A new variant of Chebyshev-Halley's method without second derivative with convergence of optimal order

Abstract

Background and Objective: The Chebyshev-Halley is an third order iterative method that be used to find the roots of a nonlinear equation. This study is presented a new variant of Chebyshev-Halley's method without second derivative with two parameters. Methodology: In order to avoid the second derivative, it is approximated by using an equality of two methods, namely, use of a circle of curvature that has the same tangent line and to equate to the Potra-Ptak's method. Results: The results show that the method requires two evaluation of functions and one its first derivative per iteration with the efficiency index equal to 4 1/3 ≈ 1.5874. The convergence 1 3 4 analysis shows that the proposed method has the fourth-order convergence for θ = 1 and β= 1 and requires three evaluation of functions per iteration. Conclusion: The final results show that the proposed methods has better performance as compared some other kind of methods. A numerical simulation is presented to show the performance of the proposed method by using several functions.