An optimized derivative-free form of the Potra-Ptak method

作者：Soleymani, F.^*; Sharma, Rajni; Li, Xiaowu; Tohidi, E.

来源：Mathematical and Computer Modelling, 2012, 56(5-6): 97-104.

DOI：10.1016/j.mcm.2011.12.005

• Optimal iterations
• Derivative-free
• Without memory
• Nonlinear equations
• The Potra-Ptak method
• Kung-Traub conjecture

In this paper, we discuss iterative methods for solving univariate nonlinear equations. First of all, we construct a family of methods with optimal convergence rate 4 based upon the Potra-Ptak scheme and provide its error equation theoretically. Second, by using this derivative-involved family, a novel derivative-free family of two-step iterations without memory is derived. This derivative-free family agrees with the Kung-Traub conjecture (1974) for building optimal multi-point iterations without memory, since it is proven that each derivative-free method of the family reaches the convergence rate 4 requiring only three function evaluations per full iteration. Finally, numerical test problems are also provided to confirm the theoretical results.

• 出版日期2012-9
• 单位贵州民族大学; 中国人民解放军信息工程大学