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#
# Author: Travis Oliphant, March 2002
#
from __future__ import division, print_function, absolute_import
__all__ = ['expm','cosm','sinm','tanm','coshm','sinhm',
'tanhm','logm','funm','signm','sqrtm',
'expm_frechet', 'expm_cond', 'fractional_matrix_power']
from numpy import (Inf, dot, diag, product, logical_not, ravel,
transpose, conjugate, absolute, amax, sign, isfinite, single)
import numpy as np
# Local imports
from .misc import norm
from .basic import solve, inv
from .special_matrices import triu
from .decomp_svd import svd
from .decomp_schur import schur, rsf2csf
from ._expm_frechet import expm_frechet, expm_cond
from ._matfuncs_sqrtm import sqrtm
eps = np.finfo(float).eps
feps = np.finfo(single).eps
_array_precision = {'i': 1, 'l': 1, 'f': 0, 'd': 1, 'F': 0, 'D': 1}
###############################################################################
# Utility functions.
def _asarray_square(A):
"""
Wraps asarray with the extra requirement that the input be a square matrix.
The motivation is that the matfuncs module has real functions that have
been lifted to square matrix functions.
Parameters
----------
A : array_like
A square matrix.
Returns
-------
out : ndarray
An ndarray copy or view or other representation of A.
"""
A = np.asarray(A)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected square array_like input')
return A
def _maybe_real(A, B, tol=None):
"""
Return either B or the real part of B, depending on properties of A and B.
The motivation is that B has been computed as a complicated function of A,
and B may be perturbed by negligible imaginary components.
If A is real and B is complex with small imaginary components,
then return a real copy of B. The assumption in that case would be that
the imaginary components of B are numerical artifacts.
Parameters
----------
A : ndarray
Input array whose type is to be checked as real vs. complex.
B : ndarray
Array to be returned, possibly without its imaginary part.
tol : float
Absolute tolerance.
Returns
-------
out : real or complex array
Either the input array B or only the real part of the input array B.
"""
# Note that booleans and integers compare as real.
if np.isrealobj(A) and np.iscomplexobj(B):
if tol is None:
tol = {0:feps*1e3, 1:eps*1e6}[_array_precision[B.dtype.char]]
if np.allclose(B.imag, 0.0, atol=tol):
B = B.real
return B
###############################################################################
# Matrix functions.
def fractional_matrix_power(A, t):
"""
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.
t : 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
Examples
--------
>>> from scipy.linalg import fractional_matrix_power
>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
>>> b = fractional_matrix_power(a, 0.5)
>>> b
array([[ 0.75592895, 1.13389342],
[ 0.37796447, 1.88982237]])
>>> np.dot(b, b) # Verify square root
array([[ 1., 3.],
[ 1., 4.]])
"""
# This fixes some issue with imports;
# this function calls onenormest which is in scipy.sparse.
A = _asarray_square(A)
import scipy.linalg._matfuncs_inv_ssq
return scipy.linalg._matfuncs_inv_ssq._fractional_matrix_power(A, t)
def logm(A, disp=True):
"""
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
disp : bool, optional
Print warning if error in the result is estimated large
instead of returning estimated error. (Default: True)
Returns
-------
logm : (N, N) ndarray
Matrix logarithm of `A`
errest : float
(if disp == False)
1-norm of the estimated error, ||err||_1 / ||A||_1
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
Examples
--------
>>> from scipy.linalg import logm, expm
>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
>>> b = logm(a)
>>> b
array([[-1.02571087, 2.05142174],
[ 0.68380725, 1.02571087]])
>>> expm(b) # Verify expm(logm(a)) returns a
array([[ 1., 3.],
[ 1., 4.]])
