Analysis of Richardson Extrapolation Method in Improving the Accuracy of Numerical Approximation in Definite Integral Problems Using Matlab

Numerical methods are essential for calculating values ??that cannot be evaluated directly. Methods such as Trapezoidal and Simpson are often used to calculate integrals numerically, but the results may not always be satisfactory in terms of accuracy. This is where the Richardson extrapolation method plays an important role. This method can improve the accuracy of numerical solutions by combining the results of calculations with different levels of accuracy, thereby reducing errors. First, we use the Taylor series to approximate the exponential function around a certain point. Next, we apply the Trapezoidal method to calculate the integral value numerically. After getting the results from the Trapezoidal method, we apply the Richardson extrapolation method to improve the accuracy of the results. This method combines the results of two calculations with different levels of accuracy to reduce errors. To verify the results and improve the efficiency of the calculation, we also performed simulations using MATLAB. And the results issued in matlab are more precise results because they are very close to the exact value, with an absolute error value of 0.0000000472. and also, with a relative error value of 0.000006316. where both of these errors are very small and very close to 0. The calculation results with manual numerical approximation, the error obtained is 0.000235, while when using Richardson extrapolation, the error obtained is 0.000006316. So, it is true that if using the Richardson extrapolation method, it can reduce the error so that it will increase the accuracy of the integral solution.