def _fractional_matrix_power(A, p):
"""
Compute the fractional power of a matrix.
See the fractional_matrix_power docstring in matfuncs.py for more info.
"""
A = np.asarray(A)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected a square matrix')
if p == int(p):
return np.linalg.matrix_power(A, int(p))
# Compute singular values.
s = svdvals(A)
# Inverse scaling and squaring cannot deal with a singular matrix,
# because the process of repeatedly taking square roots
# would not converge to the identity matrix.
if s[-1]:
# Compute the condition number relative to matrix inversion,
# and use this to decide between floor(p) and ceil(p).
k2 = s[0] / s[-1]
p1 = p - np.floor(p)
p2 = p - np.ceil(p)
if p1 * k2 ** (1 - p1) <= -p2 * k2:
a = int(np.floor(p))
b = p1
else:
a = int(np.ceil(p))
b = p2
try:
R = _remainder_matrix_power(A, b)
Q = np.linalg.matrix_power(A, a)
return Q.dot(R)
except np.linalg.LinAlgError as e:
pass
# If p is negative then we are going to give up.
# If p is non-negative then we can fall back to generic funm.
if p < 0:
X = np.empty_like(A)
X.fill(np.nan)
return X
else:
p1 = p - np.floor(p)
a = int(np.floor(p))
b = p1
R, info = funm(A, lambda x: pow(x, b), disp=False)
Q = np.linalg.matrix_power(A, a)
return Q.dot(R)
def _fractional_matrix_power(A, p):
"""
Compute the fractional power of a matrix.
See the fractional_matrix_power docstring in matfuncs.py for more info.
"""
A = np.asarray(A)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected a square matrix')
if p == int(p):
return np.linalg.matrix_power(A, int(p))
# Compute singular values.
s = svdvals(A)
# Inverse scaling and squaring cannot deal with a singular matrix,
# because the process of repeatedly taking square roots
# would not converge to the identity matrix.
if s[-1]:
# Compute the condition number relative to matrix inversion,
# and use this to decide between floor(p) and ceil(p).
k2 = s[0] / s[-1]
p1 = p - np.floor(p)
p2 = p - np.ceil(p)
if p1 * k2**(1 - p1) <= -p2 * k2:
a = int(np.floor(p))
b = p1
else:
a = int(np.ceil(p))
b = p2
try:
R = _remainder_matrix_power(A, b)
Q = np.linalg.matrix_power(A, a)
return Q.dot(R)
except np.linalg.LinAlgError as e:
pass
# If p is negative then we are going to give up.
# If p is non-negative then we can fall back to generic funm.
if p < 0:
X = np.empty_like(A)
X.fill(np.nan)
return X
else:
p1 = p - np.floor(p)
a = int(np.floor(p))
b = p1
R, info = funm(A, lambda x: pow(x, b), disp=False)
Q = np.linalg.matrix_power(A, a)
return Q.dot(R)
def fractional_matrix_power(A, p):
"""
Compute the fractional power of a matrix.
Proceeds according to the discussion in section (6) of [1]_.
Parameters
----------
A : (N, N) array_like
Matrix whose fractional power to evaluate.
p : float
Fractional power.
Returns
-------
X : (N, N) array_like
The fractional power of the matrix.
References
----------
.. [1] Nicholas J. Higham and Lijing lin (2011)
"A Schur-Pade Algorithm for Fractional Powers of a Matrix."
SIAM Journal on Matrix Analysis and Applications,
32 (3). pp. 1056-1078. ISSN 0895-4798
"""
A = np.asarray(A)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected a square matrix')
if p == int(p):
return np.linalg.matrix_power(A, int(p))
# Compute singular values.
s = svdvals(A)
# Inverse scaling and squaring cannot deal with a singular matrix,
# because the process of repeatedly taking square roots
# would not converge to the identity matrix.
if s[-1]:
# Compute the condition number relative to matrix inversion,
# and use this to decide between floor(p) and ceil(p).
k2 = s[0] / s[-1]
p1 = p - np.floor(p)
p2 = p - np.ceil(p)
if p1 * k2 ** (1 - p1) <= -p2 * k2:
a = int(np.floor(p))
b = p1
else:
a = int(np.ceil(p))
b = p2
try:
R = _remainder_matrix_power(A, b)
Q = np.linalg.matrix_power(A, a)
return Q.dot(R)
except np.linalg.LinAlgError as e:
pass
# If p is negative then we are going to give up.
# If p is non-negative then we can fall back to generic funm.
if p < 0:
X = np.empty_like(A)
X.fill(np.nan)
return X
else:
p1 = p - np.floor(p)
a = int(np.floor(p))
b = p1
R, info = funm(A, lambda x : pow(x, b), disp=False)
Q = np.linalg.matrix_power(A, a)
return Q.dot(R)
def fractional_matrix_power(A, p):
"""
Compute the fractional power of a matrix.
Proceeds according to the discussion in section (6) of [1]_.
Parameters
----------
A : (N, N) array_like
Matrix whose fractional power to evaluate.
p : float
Fractional power.
Returns
-------
X : (N, N) array_like
The fractional power of the matrix.
References
----------
.. [1] Nicholas J. Higham and Lijing lin (2011)
"A Schur-Pade Algorithm for Fractional Powers of a Matrix."
SIAM Journal on Matrix Analysis and Applications,
32 (3). pp. 1056-1078. ISSN 0895-4798
"""
A = np.asarray(A)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected a square matrix')
if p == int(p):
return np.linalg.matrix_power(A, int(p))
# Compute singular values.
s = svdvals(A)
# Inverse scaling and squaring cannot deal with a singular matrix,
# because the process of repeatedly taking square roots
# would not converge to the identity matrix.
if s[-1]:
# Compute the condition number relative to matrix inversion,
# and use this to decide between floor(p) and ceil(p).
k2 = s[0] / s[-1]
p1 = p - np.floor(p)
p2 = p - np.ceil(p)
if p1 * k2**(1 - p1) <= -p2 * k2:
a = int(np.floor(p))
b = p1
else:
a = int(np.ceil(p))
b = p2
try:
R = _remainder_matrix_power(A, b)
Q = np.linalg.matrix_power(A, a)
return Q.dot(R)
except np.linalg.LinAlgError as e:
pass
# If p is negative then we are going to give up.
# If p is non-negative then we can fall back to generic funm.
if p < 0:
X = np.empty_like(A)
X.fill(np.nan)
return X
else:
p1 = p - np.floor(p)
a = int(np.floor(p))
b = p1
R, info = funm(A, lambda x: pow(x, b), disp=False)
Q = np.linalg.matrix_power(A, a)
return Q.dot(R)