Graphical and Numerical Study of a Newly Developed Root-Finding Algorithm and Its Engineering Applications

Abstract

The primary objective of this paper is to develop a new method for root-finding by combining forward and finite-difference techniques in order to provide an efficient, derivative-free algorithm with a lower processing cost per iteration. This will be accomplished by combining forward and finite-difference techniques. We also detail the convergence criterion that was devised for the root-finding approach, and we show that the method that was recommended is quintic-order convergent. We addressed a few engineering issues in order to illustrate the validity and application of the developed root-finding algorithm. The quantitative results justified the constructed root-finding algorithm's robust performance in comparison to other quintic-order methods that can be found in the literature. For the graphical analysis, we make use of the newly discovered method to plot some novel polynomiographs that are attractive to the eye, and then we evaluate these new plots in relation to previously established quintic-order root-finding strategies. The graphic analysis demonstrates that the newly created method for root-finding has better convergence with the larger area than the other comparable methods do.INDEX TERMS Computational Algorithms; Convergence-order; Non-linearity; Halley's scheme; Dynamical aspects.which is quadratic-order Newton's algorithm for root-finding of non-linear scalar equations.Recent years have seen the development of new multistep algorithms and the application of diverse mathematical methodologies that vastly improve upon the previous approaches. Using decomposition methods, the authors of [2], [3] developed some new algorithms for root-finding of VOLUME 4, 2016 i This article has been accepted for publication in IEEE Access.

An Improved Blended Numerical Root-Solver for Nonlinear Equations

“…Being higher-order accurate and cost-efficient is crucial for a numerical approach, achieved by minimizing the number of evaluations and iterations required. The study of efficient techniques for solving non-linear equations is carried out in many studies; see for example some of the recently published research papers sania.qureshi@faculty.muet.edu.pk VFAST Transactions on Mathematics [7,15,[19][20][21]. Nonlinear equations frequently arise through the application of mathematical methods, and finding their solution is a crucial objective in numerical analysis.…”

A New Second Derivative Free Iterative Method of Fifth Order of Convergence and Its Applications

“…Many researchers have employed several techniques in order to avoid these problems. There is a vast literature on how to avoid the calculation of second derivative in an iterative methods as one can find in ( [5], [8], [9], [10], [11], [12], [13]).…”