# scipy.sparse.linalg.minres

scipy.sparse.linalg.minres(A, b, x0=None, shift=0.0, tol=1e-05, maxiter=None, M=None, callback=None, show=False, check=False)

  Use MINimum RESidual iteration to solve Ax=b

  MINRES minimizes norm(A*x - b) for a real symmetric matrix A. Unlike the Conjugate Gradient method, A can be indefinite or singular.

  If shift != 0 then the method solves (A - shift*I)x = b

  Parameters
    A{sparse matrix, dense matrix, LinearOperator}

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

    b{array, matrix}

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

  Returns
    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

  Other Parameters
    x0{array, matrix}

      Starting guess for the solution.

    tol float

      Tolerance to achieve. The algorithm terminates when the relative 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.

    M{sparse matrix, dense matrix, LinearOperator}

      Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.

    callback function

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

  References

  Solution of sparse indefinite systems of linear equations,

    C. C. Paige and M. A. Saunders (1975), SIAM J. Numer. Anal. 12(4), pp. 617-629. https://web.stanford.edu/group/SOL/software/minres/

  This file is a translation of the following MATLAB implementation:

    https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip

  Examples

      >>> import numpy as np
      >>> from scipy.sparse import csc_matrix
      >>> from scipy.sparse.linalg import minres
      >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
      >>> A = A + A.T
      >>> b = np.array([2, 4, -1], dtype=float)
      >>> x, exitCode = minres(A, b)
      >>> print(exitCode)            # 0 indicates successful convergence
      0
      >>> np.allclose(A.dot(x), b)
      True

scipy.sparse.linalg.lgmres scipy.sparse.linalg.qmr