def logm(A):
"""
Compute matrix logarithm.
The matrix logarithm is the inverse of
expm: expm(logm(`A`)) == `A`
Parameters
----------
A : (N, N) array_like
Matrix whose logarithm to evaluate
Returns
-------
logm : (N, N) ndarray
Matrix logarithm of `A`
References
----------
.. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012)
"Improved Inverse Scaling and Squaring Algorithms
for the Matrix Logarithm."
SIAM Journal on Scientific Computing, 34 (4). C152-C169.
ISSN 1095-7197
.. [2] Nicholas J. Higham (2008)
"Functions of Matrices: Theory and Computation"
ISBN 978-0-898716-46-7
.. [3] 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')
n, n = A.shape
keep_it_real = not _has_complex_dtype_char(A)
try:
if np.array_equal(A, np.triu(A)):
A_diag = np.diag(A)
if np.count_nonzero(A_diag) != n:
raise LogmError('cannot find logm of a singular matrix')
if np.min(A_diag) < 0:
A = A.astype(complex)
return _logm_triu(A)
else:
if keep_it_real:
T, Z = schur(A)
if not np.array_equal(T, np.triu(T)):
T, Z = rsf2csf(T,Z)
else:
T, Z = schur(A, output='complex')
if np.count_nonzero(np.diag(T)) != n:
raise LogmError('cannot find logm of a singular matrix')
U = _logm_triu(T)
U, Z = all_mat(U, Z)
X = (Z * U * Z.H)
return X.A
except (SqrtmError, LogmError) as e:
X = np.empty_like(A)
X.fill(np.nan)
return X
def _remainder_matrix_power(A, t):
"""
Compute the fractional power of a matrix, for fractions -1 < t < 1.
This uses algorithm (3.1) of [1]_.
The Pade approximation itself uses algorithm (4.1) of [2]_.
Parameters
----------
A : (N, N) array_like
Matrix whose fractional power to evaluate.
t : float
Fractional power between -1 and 1 exclusive.
Returns
-------
X : (N, N) array_like
The fractional power of the matrix.
References
----------
.. [1] Nicholas J. Higham and Lijing Lin (2013)
"An Improved Schur-Pade Algorithm for Fractional Powers
of a Matrix and their Frechet Derivatives."
.. [2] 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')
n, n = A.shape
# Triangularize the matrix if necessary,
# attempting to preserve dtype if possible.
if np.array_equal(A, np.triu(A)):
Z = None
T = A
else:
if not _has_complex_dtype_char(A):
T, Z = schur(A)
if not np.array_equal(T, np.triu(T)):
T, Z = rsf2csf(T, Z)
else:
T, Z = schur(A, output='complex')
# Zeros on the diagonal of the triangular matrix are forbidden,
# because the inverse scaling and squaring cannot deal with it.
T_diag = np.diag(T)
if np.count_nonzero(T_diag) != n:
raise FractionalMatrixPowerError(
'cannot use inverse scaling and squaring to find '
'the fractional matrix power of a singular matrix')
# If the triangular matrix is real and has a negative
# entry on the diagonal, then force the matrix to be complex.
if not _has_complex_dtype_char(T):
if np.min(T_diag) < 0:
T = T.astype(complex)
# Get the fractional power of the triangular matrix,
# and de-triangularize it if necessary.
U = _remainder_matrix_power_triu(T, t)
if Z is not None:
U, Z = all_mat(U, Z)
X = (Z * U * Z.H)
return X.A
else:
return U
def _remainder_matrix_power(A, t):
"""
Compute the fractional power of a matrix, for fractions -1 < t < 1.
This uses algorithm (3.1) of [1]_.
The Pade approximation itself uses algorithm (4.1) of [2]_.
Parameters
----------
A : (N, N) array_like
Matrix whose fractional power to evaluate.
t : float
Fractional power between -1 and 1 exclusive.
Returns
-------
X : (N, N) array_like
The fractional power of the matrix.
References
----------
.. [1] Nicholas J. Higham and Lijing Lin (2013)
"An Improved Schur-Pade Algorithm for Fractional Powers
of a Matrix and their Frechet Derivatives."
