Beispiel: QR-Matrixfaktorisierung

Funktionen > Vektor und Matrix > Matrixfaktorisierung > Beispiel: QR-Matrixfaktorisierung

Verwenden Sie die Funktion QR, um eine QR-Matrixfaktorisierung auszuführen.

• Um logische Konflikte zu vermeiden, wenn Sie boolesche Vergleiche ausführen, aktivieren Sie Annähernde Gleichheit in der Dropdown-Liste Berechnungsoptionen.

• Im Beispiel wird eine komplexe Matrix als Eingabe verwendet, die Funktion akzeptiert jedoch auch eine reelle Matrize als Eingabe.

QR-Faktorisierung mit Pivotisierung

1. Definieren Sie eine reelle Matrix M1 mit den Dimensionen m x n, wobei gilt: m > n.

2. Legen Sie das Argument p fest, um das Aktivieren/Deaktivieren der Pivotisierung zu steuern.

3. Verwenden Sie die Funktion QR, um eine QR-Matrixfaktorisierung der Matrix M1 durchzuführen.

Die Standardfunktion QR(M1) ist mit QR(M,1). gleichwertig.

4. Zeigen Sie, dass M1 x P1 = Q1 x R1.

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<region id="ID0E3FQFB" actualWidth="153.23666666666668" actualHeight="25.6" top="1286.4" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="20"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P1</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R1</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
Die Beziehung ist logisch wahr.

5. Verwenden Sie die Funktion submatrix, um die Matrix M2 zu extrahieren, sodass Folgendes gilt: m < n. Wenden Sie anschließend die Funktion QR an.

6. Zeigen Sie, dass M2 x P2 = Q2 x R2.

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Die Beziehung ist logisch wahr.

7. Verwenden Sie die Funktion submatrix, um die Matrix M3 zu extrahieren, sodass Folgendes gilt: m = n. Wenden Sie anschließend die Funktion QR an.

8. Zeigen Sie, dass M3 x P3 = Q3 x R3.

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Die Beziehung ist logisch wahr.

QR-Faktorisierung ohne Pivotisierung

1. Deaktivieren Sie die Pivotisierung, und wenden Sie anschließend die Funktion QR auf die Matrix M1 (m > n) an.

2. Zeigen Sie, dass M1 = Q10 x R10.

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Die Beziehung ist logisch wahr.

3. Deaktivieren Sie die Pivotisierung, und wenden Sie anschließend die Funktion QR auf die Matrix M2 (m < n) an.

4. Zeigen Sie, dass M2 = Q20 x R20.

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Die Beziehung ist logisch wahr.

5. Deaktivieren Sie die Pivotisierung, und wenden Sie anschließend die Funktion QR auf die Matrix M3 (m = n) an.

6. Zeigen Sie, dass M3 = Q30 x R30.

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Die Beziehung ist logisch wahr.

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