Sixteenth-Order Optimal Iterative Scheme Based on Inverse Interpolatory Rational Function for Nonlinear Equations

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.