Development of the Levin-type algorithms for accelerating convergence of sequences

There are two aims of this paper. Firstly we shall introduce the determinantal representations of the new Levin-type algorithms and secondly we shall demonstrate further development of the Levin-type algorithms. We consider the use of the Levin-type algorithms to accelerate the convergence of scalar sequence and their effectiveness for approximating the solution of a given power series is illustrated. In process we shall demonstrate the convergence of each of the methods considered. The approximate solution of the super enhanced Levin algorithm and the efficient Levin algorithm are found to be substantially more accurate than the Cizek, Zamastil and Skala transformation and the iterated Aitken Δ²${\mathrm{\Delta }}^2$ algorithm.