Esempio: fattorizzazione QR di matrici

Funzioni > Vettore e matrice > Fattorizzazione di matrici > Esempio: fattorizzazione QR di matrici

Utilizzare la funzione QR per eseguire la fattorizzazione QR di matrici.

• Per evitare corrispondenze errate a livello logico quando si eseguono confronti booleani, attivare l'opzione Uguaglianza approssimativa nell'elenco a discesa Opzioni calcolo.

• In questo esempio viene utilizzata una matrice complessa come input, ma la funzione accetta come input solo una matrice reale.

Fattorizzazione QR con pivot

1. Definire una matrice reale M1 di dimensioni m x n in modo che m > n.

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2. Impostare l'argomento p per il controllo dell'attivazione e della disattivazione del pivot.

3. Utilizzare la funzione QR per eseguire la fattorizzazione QR della matrice M1.

La funzione di default QR(M1) equivale a QR(M,1).

4. Mostrare che M1 x P1 = Q1 x R1.

<region id="ID0EHFQFB" actualWidth="167.83666666666665" actualHeight="81.4" top="1209.6000000000001" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="16"> <eval xmlns="http://schemas.mathsoft.com/math50"> <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> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EVFQFB" actualWidth="165.74" actualHeight="81.4" top="1209.6000000000001" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="18"> <eval xmlns="http://schemas.mathsoft.com/math50"> <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> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EIGQFB" 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>
La relazione è logicamente vera.

5. Utilizzare la funzione submatrix per estrarre la matrice M2 in modo che m < n e quindi applicare la funzione QR.

6. Mostrare che M2 x P2 = Q2 x R2.

<region id="ID0EXMQFB" actualWidth="69.890000000000015" actualHeight="36.800000000000004" top="1516.8000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="26"> <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> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EFNQFB" actualWidth="71.38666666666667" actualHeight="36.800000000000004" top="1516.8000000000002" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="28"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</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="ID0ETNQFB" actualWidth="70.816666666666677" actualHeight="36.800000000000004" top="1516.8000000000002" left="230.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="30"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</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="ID0EGOQFB" actualWidth="153.23666666666668" actualHeight="25.6" top="1555.2000000000003" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="32"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M2</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P2</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q2</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R2</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 relazione è logicamente vera.

7. Utilizzare la funzione submatrix per estrarre la matrice M3 in modo che m = n e quindi applicare la funzione QR.

8. Mostrare che M3 x P3 = Q3 x R3.

<region id="ID0ENUQFB" actualWidth="69.890000000000015" actualHeight="36.800000000000004" top="1785.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="38"> <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="ID0E2UQFB" actualWidth="71.38666666666667" actualHeight="36.800000000000004" top="1785.6000000000004" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="40"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q3</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="ID0EJVQFB" actualWidth="70.816666666666677" actualHeight="36.800000000000004" top="1785.6000000000004" left="230.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="42"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R3</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="ID0E3VQFB" actualWidth="153.23666666666668" actualHeight="25.6" top="1824.0000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="44"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M3</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P3</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q3</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R3</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 relazione è logicamente vera.

Fattorizzazione QR senza pivot

1. Disattivare il pivot e quindi applicare la funzione QR alla matrice M1 (m > n).

2. Mostrare che M1 = Q10 x R10.

<region id="ID0EQ3QFB" actualWidth="140.69333333333333" actualHeight="81.4" top="2380.8" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="52"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0E53QFB" actualWidth="170.95" actualHeight="81.4" top="2380.8" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="54"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q10</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R10</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="ID0ER4QFB" actualWidth="142.05333333333334" actualHeight="25.6" top="2457.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="56"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q10</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R10</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 relazione è logicamente vera.

3. Disattivare il pivot e quindi applicare la funzione QR alla matrice M2 (m < n).

4. Mostrare che M2 = Q20 x R20.

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<region id="ID0EXDRFB" actualWidth="142.05333333333334" actualHeight="25.6" top="2688.0000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="64"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M2</id> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q20</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R20</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 relazione è logicamente vera.

5. Disattivare il pivot e quindi applicare la funzione QR alla matrice M3 (m = n).

6. Mostrare che M3 = Q30 x R30.

<region id="ID0E2IRFB" actualWidth="87.346666666666664" actualHeight="36.800000000000004" top="2880.0000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="68"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q30</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m30</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="ID0EJJRFB" actualWidth="86.776666666666671" actualHeight="36.800000000000004" top="2880.0000000000005" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="70"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R30</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m30</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="ID0E3JRFB" actualWidth="142.05333333333334" actualHeight="25.6" top="2918.4000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="72"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M3</id> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q30</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R30</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 relazione è logicamente vera.

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