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scipy.sparse.linalg.

# gcrotmk

scipy.sparse.linalg.gcrotmk(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=1000, M=None, callback=None, m=20, k=None, CU=None, discard_C=False, truncate='oldest')

  Solve a matrix equation using flexible GCROT(m,k) algorithm.

  Parameters:
    A{sparse array, ndarray, LinearOperator}

      The real or complex N-by-N matrix of the linear system. Alternatively, A can be a linear operator which can produce `Ax` using, e.g., `LinearOperator`.

    b ndarray

      Right hand side of the linear system. Has shape (N,) or (N,1).

    x0 ndarray

      Starting guess for the solution.

    rtol, atol float, optional

      Parameters for the convergence test. For convergence, `norm(b - A @ x) <= max(rtol*norm(b), atol)` should be satisfied. The default is `rtol=1e-5` and `atol=0.0`.

    maxiter int, optional

      Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. The default is `1000`.

    M{sparse array, ndarray, LinearOperator}, optional

      Preconditioner for A. The preconditioner should approximate the inverse of A. gcrotmk is a ‘flexible’ algorithm and the preconditioner can vary from iteration to iteration. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.

    callback function, optional

      User-supplied function to call after each iteration. It is called as `callback(xk)`, where `xk` is the current solution vector.

    m int, optional

      Number of inner FGMRES iterations per each outer iteration. Default: 20

    k int, optional

      Number of vectors to carry between inner FGMRES iterations. According to [2], good values are around m. Default: m

    CU list of tuples, optional

      List of tuples `(c, u)` which contain the columns of the matrices C and U in the GCROT(m,k) algorithm. For details, see [2]. The list given and vectors contained in it are modified in-place. If not given, start from empty matrices. The `c` elements in the tuples can be `None`, in which case the vectors are recomputed via `c = A u` on start and orthogonalized as described in [3].

    discard_C bool, optional

      Discard the C-vectors at the end. Useful if recycling Krylov subspaces for different linear systems.

    truncate{‘oldest’, ‘smallest’}, optional

      Truncation scheme to use. Drop: oldest vectors, or vectors with smallest singular values using the scheme discussed in [1,2]. See [2] for detailed comparison. Default: ‘oldest’

  Returns:
    x ndarray

      The solution found.

    info int

      Provides convergence information:

      - 0 : successful exit

      - >0 : convergence to tolerance not achieved, number of iterations

  References

  [1]

  E. de Sturler, ‘’Truncation strategies for optimal Krylov subspace methods’’, SIAM J. Numer. Anal. 36, 864 (1999).

  [2] (1,2,3)

  J.E. Hicken and D.W. Zingg, ‘’A simplified and flexible variant of GCROT for solving nonsymmetric linear systems’’, SIAM J. Sci. Comput. 32, 172 (2010).

  [3]

  M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti, ‘’Recycling Krylov subspaces for sequences of linear systems’’, SIAM J. Sci. Comput. 28, 1651 (2006).

  Examples

      >>> import numpy as np
      >>> from scipy.sparse import csc_array
      >>> from scipy.sparse.linalg import gcrotmk
      >>> R = np.random.randn(5, 5)
      >>> A = csc_array(R)
      >>> b = np.random.randn(5)
      >>> x, exit_code = gcrotmk(A, b, atol=1e-5)
      >>> print(exit_code)
      0
      >>> np.allclose(A.dot(x), b)
      True