887 lines
27 KiB
Python
887 lines
27 KiB
Python
"""
|
|
Matrix functions that use Pade approximation with inverse scaling and squaring.
|
|
|
|
"""
|
|
import warnings
|
|
|
|
import numpy as np
|
|
|
|
from scipy.linalg._matfuncs_sqrtm import SqrtmError, _sqrtm_triu
|
|
from scipy.linalg._decomp_schur import schur, rsf2csf
|
|
from scipy.linalg._matfuncs import funm
|
|
from scipy.linalg import svdvals, solve_triangular
|
|
from scipy.sparse.linalg._interface import LinearOperator
|
|
from scipy.sparse.linalg import onenormest
|
|
import scipy.special
|
|
|
|
|
|
class LogmRankWarning(UserWarning):
|
|
pass
|
|
|
|
|
|
class LogmExactlySingularWarning(LogmRankWarning):
|
|
pass
|
|
|
|
|
|
class LogmNearlySingularWarning(LogmRankWarning):
|
|
pass
|
|
|
|
|
|
class LogmError(np.linalg.LinAlgError):
|
|
pass
|
|
|
|
|
|
class FractionalMatrixPowerError(np.linalg.LinAlgError):
|
|
pass
|
|
|
|
|
|
#TODO renovate or move this class when scipy operators are more mature
|
|
class _MatrixM1PowerOperator(LinearOperator):
|
|
"""
|
|
A representation of the linear operator (A - I)^p.
|
|
"""
|
|
|
|
def __init__(self, A, p):
|
|
if A.ndim != 2 or A.shape[0] != A.shape[1]:
|
|
raise ValueError('expected A to be like a square matrix')
|
|
if p < 0 or p != int(p):
|
|
raise ValueError('expected p to be a non-negative integer')
|
|
self._A = A
|
|
self._p = p
|
|
self.ndim = A.ndim
|
|
self.shape = A.shape
|
|
|
|
def _matvec(self, x):
|
|
for i in range(self._p):
|
|
x = self._A.dot(x) - x
|
|
return x
|
|
|
|
def _rmatvec(self, x):
|
|
for i in range(self._p):
|
|
x = x.dot(self._A) - x
|
|
return x
|
|
|
|
def _matmat(self, X):
|
|
for i in range(self._p):
|
|
X = self._A.dot(X) - X
|
|
return X
|
|
|
|
def _adjoint(self):
|
|
return _MatrixM1PowerOperator(self._A.T, self._p)
|
|
|
|
|
|
#TODO renovate or move this function when SciPy operators are more mature
|
|
def _onenormest_m1_power(A, p,
|
|
t=2, itmax=5, compute_v=False, compute_w=False):
|
|
"""
|
|
Efficiently estimate the 1-norm of (A - I)^p.
|
|
|
|
Parameters
|
|
----------
|
|
A : ndarray
|
|
Matrix whose 1-norm of a power is to be computed.
|
|
p : int
|
|
Non-negative integer power.
|
|
t : int, optional
|
|
A positive parameter controlling the tradeoff between
|
|
accuracy versus time and memory usage.
|
|
Larger values take longer and use more memory
|
|
but give more accurate output.
|
|
itmax : int, optional
|
|
Use at most this many iterations.
|
|
compute_v : bool, optional
|
|
Request a norm-maximizing linear operator input vector if True.
|
|
compute_w : bool, optional
|
|
Request a norm-maximizing linear operator output vector if True.
|
|
|
|
Returns
|
|
-------
|
|
est : float
|
|
An underestimate of the 1-norm of the sparse matrix.
|
|
v : ndarray, optional
|
|
The vector such that ||Av||_1 == est*||v||_1.
|
|
It can be thought of as an input to the linear operator
|
|
that gives an output with particularly large norm.
|
|
w : ndarray, optional
|
|
The vector Av which has relatively large 1-norm.
|
|
It can be thought of as an output of the linear operator
|
|
that is relatively large in norm compared to the input.
|
|
|
|
"""
|
|
return onenormest(_MatrixM1PowerOperator(A, p),
|
|
t=t, itmax=itmax, compute_v=compute_v, compute_w=compute_w)
|
|
|
|
|
|
def _unwindk(z):
|
|
"""
|
|
Compute the scalar unwinding number.
|
|
|
|
Uses Eq. (5.3) in [1]_, and should be equal to (z - log(exp(z)) / (2 pi i).
