Esempio: 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.

Fare clic per copiare questa espressione

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

<region id="ID0EUNIKB" top="1209.6000000000001" left="38.400000000000006" xml:lang="it" actualWidth="167.83666666666665" actualHeight="81.4" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="16" xml:lang="it"> <eval xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">M1</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">P1</id> </apply> <unitOverride xml:lang="it"> <placeholder xml:lang="it" /> </unitOverride> </eval> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>	<region id="ID0EHOIKB" top="1209.6000000000001" left="211.20000000000005" xml:lang="it" actualWidth="165.74" actualHeight="81.4" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="18" xml:lang="it"> <eval xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">Q1</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">R1</id> </apply> <unitOverride xml:lang="it"> <placeholder xml:lang="it" /> </unitOverride> </eval> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
<region id="ID0E5OIKB" top="1286.4" left="38.400000000000006" xml:lang="it" actualWidth="153.23666666666668" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="20" xml:lang="it"> <eval xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="it"> <equal xml:lang="it" /> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">M1</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">P1</id> </apply> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">Q1</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">R1</id> </apply> </apply> <unitOverride xml:lang="it"> <placeholder xml:lang="it" /> </unitOverride> </eval> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
La relazione è logicamente vera.

<region id="ID0EBSIKB" top="1401.6000000000001" left="38.400000000000006" xml:lang="it" actualWidth="220.22000000000003" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50">
 <math resultRef="22" xml:lang="it">
 <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50">
 <id labels="VARIABLE" xml:space="preserve" xml:lang="it">M2</id>
 <apply xml:lang="it">
 <id labels="FUNCTION" xml:space="preserve" xml:lang="it" label-is-contextual="true">submatrix</id>
 <sequence xml:lang="it">
 <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">M1</id>
 <real xml:lang="it">0</real>
 <real xml:lang="it">1</real>
 <real xml:lang="it">0</real>
 <real xml:lang="it">2</real>
 </sequence>
 </apply>
 </define>
 <resultFormat xml:lang="it">
 <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" />
 </resultFormat>
 </math>
</region>

<region id="ID0EYVIKB" top="1516.8000000000002" left="38.400000000000006" xml:lang="it" actualWidth="69.890000000000015" actualHeight="36.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="26" xml:lang="it"> <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it">P2</id> <apply xml:lang="it"> <indexer xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">m2</id> <real xml:lang="it">0</real> </apply> </define> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>	<region id="ID0ELWIKB" top="1516.8000000000002" left="134.40000000000003" xml:lang="it" actualWidth="71.38666666666667" actualHeight="36.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="28" xml:lang="it"> <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it">Q2</id> <apply xml:lang="it"> <indexer xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">m2</id> <real xml:lang="it">1</real> </apply> </define> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>	<region id="ID0E5WIKB" top="1516.8000000000002" left="230.40000000000003" xml:lang="it" actualWidth="70.816666666666677" actualHeight="36.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="30" xml:lang="it"> <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it">R2</id> <apply xml:lang="it"> <indexer xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">m2</id> <real xml:lang="it">2</real> </apply> </define> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
<region id="ID0EVXIKB" top="1555.2000000000003" left="38.400000000000006" xml:lang="it" actualWidth="153.23666666666668" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="32" xml:lang="it"> <eval xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="it"> <equal xml:lang="it" /> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">M2</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">P2</id> </apply> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">Q2</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">R2</id> </apply> </apply> <unitOverride xml:lang="it"> <placeholder xml:lang="it" /> </unitOverride> </eval> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
La relazione è logicamente vera.

<region id="ID0E21IKB" top="1670.4000000000003" left="38.400000000000006" xml:lang="it" actualWidth="220.22000000000003" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50">
 <math resultRef="34" xml:lang="it">
 <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50">
 <id labels="VARIABLE" xml:space="preserve" xml:lang="it">M3</id>
 <apply xml:lang="it">
 <id labels="FUNCTION" xml:space="preserve" xml:lang="it" label-is-contextual="true">submatrix</id>
 <sequence xml:lang="it">
 <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">M1</id>
 <real xml:lang="it">0</real>
 <real xml:lang="it">2</real>
 <real xml:lang="it">0</real>
 <real xml:lang="it">2</real>
 </sequence>
 </apply>
 </define>
 <resultFormat xml:lang="it">
 <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" />
 </resultFormat>
 </math>
</region>

