# scipy.sparse.linalg.qmr¶

scipy.sparse.linalg.qmr(A, b, x0=None, tol=1e-05, maxiter=None, M1=None, M2=None, callback=None, atol=None)

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).

Returns
xndarray

The converged solution.

infointeger
Provides convergence information:

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

Other Parameters
x0ndarray

Starting guess for the solution.

tol, atolfloat, optional

Tolerances for convergence, `norm(residual) <= max(tol*norm(b), atol)`. The default for `atol` is `'legacy'`, which emulates a different legacy behavior.

Warning

The default value for atol will be changed in a future release. For future compatibility, specify atol explicitly.

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.

Examples

```>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import qmr
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> b = np.array([2, 4, -1], dtype=float)
>>> x, exitCode = qmr(A, b)
>>> print(exitCode)            # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True
```