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bundles / scipy latest / scipy / sparse / linalg / _isolve / iterative / qmr

function

scipy.sparse.linalg._isolve.iterative:qmr

source: /scipy/sparse/linalg/_isolve/iterative.py :849

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.

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:

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)
✓

See also

LinearOperator

Aliases

  • scipy.sparse.linalg.qmr