|
|
Note that this definition differs in sign from the original definition
|
|
in equations (5, 6) in [2]_. The sign convention is justified in [3]_.
|
|
|
|
Parameters
|
|
----------
|
|
z : complex
|
|
A complex number.
|
|
|
|
Returns
|
|
-------
|
|
unwinding_number : integer
|
|
The scalar unwinding number of z.
|
|
|
|
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
|
|
|
|
.. [2] Robert M. Corless and David J. Jeffrey,
|
|
"The unwinding number." Newsletter ACM SIGSAM Bulletin
|
|
Volume 30, Issue 2, June 1996, Pages 28-35.
|
|
|
|
.. [3] Russell Bradford and Robert M. Corless and James H. Davenport and
|
|
David J. Jeffrey and Stephen M. Watt,
|
|
"Reasoning about the elementary functions of complex analysis"
|
|
Annals of Mathematics and Artificial Intelligence,
|
|
36: 303-318, 2002.
|
|
|
|
"""
|
|
return int(np.ceil((z.imag - np.pi) / (2*np.pi)))
|
|
|
|
|
|
def _briggs_helper_function(a, k):
|
|
"""
|
|
Computes r = a^(1 / (2^k)) - 1.
|
|
|
|
This is algorithm (2) of [1]_.
|
|
The purpose is to avoid a danger of subtractive cancellation.
|
|
For more computational efficiency it should probably be cythonized.
|
|
|
|
Parameters
|
|
----------
|
|
a : complex
|
|
A complex number.
|
|
k : integer
|
|
A nonnegative integer.
|
|
|
|
Returns
|
|
-------
|
|
r : complex
|
|
The value r = a^(1 / (2^k)) - 1 computed with less cancellation.
|
|
|
|
Notes
|
|
-----
|
|
The algorithm as formulated in the reference does not handle k=0 or k=1
|
|
correctly, so these are special-cased in this implementation.
|
|
This function is intended to not allow `a` to belong to the closed
|
|
negative real axis, but this constraint is relaxed.
|
|
|
|
References
|
|
----------
|
|
.. [1] Awad H. Al-Mohy (2012)
|
|
"A more accurate Briggs method for the logarithm",
|
|
Numerical Algorithms, 59 : 393--402.
|
|
|
|
"""
|
|
if k < 0 or int(k) != k:
|
|
raise ValueError('expected a nonnegative integer k')
|
|
if k == 0:
|
|
return a - 1
|
|
elif k == 1:
|
|
return np.sqrt(a) - 1
|
|
else:
|
|
k_hat = k
|
|
if np.angle(a) >= np.pi / 2:
|
|
a = np.sqrt(a)
|
|
k_hat = k - 1
|
|
z0 = a - 1
|
|
a = np.sqrt(a)
|
|
r = 1 + a
|
|
for j in range(1, k_hat):
|
|
a = np.sqrt(a)
|
|
r = r * (1 + a)
|
|
r = z0 / r
|
|
return r
|
|
|
|
|
|
def _fractional_power_superdiag_entry(l1, l2, t12, p):
|
|
"""
|
|
Compute a superdiagonal entry of a fractional matrix power.
|
|
|
|
This is Eq. (5.6) in [1]_.
|
|
|
|
Parameters
|
|
----------
|
|
l1 : complex
|
|
A diagonal entry of the matrix.
|
|
l2 : complex
|
|
A diagonal entry of the matrix.
|
|
t12 : complex
|
|
A superdiagonal entry of the matrix.
|
|
p : float
|
|
A fractional power.
|
|
|
|
Returns
|
|
-------
|
|
f12 : complex
|
|
A superdiagonal entry of the fractional matrix power.
|
|
|
|
Notes
|
|
-----
|
|
Care has been taken to return a real number if possible when
|
|
all of the inputs are real numbers.
