scipy.sparse.linalg.minres(A, b, x0=None, shift=0.0, tol=1e-05, maxiter=None, M=None, callback=None, show=False, check=False)[source]

Use MINimum RESidual iteration to solve Ax=b

MINRES minimizes norm(A*x - b) for a real symmetric matrix A. Unlike the Conjugate Gradient method, A can be indefinite or singular.

If shift != 0 then the method solves (A - shift*I)x = b

A{sparse matrix, dense matrix, LinearOperator}

The real symmetric 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{array, matrix}

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

x{array, matrix}

The converged solution.

Provides convergence information:

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

Other Parameters
x0{array, matrix}

Starting guess for the solution.


Tolerance to achieve. The algorithm terminates when the relative residual is below tol.


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

M{sparse matrix, dense matrix, LinearOperator}

Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.


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


Solution of sparse indefinite systems of linear equations,

C. C. Paige and M. A. Saunders (1975), SIAM J. Numer. Anal. 12(4), pp. 617-629.

This file is a translation of the following MATLAB implementation:


>>> import numpy as np
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import minres
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> A = A + A.T
>>> b = np.array([2, 4, -1], dtype=float)
>>> x, exitCode = minres(A, b)
>>> print(exitCode)            # 0 indicates successful convergence
>>> np.allclose(, b)