Difference between revisions of "Manuals/calci/QRDECOMPOSITION"

Revision as of 04:48, 4 September 2017

QRDECOMPOSITION (Matrix)

This function gives the value of QR Decomposition.
In $QRDECOMPOSITION(Matrix)$ , $Matrix$ 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 $A=QR$ , where $Q$ stands for orthogonal matrix and $R$ stands for Upper Triangular matrix.
An orthogonal matrix should satisfy $Q^{T}Q=I$ , where $I$ is identity or Unitary matrix.
$Q^{T}$ 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.

1. QRDECOMPOSITION([[2,6],[10,-15]])

-0.19611613513818393 -0.9805806756909202	-0.9805806756909202 0.19611613513818393
-10.19803902718557 13.5320133245347	-1.1102230246251565e-15 -8.825226081218279

2. QRDECOMPOSITION([[3,8,-5],[4,-6.3,9],[2,5,-1]])

-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

@@ Line 7: / Line 7: @@
 *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 <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.
-*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.