The Effect of Selection of Initial Values on Finding the Root of a Multiple roots – Polynomial Equation

Abstract This study aims to determine the effect of selecting the initial value on the root of a polynomial equation that has multiple roots. There are three methods used, namely the Brent method, the bisection method and the modified secant method. The Brent method is a combination of the bisection method, the IQI (Interval Quadratic Inverse) method and the secant method. Therefore, it is necessary to know the performance of the Brent method in finding the multiple roots of polynomial equations. The search for multiple roots reaches convergence faster when using the modified Newton-Raphson method or the modified secant method. There are 2 types of polynomial equations used. One equation has 3 roots and two of them have multiple roots (multiple roots). One other equation is a polynomial which has 4 equal roots and 3 of them have multiple roots with an odd number of roots. The analysis was carried out by selecting the same initial value for the three methods. The same lower bound is a = xl = xi-1 for the Brent method, the modified bisection method and the modified secant method, respectively. Likewise for the upper bound a = xu = xi. The selection of the initial value affects the final result of the root search, for the Brent method, the bisection method and the modified secant method. The comparisons made refer to the results of the root value and the speed of convergence as seen from the number of iterations of each method. From the simulation results, we propose the use of a modified secant method because it is more efficient in finding multiple roots in polynomial equations. For the Brent method, it needs to be further modified in order to get the multiple root value in the polynomial equation.