|
|
|
|
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
|
|
|
|
"""
|
|
if l1 == l2:
|
|
f12 = t12 * p * l1**(p-1)
|
|
elif abs(l2 - l1) > abs(l1 + l2) / 2:
|
|
f12 = t12 * ((l2**p) - (l1**p)) / (l2 - l1)
|
|
else:
|
|
# This is Eq. (5.5) in [1].
|
|
z = (l2 - l1) / (l2 + l1)
|
|
log_l1 = np.log(l1)
|
|
log_l2 = np.log(l2)
|
|
arctanh_z = np.arctanh(z)
|
|
tmp_a = t12 * np.exp((p/2)*(log_l2 + log_l1))
|
|
tmp_u = _unwindk(log_l2 - log_l1)
|
|
if tmp_u:
|
|
tmp_b = p * (arctanh_z + np.pi * 1j * tmp_u)
|
|
else:
|
|
tmp_b = p * arctanh_z
|
|
tmp_c = 2 * np.sinh(tmp_b) / (l2 - l1)
|
|
f12 = tmp_a * tmp_c
|
|
return f12
|
|
|
|
|
|
def _logm_superdiag_entry(l1, l2, t12):
|
|
"""
|
|
Compute a superdiagonal entry of a matrix logarithm.
|
|
|
|
This is like Eq. (11.28) in [1]_, except the determination of whether
|
|
l1 and l2 are sufficiently far apart has been modified.
|
|
|
|
Parameters
|
|
----------
|
|
l1 : complex
|
|
A diagonal entry of the matrix.
|
|
l2 : complex
|
|
A diagonal entry of the matrix.
|
|
t12 : complex
|
|
A superdiagonal entry of the matrix.
|
|
|
|
Returns
|
|
-------
|
|
f12 : complex
|
|
A superdiagonal entry of the matrix logarithm.
|
|
|
|
Notes
|
|
-----
|
|
Care has been taken to return a real number if possible when
|
|
all of the inputs are real numbers.
|
|
|
|
References
|
|
----------
|
|
.. [1] Nicholas J. Higham (2008)
|
|
"Functions of Matrices: Theory and Computation"
|
|
ISBN 978-0-898716-46-7
|
|
|
|
"""
|
|
if l1 == l2:
|
|
f12 = t12 / l1
|
|
elif abs(l2 - l1) > abs(l1 + l2) / 2:
|
|
f12 = t12 * (np.log(l2) - np.log(l1)) / (l2 - l1)
|
|
else:
|
|
z = (l2 - l1) / (l2 + l1)
|
|
u = _unwindk(np.log(l2) - np.log(l1))
|
|
if u:
|
|
f12 = t12 * 2 * (np.arctanh(z) + np.pi*1j*u) / (l2 - l1)
|
|
else:
|
|
f12 = t12 * 2 * np.arctanh(z) / (l2 - l1)
|
|
return f12
|
|
|
|
|
|
def _inverse_squaring_helper(T0, theta):
|
|
"""
|
|
A helper function for inverse scaling and squaring for Pade approximation.
|
|
|
|
Parameters
|
|
----------
|
|
T0 : (N, N) array_like upper triangular
|
|
Matrix involved in inverse scaling and squaring.
|
|
theta : indexable
|
|
The values theta[1] .. theta[7] must be available.
|
|
They represent bounds related to Pade approximation, and they depend
|
|
on the matrix function which is being computed.
|
|
For example, different values of theta are required for
|
|
matrix logarithm than for fractional matrix power.
|
|
|
|
Returns
|
|
-------
|
|
R : (N, N) array_like upper triangular
|
|
Composition of zero or more matrix square roots of T0, minus I.
|
|
s : non-negative integer
|
|
Number of square roots taken.
|
|
m : positive integer
|
|
The degree of the Pade approximation.
|
|
|
|
Notes
|
|
-----
|
|
This subroutine appears as a chunk of lines within
|
|
a couple of published algorithms; for example it appears
|
|
as lines 4--35 in algorithm (3.1) of [1]_, and
|
|
as lines 3--34 in algorithm (4.1) of [2]_.
|
|
The instances of 'goto line 38' in algorithm (3.1) of [1]_
|
|
probably mean 'goto line 36' and have been intepreted accordingly.
|
|
|
|
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] 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
|
|
|
|
"""
|
|
if len(T0.shape) != 2 or T0.shape[0] != T0.shape[1]:
|
|
raise ValueError('expected an upper triangular square matrix')
