By Josef Stoer, Roland Bulirsch
Dieses zweib?ndige Standardlehrbuch bietet einen umfassenden und aktuellen ?berblick ?ber die Numerische Mathematik. Dabei wird besonderer Wert auf solche Vorgehensweisen und Methoden gelegt, die sich durch gro?e Wirksamkeit auszeichnen. Ihr praktischer Nutzen, aber auch die Grenzen ihrer Anwendung werden vergleichend diskutiert. Zahlreiche Beispiele runden dieses unentbehrliche Buch ab. Die Neuauflage des zweiten Bandes wurde vollst?ndig ?berarbeitet und erg?nzt um eine Beschreibung weiterer Techniken im Rahmen der Mehrzielmethode zur L?sung von Randwertproblemen f?r Gew?hnliche Differentialgleichungen.
"Das Lehrbuch ... setzt Ma?st?be f?r eine Numerik-Vorlesung und ist jedem Studenten der angewandten Mathematik zu empfehlen." Die Neue Hochschule
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Extra resources for Numerische Mathematik 2: eine Einfuehrung
Example text
1 Reduktion einer Hermiteschen Matrix auf Tridiagonalgestalt. Das Verfahren von Householder Bei dem Verfahren von Householder zur Tridiagonalisierung einer Hermiteschen n × n-Matrix A H = A =: A0 werden zur Transformation Ai = Ti−1 Ai−1 Ti geeignete Householder-Matrizen [s. 7] benutzt: Ti H = Ti−1 = Ti = I − βi u i u iH . 1) Ai−1 = ⎢ δi ⎢ cH ⎣ 0 ai bereits die folgende Gestalt 0 aiH A˜ i−1 ⎤ ⎥ ⎥ ⎥ = (α j k ) ⎥ ⎦ mit Ji−1 c H δi c ⎡ δ1 ⎢ ⎢ γ2 ⎢ ⎢ =⎢ ⎢ ⎢0 ⎢ ⎣ 0 γ¯2 δ2 .. .. . · · · γi−1 ··· 0 0 ..
Im folgenden Programm [s. Golub und Van Loan (1983)], das den Lanczos-Algorithmus f¨ur eine Hermitesche n × n-Matrix A = A H realisiert, bedeuten vk , wk , k = 1, . . , n, die Komponenten dieser Vektoren. w := 0; γ1 := 1; i := 1; 1: if γi = 0 then begin if i = 1 then for k := 1 step 1 until n do begin t := vk ; vk := wk /γi ; wk := −γi t end; w := Av + w; δi := v H w; w := w − δi v; √ m := i; i := i + 1; γi := w H w ; goto 1; end; Pro Schritt i → i + 1 hat man lediglich ca. 5n Multiplikaktionen durchzuf¨uhren und einmal die Matrix A mit einem Vektor zu multiplizieren.
A′ ⎢ j1 ⎢ ⎢ . =⎢ ⎢ . ⎢ ⎢ ′ ⎢ a ⎢ k1 ⎢ ⎢ . ⎢ .. ⎢ ⎣ ′ an1 ... a1′ j .. ... ′ a1k .. ... ′ a1n .. ... a j′ j ... 0 ... ′ a jn .. ... 0 ... . an′ j .. ... ′ akk .. ... ′ akn .. ... ′ ank .. ... 2) f¨ur r = j, k, a j′ j = c2 a j j + s 2 akk + 2csa j k , ! a j′ k = ak′ j = −cs(a j j − akk ) + (c2 − s 2 )a j k = 0, ′ akk = s 2 a j j + c2 akk − 2csa j k . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. 5 Reduktion von Matrizen auf einfachere Gestalt 35 Daraus erh¨alt man f¨ur den Winkel ϕ die Bestimmungsgleichung 2a j k 2cs = , c2 − s 2 a j j − akk tg 2ϕ = |ϕ| ≤ π .