Real-World Applications of a Newly Designed Root-Finding Algorithm and Its Polynomiography

Abstract

Solving non-linear equations in different scientific disciplines is one of the most important and frequently appearing problems. A variety of real-world problems in different scientific fields can be modeled via non-linear equations. Iterative algorithms play a vital role in finding the solution of such nonlinear problems. This article aims to design a new iterative algorithm that is derivative-free and performing better. We construct this algorithm by applying the forward-and finite-difference schemes on the wellknown Traubs's method which yields us an efficient and derivative-free algorithm whose computational cost is low as per iteration. We also study the convergence criterion of the designed algorithm and prove its fifth-order convergence. To demonstrate the accuracy, validity and applicability of the designed algorithm, we consider eleven different types of numerical test examples and solve them. The considered problems also involve some real-life applications of civil and chemical engineering. The obtained numerical results of the test examples show that the newly designed algorithm is working better against the other similarorder algorithms in the literature. For the graphical analysis, we consider some different-degree complex polynomials and draw polynomiographs of the designed fifth-order algorithm and compare them with the other fifth-order methods with the help of a computer program Mathematica 12.0. The graphical results show the convergence speed and other graphical characteristics of the designed algorithm and prove its supremacy over the other comparable ones.

On new sixth and seventh order iterative methods for solving non-linear equations using homotopy perturbation technique

“…In resent time, several scientists and engineers have been focused to solve non-linear equations numerically as well as analytically. In the literature, there are several iterative methods/algorithms available which are derived from different techniques such as homotopy, interpolation, Taylor’s series, quadrature formulas, decomposition etc., and also available various modifications and improvements of the existing methods, and different hybrid iterative methods, see, for example [ 1 , 4 – 7 , 9 – 16 , 28 – 32 , 36 – 38 ]. In general, the roots of non-linear or transcendental equations cannot be expressed in closed form or cannot be computed analytically.…”

A derivative-free root-finding algorithm using exponential method and its implementation

“…Nazeer et al introduced a novel second derivative-free Householder’s method having fifth-order convergence by using finite-difference scheme [ 24 ] in 2016. Recently, in 2021, Amir et al developed an efficient and derivative-free algorithm for determining an approximate solution of the given non-linear scalar equations by applying forward- and finite-difference schemes similar to Traub’s method, see [ 25 ]. In this paper, we propose a new root- finding algorithm, which is derivative-free, using exponential method.…”