scipy.linalg.expm_frechet

scipy.linalg.expm_frechet(A, E, method=None, compute_expm=True, check_finite=True)

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

Parameters

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.

Returns

expm_Andarray

Matrix exponential of A.

expm_frechet_AEndarray

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

For compute_expm = False, only expm_frechet_AE is returned.

See also

expm

Compute the exponential of a matrix.

Notes

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.

New in version 0.13.0.

References

1

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

Examples

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

>>> import scipy.linalg
>>> rng = np.random.default_rng()
>>> A = rng.standard_normal((3, 3))
>>> E = rng.standard_normal((3, 3))
>>> expm_A, expm_frechet_AE = scipy.linalg.expm_frechet(A, E)
>>> M = np.zeros((6, 6))
>>> M[:3, :3] = A; M[:3, 3:] = E; M[3:, 3:] = A
>>> expm_M = scipy.linalg.expm(M)
>>> np.allclose(expm_A, expm_M[:3, :3])
True
>>> np.allclose(expm_frechet_AE, expm_M[:3, 3:])
True