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scipy.sparse.linalg.

# qmr

scipy.sparse.linalg.qmr(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=None, M1=None, M2=None, callback=None)

  Solve `Ax = b` with the Quasi-Minimal Residual method.

  Parameters:
    A{sparse array, 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).

    x0 ndarray

      Starting guess for the solution.

    atol, rtol float, optional

      Parameters for the convergence test. For convergence, `norm(b - A @ x) <= max(rtol*norm(b), atol)` should be satisfied. The default is `atol=0.` and `rtol=1e-5`.

    maxiter integer

      Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.

    M1{sparse array, ndarray, LinearOperator}

      Left preconditioner for A.

    M2{sparse array, 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.

  Returns:
    x ndarray

      The converged solution.

    info integer
      Provides convergence information:

        0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : parameter breakdown

  See also

  `LinearOperator`

  Examples

      >>> import numpy as np
      >>> from scipy.sparse import csc_array
      >>> from scipy.sparse.linalg import qmr
      >>> A = csc_array([[3., 2., 0.], [1., -1., 0.], [0., 5., 1.]])
      >>> b = np.array([2., 4., -1.])
      >>> x, exitCode = qmr(A, b, atol=1e-5)
      >>> print(exitCode)            # 0 indicates successful convergence
      0
      >>> np.allclose(A.dot(x), b)
      True