Abstract
In this study, we propose a modified predictor‐corrector Newton‐Halley (MPCNH) method for solving nonlinear equations. The proposed sixteenth‐order MPCNH is free of second derivatives and has a high efficiency index. The convergence analysis of the modified method is discussed. Different problems were tested to demonstrate the applicability of the proposed method. Some are real life problems such as a chemical equilibrium problem (conversion in a chemical reactor), azeotropic point of a binary solution, and volume from van der Waals equation. Several comparisons with other optimal and nonoptimal iterative techniques of equal order are presented to show the efficiency of the modified method and to clarify the question, are the optimal methods always good for solving nonlinear equations?
Highlights
IntroductionMany researchers applied the technique of updating the solution to improve the convergence order of the iterative schemes
Searching out a solution of g(z) = 0, when g(z) is nonlinear, is highly significant in mathematics; because many equations of that type are common in applied sciences and real life problems
We propose a modified predictor-corrector Newton-Halley (MPCNH) method for solving nonlinear equations
Summary
Many researchers applied the technique of updating the solution to improve the convergence order of the iterative schemes. Kung and Traub [8] conjectured that the iterative scheme with the number of functional evaluations equal to r is optimal if its order of convergence equals 2r−1. To obtain optimal methods to reduce the functional evaluations at each iteration, researchers use different approximations and interpolations. The proposed scheme has efficiency index (16)1/6 ≈ 1.587, which is better than PCNH and both Halley’s and Householder’s methods. Another advantage of this modified method is that it is second derivative free scheme. Note that this modified method is not optimal since it does not satisfy Kung-Traub conjecture
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