Sixteenth-order Methods Research Articles

Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application

The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the first derivative; the eighth order method uses three evaluations of the function and one evaluation of the first derivative; and sixteenth order method uses four evaluations of the function and one evaluation of the first derivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions are compared with the existing competitors on some standard academic problems. The conjugacy maps of the presented method and other existing eighth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. We apply the new scheme to find the optimal launch angle in a projectile motion problem and Planck’s radiation law problem as an application.

An efficient optimal family of sixteenth order methods for nonlinear models

The principle aim of this manuscript is to propose a general scheme that can be applied to any optimal iteration function of order eight whose first substep employ Newton’s method to further develop new interesting optimal scheme of order sixteen. This scheme requires four evaluations of the involved function and one evaluation of its first-order derivative at each step. So, it is being optimally consistent with the conjecture of Kung–Traub. In addition, theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence. Moreover, the conjugacy maps and the strange fixed points of some iterative methods are discussed, their basins of attractions are also given to show their dynamical behavior around the simple roots. From the numerical experiments, we find that our methods perform better than the existing ones when we checked the performance on a concrete variety of nonlinear equations.

On a general class of optimal order multipoint methods for solving nonlinear equations

We develop a class of n-point iterative methods with optimal 2n order of convergence for solving nonlinear equations. Newton's second order and Ostrowski's fourth order methods are special cases corresponding to n=1 and n=2. Eighth and sixteenth order methods that correspond to n=3 and n=4 of the class are special cases of the eighth and sixteenth order methods proposed by Sharma et al. [25]. The methodology is based on employing the previously obtained (n−1)-step scheme and modifying the n-th step by using rational Hermite interpolation. Unlike that of existing higher order techniques the proposed technique is attractive since it leads to a simple implementation. Local convergence analysis is provided to show that the iterations are locally well defined and convergent. Theoretical results are verified through numerical experimentations. The performance is also compared with already established methods in literature. It is observed that new algorithms are more accurate than existing counterparts and very effective in high precision computations.

Hydraulic Analysis of Water Supply Networks Using a Modified Hardy Cross Method

There are different methods for the hydraulic analysis of water supply networks. In the solution process of most of these methods, a large system of linear equations is solved in each iteration. This usually requires a high computational effort. Hardy Cross method is one of the approaches that do not need such a process and may converge to the solution through scalar divisions. However, this method has two shortcomings: first, initial discharges should satisfy continuity equation at each node; second a large number of iterations are required to converge to solution. In this article an algorithm is suggested for initial discharges that are close to the final results while the continuity equations are automatically established. This algorithm may be directly implemented in the Hardy Cross method. To reduce the number of iterations the Hardy Cross method is combined with third-order and sixteenth-order methods. The results of some numerical examples demonstrate that the use of the combined approach with the suggested initial guess reduces the number of iterations and hydraulic analysis time and the solutions converge with a high accuracy.

An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations

In this study, the sixteenth-order iterative scheme of Li et al. [X. Li, C. Mu, J. Ma, C. Wang, Sixteenth-order method for nonlinear equations, Appl. Math. Com. 215 2010, 3754--3758] is considered. We increase its efficiency index from 1.587 to 1.644, by reducing the number of evaluations from six to five per iteration. This goal is achieved by providing an approximation for the first-order derivative of the function in the fourth step. Error analysis will also be studied. In the sequel, some numerical instances are given to show the accuracy of the new obtained twelfth-order technique. Therein, another objective of this paper is achieved by proposing a hybrid method for finding all the real solutions of nonlinear equations.

A NOTE ON THE PAPER ENTITLED SIXTEENTH-ORDER METHOD FOR NONLINEAR EQUATIONS

The purpose of this paper is to provide some corrections regarding algebraic flaws encountered in the paper entitled "Sixteenth-order method for nonlinear equations" which was published in January of 2010 by Li et al.[9]. Further detailed comments on their error equation are stated together with convergence analysis as well as high-precision numerical experiments.