Quadrature based optimal iterative methods

Abstract

We present a simple yet powerful and applicable quadrature based scheme for constructing optimal iterative methods. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on $n+1$ evaluations could achieve a maximum convergence order of $2^n$. Through quadrature, we develop optimal iterative methods of orders four and eight. The scheme can be further applied to develop iterative methods of even higher order. Computational results demonstrate that the developed methods are efficient as compared with many well known methods.