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scipy.optimize.

# broyden2

scipy.optimize.broyden2(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)

  Find a root of a function, using Broyden’s second Jacobian approximation.

  This method is also known as “Broyden’s bad method”.

  Parameters:
    F function(x) -> f

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

      xinarray_like

        Initial guess for the solution

    alpha float, optional

      > Initial guess for the Jacobian is `(-1/alpha)`.

      reduction_methodstr 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_rankint, 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.

      verbosebool, optional

        Print status to stdout on every iteration.

      maxiterint, optional

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

      f_tolfloat, optional

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

      f_rtolfloat, optional

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

      x_tolfloat, 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_rtolfloat, optional

        Relative minimum step size. If omitted, not used.

      tol_normfunction(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’.

      callbackfunction, 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='broyden2'` in particular.

  Notes

  This algorithm implements the inverse Jacobian Quasi-Newton update

  \[H_+ = H + (dx - H df) df^\dagger / ( df^\dagger df)\]

  corresponding to Broyden’s second method.

  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.broyden2(fun, [0, 0])
      >>> sol
      array([0.84116365, 0.15883529])