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# scipy.optimize.newton_krylov

scipy.optimize.newton_krylov(F, xin, iter=None, rdiff=None, method='lgmres', inner_maxiter=20, inner_M=None, outer_k=10, verbose=False, maxiter=None, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None, tol_norm=None, line_search='armijo', callback=None, **kw)

  Find a root of a function, using Krylov approximation for inverse Jacobian.

  This method is suitable for solving large-scale problems.

  Parameters:
    F function(x) -> f

      Function whose root to find; should take and return an array-like object.

    xin array_like

      Initial guess for the solution

    rdiff float, optional

      Relative step size to use in numerical differentiation.

    method str or callable, optional

      Krylov method to use to approximate the Jacobian. Can be a string, or a function implementing the same interface as the iterative solvers in `scipy.sparse.linalg`. If a string, needs to be one of: `'lgmres'`, `'gmres'`, `'bicgstab'`, `'cgs'`, `'minres'`, `'tfqmr'`.

      The default is `scipy.sparse.linalg.lgmres`.

    inner_maxiter int, optional

      Parameter to pass to the “inner” Krylov solver: maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.

    inner_M LinearOperator or InverseJacobian

      Preconditioner for the inner Krylov iteration. Note that you can use also inverse Jacobians as (adaptive) preconditioners. For example,

          >>> from scipy.optimize import BroydenFirst, KrylovJacobian
          >>> from scipy.optimize import InverseJacobian
          >>> jac = BroydenFirst()
          >>> kjac = KrylovJacobian(inner_M=InverseJacobian(jac))

      If the preconditioner has a method named ‘update’, it will be called as `update(x, f)` after each nonlinear step, with `x` giving the current point, and `f` the current function value.

    outer_k int, optional

      Size of the subspace kept across LGMRES nonlinear iterations. See `scipy.sparse.linalg.lgmres` for details.

    inner_kwargs kwargs

      Keyword parameters for the “inner” Krylov solver (defined with method). Parameter names must start with the inner_ prefix which will be stripped before passing on the inner method. See, e.g., `scipy.sparse.linalg.gmres` for details.

    iter int, optional

      Number of iterations to make. If omitted (default), make as many as required to meet tolerances.

    verbose bool, optional

      Print status to stdout on every iteration.

    maxiter int, optional

      Maximum number of iterations to make. If more are needed to meet convergence, `NoConvergence` is raised.

    f_tol float, optional

      Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6.

    f_rtol float, optional

      Relative tolerance for the residual. If omitted, not used.

    x_tol float, optional

      Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used.

    x_rtol float, optional

      Relative minimum step size. If omitted, not used.

    tol_norm function(vector) -> scalar, optional

      Norm to use in convergence check. Default is the maximum norm.

    line_search{None, ‘armijo’ (default), ‘wolfe’}, optional

      Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to ‘armijo’.

    callback function, optional

      Optional callback function. It is called on every iteration as `callback(x, f)` where x is the current solution and f the corresponding residual.

  Returns:
    sol ndarray

      An array (of similar array type as x0) containing the final solution.

  Raises:
    NoConvergence

      When a solution was not found.

  See also

  `root`

    Interface to root finding algorithms for multivariate functions. See `method='krylov'` in particular.

  `scipy.sparse.linalg.gmres`
  `scipy.sparse.linalg.lgmres`

  Notes

  This function implements a Newton-Krylov solver. The basic idea is to compute the inverse of the Jacobian with an iterative Krylov method. These methods require only evaluating the Jacobian-vector products, which are conveniently approximated by a finite difference:

  \[J v \approx (f(x + \omega*v/|v|) - f(x)) / \omega\]

  Due to the use of iterative matrix inverses, these methods can deal with large nonlinear problems.

  SciPy’s `scipy.sparse.linalg` module offers a selection of Krylov solvers to choose from. The default here is lgmres, which is a variant of restarted GMRES iteration that reuses some of the information obtained in the previous Newton steps to invert Jacobians in subsequent steps.

  For a review on Newton-Krylov methods, see for example [1], and for the LGMRES sparse inverse method, see [2].

  References

  [1]

  C. T. Kelley, Solving Nonlinear Equations with Newton’s Method, SIAM, pp.57-83, 2003. DOI:10.1137/1.9780898718898.ch3

  [2]

  D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2004). DOI:10.1016/j.jcp.2003.08.010

  [3]

  A.H. Baker and E.R. Jessup and T. Manteuffel, SIAM J. Matrix Anal. Appl. 26, 962 (2005). DOI:10.1137/S0895479803422014

  Examples

  The following functions define a system of nonlinear equations

      >>> def fun(x):
      ...     return [x[0] + 0.5 * x[1] - 1.0,
      ...             0.5 * (x[1] - x[0]) ** 2]

  A solution can be obtained as follows.

      >>> from scipy import optimize
      >>> sol = optimize.newton_krylov(fun, [0, 0])
      >>> sol
      array([0.66731771, 0.66536458])