def _logm(A):
"""
Compute the matrix logarithm.
See the logm docstring in matfuncs.py for more info.
Notes
-----
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)
ZH = np.conjugate(Z).T
return Z.dot(U).dot(ZH)
except (SqrtmError, LogmError) as e:
X = np.empty_like(A)
X.fill(np.nan)
return X
def _logm(A):
"""
Compute the matrix logarithm.
See the logm docstring in matfuncs.py for more info.
Notes
-----
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)
ZH = np.conjugate(Z).T
return Z.dot(U).dot(ZH)
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
"""
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 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)
ZH = np.conjugate(Z).T
return Z.dot(U).dot(ZH)
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
"""
# 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 np.isrealobj(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 np.isrealobj(T) and 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:
ZH = np.conjugate(Z).T
return Z.dot(U).dot(ZH)
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)
ZH = np.conjugate(Z).T
return Z.dot(U).dot(ZH)
except (SqrtmError, LogmError) as e:
X = np.empty_like(A)
X.fill(np.nan)
return X