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Fonction newton_krylov - module scipy.optimize

Signature de la fonction newton_krylov

def 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) 

Description

newton_krylov.__doc__

    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 : {'lgmres', 'gmres', 'bicgstab', 'cgs', 'minres'} or function
        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`.

        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.nonlin import BroydenFirst, KrylovJacobian
        >>> from scipy.optimize.nonlin 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:

    .. math:: 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] D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2004).
           :doi:`10.1016/j.jcp.2003.08.010`
    .. [2] 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])