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scipy.sparse.linalg.

# funm_multiply_krylov

scipy.sparse.linalg.funm_multiply_krylov(f, A, b, *, assume_a='general', t=1.0, atol=0.0, rtol=1e-06, restart_every_m=None, max_restarts=20)

  A restarted Krylov method for evaluating `y = f(tA) b` from [1] [2].

  Parameters:
    f callable

      Callable object that computes the matrix function `F = f(X)`.

    A{sparse array, ndarray, LinearOperator}

      A real or complex N-by-N matrix. Alternatively, A can be a linear operator which can produce `Ax` using, e.g., `scipy.sparse.linalg.LinearOperator`.

    b ndarray

      A vector to multiply the `f(tA)` with.

    assume_a str, optional

      Indicate the structure of `A`. The algorithm will use this information to select the appropriated code path. The available options are ‘hermitian’/’her’ and ‘general’/’gen’. If ommited, then it is assumed that `A` has a ‘general’ structure.

    t float, optional

      The value to scale the matrix `A` with. The default is `t = 1.0`

    atol, rtol float, optional

      Parameters for the convergence test. For convergence, `norm(||y_k - y_k-1||) <= max(rtol*norm(b), atol)` should be satisfied. The default is `atol=0.` and `rtol=1e-6`.

    restart_every_m int

      If the iteration number reaches this value a restart is triggered. Larger values increase iteration cost but may be necessary for convergence. If omitted, `min(20, n)` is used.

    max_restarts int, optional

      Maximum number of restart cycles. The algorithm will stop after max_restarts cycles even if the specified tolerance has not been achieved. The default is `max_restarts=20`

  Returns:
    y ndarray

      The result of `f(tA) b`.

  Notes

  The convergence of the Krylov method heavily depends on the spectrum of `A` and the function `f`. With restarting, there are only formal proofs for functions of order 1 (e.g., `exp`, `sin`, `cos`) and Stieltjes functions [2] [3], while the general case remains an open problem.

  References

  [1]

  M. Afanasjew, M. Eiermann, O. G. Ernst, and S. Güttel, “Implementation of a restarted Krylov subspace method for the evaluation of matrix functions,” Linear Algebra and its Applications, vol. 429, no. 10, pp. 2293-2314, Nov. 2008, DOI:10.1016/j.laa.2008.06.029.

  [2] (1,2)

  M. Eiermann and O. G. Ernst, “A Restarted Krylov Subspace Method for the Evaluation of Matrix Functions,” SIAM J. Numer. Anal., vol. 44, no. 6, pp. 2481-2504, Jan. 2006, DOI:10.1137/050633846.

  [3]

  A. Frommer, S. Güttel, and M. Schweitzer, “Convergence of Restarted Krylov Subspace Methods for Stieltjes Functions of Matrices,” SIAM J. Matrix Anal. Appl., vol. 35, no. 4, pp. 1602-1624, Jan. 2014, DOI:10.1137/140973463.

  Examples

      >>> import numpy as np
      >>> from scipy.sparse import csr_array
      >>> from scipy.sparse.linalg import funm_multiply_krylov
      >>> from scipy.linalg import expm, solve
      >>> A = csr_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
      >>> b = np.array([2, 4, -1], dtype=float)
      >>> t = 0.1

  Compute `y = exp(tA) b`.

      >>> y = funm_multiply_krylov(expm, A, b, t = t)
      >>> y
      array([3.6164913 , 3.88421511 , 0.96073457])

      >>> ref = expm(t * A.todense()) @ b
      >>> err = y - ref
      >>> err
      array([4.44089210e-16 , 0.00000000e+00 , 2.22044605e-16])

  Compute \(y = (A^3 - A) b\).

      >>> poly = lambda X : X @ X @ X - X
      >>> y = funm_multiply_krylov(poly, A, b)
      >>> y
      array([132. , 24. , 70.])

      >>> ref = poly(A.todense()) @ b
      >>> err = y - ref
      >>> err
      array([ 0.00000000e+00 , 7.10542736e-15 , -2.84217094e-14])

  Compute \(y = f(tA) b\), where \(f(X) = X^{-1}(e^{X} - I)\). This is known as the “phi function” from the exponential integrator literature.

      >>> phim_1 = lambda X : solve(X, expm(X) - np.eye(X.shape[0]))
      >>> y = funm_multiply_krylov(phim_1, A, b, t = t)
      >>> y
      array([ 2.76984306 , 3.92769192 , -0.03111392])

      >>> ref = phim_1(t * A.todense()) @ b
      >>> err = y - ref
      >>> err
      array([ 0.00000000e+00 , 8.88178420e-16 , -4.60742555e-15])