Nonstationary iterative processes

TL;DR

This paper introduces s-nonstationary iterative methods that enhance efficiency for root-finding, generalizing classical methods like Newton's and Halley's, with practical algorithms and numerical demonstrations.

Contribution

It defines s-nonstationary iterative processes and proves their higher efficiency compared to traditional one-point methods, providing constructions and geometric interpretations.

Findings

Existence of higher-efficiency s-nonstationary processes

Construction methods for these processes

Numerical examples demonstrating effectiveness

Abstract

In this paper we present iterative methods of high efficiency by the criteria of J. F. Traub and A. M. Ostrowski. We define {\it s-nonstationary iterative processes} and prove that, for any one-point iterative process without memory, such as, for example, Newton's, Halley's, Chebyshev's methods, there exists an s-nonstationary process of the same order, but of higher efficiency. We supply constructions of these methods, obtain their properties and, for some of them, also their geometric interpretation. The algorithms we present can be transformed into computer programs in straight-forward manner. The methods are demonstrated by numerical examples.