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Module « scipy.sparse.linalg »

Fonction minres - module scipy.sparse.linalg

Signature de la fonction minres

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

Description

help(scipy.sparse.linalg.minres)

Use MINimum RESidual iteration to solve Ax=b

MINRES minimizes norm(Ax - 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 array, ndarray, 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 : ndarray
    Right hand side of the linear system. Has shape (N,) or (N,1).

Returns
-------
x : ndarray
    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 : ndarray
    Starting guess for the solution.
shift : float
    Value to apply to the system ``(A - shift * I)x = b``. Default is 0.
rtol : float
    Tolerance to achieve. The algorithm terminates when the relative
    residual is below ``rtol``.
maxiter : integer
    Maximum number of iterations.  Iteration will stop after maxiter
    steps even if the specified tolerance has not been achieved.
M : {sparse array, ndarray, 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.
show : bool
    If ``True``, print out a summary and metrics related to the solution
    during iterations. Default is ``False``.
check : bool
    If ``True``, run additional input validation to check that `A` and
    `M` (if specified) are symmetric. Default is ``False``.

Examples
--------
>>> import numpy as np
>>> from scipy.sparse import csc_array
>>> from scipy.sparse.linalg import minres
>>> A = csc_array([[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

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