Recursive families of higher order iterative maps

TL;DR

This paper introduces recursive families of higher-order iterative maps, including the Newton-barycentric family, to efficiently approximate roots and extrema in complex functions, demonstrated through practical examples.

Contribution

It presents a new recursive framework for higher-order iterative maps, notably the Newton-barycentric family, enhancing root-finding and extremum locating methods.

Findings

Newton-barycentric maps achieve higher convergence orders.

Applications include least squares problems and multiple extrema detection.

Demonstrated effectiveness in practical examples.

Abstract

To approximate a simple root of an equation we construct families of iterative maps of higher order of convergence. These maps are based on model functions which can be written as an inner product. The main family of maps discussed is defined recursively and is called {\it Newton-barycentric}. We illustrate the application of Newton-barycentric maps in two worked examples, one dealing with a typical least squares problem and the other showing how to locate simultaneously a great number of extrema of the Ackley's function.