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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.

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

  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.

  Added 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 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