The primary goal of this work is to provide a general optimal three-step class of iterative methods based on the schemes designed by Bi et al. (2009). Accordingly, it requires four functional evaluations per iteration with eighth-order convergence. Consequently, it satisfies Kung and Traub's conjecture relevant to construction optimal methods without memory. Moreover, some concrete methods of t...

This paper constructs a general class of two-point optimal fourthorder methods without memory for locating multiple roots of a nonlinear equation. On the basis of Kung-Traub’s conjecture, we investigate the optimal convergence as well as computational properties for the class. Explicit forms of asymptotic error constants are presented to strongly verify the convergence order. A variety of numer...

1Department of Psychiatry, National Cheng Kung University Hospital, College of Medicine, National Cheng Kung University, Tainan, Taiwan 2Institute of Behavioral Medicine, National Cheng Kung University Hospital, College of Medicine, National Cheng Kung University, Tainan, Taiwan 3Institute of Allied Health Sciences, College of Medicine and Hospital, National Cheng Kung University, Tainan, Taiwa...

In this paper we show a family of iterative schemes for solving nonlinear equations with order of convergence 2, by using n+ 1 functional evaluations per step, so these methods are optimal in the sense of the Kung-Traub’s conjecture. The family is obtained by composing n Newton’s steps and approximating the derivative by using Hermite’s interpolation polynomial. Some numerical examples are prov...

Two families of derivative free two-point iterative methods for solving nonlinear equations are constructed. These methods use a suitable parametric function and an arbitrary real parameter. It is proved that the first family has the convergence order four requiring only three function evaluations per iteration. In this way it is demonstrated that the proposed family without memory supports the...

Let S be a transformation semigroup acting on set Ω. The action of Ω can naturally extended to an all subsets We say that is ℓ-homogeneous provided it send A B for any two (not necessarily distinct) ℓ-subsets and On the condition k ≤ ℓ < + |Ω|, we show every must k-homogeneous. report other variants this result Boolean semirings affine/projective geometries. In general, poset gives rise auto...

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...