|
|
n, n = T0.shape
|
|
T = T0
|
|
|
|
# Find s0, the smallest s such that the spectral radius
|
|
# of a certain diagonal matrix is at most theta[7].
|
|
# Note that because theta[7] < 1,
|
|
# this search will not terminate if any diagonal entry of T is zero.
|
|
s0 = 0
|
|
tmp_diag = np.diag(T)
|
|
if np.count_nonzero(tmp_diag) != n:
|
|
raise Exception('Diagonal entries of T must be nonzero')
|
|
while np.max(np.absolute(tmp_diag - 1)) > theta[7]:
|
|
tmp_diag = np.sqrt(tmp_diag)
|
|
s0 += 1
|
|
|
|
# Take matrix square roots of T.
|
|
for i in range(s0):
|
|
T = _sqrtm_triu(T)
|
|
|
|
# Flow control in this section is a little odd.
|
|
# This is because I am translating algorithm descriptions
|
|
# which have GOTOs in the publication.
|
|
s = s0
|
|
k = 0
|
|
d2 = _onenormest_m1_power(T, 2) ** (1/2)
|
|
d3 = _onenormest_m1_power(T, 3) ** (1/3)
|
|
a2 = max(d2, d3)
|
|
m = None
|
|
for i in (1, 2):
|
|
if a2 <= theta[i]:
|
|
m = i
|
|
break
|
|
while m is None:
|
|
if s > s0:
|
|
d3 = _onenormest_m1_power(T, 3) ** (1/3)
|
|
d4 = _onenormest_m1_power(T, 4) ** (1/4)
|
|
a3 = max(d3, d4)
|
|
if a3 <= theta[7]:
|
|
j1 = min(i for i in (3, 4, 5, 6, 7) if a3 <= theta[i])
|
|
if j1 <= 6:
|
|
m = j1
|
|
break
|
|
elif a3 / 2 <= theta[5] and k < 2:
|
|
k += 1
|
|
T = _sqrtm_triu(T)
|
|
s += 1
|
|
continue
|
|
d5 = _onenormest_m1_power(T, 5) ** (1/5)
|
|
a4 = max(d4, d5)
|
|
eta = min(a3, a4)
|
|
for i in (6, 7):
|
|
if eta <= theta[i]:
|
|
m = i
|
|
break
|
|
if m is not None:
|
|
break
|
|
T = _sqrtm_triu(T)
|
|
s += 1
|
|
|
|
# The subtraction of the identity is redundant here,
|
|
# because the diagonal will be replaced for improved numerical accuracy,
|
|
# but this formulation should help clarify the meaning of R.
|
|
R = T - np.identity(n)
|
|
|
|
# Replace the diagonal and first superdiagonal of T0^(1/(2^s)) - I
|
|
# using formulas that have less subtractive cancellation.
|
|
# Skip this step if the principal branch
|
|
# does not exist at T0; this happens when a diagonal entry of T0
|
|
# is negative with imaginary part 0.
|
|
has_principal_branch = all(x.real > 0 or x.imag != 0 for x in np.diag(T0))
|
|
if has_principal_branch:
|
|
for j in range(n):
|
|
a = T0[j, j]
|
|
r = _briggs_helper_function(a, s)
|
|
R[j, j] = r
|
|
p = np.exp2(-s)
|
|
for j in range(n-1):
|
|
l1 = T0[j, j]
|
|
l2 = T0[j+1, j+1]
|
|
t12 = T0[j, j+1]
|
|
f12 = _fractional_power_superdiag_entry(l1, l2, t12, p)
|
|
R[j, j+1] = f12
|
|
|
|
# Return the T-I matrix, the number of square roots, and the Pade degree.
|
|
if not np.array_equal(R, np.triu(R)):
|
|
raise Exception('R is not upper triangular')
|
|
return R, s, m
|
|
|
|
|
|
def _fractional_power_pade_constant(i, t):
|
|
# A helper function for matrix fractional power.
|
|
if i < 1:
|
|
raise ValueError('expected a positive integer i')
|
|
if not (-1 < t < 1):
|
|
raise ValueError('expected -1 < t < 1')
|
|
if i == 1:
|
|
return -t
|
|
elif i % 2 == 0:
|
|
j = i // 2
|
|
return (-j + t) / (2 * (2*j - 1))
|
|
elif i % 2 == 1:
|
|
j = (i - 1) // 2
|
|
return (-j - t) / (2 * (2*j + 1))
|
|
else:
|
|
raise Exception('unnexpected value of i, i = {}'.format(i))
|
|
|
|
|
|
def _fractional_power_pade(R, t, m):
|
|
"""
|
|
Evaluate the Pade approximation of a fractional matrix power.
|
|
|
|
Evaluate the degree-m Pade approximation of R
|
|
to the fractional matrix power t using the continued fraction
|
|
in bottom-up fashion using algorithm (4.1) in [1]_.
