SciPy

scipy.sparse.linalg.minres

scipy.sparse.linalg.minres(A, b, x0=None, shift=0.0, tol=1e-05, maxiter=None, xtype=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

Parameters :

A : {sparse matrix, dense matrix, LinearOperator}

The real symmetric N-by-N matrix of the linear system

b : {array, matrix}

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

Returns :

x : {array, matrix}

The converged solution.

info : integer

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.

tol : float

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

maxiter : integer

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.

callback : function

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

xtype : {‘f’,’d’,’F’,’D’}

This parameter is deprecated – avoid using it.

The type of the result. If None, then it will be determined from A.dtype.char and b. If A does not have a typecode method then it will compute A.matvec(x0) to get a typecode. To save the extra computation when A does not have a typecode attribute use xtype=0 for the same type as b or use xtype=’f’,’d’,’F’,or ‘D’. This parameter has been superseded by LinearOperator.

Notes

THIS FUNCTION IS EXPERIMENTAL AND SUBJECT TO CHANGE!

References

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. http://www.stanford.edu/group/SOL/software/minres.html
This file is a translation of the following MATLAB implementation:
http://www.stanford.edu/group/SOL/software/minres/matlab/