarray
Second matrix.
alpha : float
The matrix product will be scaled by this constant.
structure : str, optional
A string describing the structure of both matrices `A` and `B`.
Only `upper_triangular` is currently supported.
Returns
-------
M : 2d ndarray
Matrix product of A and B.
"""
if len(A.shape) != 2:
raise ValueError('expected A to be a rectangular matrix')
if len(B.shape) != 2:
raise ValueError('expected B to be a rectangular matrix')
f = None
if structure == UPPER_TRIANGULAR:
if (not isspmatrix(A) and not isspmatrix(B)
and not is_pydata_spmatrix(A) and not is_pydata_spmatrix(B)):
f, = scipy.linalg.get_blas_funcs(('trmm',), (A, B))
if f is not None:
if alpha is None:
alpha = 1.
out = f(alpha, A, B)
else:
if alpha is None:
out = A.dot(B)
else:
out = alpha * A.dot(B)
return out
def _is_upper_triangular(A):
# This function could possibly be of wider interest.
if isspmatrix(A):
lower_part = scipy.sparse.tril(A, -1)
# Check structural upper triangularity,
# then coincidental upper triangularity if needed.
return lower_part.nnz == 0 or lower_part.count_nonzero() == 0
elif is_pydata_spmatrix(A):
import sparse
lower_part = sparse.tril(A, -1)
return lower_part.nnz == 0
else:
return not np.tril(A, -1).any()
def _expm(A, use_exact_onenorm):
# Core of expm, separated to allow testing exact and approximate
# algorithms.
# Avoid indiscriminate asarray() to allow sparse or other strange arrays.
if isinstance(A, (list, tuple, np.matrix)):
A = np.asarray(A)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected a square matrix')
# gracefully handle size-0 input,
# carefully handling sparse scenario
if A.shape == (0, 0):
out = np.zeros([0, 0], dtype=A.dtype)
if isspmatrix(A) or is_pydata_spmatrix(A):
return A.__class__(out)
return out
# Trivial case
if A.shape == (1, 1):
out = [[np.exp(A[0, 0])]]
# Avoid indiscriminate casting to ndarray to
# allow for sparse or other strange arrays
if isspmatrix(A) or is_pydata_spmatrix(A):
return A.__class__(out)
return np.array(out)
# Ensure input is of float type, to avoid integer overflows etc.
if ((isinstance(A, np.ndarray) or isspmatrix(A) or is_pydata_spmatrix(A))
and not np.issubdtype(A.dtype, np.inexact)):
A = A.astype(float)
# Detect upper triangularity.
structure = UPPER_TRIANGULAR if _is_upper_triangular(A) else None
if use_exact_onenorm == "auto":
# Hardcode a matrix order threshold for exact vs. estimated one-norms.
use_exact_onenorm = A.shape[0] < 200
# Track functions of A to help compute the matrix exponential.
h = _ExpmPadeHelper(
A, structure=structure, use_exact_onenorm=use_exact_onenorm)
# Try Pade order 3.
eta_1 = max(h.d4_loose, h.d6_loose)
if eta_1 < 1.495585217958292e-002 and _ell(h.A, 3) == 0:
U, V = h.pade3()
return _solve_P_Q(U, V, structure=structure)
# Try Pade order 5.
eta_2 = max(h.d4_tight, h.d6_loose)
if eta_2 < 2.539398330063230e-001 and _ell(h.A, 5) == 0:
U, V = h.pade5()
return _solve_P_Q(U, V, structure=structure)
# Try Pade orders 7 and 9.
eta_3 = max(h.d6_tight, h.d8_loose)
if eta_3 < 9.504178996162932e-001 and _ell(h.A, 7) == 0:
U, V = h.pade7()
return _solve_P_Q(U, V, structure=structure)
if eta_3 < 2.097847961257068e+000 and _ell(h.A, 9) == 0:
U, V = h.pade9()
return _solve_P_Q(U, V, structure=structure)
# Use Pade order 13.
eta_4 = max(h.d8_loose, h.d10_loose)
eta_5 = min(eta_3, eta_4)
theta_13 = 4.25
# Choose smallest s>=0 such that 2**(-s) eta_5 <= theta_13
if eta_5 == 0:
# Nilpotent special case
s = 0
else:
s = max(int(np.ceil(np.log2(eta_5 / theta_13))), 0)
s = s + _ell(2**-s * h.A, 13)
U, V = h.pade13_scaled(s)
X = _solve_P_Q(U, V, structure=structure)
if structure == UPPER_TRIANGULAR:
# Invoke Code Fragment 2.1.
X = _fragment_2_1(X, h.A, s)
else:
# X = r_13(A)^(2^s) by repeated squaring.
for i in range(s):
X = X.dot(X)
return X