|
|
|
|
Parameters
|
|
----------
|
|
R : (N, N) array_like
|
|
Upper triangular matrix whose fractional power to evaluate.
|
|
t : float
|
|
Fractional power between -1 and 1 exclusive.
|
|
m : positive integer
|
|
Degree of Pade approximation.
|
|
|
|
Returns
|
|
-------
|
|
U : (N, N) array_like
|
|
The degree-m Pade approximation of R to the fractional power t.
|
|
This matrix will be upper triangular.
|
|
|
|
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
|
|
|
|
"""
|
|
if m < 1 or int(m) != m:
|
|
raise ValueError('expected a positive integer m')
|
|
if not (-1 < t < 1):
|
|
raise ValueError('expected -1 < t < 1')
|
|
R = np.asarray(R)
|
|
if len(R.shape) != 2 or R.shape[0] != R.shape[1]:
|
|
raise ValueError('expected an upper triangular square matrix')
|
|
n, n = R.shape
|
|
ident = np.identity(n)
|
|
Y = R * _fractional_power_pade_constant(2*m, t)
|
|
for j in range(2*m - 1, 0, -1):
|
|
rhs = R * _fractional_power_pade_constant(j, t)
|
|
Y = solve_triangular(ident + Y, rhs)
|
|
U = ident + Y
|
|
if not np.array_equal(U, np.triu(U)):
|
|
raise Exception('U is not upper triangular')
|
|
return U
|
|
|
|
|
|
def _remainder_matrix_power_triu(T, t):
|
|
"""
|
|
Compute a fractional power of an upper triangular matrix.
|
|
|
|
The fractional power is restricted to fractions -1 < t < 1.
|
|
This uses algorithm (3.1) of [1]_.
|
|
The Pade approximation itself uses algorithm (4.1) of [2]_.
|
|
|
|
Parameters
|
|
----------
|
|
T : (N, N) array_like
|
|
Upper triangular 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
|
|
|
|
"""
|
|
m_to_theta = {
|
|
1: 1.51e-5,
|
|
2: 2.24e-3,
|
|
3: 1.88e-2,
|
|
4: 6.04e-2,
|
|
5: 1.24e-1,
|
|
6: 2.00e-1,
|
|
7: 2.79e-1,
|
|
}
|
|
n, n = T.shape
|
|
T0 = T
|
|
T0_diag = np.diag(T0)
|
|
if np.array_equal(T0, np.diag(T0_diag)):
|
|
U = np.diag(T0_diag ** t)
|
|
else:
|
|
R, s, m = _inverse_squaring_helper(T0, m_to_theta)
|
|
|
|
# Evaluate the Pade approximation.
|
|
# Note that this function expects the negative of the matrix
|
|
# returned by the inverse squaring helper.
|
|
U = _fractional_power_pade(-R, t, m)
|
|
|
|
# Undo the inverse scaling and squaring.
|
|
# Be less clever about this
|
|
# if the principal branch does not exist at T0;
|
|
# this happens when a diagonal entry of T0
|
|
# is negative with imaginary part 0.
|
|
eivals = np.diag(T0)
|
|
has_principal_branch = all(x.real > 0 or x.imag != 0 for x in eivals)
|
|
for i in range(s, -1, -1):
|
|
if i < s:
|
|
U = U.dot(U)
|
|
else:
|
|
if has_principal_branch:
|
|
p = t * np.exp2(-i)
|
|
U[np.diag_indices(n)] = T0_diag ** p
|
|
for j in range(n-1):
|
|
l1 = T0[j, j]
|
|
l2 = T0[j+1, j+1]
|
|
t12 = T0[j, j+1]
|
|
f12 = _fractional_power_superdiag_entry(l1, l2, t12, p)
|
|
U[j, j+1] = f12
|
|
if not np.array_equal(U, np.triu(U)):
|
|
raise Exception('U is not upper triangular')
|
|
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 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 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 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 _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:
|
|
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 _logm_triu(T):
|
|
"""
|
|
Compute matrix logarithm of an upper triangular matrix.
