Spectral analysis of the errors of some families of multi-step Newton-like methods

Abstract

Higher order multi-point methods are advantageous to solve scalar nonlinear equations because of their speed of convergence. The approximate solution is usually obtained in a few steps, thus saving time and computational effort. In this work, we study the Power Spectrum of the Fourier Transforms of the errors of a well-known family of multi-step Newton method. Firstly, by two numerical examples we observe that as the order p of the methods increases, the Power Spectrum of Low Frequency modes of successive approximations decreases and the Power Spectrum of the High Frequency modes increases until they become almost equal when p become large. We prove this point analytically by showing that the Power spectrum of the Fourier Transforms of the errors of the methods from the error equation is independent of the frequency and the Amplitude of the Power Spectrum tends to zero as p tends to infinity. From there, we attempt to find the best method of this family. We also compare the family with two other families of higher order multistep methods, in terms of order, efficiency index and Amplitude of Power Spectrum. Finally, we conduct more numerical experiments to find the best method of the 3 families.