Convergence and visualization of Laguerre's rootfinding algorithm

TL;DR

This paper establishes the first sufficient convergence criterion for Laguerre's rootfinding algorithm, enhancing its reliability by integrating the Sums of Powers Algorithm and visualizing convergence regions.

Contribution

It introduces a new convergence criterion for Laguerre's method applicable to simple roots of polynomials of degree > 3, and combines it with SPA to create a reliable algorithm called LaSPA.

Findings

Proves a new convergence criterion for Laguerre's method.

Develops the LaSPA algorithm combining Laguerre's method and SPA.

Visualizes convergence neighborhoods for different starting points.

Abstract

Laguerre's rootfinding algorithm is highly recommended although most of its properties are known only by empirical evidence. In view of this, we prove the first sufficient convergence criterion. It is applicable to simple roots of polynomials with degree greater than 3. The "Sums of Powers Algorithm" (SPA), which is a reliable iterative rootfinding method, can be used to fulfill the condition for each root. Therefore, Laguerre's method together with the SPA is now a reliable algorithm (LaSPA). In computational mathematics these results solve a central task which was first attacked by L. Euler 266 years ago. In order to study convergence properties, we eliminate the polynomial and its derivatives in the definition of the Laguerre iteration, replacing them by sums, only depending on the roots and the iterated values. For this iteration with roots, the above criterion of convergence represents an efficient stopping condition. In this way, we visualize convergence properties by coloring small neighbourhoods of each starting point in squares.