A variant of Steffensen-King's type family with accelerated sixth-order convergence and high efficiency index: Dynamic study and approach

Keywords: Multipoint iterative methods Steffensen's method King's family Derivative-free Efficiency index a b s t r a c t First, it is attempted to derive an optimal derivative-free Steffensen–King's type family without memory for computing a simple zero of a nonlinear function with efficiency index 4 1=3 % 1:587. Next, since our without memory family includes a parameter in which it is still possible to increase the convergence order without any new function evaluations. Therefore, we extract a new method with memory so that the convergence order rises to six without any new function evaluation and therefore reaches efficiency index 6 1=3 % 1:817. Consequently, derivative-free and high efficiency index would be the substantial contributions of this work as opposed to the classical Steffensen's and King's methods. Finally, we compare some of the convergence planes with different weight functions in order to show which are the best ones. Construction and development of Multipoint methods without memory, for computing a simple or multiple roots of a given nonlinear function, is based on Kung and Traub's conjecture [12], proved in some special cases [22,25,26]. It is supposed that any multipoint method without memory by using n function evaluations per iteration can reach optimal convergence order 2 n. The most well known optimal method is Newton's method [2] but we are interested in two-point methods. There are some legendary optimal two-point methods without memory like King's [10], Ostrowski's [16] and Jarratt's [7,8,14] methods in which they consume three function evaluations per iteration with optimal convergence four. In spite being optimal, they require evaluation of the first derivative at one or two points and hence cannot be used for nonsmooth functions. Needless to say, King's and Ostrowski's type methods have been considerably used to create higher optimal order methods without memory on the other hand, based our best knowledge, there is no higher optimal order of Jarratt's type method. In this work, King's family is modified to avoid using derivative evaluation and consequently can be applied to nonsmooth functions. To this end, its first step is replaced by an Steffensen's type method [19] and its second step is altered by considering the idea of weight function. It is attempted to preserve optimality, too. Until now, this modified family can be called a