Several improvements of the 2-point third order midpoint iterative method using weight functions

In this paper, we improve the order of the Midpoint iterative method from three to four with same number of function evaluations using weight functions to obtain a class of 2-point fourth order Jarratt-type methods. Then we prove a general result to further improve the order of the old methods through an additional function evaluation using weight functions. In this way, we develop two 3-point sixth order midpoint methods using different weights. A family of n-point I.F.s of order 2n is proposed. Using weight functions, we further improve the methods to obtain a 5-point sixteenth order midpoint method with same efficiency index as the optimal 2-point fourth order Jarratt-type methods. Two numerical examples using different starting are provided to test the robustness of the methods.