"""
A = _asarray_square(A)
# Avoid circular import ... this is OK, right?
import scipy.linalg._matfuncs_inv_ssq
F = scipy.linalg._matfuncs_inv_ssq._logm(A)
F = _maybe_real(A, F)
errtol = 1000*eps
#TODO use a better error approximation
errest = norm(expm(F)-A,1) / norm(A,1)
if disp:
if not isfinite(errest) or errest >= errtol:
print("logm result may be inaccurate, approximate err =", errest)
return F
else:
return F, errest
def expm(A):
"""
Compute the matrix exponential using Pade approximation.
Parameters
----------
A : (N, N) array_like or sparse matrix
Matrix to be exponentiated.
Returns
-------
expm : (N, N) ndarray
Matrix exponential of `A`.
References
----------
.. [1] Awad H. Al-Mohy and Nicholas J. Higham (2009)
"A New Scaling and Squaring Algorithm for the Matrix Exponential."
SIAM Journal on Matrix Analysis and Applications.
31 (3). pp. 970-989. ISSN 1095-7162
Examples
--------
>>> from scipy.linalg import expm, sinm, cosm
Matrix version of the formula exp(0) = 1:
>>> expm(np.zeros((2,2)))
array([[ 1., 0.],
[ 0., 1.]])
Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta))
applied to a matrix:
>>> a = np.array([[1.0, 2.0], [-1.0, 3.0]])
>>> expm(1j*a)
array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
>>> cosm(a) + 1j*sinm(a)
array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
"""
# Input checking and conversion is provided by sparse.linalg.expm().
import scipy.sparse.linalg
return scipy.sparse.linalg.expm(A)
def cosm(A):
"""
Compute the matrix cosine.
This routine uses expm to compute the matrix exponentials.
Parameters
----------
A : (N, N) array_like
Input array
Returns
-------
cosm : (N, N) ndarray
Matrix cosine of A
Examples
--------
>>> from scipy.linalg import expm, sinm, cosm
Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta))
applied to a matrix:
>>> a = np.array([[1.0, 2.0], [-1.0, 3.0]])
>>> expm(1j*a)
array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
>>> cosm(a) + 1j*sinm(a)
array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
"""
A = _asarray_square(A)
if np.iscomplexobj(A):
return 0.5*(expm(1j*A) + expm(-1j*A))
else:
return expm(1j*A).real
def sinm(A):
"""
Compute the matrix sine.
This routine uses expm to compute the matrix exponentials.
Parameters
----------
A : (N, N) array_like
Input array.
Returns
-------
sinm : (N, N) ndarray
Matrix sine of `A`
Examples
--------
>>> from scipy.linalg import expm, sinm, cosm
Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta))
applied to a matrix:
>>> a = np.array([[1.0, 2.0], [-1.0, 3.0]])
>>> expm(1j*a)
array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
>>> cosm(a) + 1j*sinm(a)
array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
"""
A = _asarray_square(A)
if np.iscomplexobj(A):
return -0.5j*(expm(1j*A) - expm(-1j*A))
else:
return expm(1j*A).imag
def tanm(A):
"""
Compute the matrix tangent.
This routine uses expm to compute the matrix exponentials.
Parameters
----------
A : (N, N) array_like
Input array.
Returns
-------
tanm : (N, N) ndarray
Matrix tangent of `A`
Examples
--------
>>> from scipy.linalg import tanm, sinm, cosm
>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
>>> t = tanm(a)
>>> t
array([[ -2.00876993, -8.41880636],
[ -2.80626879, -10.42757629]])
Verify tanm(a) = sinm(a).dot(inv(cosm(a)))
>>> s = sinm(a)
>>> c = cosm(a)
>>> s.dot(np.linalg.inv(c))
array([[ -2.00876993, -8.41880636],
[ -2.80626879, -10.42757629]])
"""
A = _asarray_square(A)
return _maybe_real(A, solve(cosm(A), sinm(A)))
def coshm(A):
"""
Compute the hyperbolic matrix cosine.
This routine uses expm to compute the matrix exponentials.
Parameters
----------
A : (N, N) array_like
Input array.
