An Optimal Fourth Order Derivative-Free Numerical Algorithm for Multiple Roots

A plethora of higher order iterative methods, involving derivatives in algorithms, areavailableintheliteratureforfindingmultipleroots. Contrarytothisfact,thehigherordermethods without derivatives in the iteration are difficult to construct, and hence, such methods are almost non-existent. This motivated us to explore a derivative-free iterative scheme with optimal fourth order convergence. The applicability of the new scheme is shown by testing on different functions, whichillustratestheexcellentconvergence. Moreover, thecomparisonoftheperformanceshowsthat the new technique is a good competitor to existing optimal fourth order Newton-like techniques.