<region id="ID0ES5IKB" top="1785.6000000000004" left="38.400000000000006" xml:lang="it" actualWidth="69.890000000000015" actualHeight="36.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="38" xml:lang="it"> <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it">P3</id> <apply xml:lang="it"> <indexer xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">m3</id> <real xml:lang="it">0</real> </apply> </define> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>	<region id="ID0EF6IKB" top="1785.6000000000004" left="134.40000000000003" xml:lang="it" actualWidth="71.38666666666667" actualHeight="36.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="40" xml:lang="it"> <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it">Q3</id> <apply xml:lang="it"> <indexer xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">m3</id> <real xml:lang="it">1</real> </apply> </define> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>	<region id="ID0EY6IKB" top="1785.6000000000004" left="230.40000000000003" xml:lang="it" actualWidth="70.816666666666677" actualHeight="36.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="42" xml:lang="it"> <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it">R3</id> <apply xml:lang="it"> <indexer xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">m3</id> <real xml:lang="it">2</real> </apply> </define> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
<region id="ID0EPAJKB" top="1824.0000000000002" left="38.400000000000006" xml:lang="it" actualWidth="153.23666666666668" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="44" xml:lang="it"> <eval xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="it"> <equal xml:lang="it" /> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">M3</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">P3</id> </apply> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">Q3</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">R3</id> </apply> </apply> <unitOverride xml:lang="it"> <placeholder xml:lang="it" /> </unitOverride> </eval> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
La relazione è logicamente vera.

<region id="ID0EEIJKB" top="2380.8" left="38.400000000000006" xml:lang="it" actualWidth="140.69333333333333" actualHeight="81.4" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="52" xml:lang="it"> <eval xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">M1</id> <unitOverride xml:lang="it"> <placeholder xml:lang="it" /> </unitOverride> </eval> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>	<region id="ID0EXIJKB" top="2380.8" left="211.20000000000005" xml:lang="it" actualWidth="170.95" actualHeight="81.4" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="54" xml:lang="it"> <eval xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">Q10</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">R10</id> </apply> <unitOverride xml:lang="it"> <placeholder xml:lang="it" /> </unitOverride> </eval> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
<region id="ID0EOJJKB" top="2457.6000000000004" left="38.400000000000006" xml:lang="it" actualWidth="142.05333333333334" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="56" xml:lang="it"> <eval xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="it"> <equal xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">M1</id> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">Q10</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">R10</id> </apply> </apply> <unitOverride xml:lang="it"> <placeholder xml:lang="it" /> </unitOverride> </eval> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
La relazione è logicamente vera.

<region id="ID0EBPJKB" top="2649.6000000000004" left="38.400000000000006" xml:lang="it" actualWidth="87.346666666666664" actualHeight="36.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="60" xml:lang="it"> <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it">Q20</id> <apply xml:lang="it"> <indexer xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">m20</id> <real xml:lang="it">0</real> </apply> </define> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>	<region id="ID0EUPJKB" top="2649.6000000000004" left="134.40000000000003" xml:lang="it" actualWidth="86.776666666666671" actualHeight="36.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="62" xml:lang="it"> <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it">R20</id> <apply xml:lang="it"> <indexer xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">m20</id> <real xml:lang="it">1</real> </apply> </define> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
<region id="ID0ELQJKB" top="2688.0000000000005" left="38.400000000000006" xml:lang="it" actualWidth="142.05333333333334" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="64" xml:lang="it"> <eval xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="it"> <equal xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">M2</id> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">Q20</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">R20</id> </apply> </apply> <unitOverride xml:lang="it"> <placeholder xml:lang="it" /> </unitOverride> </eval> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
La relazione è logicamente vera.

<region id="ID0E4VJKB" top="2880.0000000000005" left="38.400000000000006" xml:lang="it" actualWidth="87.346666666666664" actualHeight="36.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="68" xml:lang="it"> <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it">Q30</id> <apply xml:lang="it"> <indexer xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">m30</id> <real xml:lang="it">0</real> </apply> </define> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>	<region id="ID0EQWJKB" top="2880.0000000000005" left="134.40000000000003" xml:lang="it" actualWidth="86.776666666666671" actualHeight="36.800000000000004" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="70" xml:lang="it"> <define xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" xml:lang="it">R30</id> <apply xml:lang="it"> <indexer xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">m30</id> <real xml:lang="it">1</real> </apply> </define> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
<region id="ID0EHXJKB" top="2918.4000000000005" left="38.400000000000006" xml:lang="it" actualWidth="142.05333333333334" actualHeight="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="72" xml:lang="it"> <eval xml:lang="it" xmlns="http://schemas.mathsoft.com/math50"> <apply xml:lang="it"> <equal xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">M3</id> <apply xml:lang="it"> <mult xml:lang="it" /> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">Q30</id> <id labels="VARIABLE" xml:space="preserve" xml:lang="it" label-is-contextual="true">R30</id> </apply> </apply> <unitOverride xml:lang="it"> <placeholder xml:lang="it" /> </unitOverride> </eval> <resultFormat xml:lang="it"> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="it" /> </resultFormat> </math> </region>
La relazione è logicamente vera.