Potra-Pták Iterative Method with Memory

The problem is to extend the method proposed by Soleymani et al. (2012) to a method with memory. Following this aim, a free parameter is calculated using Newton’s interpolatory polynomial of the third degree. So the R-order of convergence is increased from 4 to 6 without any new function evaluations. Numerically the extended method is examined along with comparison to some existing methods with the similar properties. 1. Introduction Root finding is a great task in mathematics, both historically and practically. It has attracted attention of great mathematicians like Gauss and Newton. It has real major applications and because of these real features it is still alive as a research field. Kung and Traub's conjecture is the basic fact to construct optimal multipoint methods without memory [1]. On the other hand, multipoint methods with memory can increase efficiency index of an optimal method without memory without consuming any new functional evaluations and merely using accelerator parameter(s). This great power of methods with memory has not been well considered until very recently. So we have been motivated to extend modified Potra-Pták [2] to its with memory method. Traub in his book [3] introduced methods with and without memory for the first time. Moreover, he constructed a Steffensen-type method with memory using secant approach. In fact, he increased the order of convergence of the Steffensen method [4] from 2 to 2.41. This is the first method with memory based on our best knowledge. In other words, Traub changed Steffensen's method slightly as follows (see [3, pages 185–187]): The parameter is called self-accelerator and method (1) has convergence order of 2.41. It is still possible to increase the convergence order using better self-accelerator parameter based on better Newton interpolation. Free derivative can be considered as another virtue of (1). We use the symbols , , and according to the following conventions [3]. If , we write or . If , we write or . If , where is a nonzero constant, we write or . Let be a function defined on an interval , where is the smallest interval containing distinct nodes . The divided difference with th-order is defined as follows: , Moreover, we recall the definition of efficiency index (EI) as , where is the order of convergence and is the total number of function evaluations per iteration. This paper is organized as follows. Section 2 reviews modified Potra-Pták's method and we try to remodify it slightly too. Error equation for our modification is provided. In Section 3, development to with memory is carried