670 lines
18 KiB
Python
670 lines
18 KiB
Python
#
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# Author: Travis Oliphant, March 2002
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#
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from __future__ import division, print_function, absolute_import
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__all__ = ['expm','cosm','sinm','tanm','coshm','sinhm',
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'tanhm','logm','funm','signm','sqrtm',
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'expm_frechet', 'expm_cond', 'fractional_matrix_power']
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from numpy import (Inf, dot, diag, product, logical_not, ravel,
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transpose, conjugate, absolute, amax, sign, isfinite, single)
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import numpy as np
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# Local imports
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from .misc import norm
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from .basic import solve, inv
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from .special_matrices import triu
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from .decomp_svd import svd
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from .decomp_schur import schur, rsf2csf
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from ._expm_frechet import expm_frechet, expm_cond
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from ._matfuncs_sqrtm import sqrtm
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eps = np.finfo(float).eps
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feps = np.finfo(single).eps
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_array_precision = {'i': 1, 'l': 1, 'f': 0, 'd': 1, 'F': 0, 'D': 1}
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###############################################################################
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# Utility functions.
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def _asarray_square(A):
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"""
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Wraps asarray with the extra requirement that the input be a square matrix.
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The motivation is that the matfuncs module has real functions that have
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been lifted to square matrix functions.
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Parameters
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----------
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A : array_like
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A square matrix.
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Returns
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-------
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out : ndarray
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An ndarray copy or view or other representation of A.
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"""
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A = np.asarray(A)
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if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
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raise ValueError('expected square array_like input')
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return A
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def _maybe_real(A, B, tol=None):
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"""
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Return either B or the real part of B, depending on properties of A and B.
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The motivation is that B has been computed as a complicated function of A,
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and B may be perturbed by negligible imaginary components.
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If A is real and B is complex with small imaginary components,
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then return a real copy of B. The assumption in that case would be that
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the imaginary components of B are numerical artifacts.
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Parameters
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----------
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A : ndarray
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Input array whose type is to be checked as real vs. complex.
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B : ndarray
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Array to be returned, possibly without its imaginary part.
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tol : float
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Absolute tolerance.
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Returns
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-------
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out : real or complex array
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Either the input array B or only the real part of the input array B.
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"""
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# Note that booleans and integers compare as real.
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if np.isrealobj(A) and np.iscomplexobj(B):
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if tol is None:
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tol = {0:feps*1e3, 1:eps*1e6}[_array_precision[B.dtype.char]]
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if np.allclose(B.imag, 0.0, atol=tol):
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B = B.real
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return B
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###############################################################################
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# Matrix functions.
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def fractional_matrix_power(A, t):
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"""
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Compute the fractional power of a matrix.
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Proceeds according to the discussion in section (6) of [1]_.
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Parameters
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----------
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A : (N, N) array_like
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Matrix whose fractional power to evaluate.
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t : float
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Fractional power.
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Returns
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-------
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X : (N, N) array_like
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The fractional power of the matrix.
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References
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----------
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.. [1] Nicholas J. Higham and Lijing lin (2011)
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"A Schur-Pade Algorithm for Fractional Powers of a Matrix."
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SIAM Journal on Matrix Analysis and Applications,
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32 (3). pp. 1056-1078. ISSN 0895-4798
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Examples
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--------
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>>> from scipy.linalg import fractional_matrix_power
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>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
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>>> b = fractional_matrix_power(a, 0.5)
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>>> b
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array([[ 0.75592895, 1.13389342],
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[ 0.37796447, 1.88982237]])
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>>> np.dot(b, b) # Verify square root
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array([[ 1., 3.],
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[ 1., 4.]])
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"""
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# This fixes some issue with imports;
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# this function calls onenormest which is in scipy.sparse.
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A = _asarray_square(A)
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import scipy.linalg._matfuncs_inv_ssq
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return scipy.linalg._matfuncs_inv_ssq._fractional_matrix_power(A, t)
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def logm(A, disp=True):
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"""
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Compute matrix logarithm.
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The matrix logarithm is the inverse of
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expm: expm(logm(`A`)) == `A`
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Parameters
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----------
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A : (N, N) array_like
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Matrix whose logarithm to evaluate
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disp : bool, optional
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Print warning if error in the result is estimated large
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instead of returning estimated error. (Default: True)
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Returns
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-------
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logm : (N, N) ndarray
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Matrix logarithm of `A`
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errest : float
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(if disp == False)
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1-norm of the estimated error, ||err||_1 / ||A||_1
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References
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----------
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.. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012)
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"Improved Inverse Scaling and Squaring Algorithms
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for the Matrix Logarithm."
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SIAM Journal on Scientific Computing, 34 (4). C152-C169.
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ISSN 1095-7197
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.. [2] Nicholas J. Higham (2008)
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"Functions of Matrices: Theory and Computation"
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ISBN 978-0-898716-46-7
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.. [3] Nicholas J. Higham and Lijing lin (2011)
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"A Schur-Pade Algorithm for Fractional Powers of a Matrix."
