Modified Newton's method using harmonic mean for solving nonlinear equations

Abstract

Different modifications in the Newton's method with cubic convergence have become popular iterative methods to find the roots of non-linear equations. In this paper, a modified Newton's method for solving a single nonlinear equation is proposed. This method uses harmonic mean while using Simpson's integration rule, thus replacing �� ' �� in the classical Newton's method. The convergence of the proposed method is found to be order three. Numerical examples are provided to compare e the efficiency of the method with few other cubic convergent methods. In this paper,we propose amodification in the method of(3) including Harmonic mean instead of Arithmetic meanwhile using the Simpson's 1/3 rule of integration for the equation (3). The proposed new method has the advantage of evaluating only the firstderivativeand less number of iterations to achieve third order accuracy.In Section II, we present some definitions related to our study. In Section III, some known variants of Newton's method are discussed. Section IV presents the new method and its analysis of convergence. Finally, Section V gives numerical results and a discussion is carried out in section VI.