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Module « scipy.sparse.linalg »

Fonction qmr - module scipy.sparse.linalg

Signature de la fonction qmr

def qmr(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=None, M1=None, M2=None, callback=None) 

Description

help(scipy.sparse.linalg.qmr)

Use Quasi-Minimal Residual iteration to solve ``Ax = b``.

Parameters
----------
A : {sparse array, ndarray, LinearOperator}
    The real-valued N-by-N matrix of the linear system.
    Alternatively, ``A`` can be a linear operator which can
    produce ``Ax`` and ``A^T x`` using, e.g.,
    ``scipy.sparse.linalg.LinearOperator``.
b : ndarray
    Right hand side of the linear system. Has shape (N,) or (N,1).
x0 : ndarray
    Starting guess for the solution.
atol, rtol : float, optional
    Parameters for the convergence test. For convergence,
    ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied.
    The default is ``atol=0.`` and ``rtol=1e-5``.
maxiter : integer
    Maximum number of iterations.  Iteration will stop after maxiter
    steps even if the specified tolerance has not been achieved.
M1 : {sparse array, ndarray, LinearOperator}
    Left preconditioner for A.
M2 : {sparse array, ndarray, LinearOperator}
    Right preconditioner for A. Used together with the left
    preconditioner M1.  The matrix M1@A@M2 should have better
    conditioned than A alone.
callback : function
    User-supplied function to call after each iteration.  It is called
    as callback(xk), where xk is the current solution vector.

Returns
-------
x : ndarray
    The converged solution.
info : integer
    Provides convergence information:
        0  : successful exit
        >0 : convergence to tolerance not achieved, number of iterations
        <0 : parameter breakdown

See Also
--------
LinearOperator

Examples
--------
>>> import numpy as np
>>> from scipy.sparse import csc_array
>>> from scipy.sparse.linalg import qmr
>>> A = csc_array([[3., 2., 0.], [1., -1., 0.], [0., 5., 1.]])
>>> b = np.array([2., 4., -1.])
>>> x, exitCode = qmr(A, b, atol=1e-5)
>>> print(exitCode)            # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True

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