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SIAM Journal on Matrix Analysis and Applications,
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32 (3). pp. 1056-1078. ISSN 0895-4798
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Examples
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--------
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>>> from scipy.linalg import logm, expm
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>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
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>>> b = logm(a)
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>>> b
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array([[-1.02571087, 2.05142174],
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[ 0.68380725, 1.02571087]])
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>>> expm(b) # Verify expm(logm(a)) returns a
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array([[ 1., 3.],
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[ 1., 4.]])
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"""
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A = _asarray_square(A)
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# Avoid circular import ... this is OK, right?
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import scipy.linalg._matfuncs_inv_ssq
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F = scipy.linalg._matfuncs_inv_ssq._logm(A)
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F = _maybe_real(A, F)
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errtol = 1000*eps
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#TODO use a better error approximation
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errest = norm(expm(F)-A,1) / norm(A,1)
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if disp:
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if not isfinite(errest) or errest >= errtol:
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print("logm result may be inaccurate, approximate err =", errest)
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return F
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else:
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return F, errest
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def expm(A):
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"""
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Compute the matrix exponential using Pade approximation.
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Parameters
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----------
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A : (N, N) array_like or sparse matrix
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Matrix to be exponentiated.
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Returns
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-------
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expm : (N, N) ndarray
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Matrix exponential of `A`.
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References
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----------
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.. [1] Awad H. Al-Mohy and Nicholas J. Higham (2009)
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"A New Scaling and Squaring Algorithm for the Matrix Exponential."
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SIAM Journal on Matrix Analysis and Applications.
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31 (3). pp. 970-989. ISSN 1095-7162
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Examples
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--------
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>>> from scipy.linalg import expm, sinm, cosm
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Matrix version of the formula exp(0) = 1:
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>>> expm(np.zeros((2,2)))
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array([[ 1., 0.],
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[ 0., 1.]])
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Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta))
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applied to a matrix:
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>>> a = np.array([[1.0, 2.0], [-1.0, 3.0]])
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>>> expm(1j*a)
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array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
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[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
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>>> cosm(a) + 1j*sinm(a)
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array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
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[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
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"""
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# Input checking and conversion is provided by sparse.linalg.expm().
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import scipy.sparse.linalg
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return scipy.sparse.linalg.expm(A)
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def cosm(A):
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"""
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Compute the matrix cosine.
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This routine uses expm to compute the matrix exponentials.
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Parameters
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----------
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A : (N, N) array_like
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Input array
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Returns
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-------
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cosm : (N, N) ndarray
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Matrix cosine of A
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Examples
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--------
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>>> from scipy.linalg import expm, sinm, cosm
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Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta))
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applied to a matrix:
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>>> a = np.array([[1.0, 2.0], [-1.0, 3.0]])
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>>> expm(1j*a)
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array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
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[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
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>>> cosm(a) + 1j*sinm(a)
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array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
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[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
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"""
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A = _asarray_square(A)
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if np.iscomplexobj(A):
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return 0.5*(expm(1j*A) + expm(-1j*A))
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else:
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return expm(1j*A).real
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def sinm(A):
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"""
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Compute the matrix sine.
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This routine uses expm to compute the matrix exponentials.
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Parameters
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----------
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A : (N, N) array_like
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Input array.
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Returns
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-------
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sinm : (N, N) ndarray
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Matrix sine of `A`
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Examples
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--------
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>>> from scipy.linalg import expm, sinm, cosm
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Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta))
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applied to a matrix:
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>>> a = np.array([[1.0, 2.0], [-1.0, 3.0]])
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>>> expm(1j*a)
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array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
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[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
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>>> cosm(a) + 1j*sinm(a)
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array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j],
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[ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
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"""
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A = _asarray_square(A)
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if np.iscomplexobj(A):
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return -0.5j*(expm(1j*A) - expm(-1j*A))
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else:
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return expm(1j*A).imag
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def tanm(A):
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"""
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Compute the matrix tangent.
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This routine uses expm to compute the matrix exponentials.
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Parameters
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----------
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A : (N, N) array_like
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Input array.
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Returns
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-------
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tanm : (N, N) ndarray
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Matrix tangent of `A`
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Examples
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--------
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>>> from scipy.linalg import tanm, sinm, cosm
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>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
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>>> t = tanm(a)
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>>> t
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array([[ -2.00876993, -8.41880636],
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[ -2.80626879, -10.42757629]])
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Verify tanm(a) = sinm(a).dot(inv(cosm(a)))
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>>> s = sinm(a)
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>>> c = cosm(a)
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>>> s.dot(np.linalg.inv(c))
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array([[ -2.00876993, -8.41880636],
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[ -2.80626879, -10.42757629]])
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"""
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A = _asarray_square(A)
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return _maybe_real(A, solve(cosm(A), sinm(A)))
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def coshm(A):
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"""
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Compute the hyperbolic matrix cosine.
