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scipy.optimize.

# anderson

scipy.optimize.anderson(F, xin, iter=None, alpha=None, w0=0.01, M=5, 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 (extended) Anderson mixing.

  The Jacobian is formed by for a ‘best’ solution in the space spanned by last M vectors. As a result, only a MxM matrix inversions and MxN multiplications are required. [Ey]

  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).

    M float, optional

      Number of previous vectors to retain. Defaults to 5.

    w0 float, optional

      Regularization parameter for numerical stability. Compared to unity, good values of the order of 0.01.

    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='anderson'` in particular.

  References

  [Ey]

  22. Eyert, J. Comp. Phys., 124, 271 (1996).

  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.anderson(fun, [0, 0])
      >>> sol
      array([0.84116588, 0.15883789])