Some New Higher Order Weighted Newton Methods for Solving Nonlinear Equation with Applications

Abstract

Three new iterative methods for solving scalar nonlinear equations using weight function technique are presented. The first one is a two-step fifth order method with four function evaluations which is improved from a two-step Newton’s method having same number of function evaluations. By this, the efficiency index of the new method is improved from 1.414 to 1.495. The second one is a three step method with one additional function evaluation producing eighth order accuracy with efficiency index 1.516. The last one is a new fourth order optimal two-step method with efficiency index 1.587. All these three methods are better than Newton’s method and many other equivalent higher order methods. Convergence analyses are established so that these methods have fifth, eighth and fourth order respectively. Numerical examples ascertain that the proposed methods are efficient and demonstrate better performance when compared to some equivalent and optimal methods. Seven application problems are solved to illustrate the efficiency and performance of the proposed methods.

A new fifth-order iterative method for solving non-linear equations using weight function technique and the basins of attraction

“…This polynomial has three roots as {1, −0.5 − 0.866025i, −0.5 + 0.866025i}. In Figure 1 we plotted the fifth order methods, as PJ in [11], FLM in [7], M2 in [10], SH3 in [14], and the new method AMK1. We compare from where the number of black points NB, the brightness index BI, the time taken to reach the solution T , the average of iterations per point until the method decides that a root has been reached I/P.…”

“…We present five numerical examples in real domain to illustrate efficiency of the new fifth-order iterative method described as AMK1, comparing it with some fifth-order iterative methods such as FLM method in [7], M2 method in [10], PJ method in [11], and SH3 method in [14]. We performed all the calculations in Table 7 by using the Mathematica 11 with 128 significant digits, where 10 digits are displayed for x n , we use the number of iteration n = 4 and with precision ε = 10 −50 .…”

“…While Newton's method was the first attempt to solve such equation, which converges quadratically, in recent years, researchers have developed Newton's method to achieve higher convergence and more accurate results as, see for example the method shown in [13] has the fourth order and was the first proposed multi-point method of this order, method shown in [8] has fifth order, method shown in [10] has the sixth order and method shown in [1] has eighth order and references therein. In this research, we are interested to find iterative method for solving non-linear equations of order five where methods of this order have been found in several researches as in [7,8,10,11,14]. These iterative methods have been constructed by using different techniques for solving the non-linear equations such as the variational iteration technique and weight function and etc.…”

“…Its roots are {−0.759845 + 0.592595i, −0.532605 − 1.08829i, 0.181924 + 0.732098i, 1.11052 − 0.236405i}. We compared the new iterative method AMK1 with four different methods such as PJ in [11], FLM in [7], SH3 in [14], and in [10], the new method AMK1 has less number of black points and higher brightness.…”

An optimal fourth order method for solving nonlinear equations

“…Numerical analysis has interesting applications in several branches of pure and applied science that can be studied in the general framework of the non-linear equations [2,12,17,26,32]. Searching out a solution for non-linear equations is highly significant.…”