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scipy.sparse.linalg.

# gcrotmk

scipy.sparse.linalg.gcrotmk(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=1000, M=None, callback=None, m=20, k=None, CU=None, discard_C=False, truncate='oldest')

  Solve `Ax = b` with the flexible GCROT(m,k) algorithm.

  Parameters:
    A{sparse array, ndarray, LinearOperator}

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

    bndarray

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

    x0ndarray

      Starting guess for the solution.

    rtol, atolfloat, optional

      Parameters for the convergence test. For convergence, `norm(b - A @ x) <= max(rtol*norm(b), atol)` should be satisfied. The default is `rtol=1e-5` and `atol=0.0`.

    maxiterint, optional

      Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. The default is `1000`.

    M{sparse array, ndarray, LinearOperator}, optional

      Preconditioner for A. The preconditioner should approximate the inverse of A. gcrotmk is a ‘flexible’ algorithm and the preconditioner can vary from iteration to iteration. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.

    callbackfunction, optional

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

    mint, optional

      Number of inner FGMRES iterations per each outer iteration. Default: 20

    kint, optional

      Number of vectors to carry between inner FGMRES iterations. According to [2], good values are around m. Default: m

    CUlist of tuples, optional

      List of tuples `(c, u)` which contain the columns of the matrices C and U in the GCROT(m,k) algorithm. For details, see [2]. The list given and vectors contained in it are modified in-place. If not given, start from empty matrices. The `c` elements in the tuples can be `None`, in which case the vectors are recomputed via `c = A u` on start and orthogonalized as described in [3].

    discard_Cbool, optional

      Discard the C-vectors at the end. Useful if recycling Krylov subspaces for different linear systems.

    truncate{‘oldest’, ‘smallest’}, optional

      Truncation scheme to use. Drop: oldest vectors, or vectors with smallest singular values using the scheme discussed in [1,2]. See [2] for detailed comparison. Default: ‘oldest’

  Returns:
    xndarray

      The solution found.

    infoint

      Provides convergence information:

      - 0 : successful exit

      - >0 : convergence to tolerance not achieved, number of iterations

  References

  [1]

  E. de Sturler, ‘’Truncation strategies for optimal Krylov subspace methods’’, SIAM J. Numer. Anal. 36, 864 (1999).

  [2] (1,2,3)

  J.E. Hicken and D.W. Zingg, ‘’A simplified and flexible variant of GCROT for solving nonsymmetric linear systems’’, SIAM J. Sci. Comput. 32, 172 (2010).

  [3]

  M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti, ‘’Recycling Krylov subspaces for sequences of linear systems’’, SIAM J. Sci. Comput. 28, 1651 (2006).

  Examples

      >>> import numpy as np
      >>> from scipy.sparse import csc_array
      >>> from scipy.sparse.linalg import gcrotmk
      >>> R = np.random.randn(5, 5)
      >>> A = csc_array(R)
      >>> b = np.random.randn(5)
      >>> x, exit_code = gcrotmk(A, b, atol=1e-5)
      >>> print(exit_code)
      0
      >>> np.allclose(A.dot(x), b)
      True