In this paper, a family of Steffensen type methods of fourth-order convergence for solving nonlinear smooth equations is suggested. In the proposed methods, a linear combination of divided diferences is used to get a better approximation to the derivative of the given function. Each derivative-free member of the family requires only three evaluations of the given function per iteration. Therefo...

In this paper, a family of Steffensen-type methods of optimal order of convergence with two parameters is constructed by direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub (J. Assoc. Comput. Math. 1974, 21, 634–651) that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2m−1. Furthermore, ...

The problem is to calculate a simple zero of a non-linear function f. We consider rational iterations without memory which use two evaluations of f or its derivatives. It is shown that the optimal order is 2. This settles a conjecture of Kung and Traub that an iteration using n evaluations without memory is of order at most 2 "^, for the case n = 2. Furthermore we show that any rational two-eva...

A class of three-step eighth-order root solvers is constructed in this study. Our aim is fulfilled by using an interpolatory rational function in the third step of a three-step cycle. Each method of the class reaches the optimal efficiency index according to the Kung-Traub conjecture concerning multipoint iterative methods without memory. Moreover, the class is free from derivative calculation ...

In this paper, three new families of eighth-order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods. Per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative. This implies that the efficiency index of the developed methods is 1.682, which is optimal according to Kung ...

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

Chopping and beating processes were used as meat-cutting methods in preparing kung-wan to produce low-salt products while retaining or improving the emulsion stability, sensory evaluation, and physico-chemical properties of the standard high-salt formulation. Increased salt content improved emulsion stability and dynamic rheology. However, 3% salt content decreased the overall acceptance of kun...

In this paper, a general family of n-point Newton type iterative methods for solving nonlinear equations is constructed by using direct Hermite interpolation. The order of convergence of the new n-point iterative methods without memory is 2 requiring the evaluations of n functions and one first-order derivative in per full iteration, which implies that this family is optimal according to Kung a...