.. [2] 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
"""
# This code block is copied from numpy.matrix_power().
A = np.asarray(A)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError('input must be a square array')
# Get the number of rows and columns.
n, n = A.shape
# Triangularize the matrix if necessary,
# attempting to preserve dtype if possible.
if np.array_equal(A, np.triu(A)):
Z = None
T = A
else:
if not _has_complex_dtype_char(A):
T, Z = schur(A)
if not np.array_equal(T, np.triu(T)):
T, Z = rsf2csf(T, Z)
else:
T, Z = schur(A, output='complex')
# Zeros on the diagonal of the triangular matrix are forbidden,
# because the inverse scaling and squaring cannot deal with it.
T_diag = np.diag(T)
if _count_nonzero(T_diag) != n:
raise FractionalMatrixPowerError(
'cannot use inverse scaling and squaring to find '
'the fractional matrix power of a singular matrix')
# If the triangular matrix is real and has a negative
# entry on the diagonal, then force the matrix to be complex.
if not _has_complex_dtype_char(T):
if np.min(T_diag) < 0:
T = T.astype(complex)
# Get the fractional power of the triangular matrix,
# and de-triangularize it if necessary.
U = _remainder_matrix_power_triu(T, t)
if Z is not None:
U, Z = all_mat(U, Z)
X = (Z * U * Z.H)
return X.A
else:
return U
def logm(A):
"""
Compute matrix logarithm.
The matrix logarithm is the inverse of
expm: expm(logm(`A`)) == `A`
Parameters
----------
A : (N, N) array_like
Matrix whose logarithm to evaluate
Returns
-------
logm : (N, N) ndarray
Matrix logarithm of `A`
References
----------
.. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012)
"Improved Inverse Scaling and Squaring Algorithms
for the Matrix Logarithm."
SIAM Journal on Scientific Computing, 34 (4). C152-C169.
ISSN 1095-7197
.. [2] Nicholas J. Higham (2008)
"Functions of Matrices: Theory and Computation"
ISBN 978-0-898716-46-7
.. [3] 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')
n, n = A.shape
keep_it_real = not _has_complex_dtype_char(A)
try:
if np.array_equal(A, np.triu(A)):
A_diag = np.diag(A)
if _count_nonzero(A_diag) != n:
raise LogmError('cannot find logm of a singular matrix')
if np.min(A_diag) < 0:
A = A.astype(complex)
return _logm_triu(A)
else:
if keep_it_real:
T, Z = schur(A)
if not np.array_equal(T, np.triu(T)):
T, Z = rsf2csf(T, Z)
else:
T, Z = schur(A, output='complex')
if _count_nonzero(np.diag(T)) != n:
raise LogmError('cannot find logm of a singular matrix')
U = _logm_triu(T)
U, Z = all_mat(U, Z)
X = (Z * U * Z.H)
return X.A
except (SqrtmError, LogmError) as e:
X = np.empty_like(A)
X.fill(np.nan)
return X
def logm(A):
"""
Compute matrix logarithm.
The matrix logarithm is the inverse of
expm: expm(logm(`A`)) == `A`
Parameters
----------
A : (N, N) array_like
Matrix whose logarithm to evaluate
Returns
-------
logm : (N, N) ndarray
Matrix logarithm of `A`
References
----------
.. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012)
"Improved Inverse Scaling and Squaring Algorithms
for the Matrix Logarithm."
SIAM Journal on Scientific Computing, 34 (4). C152-C169.
ISSN 1095-7197
.. [2] Nicholas J. Higham (2008)
"Functions of Matrices: Theory and Computation"
ISBN 978-0-898716-46-7
.. [3] 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
"""
# In this function we look at triangular matrices that are similar
# to the input matrix. If any diagonal entry of such a triangular matrix
# is exactly zero then the original matrix is singular.
# The matrix logarithm does not exist for such matrices,
# but in such cases we will pretend that the diagonal entries that are zero
# are actually slightly positive by an ad-hoc amount, in the interest
# of returning something more useful than NaN. This will cause a warning.
A = np.asarray(A)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected a square matrix')
n = A.shape[0]
# If the input matrix dtype is integer then copy to a float dtype matrix.
if issubclass(A.dtype.type, np.integer):
A = np.asarray(A, dtype=float)
keep_it_real = np.isrealobj(A)
try:
if np.array_equal(A, np.triu(A)):
A = _logm_force_nonsingular_triangular_matrix(A)
if np.min(np.diag(A)) < 0:
A = A.astype(complex)
return _logm_triu(A)
else:
if keep_it_real:
T, Z = schur(A)
if not np.array_equal(T, np.triu(T)):
T, Z = rsf2csf(T,Z)
else:
T, Z = schur(A, output='complex')
T = _logm_force_nonsingular_triangular_matrix(T, inplace=True)
U = _logm_triu(T)
U, Z = all_mat(U, Z)
X = (Z * U * Z.H)
return X.A
except (SqrtmError, LogmError) as e:
X = np.empty_like(A)
X.fill(np.nan)
return X
