By K. Wendler
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Additional resources for Hauptaustauschschritte (Principal Pivoting)
H. es handelt sich um konvexe polyedrische Kegel. 4. 1)nicht Einmundungenl - 49 - 6. Matrizen mit nichtnegativen (positiven) Hauptminoren In diesem Abschnitt besprechen wir eine Verallgemeinerung des Begriffs der Semidefinitheit von Matrizen, die im Zusammenhang mit Hauptaustauschschritten eine Rolle spielt und im n~chsten Abschnitt bei der Behand- lung des sog. Fundamentalproblems gebraucht wird. 8 ist eine symmetrische Matrix genau dann positiv semidefinit, wenn ihre Hauptminoren ~ 0 s~mtlichen sind und genau dann negativ semidefinit, wenn die Haupt- minoren ungerader Reihenzahl ~ 0 und dieJenigen gerader Reihenzahl ~ 0 sind.
4 ~ Ist A streng negativ definit, so hat das Fundamentalproblem genau eine LBsung. Beweis: Seien t;: 1 l;: ) und LBsungen von (51), (52). Dann folgt xl 'A xl = x l 'a x l 'yl x 2 'A x 2 = x 2 'a _ x 2 'y2 x 2 'A xl = x 2 'a x 2 'yl - 59 - Damit wird Andererseits gilt aber, weil A streng negativ definit ist, fUr xl ~ x 2 • Also mue also auch xl = x 2 sein. 5 Satz: 1st A streng negativ quasidefinit, so ist das Fundamental- problem (51), (52) fUr jeden Vektor a lBsbar. Die Losung ist eindeutig bestimmt.
Aber die Beziehungen (41) bis (44) gel ten nun auch fUr die obigen (entarteten) F~lle. Unabh~ngig von Mehrdeutigkeiten bei - 44 der Indexwahl gel ten (41) und (42) stets fUr gelten immer dann, wenn Ii n 12 ~ 0 (bzw. A > 0 K1 ~ und (43) und (44) , K2 ~ 0 ), wobei Ii (bzw. K1 ) die Indexmenge sein solI, fUr die das Maximum bezUglich s1 und 12 (bzw. K2 ) die Indexmenge sein solI, fUr die das Maximum bezUgs1 + s2 lich s2 angenommen wird. Das Maximum bezUglich der Summe wird dann fUr aIle iGI 1 ~ 12 (bzw.
Hauptaustauschschritte (Principal Pivoting) by K. Wendler