The simplified topological epsilon-algorithms for accelerating sequences in a vector space

When a sequence of numbers is slowly converging, it can be transformed into a new sequence which, under some assumptions, converges faster to the same limit. One of the best--known sequence transformations is Shanks transformation which can be recursively implemented by the epsilon-algorithm of Wynn. This transformation and this algorithm have been extended (in two different ways) to sequence of elements of a topological vector space E. In this paper, we present new algorithms of implementing these topological Shanks transformations. They no longer require the manipulation of elements of the algebraic dual space E^* of E, nor the use of the duality product inside the rules of the algorithms, they need the storage of less elements of E, and the stability is improved. They also allow us to prove convergence and acceleration results for some types of sequences. Various applications involving sequences of vectors or matrices show the interest of the new algorithms.