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:

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 This parameter has been superseded by LinearOperator.

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/