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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.