Module « scipy.sparse.linalg »

Fonction minres - module scipy.sparse.linalg

Signature de la fonction minres

def minres(A, b, x0=None, shift=0.0, tol=1e-05, maxiter=None, M=None, callback=None, show=False, check=False) 

Description

minres.__doc__

    Use MINimum RESidual iteration to solve Ax=b

    MINRES minimizes norm(A*x - b) for a real symmetric matrix A.  Unlike
    the Conjugate Gradient method, A can be indefinite or singular.

    If shift != 0 then the method solves (A - shift*I)x = b

    Parameters
    ----------
    A : {sparse matrix, dense matrix, LinearOperator}
        The real symmetric 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).

    Returns
    -------
    x : {array, matrix}
        The converged solution.
    info : integer
        Provides convergence information:
            0  : successful exit
            >0 : convergence to tolerance not achieved, number of iterations
            <0 : illegal input or breakdown

    Other Parameters
    ----------------
    x0  : {array, matrix}
        Starting guess for the solution.
    tol : float
        Tolerance to achieve. The algorithm terminates when the relative
        residual is below `tol`.
    maxiter : integer
        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}
        Preconditioner for A.  The preconditioner should approximate the
        inverse of A.  Effective preconditioning dramatically improves the
        rate of convergence, which implies that fewer iterations are needed
        to reach a given error tolerance.
    callback : function
        User-supplied function to call after each iteration.  It is called
        as callback(xk), where xk is the current solution vector.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.sparse import csc_matrix
    >>> from scipy.sparse.linalg import minres
    >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
    >>> A = A + A.T
    >>> b = np.array([2, 4, -1], dtype=float)
    >>> x, exitCode = minres(A, b)
    >>> print(exitCode)            # 0 indicates successful convergence
    0
    >>> np.allclose(A.dot(x), b)
    True

    References
    ----------
    Solution of sparse indefinite systems of linear equations,
        C. C. Paige and M. A. Saunders (1975),
        SIAM J. Numer. Anal. 12(4), pp. 617-629.
        https://web.stanford.edu/group/SOL/software/minres/

    This file is a translation of the following MATLAB implementation:
        https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip