scipy.sparse.linalg.qmr¶

scipy.sparse.linalg.qmr(A, b, x0=None, tol=1e-05, maxiter=None, xtype=None, M1=None, M2=None, callback=None)[source]¶

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

Parameters:

Parameters:	A : {sparse matrix, dense matrix, LinearOperator} The real-valued N-by-N matrix of the linear system. It is required that the linear operator can produce `Ax` and `A^T x`. 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. M1 : {sparse matrix, dense matrix, LinearOperator} Left preconditioner for A. M2 : {sparse matrix, dense matrix, LinearOperator} Right preconditioner for A. Used together with the left preconditioner M1. The matrix M1AM2 should have better conditioned than A alone. 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 This parameter has been superseded by LinearOperator.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-valued N-by-N matrix of the linear system. It is required that the linear operator can produce Ax and A^T x.

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.

M1 : {sparse matrix, dense matrix, LinearOperator}

Left preconditioner for A.

M2 : {sparse matrix, dense matrix, LinearOperator}

Right preconditioner for A. Used together with the left preconditioner M1. The matrix M1*A*M2 should have better conditioned than A alone.

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.