Pengujian Metode Ekstrapolasi Richardson Untuk Meningkatkan Akurasi Turunan Numerik

Abstract

The Richardson extrapolation method, named after the English mathematician Lewis Fry Richardson (1881-1953), is a method of sequential acceleration to increase the rate of convergence of a series of approximate values ​​of a constant. If an approximation formula N1(h) is valid for all step sizes  then a new approximation formula N2(h) can be constructed which has a better truncation error than the truncation error of N1(h) . From formula N2(h) then formula N3(h) can be constructed with a better runcation error and so on. The formulation of this new approximation formula was carried out by assuming that the approximation error of N1(h) follows a certain form. In this paper, we discuss testing for increasing the accuracy of calculating of functions derivatives numerically using the Richardson extrapolation method. The three numerical derivative formulas tested are the forward-difference, the backward-difference, and the central-difference formula. The first two formulas have a truncation error of O(h) while the last formula has a truncation error of O(h^2). From the five test functions used, the results showed that the Richardson extrapolation method was able to increase the accuracy of the approximation of the three formulas. The best results were obtained in the central-difference formula regarding its truncation error which only contains even powers.