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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`.

    bndarray

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

    x0ndarray

      Starting guess for the solution.

    atol, rtolfloat, 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`.

    maxiterinteger

      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.

    callbackfunction

      User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

  Returns:
    xndarray

      The converged solution.

    infointeger
      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