By G. Alefeld (auth.), R. Bulirsch, R. D. Grigorieff, J. Schröder (eds.)

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Extra info for Numerical Treatment of Differential Equations: Proceedings of a Conference, Held at Oberwolfach, July 4–10, 1976

Example text

9) e r g e b e n sich die F o r d e r u n g e n mb ~ p+l. Da viele der B e d i n g u n g s g l e i c h u n g e n linear a b h 6 n g i g sind, viele U n b e k a n n t e U wie G l e i c h u n g e n G einzuf6hren: ist es aber nicht n@tig, gleich Zuerst wird U auf die Anzahl G der sich aus T a b e l l e n 1 und 2 e r g e b e n d e n G l e i c h u n g e n festgelegt. 10) resp. 11) e r f ~ l l t wird. 9) e r g e b e n d e n G l e i c h u n g e n erweitert. F~r G l e i c h u n g s s y s t e m e , die nicht den v o l l e n Rang U aufwiesen, den U n b e k a n n t e n Null gesetzt.

Since we vector | -I gOT ( S h U _ w = ~gOS2h to be an approximation f) = K u of 6u, - Kf f ( Kf = ~gOS2h I -1 ~0T, the mapping u~-+u-w is called K = KfS h ) "correction by app~o~ma- tion ". The combination iteration of ~ smoothing steps with the correction explained above yields step: u (~-I) ~-~ ~u = G~u (~-I) + Gf, w f ~-~ u (~) = (l-K)u + Kff = ~w u(~-1) + Mf, where ~ = (I-K)G w. 5, m M h are bounded < 1 of convergence by (~ > o), are independent converge properties of h. to zero as I/v. 15o2, m ~ (ev) -I ).

13 i m Eine der zahlreichen M~glichkeiten besteht darin, das vollst~ndige Eigensystem der Matrix T zu b e s t i m m e n . nen FORTRAN Subroutinen Die erlauben T = Q R D (QR) wobei "fast" Q eine eine konjugiert F~r n= -i orthonormierte 3 also ffir T d i e [a - b 10 in E I S P A C K (Smith et al (1976)) vorhande- Zerlegung , und Einem R eine reellen Eigenwertpaar zum Beispiel D = Beispiel Matrix Diagonalmatrix: komplexen zum eine (Eigenwerte b 0l a 0 0 c] Rechtsdreiecksmatrix Eigenwert 2x2 M a t r i x bedeuten.

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