|
|
|
|
The matrix logarithm is the inverse of
|
|
expm: expm(logm(`T`)) == `T`
|
|
|
|
Parameters
|
|
----------
|
|
T : (N, N) array_like
|
|
Upper triangular matrix whose logarithm to evaluate
|
|
|
|
Returns
|
|
-------
|
|
logm : (N, N) ndarray
|
|
Matrix logarithm of `T`
|
|
|
|
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
|
|
|
|
"""
|
|
T = np.asarray(T)
|
|
if len(T.shape) != 2 or T.shape[0] != T.shape[1]:
|
|
raise ValueError('expected an upper triangular square matrix')
|
|
n, n = T.shape
|
|
|
|
# Construct T0 with the appropriate type,
|
|
# depending on the dtype and the spectrum of T.
|
|
T_diag = np.diag(T)
|
|
keep_it_real = np.isrealobj(T) and np.min(T_diag) >= 0
|
|
if keep_it_real:
|
|
T0 = T
|
|
else:
|
|
T0 = T.astype(complex)
|
|
|
|
# Define bounds given in Table (2.1).
|
|
theta = (None,
|
|
1.59e-5, 2.31e-3, 1.94e-2, 6.21e-2,
|
|
1.28e-1, 2.06e-1, 2.88e-1, 3.67e-1,
|
|
4.39e-1, 5.03e-1, 5.60e-1, 6.09e-1,
|
|
6.52e-1, 6.89e-1, 7.21e-1, 7.49e-1)
|
|
|
|
R, s, m = _inverse_squaring_helper(T0, theta)
|
|
|
|
# Evaluate U = 2**s r_m(T - I) using the partial fraction expansion (1.1).
|
|
# This requires the nodes and weights
|
|
# corresponding to degree-m Gauss-Legendre quadrature.
|
|
# These quadrature arrays need to be transformed from the [-1, 1] interval
|
|
# to the [0, 1] interval.
|
|
nodes, weights = scipy.special.p_roots(m)
|
|
nodes = nodes.real
|
|
if nodes.shape != (m,) or weights.shape != (m,):
|
|
raise Exception('internal error')
|
|
nodes = 0.5 + 0.5 * nodes
|
|
weights = 0.5 * weights
|
|
ident = np.identity(n)
|
|
U = np.zeros_like(R)
|
|
for alpha, beta in zip(weights, nodes):
|
|
U += solve_triangular(ident + beta*R, alpha*R)
|
|
U *= np.exp2(s)
|
|
|
|
# Skip this step if the principal branch
|
|
# does not exist at T0; this happens when a diagonal entry of T0
|
|
# is negative with imaginary part 0.
|
|
has_principal_branch = all(x.real > 0 or x.imag != 0 for x in np.diag(T0))
|
|
if has_principal_branch:
|
|
|
|
# Recompute diagonal entries of U.
|
|
U[np.diag_indices(n)] = np.log(np.diag(T0))
|
|
|
|
# Recompute superdiagonal entries of U.
|
|
# This indexing of this code should be renovated
|
|
# when newer np.diagonal() becomes available.
|
|
for i in range(n-1):
|
|
l1 = T0[i, i]
|
|
l2 = T0[i+1, i+1]
|
|
t12 = T0[i, i+1]
|
|
U[i, i+1] = _logm_superdiag_entry(l1, l2, t12)
|
|
|
|
# Return the logm of the upper triangular matrix.
|
|
if not np.array_equal(U, np.triu(U)):
|
|
raise Exception('U is not upper triangular')
|
|
return U
|
|
|
|
|
|
def _logm_force_nonsingular_triangular_matrix(T, inplace=False):
|
|
# The input matrix should be upper triangular.
|
|
# The eps is ad hoc and is not meant to be machine precision.
|
|
tri_eps = 1e-20
|
|
abs_diag = np.absolute(np.diag(T))
|
|
if np.any(abs_diag == 0):
|
|
exact_singularity_msg = 'The logm input matrix is exactly singular.'
|
|
warnings.warn(exact_singularity_msg, LogmExactlySingularWarning)
|
|
if not inplace:
|
|
T = T.copy()
|
|
n = T.shape[0]
|
|
for i in range(n):
|
|
if not T[i, i]:
|
|
T[i, i] = tri_eps
|
|
elif np.any(abs_diag < tri_eps):
|
|
near_singularity_msg = 'The logm input matrix may be nearly singular.'
|
|
warnings.warn(near_singularity_msg, LogmNearlySingularWarning)
|
|
return T
|
|
|
|
|
|
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')
|
|
|
|
# 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):
|
|
X = np.empty_like(A)
|
|
X.fill(np.nan)
|
|
return X
|