TAILIEUCHUNG - A family of modified Newton iteration method for solving nonlinear algebraic equations

In this study, a modified Newton iteration version for solving nonlinear algebraic equations is formulated using a correction function derived from convergence order condition of iteration. | 34 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, 2017 A family of modified Newton iteration method for solving nonlinear algebraic equations Nghiem Xuan Luc, Nguyen Nhu Hieu Abstract— In this study, a modified Newton iteration version for solving nonlinear algebraic equations is formulated using a correction function derived from convergence order condition of iteration. If the second order of convergence is selected, we get a family of the modified Newton iteration method. Several forms of the correction function are proposed in checking the effectiveness and accuracy of the present iteration method. For illustration, approximate solutions of four examples of nonlinear algebraic equations are obtained and then compared with those obtained from the classical Newton iteration method. Index Terms—nonlinear algebraic equation, modified Newton iteration, correction function. 1 INTRODUCTION F inding solutions of nonlinear algebraic equation is one of the most important tasks in computations and analysis of applied mathematical and engineering problems [1,2]. The iteration algorithm for nonlinear algebraic systems can be classified into two main groups: bracketing techniques and fixed point methods. The bracketing techniques can be addressed as the well-known bisection [3,4], Regula Falsi method [5], Cox method [6]. The group of fixed point methods includes a long list of research contributions, among them are Halley method [7], Jaratt method [8], King's method [9]. The Newton method is a well-known technique for solving non-linear equations. It can be considered as an improved version of the classical fixed point method with iteration function containing the information of derivative at each iteration step. The Newton method has a fast convergence rate of iteration process when a starting point is on the neighborhood of the exact solution of equation under consideration. The development contributions of Newton method are archived based on the improvement of

• Vật lý
• A family of modified Newton iteration method
• Solving nonlinear algebraic equations
• Convergence order condition of iteration
• Correction function derived