# scipy.sparse.linalg.qmr¶

scipy.sparse.linalg.qmr(A, b, x0=None, tol=1e-05, maxiter=None, M1=None, M2=None, callback=None, atol=None)

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, atol : float, optional Tolerances for convergence, norm(residual) <= max(tol*norm(b), atol). The default for atol is 'legacy', which emulates a different legacy behavior. Warning The default value for atol will be changed in a future release. For future compatibility, specify atol explicitly. 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.

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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