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 non-linear 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 well-known 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 similar-order 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.
Highlights
I n computational and applied mathematics, iterative algorithms for calculating approximate zeros of nonlinear scalar equations are of significant importance due to their wide applications in many branches of modern science such as Engineering, Mathematical Chemistry, Biomathematics, Physics, Statistics, etc
Multi-step algorithms possess greater convergence with higher computational cost due to the involvement of higher-order derivatives which is the downside of these methods
POLYNOMIOGRAPHY The term polynomiography was first introduced in 2005 by Bahman Kalantari [24], [25] who defined it as a process of drawing aesthetically pleasing graphical objects by employing the mathematical convergence properties of iteration functions
Summary
I n computational and applied mathematics, iterative algorithms for calculating approximate zeros of nonlinear scalar equations are of significant importance due to their wide applications in many branches of modern science such as Engineering, Mathematical Chemistry, Biomathematics, Physics, Statistics, etc. The researchers improved the order and efficiency of the existing methods and introduced multi-step algorithms. Noor et al [13] in 2007, introduced the fifth-order second derivative-free algorithm by employing the finite dif-. Nazeer et al [16] in 2016, introduced a novel second derivative-free Householder’s method with fifth-order convergence by employing finite-difference scheme, having the following form:. We introduced a novel efficient and derivative-free algorithm for determining the approximate solution of the non-linear scalar equations. We derived this iteration scheme by applying forward- and finite-difference schemes on Traub’s method.
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