Difference between revisions of "Manuals/calci/QRDECOMPOSITION"

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==Description==
 
==Description==
 
*This function gives the value of QR Decomposition.
 
*This function gives the value of QR Decomposition.
−
*In <math>QRDECOMPOSITION (Matrix)</math>,<math>Matrix</math> is any matrix.
+
*In <math>QRDECOMPOSITION (Matrix)</math>, <math>Matrix</math> is any matrix.
 
*QR Decomposition is also called QR Factorization.
 
*QR Decomposition is also called QR Factorization.
 
*QR Decomposition is defined by the product of Orthogonal matrix and Upper Triangular matrix.
 
*QR Decomposition is defined by the product of Orthogonal matrix and Upper Triangular matrix.
−
*Consider any square matrix A may be decomposed as <math>A=QR</math>,where <math>Q</math> stands for orthogonal matrix and <math>R</math> stands for Upper Triangular matrix.
+
*Consider any square matrix A may be decomposed as <math>A=QR</math>, where <math>Q</math> stands for orthogonal matrix and <math>R</math> stands for Upper Triangular matrix.
−
*An orthogonal matrix should satisfy <math>Q^TQ=I</math>,where <math>I</math> is identity or Unitary matrix.
+
*An orthogonal matrix should satisfy <math>Q^TQ=I</math>, where <math>I</math> is identity or Unitary matrix.
 
*<math>Q^T</math> is the transpose matrix of Q.  
 
*<math>Q^T</math> is the transpose matrix of Q.  
−
*If the given matrix A is non singular, then this factorization is unique.
+
*If the given matrix A is non-singular, then this factorization is unique.
 
*Gram-Schmidt process is one of the process of computing decomposition in QR Decomposition method.
 
*Gram-Schmidt process is one of the process of computing decomposition in QR Decomposition method.
  
 
==Examples==
 
==Examples==
−
1. QRDECOMPOSITION([[2,6],[10,-15]])
+
{| class="wikitable"
 +
|+Spreadsheet
 +
|-
 +
! !! A !! B     
 +
|-
 +
! 1
 +
| 2 || 6
 +
|-
 +
!2
 +
| 10 || -15
 +
|}
 +
=QRDECOMPOSITION(A1:B2)
 
{| border="1" cellpadding="5" cellspacing="0"
 
{| border="1" cellpadding="5" cellspacing="0"
 
|-
 
|-
 
|
 
|
 
   -0.19611613513818393  -0.9805806756909202
 
   -0.19611613513818393  -0.9805806756909202
−
|| -0.9805806756909202 0.19611613513818393
+
||  
 +
-0.9805806756909202 0.19611613513818393
 
|-
 
|-
 
|
 
|
 
  -10.19803902718557 13.5320133245347
 
  -10.19803902718557 13.5320133245347
−
|| -1.1102230246251565e-15 -8.825226081218279
+
||  
 +
-1.1102230246251565e-15 -8.825226081218279
 +
|}
 +
 
 +
{| class="wikitable"
 +
|+Spreadsheet
 +
|-
 +
! !! A !! B !! C     
 +
|-
 +
! 1
 +
| 3 || 8 || -5
 +
|-
 +
!2
 +
|4 || -6.3 || 9
 +
|-
 +
!3
 +
|2 || 5 || -1
 
|}
 
|}
−
2. QRDECOMPOSITION([[3,8,-5],[4,-6.3,9],[2,5,-1]])
+
=QRDECOMPOSITION(A1:C3)
 
{| border="1" cellpadding="5" cellspacing="0"
 
{| border="1" cellpadding="5" cellspacing="0"
 
|-
 
|-
 
|
 
|
 
   -0.5570860145311556 0.631547425332445 -0.5392615524675877
 
   -0.5570860145311556 0.631547425332445 -0.5392615524675877
−
|| -0.7427813527082074 -0.669329688618384 -0.01654176541311622
+
||  
−
|| -0.3713906763541037 0.3913382392381005 0.841975859527614
+
-0.7427813527082074 -0.669329688618384 -0.01654176541311622
 +
||  
 +
-0.3713906763541037 0.3913382392381005 0.841975859527614
 
|-
 
|-
−
| -5.385164807134504 -1.634118975958056 -3.528211425363985
+
|  
 +
-5.385164807134504 -1.634118975958056 -3.528211425363985
 
||
 
||
 
  -2.1551618871879059e-16 11.22584763714588 -9.573042563465782
 
  -2.1551618871879059e-16 11.22584763714588 -9.573042563465782
−
||
+
||  
 
  5.3446973501217775e-17   0 1.7054560140922779
 
  5.3446973501217775e-17   0 1.7054560140922779
 
|}
 
|}
 +
 +
==Related Videos==
 +
 +
{{#ev:youtube|v=J41Ypt6Mftc|280|center|QR Decomposition}}
 +
 +
 +
==See Also==
 +
*[[Manuals/calci/LUDECOMPOSITION| LUDECOMPOSITION]]
 +
*[[Manuals/calci/CHOLESKYFACTORIZATION| CHOLESKYFACTORIZATION]]
 +
*[[Manuals/calci/CONFERENCE| CONFERENCE]]
 +
 +
==References==
 +
*[https://en.wikipedia.org/wiki/LU_decomposition  LU Decomposition]
 +
 +
*[[Z_API_Functions | List of Main Z Functions]]
 +
 +
*[[ Z3 |  Z3 home ]]

Latest revision as of 13:30, 2 May 2019

QRDECOMPOSITION (Matrix)


  • is the set of values.

Description

  • This function gives the value of QR Decomposition.
  • In , is any matrix.
  • QR Decomposition is also called QR Factorization.
  • QR Decomposition is defined by the product of Orthogonal matrix and Upper Triangular matrix.
  • Consider any square matrix A may be decomposed as , where stands for orthogonal matrix and stands for Upper Triangular matrix.
  • An orthogonal matrix should satisfy , where is identity or Unitary matrix.
  • is the transpose matrix of Q.
  • If the given matrix A is non-singular, then this factorization is unique.
  • Gram-Schmidt process is one of the process of computing decomposition in QR Decomposition method.

Examples

Spreadsheet
A B
1 2 6
2 10 -15

=QRDECOMPOSITION(A1:B2) 

 -0.19611613513818393   -0.9805806756909202

-0.9805806756909202	0.19611613513818393

-10.19803902718557	 13.5320133245347

-1.1102230246251565e-15	 -8.825226081218279
Spreadsheet
A B C
1 3 8 -5
2 4 -6.3 9
3 2 5 -1

=QRDECOMPOSITION(A1:C3) 

 -0.5570860145311556	0.631547425332445	-0.5392615524675877

-0.7427813527082074	-0.669329688618384	-0.01654176541311622

-0.3713906763541037	0.3913382392381005	0.841975859527614

-5.385164807134504	-1.634118975958056	-3.528211425363985

-2.1551618871879059e-16	11.22584763714588	-9.573042563465782

5.3446973501217775e-17	   0	1.7054560140922779

Related Videos

QR Decomposition


See Also

References

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