Module « scipy.sparse.linalg »

Fonction gcrotmk - module scipy.sparse.linalg

Signature de la fonction gcrotmk

def gcrotmk(A, b, x0=None, tol=1e-05, maxiter=1000, M=None, callback=None, m=20, k=None, CU=None, discard_C=False, truncate='oldest', atol=None) 

Description

gcrotmk.__doc__

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

    Parameters
    ----------
    A : {sparse matrix, dense matrix, 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.,
        ``scipy.sparse.linalg.LinearOperator``.
    b : {array, matrix}
        Right hand side of the linear system. Has shape (N,) or (N,1).
    x0  : {array, matrix}
        Starting guess for the solution.
    tol, atol : float, optional
        Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``.
        The default for ``atol`` is `tol`.

        .. warning::

           The default value for `atol` will be changed in a future release.
           For future compatibility, specify `atol` explicitly.
    maxiter : int, optional
        Maximum number of iterations.  Iteration will stop after maxiter
        steps even if the specified tolerance has not been achieved.
    M : {sparse matrix, dense matrix, 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 : array or matrix
        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] 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).