Download e-book for iPad: Analyse numérique et équations différentielles by Demailly Jean-Pierre

By Demailly Jean-Pierre

Demailly J.-P. examine numerique et equations differentielles (EDP Sciences, 2006)(ISBN 286883891X)(fr)(345s)_MN_

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A C1 δn (z)A(z)n ≤ |πn+1 (z)| ≤ C2 nδn (z)A(z)n . On voit donc que le terme dominant du comportement de |πn+1 (z)| est le facteur exponentiel A(z)n . Pour ´evaluer πn+1 [a,b] , il suffit de calculer A(x) lorsque x ∈ [a, b] : b 1 ln |x − t|dt . A(x) = exp b−a a La fonction t → ln |t − x| est discontinue en t = x, mais le lecteur pourra s’assurer que l’int´egration par parties suivante est l´egitime : b b dt t − x a = (b − x) ln (b − x) + (x − a) ln (x − a) − (b − a), ln |t − x|dt = [(t − x) ln |t − x|]ba − a (t − x) car la fonction t → (t − x) ln |t − x| est continue sur [a, b] et on peut passer a` la limite sur chacun des intervalles [a, x − ε] et [x + ε, b].

On a f − rn 2 ≤ f − rα,n 2 ≤ f − f χα 2 + f χα − rα,n 2 . Soit ε > 0 fix´e. On peut d’abord choisir α > 0 tel que f − f χα 2 < 2ε ; α ´etant ainsi fix´e, on peut choisir n0 tel que n > n0 entraˆıne f χα − rα,n 2 < 2ε et donc f − rn 2 < ε. Mise en œuvre num´ erique – Si les polynˆomes pn sont connus, le calcul des rn est possible d`es lors qu’on sait ´evaluer les int´egrales f, pk : les m´ethodes d’int´egration num´erique feront pr´ecis´ement l’objet du prochain chapitre. Si les polynˆ omes pn ne sont pas connus, on peut les calculer num´eriquement par la formule de r´ecurrence du th´eor`eme 2.

Le quotient est un binˆ ome c0 + c1 x, pair si n est impair, impair si n est pair. Par cons´equent 1 − (x2 + α2 )pn (x) = c0 · πn+1 (x) c1 x · πn+1 (x) si n est impair, si n est pair. En substituant x = iα, on trouve c0 = 1/πn+1 (iα) et c1 = 1/iαπn+1 (iα), d’o` u  1 πn+1 (x)   si n est impair,  x2 + α 2 π n+1 (iα) fα (x) − pn (x) =  x πn+1 (x)   si n est pair. 2. ´ Etude de la convergence ponctuelle de la suite pn(x) 1 Si x = ±1, pn (x) = pn (±1) = fα (±1) = 1+α 2 est une suite constante.

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Analyse numérique et équations différentielles by Demailly Jean-Pierre


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