Accélération de la Convergence en Analyse Numérique by Claude Brezinski

By Claude Brezinski

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Extra resources for Accélération de la Convergence en Analyse Numérique

Example text

K2Sn+l . . A2Sn+k'''&2Sn+2k- 1 k2Sn+k-I gSn+l "'" kSn+k ASh k2Sn+l''" k2Sn+k A2Sn . . . k2Sn+k . . . . . . . . A2Sn+k-I "" &2Sn+2k-2 &Sn+ 1 ..... &Sn+ k 1 &2Sn+ I .... &2Sn+ k 12Sn~7]-i2-S~2k~-&2Sn+k - A2Sn+k_ l .. isant un d~veloppement de Schweins et en inversant les rapports : 1 ....... t ~Sn+ l .... ASn+ k AS n A2Sn+ 1 ... A2Sn+k . . . . . . . . . &2Sn+ 1 ....... A2Sn+k . &2S n . . . . g2Sn+2k_2 A2Sn+k_ 2 &2 S &2 S n+k ....... n+2k-I ASn+ I .... &Sn+ k ASn ASn+ 1 ........

Avec les conditions du thgor~me n > N car la condition A2 S n Th~or~me n a~ n) peuvent @tre calcul~s ~ [~,B] Yn > N impose que AS n+l # AS n pour et donc que lim n ->~ le proc~d~ Sn+! - S S - S n converge d~monstration lim n -~° de la convergence on a l e : S S &2 d'Aitken ~ une suite {S } qui converge n vers S e t si : &Sn+| ~S = p # 1 alors n vers S plus vite que {Sn+ 1} : en utilisant la d~finition • (n) que i~ 2 } converge imm~diatement lim n -~o l'acc~l~ration 32 : Si on applique voit ASn+ I / AS termes # O pour tout n > N.

Si l'on pose : B(n) = co(n) e(n) on trouve, en utilisant pr~c~dente s'~crit _ ¢(in) ¢~n) (n) (n) + " ' " + ¢2k-2 ¢2k-I (n) le th~or~me 35 et le fait que ~2k = 0 ~n, que l a c o r r b i n a i s o n : B(n) - B(n+l) = 0 J¢n ce qui d~montre que B(n) = constante Yn. Cette premigre partie de la dgmonstration ~t~ obtenue par Bauer obtenue p a r W y n m a [15]. La valeur de la constante g laquelle est ggale B(n) a ~t~ La d~monstration est trop longue et trop technique_pour ~tre donn~e ici. En effectuant des ~liminations dans la relation de l'¢-algorithme, Wynn[212] obtenu la : Propri~t~ 17 : [ek(+~')_ ek(n)] -1 d~monstration -1 _ [ek(n) ek(n2l)] -1 - = r (n+1) : on a : -I ~k(:l 1) ~(~)k+l- Ck-(n)=1 [~k(n) - Ck( n - l ) ] -1 ~(n+1)k_1 = [~(k +l) soustrayons %(~)] - et rgarrangeons - _ ¢k+ 1 _ les termes ~ gauche du signe ~gal : -I (~+~ - - d'oO la relation cherch~e puisque -I (n) _ c ( n + l ) ek k-2 = [~k(nll) - (n)~ ak- 1~

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