Returns
-------
coshm : (N, N) ndarray
Hyperbolic matrix cosine of `A`
Examples
--------
>>> from scipy.linalg import tanhm, sinhm, coshm
>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
>>> c = coshm(a)
>>> c
array([[ 11.24592233, 38.76236492],
[ 12.92078831, 50.00828725]])
Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))
>>> t = tanhm(a)
>>> s = sinhm(a)
>>> t - s.dot(np.linalg.inv(c))
array([[ 2.72004641e-15, 4.55191440e-15],
[ 0.00000000e+00, -5.55111512e-16]])
"""
A = _asarray_square(A)
return _maybe_real(A, 0.5 * (expm(A) + expm(-A)))
def sinhm(A):
"""
Compute the hyperbolic matrix sine.
This routine uses expm to compute the matrix exponentials.
Parameters
----------
A : (N, N) array_like
Input array.
Returns
-------
sinhm : (N, N) ndarray
Hyperbolic matrix sine of `A`
Examples
--------
>>> from scipy.linalg import tanhm, sinhm, coshm
>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
>>> s = sinhm(a)
>>> s
array([[ 10.57300653, 39.28826594],
[ 13.09608865, 49.86127247]])
Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))
>>> t = tanhm(a)
>>> c = coshm(a)
>>> t - s.dot(np.linalg.inv(c))
array([[ 2.72004641e-15, 4.55191440e-15],
[ 0.00000000e+00, -5.55111512e-16]])
"""
A = _asarray_square(A)
return _maybe_real(A, 0.5 * (expm(A) - expm(-A)))
def tanhm(A):
"""
Compute the hyperbolic matrix tangent.
This routine uses expm to compute the matrix exponentials.
Parameters
----------
A : (N, N) array_like
Input array
Returns
-------
tanhm : (N, N) ndarray
Hyperbolic matrix tangent of `A`
Examples
--------
>>> from scipy.linalg import tanhm, sinhm, coshm
>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
>>> t = tanhm(a)
>>> t
array([[ 0.3428582 , 0.51987926],
[ 0.17329309, 0.86273746]])
Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))
>>> s = sinhm(a)
>>> c = coshm(a)
>>> t - s.dot(np.linalg.inv(c))
array([[ 2.72004641e-15, 4.55191440e-15],
[ 0.00000000e+00, -5.55111512e-16]])
"""
A = _asarray_square(A)
return _maybe_real(A, solve(coshm(A), sinhm(A)))
def funm(A, func, disp=True):
"""
Evaluate a matrix function specified by a callable.
Returns the value of matrix-valued function ``f`` at `A`. The
function ``f`` is an extension of the scalar-valued function `func`
to matrices.
Parameters
----------
A : (N, N) array_like
Matrix at which to evaluate the function
func : callable
Callable object that evaluates a scalar function f.
Must be vectorized (eg. using vectorize).
disp : bool, optional
Print warning if error in the result is estimated large
instead of returning estimated error. (Default: True)
Returns
-------
funm : (N, N) ndarray
Value of the matrix function specified by func evaluated at `A`
errest : float
(if disp == False)
1-norm of the estimated error, ||err||_1 / ||A||_1
Examples
--------
>>> from scipy.linalg import funm
>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
>>> funm(a, lambda x: x*x)
array([[ 4., 15.],
[ 5., 19.]])
>>> a.dot(a)
array([[ 4., 15.],
[ 5., 19.]])
Notes
-----
This function implements the general algorithm based on Schur decomposition
(Algorithm 9.1.1. in [1]_).
If the input matrix is known to be diagonalizable, then relying on the
eigendecomposition is likely to be faster. For example, if your matrix is
Hermitian, you can do
>>> from scipy.linalg import eigh
>>> def funm_herm(a, func, check_finite=False):
... w, v = eigh(a, check_finite=check_finite)
... ## if you further know that your matrix is positive semidefinite,
... ## you can optionally guard against precision errors by doing
... # w = np.maximum(w, 0)
... w = func(w)
... return (v * w).dot(v.conj().T)
References
----------
.. [1] Gene H. Golub, Charles F. van Loan, Matrix Computations 4th ed.
