A New Sixth-Order Steffensen-Type Iterative Method for Solving Nonlinear Equations

Abstract

Based on iterative method proposed by Basto et al. (2006), we present a new derivative-free iterative method for solving nonlinear equations. The aim of this paper is to develop a new method to find the approximation of the root α of the nonlinear equation f(x)=0. This method has the efficiency index which equals 61/4=1.5651. The benefit of this method is that this method does not need to calculate any derivative. Several examples illustrate that the efficiency of the new method is better than that of previous methods.

On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations

“…Now, we replace constant parameters in the iterative formula (5) by the varying defined by (6), (7), and (8). Then, the multipoint methods with memory, following from (5), become…”

“…Positive solution of this system is = (1/3)(1 + 2 √ 7), = (2/3)(2+ √ 7), = (4/3)(2+ √ 7), and = 2(3+ √ 7) = 11.2915. Therefore, the -order of the methods with memory (12), when is calculated by (8), is at least 11.2915.…”

“…Let the function ( ) be sufficiently differentiable in a neighborhood of its simple zero . If an initial approximation 0 is sufficiently close to and the parameter in (12) is recursively calculated by the forms given in (6)-(8), where for = 2 is defined by (3), then the -order of convergence of the Steffensen-like method with memory (12) with the corresponding expressions (6)- (8) of is at least 10, 10.2426, and 10.4721, respectively. Proof. The proof of this theorem is similar to the proof of Theorem 1; hence, it is omitted.…”

“…The choice = 1 produces the well-known Steffensen method [2]. To improve the local order of convergence, many modified methods have been proposed in the open literature; see [3][4][5][6][7][8][9][10][11] and references therein. Thukral developed a scheme of optimal order of convergence eight [12], constructing a weight function as well in the following form: ) , = 1, 2, 3,…”