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XLeratorDB/math Documentation

SQL Server QR decomposition


QRdecomp_q

Updated: 31 January 2012


Use QRdecomp_q for decomposing a de-normalized N x N matrix A into the product of an upper triangular matrix R and an orthogonal matrix Q, such that
                A=QR.
Syntax
SELECT * FROM [wctMath].[wct].[QRdecomp_q](
   <@Matrix_RangeQuery, nvarchar(max),>)
Arguments
@Matrix_RangeQuery
the SELECT statement, as text, used to determine the square (N x N) matrix to be used in this function. The SELECT statement specifies the column names from the table or view or can be used to enter the matrix values directly. Data returned from the @Matrix_RangeQuery select must be of the type float or of a type that implicitly converts to float.
Return Types
TABLE (
      [Matrix] [nvarchar](2) NULL,
      [RowNum] [int] NULL,
      [ColNum] [int] NULL,
      [ItemValue] [float] NULL
Remarks
·         The number of columns in the matrix must be equal to the number of rows or an error will be returned.
·         Use the QRdecomp function for simpler queries.
·         Use QR to operate on a string representation of the matrix.
·         The function returns an error if the array contains a non-numeric value.
Examples
In this example, we will enter the matrix directly into the function without populating a table.
SELECT *
FROM wct.QRDECOMP_q('
      SELECT 12,-51,4 UNION ALL
      SELECT 6,167,-68 UNION ALL
      SELECT -4,24,-41'
      )    
This produces the following result.
Matrix      RowNum      ColNum              ItemValue
------ ----------- ----------- ----------------------
Q                0           0     -0.857142857142857
Q                0           1      0.394285714285714
Q                0           2     -0.331428571428571
Q                1           0     -0.428571428571429
Q                1           1     -0.902857142857142
Q                1           2     0.0342857142857143
Q                2           0      0.285714285714286
Q                2           1     -0.171428571428571
Q                2           2     -0.942857142857142
R                0           0                    -14
R                0           1                    -21
R                0           2                     14
R                1           0                      0
R                1           1                   -175
R                1           2                     70
R                2           0                      0
R                2           1                      0
R                2           2                     35
Note that the results are returned in third-normal form.
If the matrix values had been in a table, could have simply changed the SQL to SELECT from the table.
SELECT *
INTO #m
FROM (
      SELECT 12,-51,4 UNION ALL
      SELECT 6,167,-68 UNION ALL
      SELECT -4,24,-41
      ) n(a,b,c) 
 
SELECT *
FROM wct.QRDECOMP_q('
      SELECT * FROM #m'
      )
If we calculate Q * R we should get the original matrix, A, returned, within the limits of floating point arithmetic.
SELECT *
INTO #n
FROM wct.QRdecomp_q('
      SELECT 12,-51,4 UNION ALL
      SELECT 6,167,-68 UNION ALL
      SELECT -4,24,-41'
      )
SELECT RowNum
,ColNum
,ROUND(ItemValue, 0) as ItemValue
FROM wct.MMULTN_q(
'Select RowNum, ColNum, ItemValue FROM #n WHERE MATRIX = ' + CHAR(39) + 'Q' + CHAR(39),
'Select RowNum, ColNum, ItemValue FROM #n WHERE MATRIX = ' + CHAR(39) + 'R' + CHAR(39)
      )
This produces the following result.
     RowNum      ColNum              ItemValue
----------- ----------- ----------------------
          0           0                     12
          0           1                    -51
          0           2                      4
          1           0                      6
          1           1                    167
          1           2                    -68
          2           0                     -4
          2           1                     24
          2           2                    -41
 
We can also calculate QTQ, verifying that this will return the identity matrix.
SELECT *
INTO #n
FROM wct.QRdecomp_q('
      SELECT 12,-51,4 UNION ALL
      SELECT 6,167,-68 UNION ALL
      SELECT -4,24,-41'
      )
SELECT RowNum
,ColNum
,ROUND(ItemValue, 0) as ItemValue
FROM wct.MMULTN_q(
'Select ColNum, RowNum, ItemValue FROM #n WHERE MATRIX = ' + CHAR(39) + 'Q' + CHAR(39),
'Select RowNum, ColNum, ItemValue FROM #n WHERE MATRIX = ' + CHAR(39) + 'Q' + CHAR(39)
      )
This returns the following result.
     RowNum      ColNum              ItemValue
----------- ----------- ----------------------
          0           0                      1
          0           1                      0
          0           2                      0
          1           0                      0
          1           1                      1
          1           2                      0
          2           0                      0
          2           1                      0
          2           2                      1
 


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