Introduction to the improved Levin-type algorithms for accelerating convergence of sequence

摘要

We describe two new algorithms for accelerating the convergence of scalar sequence. We express these new algorithms as Levin transformation. The definitions of these new algorithms, namely the improved Levin algorithm and the modified Levin algorithm, are given. We examine the effectiveness of the improved Levin algorithm and the modified Levin algorithm, together with the well established methods, namely Wynn’s ρ algorithm and Brezinski’s θ algorithm, for approximating the partial sum of a given power series. Estimates of the partial sum produced by the improved Levin algorithm are found to be substantially more accurate than the original Levin transformation and the Wynn’s ρ algorithm and it is similar to the modified Levin algorithm and the Brezinski’s θ algorithm.