Konditionszahl Berechnen Beispiel Essay

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In der numerischen Mathematik beschreibt man mit der Kondition die Abhängigkeit der Lösung eines Problems von der Störung der Eingangsdaten. Die Konditionszahl stellt ein Maß für diese Abhängigkeit dar; sie beschreibt den Faktor, um den der Eingangsfehler im ungünstigsten Fall verstärkt wird. Sie ist unabhängig von konkreten Lösungsverfahren, aber abhängig vom mathematischen Problem.

Einleitung[Bearbeiten | Quelltext bearbeiten]

In der Numerik unterscheidet man zwischen den drei Größen eines Verfahrens: Kondition, Stabilität und Konsistenz, die untereinander stark verwandt sind. Die Beziehung zwischen der Kondition eines Problems und der Konsistenz des Algorithmus lässt sich wie folgt modellieren:

Es sei das mathematische Problem in Abhängigkeit von der Eingabe , und es sei der numerische Algorithmus sowie die gestörten Eingabedaten. So möchte man den folgenden Fehler - auch als Konvergenz bezeichnet - abschätzen:

.

Mit der Dreiecksungleichung gilt:

Hierbei bezeichnet man mit die Kondition des Problems und mit die Konsistenz des Algorithmus.

Absolute Kondition[Bearbeiten | Quelltext bearbeiten]

Die absolute Kondition von  am Punkt  wird definiert als

Also ist die absolute Kondition genau die kleinste Zahl, für die gilt:

Relative Kondition[Bearbeiten | Quelltext bearbeiten]

Die relative Kondition von  am Punkt  wird definiert als kleinste Zahl mit der Eigenschaft: Es gibt ein , so dass für alle mit die Ungleichung

gilt.

Dabei ist die relative Änderung des Funktionswertes und die relative Änderung der Eingabedaten. Diese Definition lässt sich auch schreiben als

. Ist an der Stelle differenzierbar, dann folgt
.

Dabei ist die Jacobi-Matrix von an der Stelle und die Norm eine mit der verwendeten Vektornorm verträgliche Matrixnorm.

Herleitung der relativen Konditionszahl aus der Taylorreihe[Bearbeiten | Quelltext bearbeiten]

Lässt man für eine Funktion in der Taylorreihe Terme höherer Ordnung unberücksichtigt, so ergibt sich

,

folglich

Hierbei stellt den absoluten Fehler in der Ausgabe dar. Durch Division durch ergibt sich sofort der relative Ausgabefehler:

Um den relativen Fehler in der Eingabe auf der rechten Seite sichtbar zu machen, wird nun noch mit erweitert:

Somit ist alleine aus der Taylorreihe ersichtlich, dass die Fehlerverstärkung durch

in guter Näherung (Terme höherer Ordnung wurden vernachlässigt!) beschrieben ist.

Kondition von linearen Abbildungen und linearen Gleichungssystemen[Bearbeiten | Quelltext bearbeiten]

Lineare Gleichungssysteme treten häufig in der Numerik auf und daher möchte man gerne die Kondition dieser gut bestimmen können. Es können verschiedene Spezialfälle einzeln betrachtet werden.

Invertierbare Matrix[Bearbeiten | Quelltext bearbeiten]

Für invertierbar ist die (relative) Kondition definiert als:

wobei die Kondition von der gewählten Matrixnorm abhängt, also: .

Störung in der rechten Seite[Bearbeiten | Quelltext bearbeiten]

Sei die exakte Lösung von und eine Näherungslösung für eine gestörte rechte Seite , also:

Nun kann man den Fehler und den Fehler im Bildraum (Residuum) definieren:

Für den relativen Fehler kann man nun die folgenden Schranken angeben:

Störung in der Matrix[Bearbeiten | Quelltext bearbeiten]

Sei die exakte Lösung von und eine Näherungslösung für eine gestörte Koeffizientenmatrix , also:

Nun kann man den relativen Fehler nur noch nach Ordnungen in entwickeln. Als Abschätzung erhält man:

Matrix mit vollem Rang[Bearbeiten | Quelltext bearbeiten]

Für

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