Application of The Bisection Method and The Dynamic Spiral Algorithm in Determining Solutions of Systems of Non-Linear Equations

Abstract

Finding or determining the solution of an equation, either a linear equation or a non-linear equation, is a process of finding or determining the value of an independent variable so that if its value is substituted into the equation, it produces a value of zero. The solution of the equation is called the roots of the equation or zero values. There are three ways or methods to solve the equation, the first is analytic method, the second is numerical method, and the third is using numerical approach method of metaheuristic algorithm. Analytical method is a method of solving mathematical models by applying familiar algebraic formulas, while numerical methods and metaheuristic algorithms are used if the mathematical model used is complex and difficult to solve using analytical methods. Solutions obtained using analytical methods are called true solutions, while solutions from numerical methods and metaheuristic algorithms are approximations or approximate solutions. Although the solution from using numerical methods or metaheuristic algorithms is an approximate solution, we can make this approximate solution as accurate as possible so that the difference or error from the true solution is very small. In this study, the two methods used to solve non-linear equations are the bisection method and the numerical approximation method using the dynamic spiral metaheuristic algorithm. The aim of this research is to find the roots of non-linear equations using two numerical approximation methods.

Keywords : Solution; Non-Linear Equation; Numerical Methods.