Advancing convergence analysis: extending the scope of a sixth order method

Abstract

In this article, we aim to emphasize the critical role of extended convergence analysis in advancing research and understanding in the interdisciplinary fields of Applied and Computational Mathematics, Physics, Engineering, and Chemistry. By gaining a comprehensive understanding of the convergence behavior of numerical methods, one can make informed decisions regarding algorithm selection, optimization, and convergence domains, leading to more accurate and reliable scientific results in diverse applications. The conventional approach to assessing the convergence order of higher order methods for solving systems of non-linear equations relied on the Taylor series expansion, necessitating the computation of higher order derivatives that were typically absent in the method. This limitation not only constrained the method’s applicability but also increased the computational cost of solving the problem. In contrast, our study introduces a unique and innovative approach, where we demonstrate the improvised convergence of the method using only first order derivatives. Our new method offers several advantages over the traditional approach, providing valuable information regarding the radii of the convergence region and precise estimates of error boundaries. Furthermore, we establish the notion of semi-local convergence, which proves to be particularly significant as it allows for the identification of the specific domain in which the iterates converge. We have validated the convergence requirements through carefully selected numerical examples.