Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations

Abstract

Solution methods are the major tools in research especially in the area of applied mathematics. This is because, most real-life problems result into system of nonlinear equations, and the right solution method with less computational error is required to obtain an approximated solution to these system of nonlinear equations. The introduction of the Broyden method set the groundwork for the development of several other methods, many of which are referred to as Broyden-like approaches by various researchers. In most cases, these methods have proven to be superior to the original classical Broyden method in terms of the number of iterations and CPU time needed to acquire a solution. Using the solutions of the traditional Broyden method as a point of comparison, this study aimed to examine the error associated with two newly developed numerical methods, the Trapezoidal-Simpson-3/8 (TS-3/8) and Midpoint-Simpson-3/8 (MS-3/8) methods. Results gathered after applying the classical Broyden, MS-3/8 and TS-3/8 methods to solve some bench-mark problems involving system of nonlinear equations and estimating the errors associated with each of the methods considered in the study, using the formula of the approximate error, showed that the error associated with the MS-3/8 method was minimal compared to that of the Broyden and the TS-3/8 methods. At the end of the study, the results gathered suggested the MS-3/8 technique as the most highly advised numerical approach among the other methods. This means that, MS-3/8 method is a more accurate solution method for solving system of nonlinear equations considering the results in this paper.