From third to fourth order variant of Newton’s method for simple roots

In this paper, we show one way of deriving a fourth order scheme from any third order Newton type method without any additional evaluation of the function or its derivatives. For this purpose, the rectangular rule may be employed in Newton’s theorem about a special point. This special point may be selected from a variant of any third order scheme. Two such cases are illustrated here. The first case leads to a new fourth order scheme while the second case leads to the well known Jarratt’s method. The efficiency index of the proposed scheme is 1.587. In order to compare its performance with some of the existing third and fourth order schemes several numerical examples are furnished here.