On root-ratio multipoint methods for finding multiple zeros of univariate functions

TL;DR

This paper introduces a natural construction for root-ratio multipoint methods to find multiple zeros of univariate functions, derives new methods, and analyzes their computational efficiency, concluding they are generally inefficient in practice.

Contribution

The paper presents a simple, natural way to construct root-ratio multipoint methods from existing simple zero methods, enabling the creation of many new algorithms.

Findings

Derived four new root-ratio methods for multiple zeros.

Found that root-ratio methods are computationally inefficient.

Compared CPU time for root extraction, highlighting practical limitations.

Abstract

Several root-ratio multipoint methods for finding multiple zeros of univariate functions were recently presented. The characteristic of these methods is that they deal with $m$-th root of ratio of two functions (hence the name root-ratio methods), where $m$ is the multiplicity of the sought zero, known in advance. Some of these methods were presented without any derivation and motivation, it could be said, out of the blue. In this paper we present an easy and entirely natural way for constructing root-ratio multipoint iterative methods starting from multipoint methods for finding simple zeros. In this way, a vast number of root-ratio multipoint methods for multiple zeros, existing as well new ones, can be constructed. For demonstration, we derive four root-ratio methods for multiple zeros. Besides, we study computational cost of the considered methods and give a comparative analysis that involves CPU time needed for the extraction of the $m$-th root. This analysis shows that root-ratio methods are pretty inefficient from the computational point of view and, thus, not suitable in practice. A general discussion on a practical need for multipoint methods of very high order is also considered.