Event
Claude Brezinski, Universit茅 de Lille - Sciences et Techniques, France
Tuesday, October 10, 2017 15:30to16:30
Room 4336, Pav. Andr茅-Aisenstadt, 2920, ch. de la Tour, Montreal, QC, CA
Shanks sequence transformations and the $\varepsilon$-algorithms. Theory and applications.
When a sequence of numbers is slowly converging and when it is impossible to have access to the process producing it, it can be transformed, by a {it sequence transformation}, into a new sequence which, under some assumptions, converges faster to the same limit. Among these general techniques is Shanks' transformation (Shanks, 1949, 1955) which is arguably the best all-purpose method for accelerating convergence of sequences. First, this transformation will be explained. Then, we will see how it can be recursively implemented by the $arepsilon$-algorithm of Wynn (1956). This algorithm can be transformed to treat sequences of vectors (Wynn, 1962). But, since its algebraic theory is quite complicated, another way to extend Shanks transformation to sequences of elements of a general vector space $E$ was proposed (C.B., 1975). This topological Shanks transformation can be recursively implemented by the topological $arepsilon$-algorithm. The rules of this algorithm are quite complicated and difficult to implement since elements of E*, the algebraic dual space of E, recursively intervene in them. Recently, these rules were greatly simplified, thus leading to the simplified topological $arepsilon$-algorithm (C.B., M.R.-Z., 2014). First, we will show how its recursive rule was derived from the old rules. Then, this new algorithm and its implementation will be discussed. We will see the simplification it brought in terms of arithmetical operations and storage. This algorithm will then be applied to the solution of systems of linear and nonlinear vector and matrix equations, the computation of matrix functions, and the solution of Fredholm integral equations of the second kind (C.B., M.R.-Z., 2017). These results were obtained by the freely available corresponding software (C.B., M.R.-Z., 2017). Travail conjoint avec Michela Redivo-Zaglia, Universit脿 di Padova, Italia. Les transparents de la pr茅sentation seront en anglais; la conf茅rence sera donn茅e en anglais ou en fran莽ais, selon l'audience.