Family Of Multipoint Iterative Methods Research Articles

A new family of multi-point iterative methods for finding multiple roots of nonlinear equations

In this paper, new three-point eighth-order iterative methods for solving nonlinear equations are constructed. It is proved that these methods have the convergence order of eight requiring only four function evaluations per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on \(n\) evaluations, could achieve optimal convergence order \(2^{n-1} .\) Thus, we present new iterative methods which agree with the Kung and Traub conjecture for \(n=4\). Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed methods using only a few function evaluations.

Iterative methods for use with nonlinear discrete algebraic models

A family of multi-point iterative methods for solving nonlinear equations were described in Cordero and Torregrosa (2008) [4], and a general error analysis was given, always for a simple root. Here we study the order of convergence of such methods when we have multiple roots. We prove that the order of convergence goes down to 1 but, when the multiplicity n is known, it may be raised to 2 by using different types of correction. For n unknown, we show that some methods of this family converge faster than the classical Newton’s method. In addition, we provide various numerical tests which confirm or improve on theoretical results and allow us to compare some methods of the aforementioned family.

A family of multi-point iterative methods for solving systems of nonlinear equations

We extend to n -dimensional case a known multi-point family of iterative methods for solving nonlinear equations. This family includes as particular cases some well known and also some new methods. The main advantage of these methods is they have order three or four and they do not require the evaluation of any second or higher order Fréchet derivatives. A local convergence analysis and numerical examples are provided.

A class of multi-point iterative methods for nonlinear equations

A family of multi-point iterative methods for solving nonlinear equations is described and the order of convergence of its elements is studied. The computational efficiency of some elements of this family, in terms of the number of function evaluations, is also provided. In addition, we present different numerical tests, which confirm or improve the theoretical results and allow us to compare some methods of this family.

On a family of iterative methods for simultaneous extraction of all roots of algebraic polynomial

A family of multi-point iterative methods for simultaneous determination of all roots of polynomial equation is obtained. This family is corresponding to a family of multi-point methods for approximation of only one root of nonlinear equations. The convergence analysis and numerical examples are given.