Module « scipy.optimize »

Fonction broyden1 - module scipy.optimize

Signature de la fonction broyden1

def broyden1(F, xin, iter=None, alpha=None, reduction_method='restart', max_rank=None, 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

broyden1.__doc__

    Find a root of a function, using Broyden's first Jacobian approximation.

    This method is also known as \"Broyden's good method\".

    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
    alpha : float, optional
        Initial guess for the Jacobian is ``(-1/alpha)``.
    reduction_method : str or tuple, optional
        Method used in ensuring that the rank of the Broyden matrix
        stays low. Can either be a string giving the name of the method,
        or a tuple of the form ``(method, param1, param2, ...)``
        that gives the name of the method and values for additional parameters.

        Methods available:

            - ``restart``: drop all matrix columns. Has no extra parameters.
            - ``simple``: drop oldest matrix column. Has no extra parameters.
            - ``svd``: keep only the most significant SVD components.
              Takes an extra parameter, ``to_retain``, which determines the
              number of SVD components to retain when rank reduction is done.
              Default is ``max_rank - 2``.

    max_rank : int, optional
        Maximum rank for the Broyden matrix.
        Default is infinity (i.e., no rank reduction).
    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=='broyden1'`` in particular.

    Notes
    -----
    This algorithm implements the inverse Jacobian Quasi-Newton update

    .. math:: H_+ = H + (dx - H df) dx^\dagger H / ( dx^\dagger H df)

    which corresponds to Broyden's first Jacobian update

    .. math:: J_+ = J + (df - J dx) dx^\dagger / dx^\dagger dx


    References
    ----------
    .. [1] B.A. van der Rotten, PhD thesis,
       \"A limited memory Broyden method to solve high-dimensional
       systems of nonlinear equations\". Mathematisch Instituut,
       Universiteit Leiden, The Netherlands (2003).

       https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf

    Examples
    --------
    The following functions define a system of nonlinear equations

    >>> def fun(x):
    ...     return [x[0]  + 0.5 * (x[0] - x[1])**3 - 1.0,
    ...             0.5 * (x[1] - x[0])**3 + x[1]]

    A solution can be obtained as follows.

    >>> from scipy import optimize
    >>> sol = optimize.broyden1(fun, [0, 0])
    >>> sol
    array([0.84116396, 0.15883641])