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Module « scipy.sparse.linalg »

Fonction tfqmr - module scipy.sparse.linalg

Signature de la fonction tfqmr

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

Description

help(scipy.sparse.linalg.tfqmr)

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

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.,
    `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.
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``, the default for ``atol`` is ``0.0``.
maxiter : int, optional
    Maximum number of iterations.  Iteration will stop after maxiter
    steps even if the specified tolerance has not been achieved.
    Default is ``min(10000, ndofs * 10)``, where ``ndofs = A.shape[0]``.
M : {sparse array, ndarray, LinearOperator}
    Inverse of the preconditioner of A.  M should approximate the
    inverse of A and be easy to solve for (see Notes).  Effective
    preconditioning dramatically improves the rate of convergence,
    which implies that fewer iterations are needed to reach a given
    error tolerance.  By default, no preconditioner is used.
callback : function, optional
    User-supplied function to call after each iteration.  It is called
    as ``callback(xk)``, where ``xk`` is the current solution vector.
show : bool, optional
    Specify ``show = True`` to show the convergence, ``show = False`` is
    to close the output of the convergence.
    Default is `False`.

Returns
-------
x : ndarray
    The converged solution.
info : int
    Provides convergence information:

        - 0  : successful exit
        - >0 : convergence to tolerance not achieved, number of iterations
        - <0 : illegal input or breakdown

Notes
-----
The Transpose-Free QMR algorithm is derived from the CGS algorithm.
However, unlike CGS, the convergence curves for the TFQMR method is
smoothed by computing a quasi minimization of the residual norm. The
implementation supports left preconditioner, and the "residual norm"
to compute in convergence criterion is actually an upper bound on the
actual residual norm ``||b - Axk||``.

References
----------
.. [1] R. W. Freund, A Transpose-Free Quasi-Minimal Residual Algorithm for
       Non-Hermitian Linear Systems, SIAM J. Sci. Comput., 14(2), 470-482,
       1993.
.. [2] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition,
       SIAM, Philadelphia, 2003.
.. [3] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations,
       number 16 in Frontiers in Applied Mathematics, SIAM, Philadelphia,
       1995.

Examples
--------
>>> import numpy as np
>>> from scipy.sparse import csc_array
>>> from scipy.sparse.linalg import tfqmr
>>> A = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> b = np.array([2, 4, -1], dtype=float)
>>> x, exitCode = tfqmr(A, b, atol=0.0)
>>> print(exitCode)            # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True

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