Konvergenzproblem bei Gauß Seidel Verfahren
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27.10.2009, 01:38 tegg Auf diesen Beitrag antworten »
Konvergenzproblem bei Gauß Seidel Verfahren
Hi.

ich versuche nunerisch ein GLS zu lösen, die Werte der Matrix sind ausser i,-i i,i und i+1 =0. i,i ist immer =1
Daraus ergibt sich ja als Konvergenzprüfung dass i,i-1 + i,i+1 > 1 sein sollen, das ist jedoch bei einigen Zeilen nicht der Fall

Einige Werte springen gegen positiv unendlich, somit überträgt sich das bei jedem Schritt in die Ergebnisse der umliegenden x Werte, so dass bei genügen Schritt alle x Werte gegen positiv unendlich gehen....

Gibts da eine Möglichkeit die Konvergenz zu "erzwingen" oder anderes Verfahren ?
27.10.2009, 10:43 lgrizu Auf diesen Beitrag antworten »
RE: Konvergenzproblem bei Gauß Seidel Verfahren
das gauß-seidel verfahren konvergiert ja nur, wenn die matrix dem starken zeilensummenkriterium genügt, konvergenz erzwingen geht nicht, entweder es konvergiert und das tut es wenn das starke zeilensummenkriterium erfüllt ist oder es konvergiert nicht, dann ist das kriterium auch nicht erfüllt.
andere verfahren zur berechnung von lösungsvektoren sind zum beispiel gauß elimination; ich glaube, alle einzel- und gesamtschrittverfahren setzen voraus, dass das starke zeilensummenkriterium erfüllt sein, damit die verfahren konvergieren.
cholesky-zerlegung setzt voraus, dass die matrix positv definit ist, also nur positive eigenwerte hat.
wenn deine matrix regulär ist, so kannst du QR-zerlegung anwenden.
wie sieht deine matrix denn aus?
27.10.2009, 12:31 tegg Auf diesen Beitrag antworten »

unterschiedlich, wird aus anderen Werten berechnet, hier mal ein Beispiel, jedoch ohne b Vektor einberechnet


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27.10.2009, 12:35 tigerbine Auf diesen Beitrag antworten »

Nur ein Google-Tipp: Numerik großer Gleichungssysteme. Die Matrix ist ja sehr schwach besetzt, erfüllt aber ws noch nicht einmal das schwache Zeilensummen-Kriterium.

Woher stammt die? Aus der numerischen Lösung einer DGL. Mit fällt als Stichwort noch ein: Finite-Elemente-Methode, die ich dir aber im Moment noch nicht erklären kann.

Wink
28.10.2009, 00:19 tegg Auf diesen Beitrag antworten »

die FEM Methoden beschreiben ja auch nur die Randbedingungen zwischen den einzelnen Elementen und beschränken diese auf eine fintite Menge....sonst benutzen die ja auch numerische Methoden ....

was mir jedoch noch auffiel



kann man durch



umrechnen...das funktioniert sogar rein von den Ausgerechneten Werten.... ist das auch mathematisch richtig ?
28.10.2009, 00:44 tigerbine Auf diesen Beitrag antworten »

Die Inverse muss links von b stehen. Aber das Invertieren ist i.A. keine angenehme Rechnung und wird nicht gemacht. Warum sollte das bei dir so einfach gehen?

Ist das bei dir eine Tridiagonalmatrix ?
 Anzeige 
 
28.10.2009, 13:22 tegg Auf diesen Beitrag antworten »

ja ist eine, ich hab den c code in matlab umgeschrieben und da kam was annehmbares raus bei B*inv(A), aber auch B*inv(A) ist gut smile
28.10.2009, 17:18 tigerbine Auf diesen Beitrag antworten »

Naja, man sollte schon "richtig" mit Matrizen rechnen.
29.10.2009, 00:48 tegg Auf diesen Beitrag antworten »

alles klar, habs auch grad nachgeschaut ... von "links" invertieren, sonst krieg ich keine Einheitsmatrix raus, damit dann x= dasteht...joa, dann erkläre ich das Problem in meinem speziellen Fall als gelöst smile
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