Chapter 65 - Computer art from Newton's, Ssecant, and Richardson's methods

Abstract

Newton's Method produces beautiful fractal patterns. The equation first used by him to demonstrate his method is z3 — 2z — 5 = 0.. This chapter solves this to create fractal patterns, which show bands of color representing areas of constant iteration number, in three ways. First is Newton's method, but with the derivative perturbed by parameter P1:f'(z) + P1[4]. Patterns produced this way serve as reference patterns for comparison with those produced using numerical approximations to the derivative. Second is the Secant Method, where f'(z) is approximated by [f(z + h) —f(z)]/h, with an error of the order of h. This error is a useful tool for creating unusual patterns. And, third is Richardson's Extrapolation for f'(z), which approximates f'(z) with an error of the order of h4, or better. This method is vastly superior to the Secant Method at higher magnifications, (for example 6 million), when the small values of h (such as 10-8) necessary using the Secant method to compute a sufficiently accurate numerical derivative suffer from computer round off.