The Simplified Topological $\varepsilon$-Algorithms for Accelerating Sequences in a Vector Space
Claude Brezinski, Michela Redivo‐Zaglia · SIAM Journal on Scientific Computing · 2014
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 the Shanks transformation, which can be recursively implemented by the $\varepsilon$-algorithm of Wynn. This transformation and this algorithm have been extended (in two different ways) to a sequence of elements of a topological vector space $E$. In this paper, we present new algorithms for implementing these topological Shanks transformations. They no longer require the manipulation of elements of the algebraic dual space $E^*$ of $E$, nor do they use the duality product inside the rules of the algorithms; they need the storage of fewer 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.