On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations

Abstract

In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet derivatives for each iteration. The multi-step version requires one more function evaluation for each iteration. The proposed new scheme does not require the evaluation of second or higher order Fréchet derivatives and still reaches fourth order convergence. The multi-step version converges with order 2r + 4, where r is a positive integer and r ≥ 1. We have proved that the root α is a point of attraction for a general iterative function, whereas the proposed new schemes also satisfy this result. Numerical experiments including an application to 1-D Bratu problem are given to illustrate the efficiency of the new methods. Also, the new methods are compared with some existing methods.

Multistep Quadrature Based Methods for Nonlinear System of Equations with Singular Jacobian

“…Equation (5) is consistency conditions, [34]. The (4) is expressed into coupled equation given in (6) and (7).…”

“…Since 0 X is initial guess, setting 0 X in (7) yields Journal of Applied Mathematics and Physics ( ) 0 0 H X = (16) From (9)and (15),…”

“…is singular [7] [8]. The development of new iterative methods for approximating the solution of (1) has attracted the attention of researchers in recent years, as evident in the literature such as [9] [10] [11] [12].…”

An improvement to double-step Newton method and its multi-step version for solving system of nonlinear equations and its applications

“…We state below a general result which has been proved in Babajee et al [4], showing that α is a point of attraction for a general iteration function…”

“…It is straightforward to see that this method requires the evaluation of one function, one first derivative, and one matrix inversion per iteration. Traub [16] suggested that multi-step iterative methods are better way to improve the order of convergence free from second derivatives, such modifications of Newton's method have been proposed in the literature; for example see [1,4,6,11,15] and references therein. The double-step third and fourth order Newton's methods have been proposed in the recent literature, see [9][10][11]15].…”