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# 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.

  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 [R83]. 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

  [R83] | (1, 2) 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