scipy.sparse.linalg.qmr#

scipy.sparse.linalg.qmr(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=None, M1=None, M2=None, callback=None)[source]#

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

Parameters:
A{sparse matrix, 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`.

bndarray

Right hand side of the linear system. Has shape (N,) or (N,1).

x0ndarray

Starting guess for the solution.

atol, rtolfloat, 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`.

maxiterinteger

Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.

M1{sparse matrix, ndarray, LinearOperator}

Left preconditioner for A.

M2{sparse matrix, 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.

callbackfunction

User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

Returns:
xndarray

The converged solution.

infointeger
Provides convergence information:

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

Examples

```>>> import numpy as np
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import qmr
>>> A = csc_matrix([[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
```