Abstract
The principal motivation of this paper is to propose a general scheme that is applicable to every existing multi-point optimal eighth-order method/family of methods to produce a further sixteenth-order scheme. By adopting our technique, we can extend all the existing optimal eighth-order schemes whose first sub-step employs Newton’s method for sixteenth-order convergence. The developed technique has an optimal convergence order regarding classical Kung-Traub conjecture. In addition, we fully investigated the computational and theoretical properties along with a main theorem that demonstrates the convergence order and asymptotic error constant term. By using Mathematica-11 with its high-precision computability, we checked the efficiency of our methods and compared them with existing robust methods with same convergence order.
Highlights
The formation of high-order multi-point iterative techniques for the approximate solution of nonlinear equations has always been a crucial problem in computational mathematics and numerical analysis
A certain recognition has been given to the construction of sixteenth-order iterative methods in the last two decades
We have a handful of optimal iterative methods of order sixteen [3,4,5,6,7,8,9]
Summary
The formation of high-order multi-point iterative techniques for the approximate solution of nonlinear equations has always been a crucial problem in computational mathematics and numerical analysis. Optimal schemes suitable to every iterative method of particular order to obtain further high-order methods have more importance than obtaining a high-order version of a native method Finding such general schemes are a more attractive and harder chore in the area of numerical analysis. In this manuscript we pursue the development of a scheme that is suitable to every optimal 8-order scheme whose first sub-step should be the classical Newton’s method, in order to have further optimal 16-order convergence, rather than applying the technique only to a certain method. The main advantage of the constructed technique is that it is suitable to every optimal 8-order scheme whose first sub-step employs Newton’s method. The effectiveness of our technique is illustrated by several numerical examples and it is found that our methods execute superior results than the existing optimal methods with the same convergence order
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