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# scipy.optimize.broyden1

scipy.optimize.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)

  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

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

  which corresponds to Broyden’s first Jacobian update

  \[J_+ = J + (df - J dx) dx^\dagger / dx^\dagger dx\]

  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])