Let $\left(S_n\right)$ be some vector sequence, converging to S, satisfying $S_n-S\sim {\varrho }^n{n}^{\theta }({\beta }_0+{\beta }_1{n}^{-1}+{\beta }_2{n}^{-2}+...),0\left|\varrho \right|1,\theta 0$, where ${\beta }_0(\ne 0),{\beta }_1,...$ are constant vectors independent of n. The purpose of this paper is to provide acceleration methods for these vector sequences. Comparisons are made with some known algorithms. Numerical examples are also given.

The author proves the existence of the multi-parameter asymptotic error expansion to the five-point difference scheme for Dirichlet problems for the linear and semilinear elliptic PDE on general domains. By Richardson extrapolation, this expansion leads to a simple process for accelerating the convergence of the method.

Some new results on convergence acceleration for the E-algorithm which is a general extrapolation method are obtained. A technique for avoiding numerical instability is proposed. Some applications are given. Theoretical results are illustrated by numerical experiments