Convergence Order Of New Methods Research Articles

Numerical Solution of Nonlinear Equations by a Twelth-Order Iterative Method with Memory

Some real life mathematical problems can be converted in the form of nonlinear equations. Solving such problems by analytical approaches is dificult in many situations. Hence numerical solution is the best way in this case. In this paper, a twelth-order iterative scheme for solvingnonlinear equations is presented and analyzed in terms of eciency. The new scheme is derived from the well-known King's method with order of convergence eight. We extend eighth-order King's method to an iterative method with memory of order 12:16 by using famous Newton'sinterpolating polynomial of degree 6 to avoid the derivative used in King's method. The new derived method is a three-step and is totally derivative free with twelth order of convergence. The method requires four functional evaluations at each iteration introducing high efficiency index of (12:16) 1/4 = 1:8673: Convergence order of new method is also studied. It is achieved by using matrix method of Herzburger. Numerical results are also provided to support theoretical analysis. Comparison of the derived scheme with previously well-known iterative schemes of the same order is also presented. As diffierent schemes of same order has efficiency index of (12)1/6 = 1:5131 because they requires six functional evaluations at each iteration, hence the proposed scheme is better than other schemes.

Two Iterative Methods with Memory Constructed by the Method of Inverse Interpolation and Their Dynamics

In this paper, we obtain two iterative methods with memory by using inverse interpolation. Firstly, using three function evaluations, we present a two-step iterative method with memory, which has the convergence order 4.5616. Secondly, a three-step iterative method of order 10.1311 is obtained, which requires four function evaluations per iteration. Herzberger’s matrix method is used to prove the convergence order of new methods. Finally, numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the presented methods.

A Modified Ren’s Method with Memory Using a Simple Self-Accelerating Parameter

In this paper, a self-accelerating type method is proposed for solving nonlinear equations, which is a modified Ren’s method. A simple way is applied to construct a variable self-accelerating parameter of the new method, which does not increase any computational costs. The highest convergence order of new method is 2 + 6 ≈ 4.4495 . Numerical experiments are made to show the performance of the new method, which supports the theoretical results.

New Eighth and Sixteenth Order Iterative Methods to Solve Nonlinear Equations

In this work we proposed two new higher order iterative methods to solve nonlinear equations. These methods based on the method Rafiullah (Numer Anal Appl 4(3):239–243, 2011) which is fifth-order. The Lagrange interpolation is used to improve the convergence order and efficiency index of the method. Convergence order of new methods are proved analytically. Some test problems are given to show the efficiency of the proposed methods.

Two-step Iterative Methods with Sixth-order Convergence for Solving Nonlinear Equations

The aim of this paper is to construct an e¢ cient iterative method to solve nonlinear equations. This method is obtained from M. Javidi’s method (Appl. Math. Comput. 193 (2007) 360-365), which is third-order. The convergence order of new method is established to six and the e¢ ciency index is 1.5651. The Proposed method is compared with the second, third and sixth order methods. Some numerical test problems are given to show the accuracy and fast convergence of the proposed method. Mathematics Subject Classi…cation: 65N99