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scipy.sparse.linalg.

# minres

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

  Solve `Ax = b` with the MINimum RESidual method, for a symmetric A.

  MINRES minimizes norm(Ax - 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 array, ndarray, 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 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.

    shift float

      Value to apply to the system `(A - shift * I)x = b`. Default is 0.

    rtol float

      Tolerance to achieve. The algorithm terminates when the relative residual is below `rtol`.

    maxiter integer

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

    M{sparse array, ndarray, 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.

    show bool

      If `True`, print out a summary and metrics related to the solution during iterations. Default is `False`.

    check bool

      If `True`, run additional input validation to check that A and M (if specified) are symmetric. Default is `False`.

  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_array
      >>> from scipy.sparse.linalg import minres
      >>> A = csc_array([[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