This paper deals with multipoint iterations without memory for the solution of the nonlinear scalar equation (m) f (x) = 0, m ^ 0. Let P n( ) be the maximal order of iterations which use n evaluations of the function or its derivatives per step. We prove the Kung and Traub conjecture p (0) = 2 n ^ for Hermitian information. We show p (m+l)^p (m) and conjecture P n( ) = 2 . The problem of the ma...

Abstract Many multipoint iterative methods without memory for solving non-linear equations in one variable are found the literature. In particular, there that provide fourth-order, eighth-order or sixteenth-order convergence using only, respectively, three, four five function evaluations per iteration step, thus supporting Kung-Traub conjecture on optimal order of convergence. This paper shows ...

A class of three-pointmethods for solving nonlinear equations of eighth order is constructed. These methods are developed by combining two-point Ostrowski’s fourth-ordermethods and amodified Newton’s method in the third step, obtained by a suitable approximation of the first derivative using the product of three weight functions. The proposed three-step methods have order eight costing only fou...

Making use of last derivative approximation and weight function approach, we construct an eighth-order class of three-step methods, which are consistent with the optimality conjecture of Kung-Traub for constructing multi-point methods without memory. Per iteration, any method of the developed class is totally free from derivative evaluation. Such classes of schemes are more practical when the c...

A class of derivative-free methods without memory for approximating a simple zero of a nonlinear equation is presented. The proposed class uses four function evaluations per iteration with convergence order eight. Therefore, it is an optimal three-step scheme without memory based on Kung-Traub conjecture. Moreover, the proposed class has an accelerator parameter with the property that it can in...

We have given a four-step, multipoint iterative method without memory for solving nonlinear equations. The method is constructed by using quasi-Hermite interpolation and has order of convergence sixteen. As this method requires four function evaluations and one derivative evaluation at each step, it is optimal in the sense of the Kung and Traub conjecture. The comparisons are given with some ot...

There is no doubt that the fourth-order King’s family one of important ones among its counterparts. However, it has two major problems: first calculation first-order derivative; secondly, a linear order convergence in case multiple roots. In to improve these complications, we suggested new iterative methods. The main features our scheme are optimal order, being free from derivatives, and workin...

Kung-Traub’s conjecture states that an optimal iterative method based on d function evaluations for finding a simple zero of a nonlinear function could achieve a maximum convergence order of 2d−1. During the last years, many attempts have been made to prove this conjecture or develop optimal methods which satisfy the conjecture. We understand from the conjecture that the maximum order reached b...

A five-point thirty-two convergence order derivative-free iterative method to find simple roots of non-linear equations is constructed. Six function evaluations are performed achieve optimal 26-1 = 32 conjectured by Kung and Traub [1]. Secant approximation the derivative computed around initial guess. High attained constructing polynomials quotients for functional values.