`scipy.sparse.linalg._isolve.iterative:qmr`

source: /scipy/sparse/linalg/_isolve/iterative.py:855

Signature

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

Summary

Solve Ax = b with the Quasi-Minimal Residual method.

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.
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 atol=0. and rtol=1e-5.
maxiter : int: 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 : int

Provides convergence information:: 0successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : parameter breakdown

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
np.allclose(A.dot(x), b)

✓

True

Aliases