scipy.linalg.expm_frechet(A, E, method=None, compute_expm=True, check_finite=True)[source]#

Frechet derivative of the matrix exponential of A in the direction E.

A(N, N) array_like

Matrix of which to take the matrix exponential.

E(N, N) array_like

Matrix direction in which to take the Frechet derivative.

methodstr, optional

Choice of algorithm. Should be one of

  • SPS (default)

  • blockEnlarge

compute_expmbool, optional

Whether to compute also expm_A in addition to expm_frechet_AE. Default is True.

check_finitebool, optional

Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.


Matrix exponential of A.


Frechet derivative of the matrix exponential of A in the direction E.

For compute_expm = False, only expm_frechet_AE is returned.

See also


Compute the exponential of a matrix.


This section describes the available implementations that can be selected by the method parameter. The default method is SPS.

Method blockEnlarge is a naive algorithm.

Method SPS is Scaling-Pade-Squaring [1]. It is a sophisticated implementation which should take only about 3/8 as much time as the naive implementation. The asymptotics are the same.

Added in version 0.13.0.



Awad H. Al-Mohy and Nicholas J. Higham (2009) Computing the Frechet Derivative of the Matrix Exponential, with an application to Condition Number Estimation. SIAM Journal On Matrix Analysis and Applications., 30 (4). pp. 1639-1657. ISSN 1095-7162


>>> import numpy as np
>>> from scipy import linalg
>>> rng = np.random.default_rng()
>>> A = rng.standard_normal((3, 3))
>>> E = rng.standard_normal((3, 3))
>>> expm_A, expm_frechet_AE = linalg.expm_frechet(A, E)
>>> expm_A.shape, expm_frechet_AE.shape
((3, 3), (3, 3))

Create a 6x6 matrix containing [[A, E], [0, A]]:

>>> M = np.zeros((6, 6))
>>> M[:3, :3] = A
>>> M[:3, 3:] = E
>>> M[3:, 3:] = A
>>> expm_M = linalg.expm(M)
>>> np.allclose(expm_A, expm_M[:3, :3])
>>> np.allclose(expm_frechet_AE, expm_M[:3, 3:])