# 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:
    A{sparse matrix, ndarray, LinearOperator}

      The real-valued N-by-N matrix of the linear system. Alternatively, `A` can be a linear operator which can produce `Ax` and `A^T x` using, e.g., `scipy.sparse.linalg.LinearOperator`.

    b ndarray

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

  Returns:
    x ndarray

      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 ndarray

      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, ndarray, LinearOperator}

      Left preconditioner for A.

    M2{sparse matrix, ndarray, 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.

  See also

  `LinearOperator`

  Examples

      >>> import numpy as np
      >>> 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