On the Adimensional Scale Invariant Steffensen (ASIS) Method

TL;DR

This paper introduces the Adimensional Scale Invariant Steffensen (ASIS) method, which is designed to be scale invariant by using an adimensional form of functions, improving robustness in solving non-linear equations.

Contribution

The paper proposes a novel scale invariant version of Steffensen's iterative method using adimensional functions to enhance robustness and address dimensional inconsistencies.

Findings

ASIS method is scale invariant and adimensional.

It corrects pathological features of classical Steffensen's method.

The method improves robustness in solving non-linear equations.

Abstract

Dimensionality of parameters and variables is a fundamental issue in physics but mostly ignored from a mathematical point of view. Diffculties arising from dimensional inconsistence are overcome by scaling analysis and, often, both concepts, dimensionality and scaling, are confused. In the particular case of iterative methods for solving non-linear equations, dimensionality and scaling affects their robutness: while some classical methods, such as Newton, are adimensional and scale independent, some other iterations as Steffensen's are not; their convergence depends on the scaling, and their evaluation needs a dimensional congruence. In this paper we introduce the concept of adimensional form of a function in order to study the behavior of iterative methods, thus correcting, if possible, some pathological features. From this adimensional form we will devise an adimensional and scale invariant method based on Steffensen's which we will call ASIS method.