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Module « scipy.optimize »

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

help(scipy.optimize.newton_krylov)

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:

.. 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] 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])

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