Integrasi Numerik Fungsi Eksponensial dengan Metode Romberg dan Gauss-Legendre

Abstract

Numerical integration is a method of integrating a function that produces an approximation to its exact value. Sometimes a function that has a complex shape will be very difficult to solve using analytical integration techniques with standard forms, so in this case numerical integration is needed to determine its value. There are two approaches to numerical integration, namely the Newton-Coates (equally space) and Gauss-Quadrature (unequally space) methods. One of the Newton-Coates methods that has good accuracy (smaller error) is the Romberg method, this method is obtained from Richardson's extrapolation which is applied continuously from the Simpson 1/3 method, Simpson 3/8 method, and the Boole method, so we get the Romberg method. Meanwhile, the Gauss-Quadrature method which is considered to have good accuracy is the Gauss-Legendre method, this method transforms the limit of function integration [a,b] into limit [-1,1]. To determine the integration value in Gauss-Legendre, several evaluation points (fixed points)  with i=0,1,2,…,n-1 and a weighting function  with i=0,1,2,…,n-1 . The more evaluation points used, the more accurate the integration results will be. In this article, we will examine the comparison of the accuracy of the numerical integration of the two methods, namely the Romberg and Gauss-Legendre methods which will be applied to solve the modified exponential function.