Abstract
A number of optimal order multiple root techniques that require derivative evaluations in the formulas have been proposed in literature. However, derivative-free optimal techniques for multiple roots are seldom obtained. By considering this factor as motivational, here we present a class of optimal fourth order methods for computing multiple roots without using derivatives in the iteration. The iterative formula consists of two steps in which the first step is a well-known Traub–Steffensen scheme whereas second step is a Traub–Steffensen-like scheme. The Methodology is based on two steps of which the first is Traub–Steffensen iteration and the second is Traub–Steffensen-like iteration. Effectiveness is validated on different problems that shows the robust convergent behavior of the proposed methods. It has been proven that the new derivative-free methods are good competitors to their existing counterparts that need derivative information.
Highlights
Finding root of a nonlinear equation ψ(u) = 0 is a very important and interesting problem in many branches of science and engineering
In order to validate of theoretical results that have been shown in previous sections, the new methods Method 1 (M1), Method 2 (M2), Method 3 (M3), and Method 4 (M4) are tested numerically by implementing them on some nonlinear equations
These are compared with some existing optimal fourth order Newton-like methods
Summary
Higher order methods without derivatives to calculate multiple roots are yet to be examined These methods are very useful in the problems where the derivative ψ0 is cumbersome to evaluate or is costly to compute. In terms of computational cost, the methods of [16] use three function evaluations per iteration and possess optimal fourth order convergence according to Kung–Traub conjecture (see [17]). Our aim in this work is to develop derivative-free multiple root methods of good computational efficiency, which is to say, the methods of higher convergence order with the amount of computational work as small as we please. Traub–Steffensen-like derivative-free fourth order methods that require three new pieces of information of the function ψ and have optimal fourth order convergence according to Kung–Traub conjecture.
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