"""
A = _asarray_square(A)
# Perform Shur decomposition (lapack ?gees)
T, Z = schur(A)
T, Z = rsf2csf(T,Z)
n,n = T.shape
F = diag(func(diag(T))) # apply function to diagonal elements
F = F.astype(T.dtype.char) # e.g. when F is real but T is complex
minden = abs(T[0,0])
# implement Algorithm 11.1.1 from Golub and Van Loan
# "matrix Computations."
for p in range(1,n):
for i in range(1,n-p+1):
j = i + p
s = T[i-1,j-1] * (F[j-1,j-1] - F[i-1,i-1])
ksl = slice(i,j-1)
val = dot(T[i-1,ksl],F[ksl,j-1]) - dot(F[i-1,ksl],T[ksl,j-1])
s = s + val
den = T[j-1,j-1] - T[i-1,i-1]
if den != 0.0:
s = s / den
F[i-1,j-1] = s
minden = min(minden,abs(den))
F = dot(dot(Z, F), transpose(conjugate(Z)))
F = _maybe_real(A, F)
tol = {0:feps, 1:eps}[_array_precision[F.dtype.char]]
if minden == 0.0:
minden = tol
err = min(1, max(tol,(tol/minden)*norm(triu(T,1),1)))
if product(ravel(logical_not(isfinite(F))),axis=0):
err = Inf
if disp:
if err > 1000*tol:
print("funm result may be inaccurate, approximate err =", err)
return F
else:
return F, err
def signm(A, disp=True):
"""
Matrix sign function.
Extension of the scalar sign(x) to matrices.
Parameters
----------
A : (N, N) array_like
Matrix at which to evaluate the sign function
disp : bool, optional
Print warning if error in the result is estimated large
instead of returning estimated error. (Default: True)
Returns
-------
signm : (N, N) ndarray
Value of the sign function at `A`
errest : float
(if disp == False)
1-norm of the estimated error, ||err||_1 / ||A||_1
Examples
--------
>>> from scipy.linalg import signm, eigvals
>>> a = [[1,2,3], [1,2,1], [1,1,1]]
>>> eigvals(a)
array([ 4.12488542+0.j, -0.76155718+0.j, 0.63667176+0.j])
>>> eigvals(signm(a))
array([-1.+0.j, 1.+0.j, 1.+0.j])
"""
A = _asarray_square(A)
def rounded_sign(x):
rx = np.real(x)
if rx.dtype.char == 'f':
c = 1e3*feps*amax(x)
else:
c = 1e3*eps*amax(x)
return sign((absolute(rx) > c) * rx)
result, errest = funm(A, rounded_sign, disp=0)
errtol = {0:1e3*feps, 1:1e3*eps}[_array_precision[result.dtype.char]]
if errest < errtol:
return result
# Handle signm of defective matrices:
# See "E.D.Denman and J.Leyva-Ramos, Appl.Math.Comp.,
# 8:237-250,1981" for how to improve the following (currently a
# rather naive) iteration process:
# a = result # sometimes iteration converges faster but where??
# Shifting to avoid zero eigenvalues. How to ensure that shifting does
# not change the spectrum too much?
vals = svd(A, compute_uv=0)
max_sv = np.amax(vals)
# min_nonzero_sv = vals[(vals>max_sv*errtol).tolist().count(1)-1]
# c = 0.5/min_nonzero_sv
c = 0.5/max_sv
S0 = A + c*np.identity(A.shape[0])
prev_errest = errest
for i in range(100):
iS0 = inv(S0)
S0 = 0.5*(S0 + iS0)
Pp = 0.5*(dot(S0,S0)+S0)
errest = norm(dot(Pp,Pp)-Pp,1)
if errest < errtol or prev_errest == errest:
break
prev_errest = errest
if disp:
if not isfinite(errest) or errest >= errtol:
print("signm result may be inaccurate, approximate err =", errest)
return S0
else:
return S0, errest