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# scipy.sparse.linalg.minres

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

  Use MINimum RESidual iteration to solve Ax=b

  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 matrix, 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 matrix, 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`.

    tol float, optional, deprecated

      Deprecated since version 1.12.0: `minres` keyword argument `tol` is deprecated in favor of `rtol` and will be removed in SciPy 1.14.0.

  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