By Peter Knabner
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Extra info for Numerische Mathematik II
M , y˙ m+1 = 1 , yi (t0 ) = y0i , i = 1, . . , m , ym+1 (t0 ) = t0 . Eine AWA f¨ ur eine skalare GDG n-ter Ordnung y (n) = f (t, y (0) , y (1) , y (2) , . . , y (n−1) ) , wobei hier dy , dt d2 y dk y y (2) := 2 , y (k) := k , dt dt l¨aßt sich auf ein System von n GDG 1. Ordnung reduzieren durch: y (0) := y , y (1) := pi (t) := y (i−1) (t) , ⇒ i = 1, . . , n . p′1 (t) = y ′(t) = p2 (t) p′i = pi+1 , i = 1, . . , n − 1 p′n = y (n) = f (t, p1 , p2 , . . 14) also p2 .. p′ = f˜(t, p) := .
Impliziert ϕ(t) ≤ g(t) exp(L(t − t0 )) f¨ ur t ∈ [t0 , TM ] und damit y−z ∞ ≤ exp(L(TM − t0 )) δ0 + TM t0 δ1 (s) ds ≤ exp(L(TM − t0 )) max(1, TM − t0 )( δ0 + δ1 ∞) . h. es gibt sogar eine Lipschitzstetige Abh¨angigkeit zwischen St¨orung und L¨osung und auch die Norm f¨ ur δ1 k¨onnte zu einer L1 -Norm in t abgeschw¨acht werden. 3 δ i. Allg. von TM − t0 und zwar sogar exponentiell abh¨angt. Das kann die Absch¨atzung f¨ ur große L und TM − t0 aussagelos machen: Beispiel: Chaotische Systeme E.
Somit ist C symmetrisch und positiv definit. Die Aufl¨osung der Hilfsgleichungssysteme erfordert also nur Vorw¨arts- und R¨ uckw¨artssubstitution bei gleicher Besetzungsstruktur wie bei der Systemmatrix, so dass auch die Forderung der geringeren Komplexit¨at erf¨ ullt ist. Aus einer genauen Absch¨atzung von κ(C −1 A) sieht man (siehe S. , Axelsson/Barker): Unter gewissen Bedingungen an A, die also Bedingungen an die Randwertaufgabe und die Diskretisierung widerspiegeln, ergibt sich eine erhebliche Konditionsverbesserung in Form einer Absch¨atzung vom Typ κ(C −1 A) ≤ const(κ(A)1/2 + 1) .