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Module « scipy.linalg »

Fonction expm_frechet - module scipy.linalg

Signature de la fonction expm_frechet

def expm_frechet(A, E, method=None, compute_expm=True, check_finite=True) 

Description

help(scipy.linalg.expm_frechet)

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.
method : str, optional
    Choice of algorithm. Should be one of

    - `SPS` (default)
    - `blockEnlarge`

compute_expm : bool, optional
    Whether to compute also `expm_A` in addition to `expm_frechet_AE`.
    Default is True.
check_finite : bool, 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_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.

.. versionadded:: 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 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])
True
>>> np.allclose(expm_frechet_AE, expm_M[:3, 3:])
True

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