Dimensionality of Iterative Methods: The Adimensional Scale Invariant Steffensen (ASIS) Method

Abstract

The dimensionality of parameters and variables is a fundamental issue in physics but is mostly ignored from a mathematical point of view. Difficulties arising from dimensional inconsistency are overcome by scaling analysis and, often, both concepts, dimensionality and scaling, are confused. In the particular case of iterative methods for solving nonlinear equations, dimensionality and scaling affect their robustness: While some classical methods, such as Newton’s, are adimensional and scale independent, some other iterations such as Steffensen’s are not; their convergence depends on scaling, and their evaluation needs a dimensional congruence. In this paper, we introduce the concept of an 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 the ASIS method.