Beispiel: LU-Matrixfaktorisierung

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Beispiel: LU-Matrixfaktorisierung

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

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

LU-Faktorisierung einer reellen Matrix

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

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2. Verwenden Sie die Funktion LU, um eine LU-Matrixfaktorisierung der Matrix M1 durchzuführen.

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3. Zeigen Sie, dass P1 x M1 = L1 x U1.

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<region id="ID0EPDNFB" actualWidth="151.08" actualHeight="25.6" top="998.40000000000009" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="93"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">L1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">U1</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>

Die Beziehung ist logisch wahr.

4. Verwenden Sie die Funktion submatrix, um die Matrix M2 zu extrahieren, wobei Folgendes gilt: m < n.

5. Zeigen Sie, dass P2 x M2 = L2 x U2.

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Die Beziehung ist logisch wahr.

6. Verwenden Sie die Funktion submatrix, um die Matrix M3 zu extrahieren, wobei Folgendes gilt: m = n.

7. Zeigen Sie, dass P3 x M3 = L3 x U3.

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Die Beziehung ist logisch wahr.

LU-Faktorisierung einer komplexen Matrix

1. Definieren Sie eine komplexe Matrix C1 mit den Dimensionen m x n, wobei gilt: m > n.

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  <math resultRef="119">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <id labels="VARIABLE" xml:space="preserve">C1</id>
      <matrix rows="4" cols="3">
        <real>3</real>
        <apply>
          <minus />
          <real>1</real>
          <imag symbol="i">3</imag>
        </apply>
        <apply>
          <minus />
          <imag symbol="i">2</imag>
          <real>3</real>
        </apply>
        <real>7</real>
        <imag symbol="i">-1</imag>
        <real>2</real>
        <real>-2</real>
        <real>8</real>
        <real>2</real>
        <real>3</real>
        <real>-1</real>
        <apply>
          <plus />
          <imag symbol="i">9</imag>
          <real>1</real>
        </apply>
      </matrix>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

2. Verwenden Sie die Funktion LU, um eine LU-Matrixfaktorisierung der Matrix C1 durchzuführen.

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3. Zeigen Sie, dass P4 x C1 = L4 x U4.

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Die Beziehung ist logisch wahr.

4. Verwenden Sie die Funktion submatrix, um die Matrix C2 zu extrahieren, wobei Folgendes gilt: m < n.

5. Zeigen Sie, dass P5 x C2 = L5 x U5.

Die Beziehung ist logisch wahr.

6. Verwenden Sie die Funktion submatrix, um die Matrix C3 zu extrahieren, wobei Folgendes gilt: m = n.

7. Zeigen Sie, dass P6 x C3 = L6 x U6.

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Die Beziehung ist logisch wahr.

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