Design and multidimensional extension of iterative methods for solving nonlinear problems

Abstract

In this paper, a three-step iterative method with sixth-order local convergence for approximating the solution of a nonlinear system is presented. From Ostrowski's scheme adding one step of Newton with 'frozen' derivative and by using a divided difference operator we construct an iterative scheme of order six for solving nonlinear systems. The computational efficiency of the new method is compared with some known ones, obtaining good conclusions. Numerical comparisons are made with other existing methods, on standard nonlinear systems and the classical 1D-Bratu problem by transforming it in a nonlinear system by using finite differences. From this numerical examples, we confirm the theoretical results and show the performance of the presented scheme.

On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems

“…In addition, to compare the above five methods, we use the computational cost which is measured by C = d + op (see [20]), where d is the number of function evaluations per iteration and op denotes the number of operations (e.g., products and quotients) needed per iteration. The various evaluations and operations that contribute towards the total computational cost for a system of m nonlinear equations in m unknowns are explained as follows.…”

“…Sharma and Arora proposed a fourth order method in [19] requiring one F and two F in each iteration. Apart from these third and fourth-order methods, researchers have also proposed some higher order methods, see, for example [11,13,15,17,[20][21][22][23][24] and references therein.…”

On the design and analysis of high‐order Weerakoon‐Fernando methods based on weight functions

“…Then in view of Taylor's series expansion, we have f (x n ) = f ′ ( )(e n + c 2 e 2 n + c 3 e 3 n + c 4 e 4 n + O(e 5 n )). (6) and f ′ (x n ) = f ′ ( )(1 + 2c 2 e n + 3c 3 e 2 n + 4c 4 e 3 n ) + O(e 4 n ). (7) so that,…”

“…One of the most effective techniques is the use weight functions. This technique can be applied both on solving nonlinear equations [6][7][8][9] and systems of nonlinear equations. [10][11][12] In Reference 7, authors proposed the following method:…”

A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems

“…Among others, Sharma et al in [18] designed a scheme with fourth-order of convergence by using this procedure and, more recently, Artidiello et al constructed in [19,20] several classes of high-order schemes by means of matrix weight functions.…”