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# scipy.sparse.linalg.gmres

scipy.sparse.linalg.gmres(A, b, x0=None, *, tol=<object object>, restart=None, maxiter=None, M=None, callback=None, restrt=<object object>, atol=0.0, callback_type=None, rtol=1e-05)

  Use Generalized Minimal RESidual iteration to solve `Ax = b`.

  Parameters:
    A{sparse matrix, 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., `scipy.sparse.linalg.LinearOperator`.

    b ndarray

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

    x0 ndarray

      Starting guess for the solution (a vector of zeros by default).

    atol, rtol float

      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`.

    restart int, optional

      Number of iterations between restarts. Larger values increase iteration cost, but may be necessary for convergence. If omitted, `min(20, n)` is used.

    maxiter int, optional

      Maximum number of iterations (restart cycles). Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. See callback_type.

    M{sparse matrix, ndarray, LinearOperator}

      Inverse of the preconditioner of A. M should approximate the inverse of A and be easy to solve for (see Notes). Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. By default, no preconditioner is used. In this implementation, left preconditioning is used, and the preconditioned residual is minimized. However, the final convergence is tested with respect to the `b - A @ x` residual.

    callback function

      User-supplied function to call after each iteration. It is called as callback(args), where args are selected by callback_type.

    callback_type{‘x’, ‘pr_norm’, ‘legacy’}, optional
      Callback function argument requested:

        - `x`: current iterate (ndarray), called on every restart

        - `pr_norm`: relative (preconditioned) residual norm (float), called on every inner iteration

        - `legacy` (default): same as `pr_norm`, but also changes the meaning of maxiter to count inner iterations instead of restart cycles.

      This keyword has no effect if callback is not set.

    restrt int, optional, deprecated

      Deprecated since version 0.11.0: `gmres` keyword argument `restrt` is deprecated in favor of `restart` and will be removed in SciPy 1.14.0.

    tol float, optional, deprecated

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

  Returns:
    x ndarray

      The converged solution.

    info int
      Provides convergence information:

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

  See also

  `LinearOperator`

  Notes

  A preconditioner, P, is chosen such that P is close to A but easy to solve for. The preconditioner parameter required by this routine is `M = P^-1`. The inverse should preferably not be calculated explicitly. Rather, use the following template to produce M:

      # Construct a linear operator that computes P^-1 @ x.
      import scipy.sparse.linalg as spla
      M_x = lambda x: spla.spsolve(P, x)
      M = spla.LinearOperator((n, n), M_x)

  Examples

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