# 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: Returns: 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). 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 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 This parameter has been superseded by LinearOperator.

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


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