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DTSTAMP:20251122T051847Z
DESCRIPTION:Shanks sequence transformations and the $\varepsilon$-algorithm
 s. Theory and applications.\n\nWhen a sequence of numbers is slowly conver
 ging 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 s
 equence which\, under some assumptions\, converges faster to the same limi
 t. Among these general techniques is Shanks' transformation (Shanks\, 1949
 \, 1955) which is arguably the best all-purpose method for accelerating co
 nvergence of sequences. First\, this transformation will be explained. The
 n\, we will see how it can be recursively implemented by the $arepsilon$-a
 lgorithm of Wynn (1956). This algorithm can be transformed to treat sequen
 ces of vectors (Wynn\, 1962). But\, since its algebraic theory is quite co
 mplicated\, another way to extend Shanks transformation to sequences of el
 ements of a general vector space $E$ was proposed (C.B.\, 1975). This topo
 logical Shanks transformation can be recursively implemented by the topolo
 gical $arepsilon$-algorithm. The rules of this algorithm are quite complic
 ated and difficult to implement since elements of E*\, the algebraic dual 
 space of E\, recursively intervene in them. Recently\, these rules were gr
 eatly 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 imp
 lementation will be discussed. We will see the simplification it brought i
 n terms of arithmetical operations and storage. This algorithm will then b
 e applied to the solution of systems of linear and nonlinear vector and ma
 trix equations\, the computation of matrix functions\, and the solution of
  Fredholm integral equations of the second kind (C.B.\, M.R.-Z.\, 2017). T
 hese results were obtained by the freely available corresponding software 
 (C.B.\, M.R.-Z.\, 2017). Travail conjoint avec Michela Redivo-Zaglia\, Uni
 versità di Padova\, Italia. Les transparents de la présentation seront en 
 anglais\; la conférence sera donnée en anglais ou en français\, selon l'au
 dience.\n
DTSTART:20171010T193000Z
DTEND:20171010T203000Z
LOCATION:Room 4336\, CA\, QC\, Montreal\, Pav. André-Aisenstadt\, 2920\, ch
 . de la Tour
SUMMARY:Claude Brezinski\, Université de Lille - Sciences et Techniques\, F
 rance
URL:/mathstat/channels/event/claude-brezinski-universi
 te-de-lille-sciences-et-techniques-france-272979
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