Householder's Method

A root-finding algorithm based on the iteration formula

This method, like Newton's method, has poor convergence properties near any point where the derivative .

A fractal is obtained by applying Householders's method to finding a root of . Coloring the basin of attraction (the set of initial points  which converge to the same root) for each root a different color then gives the above plots.

REFERENCES:

Gourdon, X. and Sebah, P. "Newton's Iteration." http://numbers.computation.free.fr/Constants/Algorithms/newton.html.

Householder, A. S. The Numerical Treatment of a Single Nonlinear Equation. New York: McGraw-Hill, 1970.

Ortega, J. M. and Rheinboldt, W. C. Iterative Solution of Nonlinear Equations in Several Variables. Philadelphia, PA: SIAM, 2000.