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This routine uses expm to compute the matrix exponentials.
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Parameters
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----------
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A : (N, N) array_like
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Input array.
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Returns
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-------
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coshm : (N, N) ndarray
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Hyperbolic matrix cosine of `A`
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Examples
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--------
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>>> from scipy.linalg import tanhm, sinhm, coshm
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>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
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>>> c = coshm(a)
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>>> c
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array([[ 11.24592233, 38.76236492],
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[ 12.92078831, 50.00828725]])
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Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))
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>>> t = tanhm(a)
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>>> s = sinhm(a)
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>>> t - s.dot(np.linalg.inv(c))
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array([[ 2.72004641e-15, 4.55191440e-15],
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[ 0.00000000e+00, -5.55111512e-16]])
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"""
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A = _asarray_square(A)
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return _maybe_real(A, 0.5 * (expm(A) + expm(-A)))
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def sinhm(A):
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"""
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Compute the hyperbolic matrix sine.
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This routine uses expm to compute the matrix exponentials.
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Parameters
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----------
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A : (N, N) array_like
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Input array.
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Returns
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-------
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sinhm : (N, N) ndarray
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Hyperbolic matrix sine of `A`
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Examples
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--------
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>>> from scipy.linalg import tanhm, sinhm, coshm
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>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
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>>> s = sinhm(a)
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>>> s
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array([[ 10.57300653, 39.28826594],
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[ 13.09608865, 49.86127247]])
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Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))
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>>> t = tanhm(a)
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>>> c = coshm(a)
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>>> t - s.dot(np.linalg.inv(c))
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array([[ 2.72004641e-15, 4.55191440e-15],
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[ 0.00000000e+00, -5.55111512e-16]])
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"""
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A = _asarray_square(A)
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return _maybe_real(A, 0.5 * (expm(A) - expm(-A)))
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def tanhm(A):
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"""
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Compute the hyperbolic matrix tangent.
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This routine uses expm to compute the matrix exponentials.
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Parameters
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----------
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A : (N, N) array_like
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Input array
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Returns
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-------
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tanhm : (N, N) ndarray
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Hyperbolic matrix tangent of `A`
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Examples
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--------
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>>> from scipy.linalg import tanhm, sinhm, coshm
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>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
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>>> t = tanhm(a)
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>>> t
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array([[ 0.3428582 , 0.51987926],
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[ 0.17329309, 0.86273746]])
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Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))
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>>> s = sinhm(a)
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>>> c = coshm(a)
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>>> t - s.dot(np.linalg.inv(c))
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array([[ 2.72004641e-15, 4.55191440e-15],
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[ 0.00000000e+00, -5.55111512e-16]])
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"""
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A = _asarray_square(A)
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return _maybe_real(A, solve(coshm(A), sinhm(A)))
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def funm(A, func, disp=True):
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"""
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Evaluate a matrix function specified by a callable.
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Returns the value of matrix-valued function ``f`` at `A`. The
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function ``f`` is an extension of the scalar-valued function `func`
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to matrices.
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Parameters
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----------
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A : (N, N) array_like
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Matrix at which to evaluate the function
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func : callable
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Callable object that evaluates a scalar function f.
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Must be vectorized (eg. using vectorize).
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disp : bool, optional
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Print warning if error in the result is estimated large
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instead of returning estimated error. (Default: True)
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Returns
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-------
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funm : (N, N) ndarray
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Value of the matrix function specified by func evaluated at `A`
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errest : float
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(if disp == False)
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1-norm of the estimated error, ||err||_1 / ||A||_1
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Examples
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--------
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>>> from scipy.linalg import funm
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>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
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>>> funm(a, lambda x: x*x)
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array([[ 4., 15.],
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[ 5., 19.]])
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>>> a.dot(a)
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array([[ 4., 15.],
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[ 5., 19.]])
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Notes
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-----
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This function implements the general algorithm based on Schur decomposition
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(Algorithm 9.1.1. in [1]_).
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If the input matrix is known to be diagonalizable, then relying on the
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eigendecomposition is likely to be faster. For example, if your matrix is
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Hermitian, you can do
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>>> from scipy.linalg import eigh
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>>> def funm_herm(a, func, check_finite=False):
|
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... w, v = eigh(a, check_finite=check_finite)
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... ## 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)
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|
... return (v * w).dot(v.conj().T)
|
|
|
|
References
|
|
----------
|
|
.. [1] Gene H. Golub, Charles F. van Loan, Matrix Computations 4th ed.
|
|
|
|
"""
|
|
A = _asarray_square(A)
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|
# Perform Shur decomposition (lapack ?gees)
|
|
T, Z = schur(A)
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|
T, Z = rsf2csf(T,Z)
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|
n,n = T.shape
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|
F = diag(func(diag(T))) # apply function to diagonal elements
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|
F = F.astype(T.dtype.char) # e.g. when F is real but T is complex
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|
|
|
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
|