Abstract
In this article, we design a novel fourth-order and derivative free root-finding algorithm. We construct this algorithm by applying the finite difference scheme on the well-known Ostrowski’s method. The convergence analysis shows that the newly designed algorithm possesses fourth-order convergence. To demonstrate the applicability of the designed algorithm, we consider five real-life engineering problems in the form of nonlinear scalar functions and then solve them via computer tools. The numerical results show that the new algorithm outperforms the other fourth-order comparable algorithms in the literature in terms of performance, applicability, and efficiency. Finally, we present the dynamics of the designed algorithm via computer tools by examining certain complex polynomials that depict the convergence and other graphical features of the designed algorithm.
Highlights
Complexity iterative algorithms and studied their dynamics
To execute an iterative algorithm, we always need a starting point which is refined after every iteration, and we find the approximated root up to the required accuracy after some finite iterations. e convergence rate and convergence order of an iterative algorithm are relied upon the selection of that starting point
We certify that the designed algorithm has fourth-order convergence. e designed algorithm is applied to some real-world engineering problems for certifying its better performance and applicability among the other fourth-order algorithms in the literature. e dynamical comparison of the designed algorithm with the other comparable ones has been presented via the computer program Mathematica 12.0
Summary
Where ψ is a real-valued function with an open interval domain. Suppose that α is a root of (1) with u0 as an initial guess near to the exact root α, the implication of Taylor’s series around u0 for (1) gives us ψ u0 −. Which is Newton’s root-finding algorithm [1, 2] for scalar nonlinear functions By taking it as a predictor, Ostrowski designed the following two-step iterative algorithm: vi (4). E main characteristic of the suggested algorithm is that it is derivative free and applicable to all those scalar functions whose derivatives become undefined within the domain. Algorithm 1 is a new iteration scheme for calculating the approximated roots of scalar nonlinear equations and needs only four evaluations per iteration. In this sense, the proposed algorithm’s computing cost is minimal which results in a higher efficiency index
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