Computational Costs of Optimal Multipoint Methods without Memory for Solving Nonlinear Scalar Equations by Multiple-Precision Arithmetic

Abstract

Abstract. For solving nonlinear scalar equations, multipoint iterative methods without memory are the most commonly used algorithms. For these methods, there exists a well-known conjecture (Kung-Traub's conjecture): the methods requiring m+1 (m ≥ 1) function evaluations per iteration have order of convergence at most 2^m. This paper shows that among the maximum order methods those of m=1 are being most effective, if the methods are executed by variable multiple precision arithmetic. As a result, the methods of 2nd order, including Newton's method, are fastest. On the contrary, the methods of larger m are shown to be faster when the precision is fixed.