Maps for global separation of roots

TL;DR

The paper introduces quasi-step maps, a new class of functions designed to globally separate roots of equations by analyzing fixed points of iteration maps like Newton and Halley, with theoretical properties and practical examples.

Contribution

It proposes the concept of quasi-step maps and demonstrates their effectiveness in globally separating roots of iteration maps, enhancing root-finding methods.

Findings

Quasi-step maps can effectively separate fixed points of iteration functions.

Theoretical properties of quasi-step maps are established.

Worked examples illustrate their application to Newton and Halley methods.

Abstract

Two simple predicates are adopted and certain real-valued piecewise continuous functions are constructed from them. This type of maps will be called quasi-step maps and aim to separate the fixed points of an iteration map in an interval. The main properties of these maps are studied. Several worked examples are given where appropriate quasi-step maps for Newton and Halley iteration maps illustrate the main features of quasi-step maps as tools for global separation of roots.