Higher-order root distillers

TL;DR

The paper introduces a high-order root distiller method that uses recursive maps to accurately identify and compute real roots of functions, especially high-degree polynomials, through a global filtering process.

Contribution

It presents a novel root distiller approach leveraging high-order convergence maps for global root separation and computation, demonstrated with high-degree polynomial examples.

Findings

Effective root separation for high-degree polynomials

Accurate root computation over entire intervals

Demonstrated with Chebyshev polynomial examples

Abstract

Recursive maps of high order of convergence $m$ (say $m=2^{10}$ or $m=2^{20}$) induce certain monotone step functions from which one can filter relevant information needed to globally separate and compute the real roots of a function on a given interval $[a,b]$. The process is here called a root distiller. A suitable root distiller has a powerful preconditioning effect enabling the computation, on the whole interval, of accurate roots of an high degree polynomial. Taking as model high-degree inexact Chebyshev polynomials and using the {\sl Mathematica} system, worked numerical examples are given detailing our distiller algorithm.