Exemple : Factorisation de matrice LU

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Exemple : Factorisation de matrice LU

Utilisez la fonction LU pour effectuer la factorisation de matrice LU.

Pour éviter les incohérences lors de comparaisons booléennes, activez Egalité approximative dans la liste déroulante Options de calcul.

Factorisation LU d'une matrice réelle

1. Définissez une matrice réelle M1 de dimensions m x n de sorte que m > n.

Cliquez pour copier cette expression

2. Utilisez la fonction LU pour effectuer la factorisation de matrice LUM1.

Cliquez pour copier cette expression

3. Affichez que P1 x M1 = L1 x U1.

<region id="ID0EAW5EB" actualWidth="165.19" actualHeight="81.4" top="614.40000000000009" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="83"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">P1</id> <eval> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m1</id> <real>0</real> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </define> </math> </region>	<region id="ID0EQW5EB" actualWidth="255.11666666666667" actualHeight="81.4" top="710.40000000000009" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="85"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">L1</id> <eval> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m1</id> <real>1</real> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="true"> <nestedMatrixFormats> <nestedMatrixFormat size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="true" position="0,0" /> </nestedMatrixFormats> </matrix> </resultFormat> </math> </region>	<region id="ID0EAX5EB" actualWidth="228.27" actualHeight="62.800000000000004" top="787.20000000000016" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="87"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">U1</id> <eval> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m1</id> <real>2</real> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="true"> <nestedMatrixFormats> <nestedMatrixFormat size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="true" position="0,0" /> </nestedMatrixFormats> </matrix> </resultFormat> </math> </region>
<region id="ID0EUX5EB" actualWidth="158.74333333333334" actualHeight="81.4" top="921.60000000000014" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="89"> <eval xmlns="http://schemas.mathsoft.com/math50"> <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> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>	<region id="ID0EEY5EB" actualWidth="154.49" actualHeight="81.4" top="921.60000000000014" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="91"> <eval xmlns="http://schemas.mathsoft.com/math50"> <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> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>
<region id="ID0EAZ5EB" 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>

La relation est logiquement vraie.

4. Utilisez la fonction M2 pour extraire la matrice submatrix comme m < n.

5. Affichez que P2 x M2 = L2 x U2.

<region id="ID0EHA6EB" actualWidth="69.890000000000015" actualHeight="36.800000000000004" top="1228.8000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="99"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">P2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</id> <real>0</real> </apply> </define> </math> </region>	<region id="ID0EVA6EB" actualWidth="69.276666666666671" actualHeight="36.800000000000004" top="1228.8000000000002" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="101"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">L2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</id> <real>1</real> </apply> </define> </math> </region>	<region id="ID0EDB6EB" actualWidth="70.77000000000001" actualHeight="36.800000000000004" top="1228.8000000000002" left="230.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="103"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">U2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</id> <real>2</real> </apply> </define> </math> </region>
<region id="ID0EWB6EB" actualWidth="151.08" actualHeight="25.6" top="1267.2000000000003" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="105"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P2</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M2</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">L2</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">U2</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>

La relation est logiquement vraie.

6. Utilisez la fonction submatrix pour extraire la matrice M3 de sorte que m = n.

7. Affichez que P3 x M3 = L3 x U3.

<region id="ID0EKI6EB" actualWidth="69.890000000000015" actualHeight="36.800000000000004" top="1497.6000000000001" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="111"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">P3</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m3</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0E1I6EB" actualWidth="69.276666666666671" actualHeight="36.800000000000004" top="1497.6000000000001" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="113"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">L3</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m3</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EKJ6EB" actualWidth="70.77000000000001" actualHeight="36.800000000000004" top="1497.6000000000001" left="230.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="115"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">U3</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m3</id> <real>2</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0E5J6EB" actualWidth="151.08" actualHeight="25.6" top="1536.0000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="117"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P3</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M3</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">L3</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">U3</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>

La relation est logiquement vraie.

Factorisation LU d'une matrice complexe

1. Définissez une matrice complexe C1 de dimensions m x n de sorte que m > n.

Cliquez pour copier cette expression

<region id="ID0E4M6EB" actualWidth="169.16" actualHeight="81.4" top="1708.8000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <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. Utilisez la fonction LU pour effectuer la factorisation de matrice LUC1.

Cliquez pour copier cette expression

3. Affichez que P4 x C1 = L4 x U4.

<region id="ID0EQR6EB" actualWidth="158.44333333333333" actualHeight="81.4" top="2073.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="123"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">P4</id> <eval> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c1</id> <real>0</real> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EAS6EB" actualWidth="425.96999999999997" actualHeight="81.4" top="2169.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="125"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">L4</id> <eval> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c1</id> <real>1</real> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="true"> <nestedMatrixFormats> <nestedMatrixFormat size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="true" position="0,0" /> </nestedMatrixFormats> </matrix> </resultFormat> </math> </region>	<region id="ID0EQS6EB" actualWidth="293.66666666666663" actualHeight="62.800000000000004" top="2246.4000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="127"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">U4</id> <eval> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c1</id> <real>2</real> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="true"> <nestedMatrixFormats> <nestedMatrixFormat size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="true" position="0,0" /> </nestedMatrixFormats> </matrix> </resultFormat> </math> </region>
<region id="ID0EET6EB" actualWidth="225.19000000000003" actualHeight="81.4" top="2323.2000000000003" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="129"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P4</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">C1</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EUT6EB" actualWidth="224.34000000000003" actualHeight="81.4" top="2323.2000000000003" left="268.80000000000007" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="131"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">L4</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">U4</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <decimal precision="3" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EQU6EB" actualWidth="147.67666666666668" actualHeight="25.6" top="2380.8" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="133"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P4</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">C1</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">L4</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">U4</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>

La relation est logiquement vraie.

4. Utilisez la fonction submatrix pour extraire la matrice C2 de sorte que m < n.

5. Affichez que P5 x C2 = L5 x U5.

La relation est logiquement vraie.

6. Utilisez la fonction submatrix pour extraire la matrice C3 de sorte que m = n.

7. Affichez que P6 x C3 = L6 x U6.

<region id="ID0EMCAFB" actualWidth="63.143333333333345" actualHeight="36.800000000000004" top="2880.0000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="151"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">P6</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c3</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0E3CAFB" actualWidth="62.53" actualHeight="36.800000000000004" top="2880.0000000000005" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="153"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">L6</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c3</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EMDAFB" actualWidth="64.023333333333341" actualHeight="36.800000000000004" top="2880.0000000000005" left="230.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="155"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">U6</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c3</id> <real>2</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EAEAFB" actualWidth="147.67666666666668" actualHeight="25.6" top="2918.4000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="157"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P6</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">C3</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">L6</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">U6</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>

La relation est logiquement vraie.

Rubriques associées

A propos des fonctions de factorisation de matrice

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