jax.scipy.linalg.expm_frechet¶

jax.scipy.linalg.expm_frechet(A, E, *, method=None, compute_expm=True)[source]¶

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

LAX-backend implementation of expm_frechet().

Does not currently support the Scipy argument jax.numpy.asarray_chkfinite, because jax.numpy.asarray_chkfinite does not exist at the moment. Does not support the method='blockEnlarge' argument.

Original docstring below.

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.
method (str, optional) – Choice of algorithm. Should be one of
compute_expm (bool, optional) – Whether to compute also expm_A in addition to expm_frechet_AE. Default is True.

Returns

expm_A (ndarray) – Matrix exponential of A.
expm_frechet_AE (ndarray) – 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
>>> A = np.random.randn(3, 3)
>>> E = np.random.randn(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
>>> A = np.random.randn(3, 3)
>>> E = np.random.randn(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