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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.

  x0 : 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 (ie., 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.

  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

  [R163] | 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). http://www.math.leidenuniv.nl/scripties/Rotten.pdf