A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture

Chein-Shan Liu,Tsung-Lin Lee,Xiaolong Qin

doi:10.1155/2021/5516694

Chein-Shan Liu, Tsung-Lin Lee + Show 1 more

Open Access

https://doi.org/10.1155/2021/5516694

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Abstract

Kung and Traub conjectured that a multipoint iterative scheme without memory based on m evaluations of functions has an optimal convergence order p = 2 m − 1 . In the paper, we first prove that the two-step fourth-order optimal iterative schemes of the same class have a common feature including a same term in the error equations, resorting on the conjecture of Kung and Traub. Based on the error equations, we derive a constantly weighting algorithm obtained from the combination of two iterative schemes, which converges faster than the departed ones. Then, a new family of fourth-order optimal iterative schemes is developed by using a new weight function technique, which needs three evaluations of functions and whose convergence order is proved to be p = 2 3 − 1 = 4 .

Highlights

Kung and Traub conjectured that a multipoint iterative scheme without memory based on m evaluations of functions has an optimal convergence order p 2m− 1
Definition 5. e iterative schemes are of the same class, if they are of the same order p and have the same m evaluations of the same functions
We begin with the error equation of the Newton method (NM): en+1 c2e2n − A3e3n − A4e4n + · · ·, (8)

Summary

Preliminaries

Before deriving the main results we begin with some standard terminologies. P is the order of convergence and C is the asymptotic error constant. Is called the error equation of an iterative scheme. An iterative scheme is said to have the optimal order p, if p 2m− 1 where m is the number of evaluations of functions (including derivatives). E conjecture of Kung and Traub asserted that a multipoint iteration without memory based on m evaluations of functions has an optimal order p 2m− 1 of convergence [11]. It indicates that the upper bound of the efficiency index is 2(1− 1/m) < 2. Definition 5. e iterative schemes are of the same class, if they are of the same order p and have the same m evaluations of the same functions

Main Results

Combinations of Iterative Schemes

Second Family of Optimal Fourth-Order Iterative Schemes

Numerical Experiments

Conclusions