By Euler L.
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Additional info for Observations about two biquadratics, of which the sum is able to be resolved into two other biquadratics
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N~l Folglich ist die rnaximale Zeilenbetragssl1mrne eine der Maximumfiorm zugeordnete Matrixnorm: IIAlloo := max k z", N z" : = 2: laknl (ZeilenSl1rnrnennorrn) . 27), N IIAlll := max 8", n 8" := 2: lalcnl (8palten811rnrnennorrn) . (24) k~l Zur Herleitung einer zur Euklidischen Vektornorm zugeordneten Matrixnonn benotigt man einige Hilfsmittel aus dem Gebiet der Matrizeneigenwertprobleme (Kap. 4). Geniigen die (reeHe oder komplexe) Zahl LX und der Nichtnullvektor y, y E IS/', der Gleichung Ay =LXy, (25) so heiBt ex Eigenwert und y Eigenvektor der N X N-Matrix A.
13. Man gebe Matrizen an, deren Struktur durch die Graphen in Abb. 6 beschrieben wird. Man transformiere die Matrix des "Geographen" durch Zeilen- und n (B C) . 14. 15. Man zeige: ist A. positiv definit, so sind die Diagonalelemente von A positiv. ;n = 0, k = 1(I)N. 1:16. Man zeige,: 1st die N X N-Matrix A symmetrisch und positiv semidefinit und ist der Hauptminor B : = AN -1, N -1 nichtsingular, so besitzt die Matrix A die Darstellung By ) yTB yTBy , A_ (B wo yeinen (N - l)dimensionalen Vektor bezeichnet.
6. 11. 6 nicht erfiillt. 12. 4. 13. ManberechneddsSkalarproduktvonuundvmitun := Ifnundvn := 1f(n+ 1) fiir n = 1(1)100 in m-stelliger und 2m-stelliger Gleitpunktarithmetik und mit Hilfe des Kahan-BabuSka-Tricks. 14. 13. 4. Intervallrechnung Die meisten Fehleraussagen des vorigen Abschnitts wurden unter VernachHissigung von Gliedern hoherer Ordhung hergeleitet, sie galten nur in erster Naherung. Solche Ergebnisse sind fur den strengen Mathematiker unbefriedigend. Korrekte Fehlerein'SchlieBungen ergeben ·sich, wenn man den gesamten Algorithmus "zweigleisig" durchfUhrt, d.