# SQL Server LU decomposition

LUdecompN_q

Updated: 19 January 2015

Use the table-value function LUdecompN_q to calculate the LU factorization of an N x N matrix A in 3rd normal form using partial pivoting. LUdecompN_q returns a lower triangular matrix L, an upper triangular matrix U, and a permutation matrix P such that,

LU = PA

For a 3 x 3 matrix this becomes:

Syntax
SELECT * FROM [wct].[LUdecompN_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 (
[RowNum] [int] NULL,
[ColNum] [int] NULL,
[Value] [float] NULL,
[Type] [nvarchar](2) 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 LUdecompN function for simpler queries.
·         Use LUdecomp_q for a table not in third-normal form.
·         The function returns an error if the array contains a non-numeric value.
·         The returned Type column contains 'L', 'U', or 'P'
Examples
In this example, we calculate the LU decomposition directly from the SELECT statement.
SELECT
*
FROM
wct.LUdecompN_q('
SELECT
*
FROM (VALUES
(0,0,0.002),
(0,1,1.231),
(0,2,2.471),
(1,0,1.196),
(1,1,3.165),
(1,2,2.54),
(2,0,1.475),
(2,1,4.271),
(2,2,2.142)
) m(r,c,x)'
)

This produces the following result.
RowNum      ColNum      Value                  Type
----------- ----------- ---------------------- ----
0           0           1                      L
0           1           0                      L
0           2           0                      L
1           0           0.00167224080267559    L
1           1           1                      L
1           2           0                      L
2           0           1.23327759197324       L
2           1           0.299970803835884      L
2           2           1                      L
0           0           1.196                  U
0           1           3.165                  U
0           2           2.54                   U
1           0           0                      U
1           1           1.22570735785953       U
1           2           2.4667525083612        U
2           0           0                      U
2           1           0                      U
2           2           -1.73047881640933      U
0           0           0                      P
0           1           1                      P
0           2           0                      P
1           0           1                      P
1           1           0                      P
1           2           0                      P
2           0           0                      P
2           1           0                      P
2           2           1                      P

Note that the results are returned in third-normal form. If we wanted to a more traditional (de-normalized) presentation of the results, we can us the PIVOT function.
SELECT
Type,[0],[1],[2]
FROM (
SELECT
*
FROM
wct.LUdecompN_q('
SELECT
*
FROM (VALUES
(0,0,0.002),
(0,1,1.231),
(0,2,2.471),
(1,0,1.196),
(1,1,3.165),
(1,2,2.54),
(2,0,1.475),
(2,1,4.271),
(2,2,2.142)
) m(r,c,x)'
)
) d
PIVOT(
SUM(Value) FOR ColNum in([0],[1],[2])
) p

This produces the following result.
Type 0                      1                      2
---- ---------------------- ---------------------- ----------------------
L    1                      0                      0
L    0.00167224080267559    1                      0
L    1.23327759197324       0.299970803835884      1
P    0                      1                      0
P    1                      0                      0
P    0                      0                      1
U    1.196                  3.165                  2.54
U    0                      1.22570735785953       2.4667525083612
U    0                      0                      -1.73047881640933

In this example, we demonstrate how to reconstruct the input matrix using the calculation P'LU.
SELECT
k.*
FROM (
SELECT
Type as MatrixType,
wct.NMATRIX2STRING(RowNum, ColNum, Value) as Matrix
FROM
wct.LUdecompN_q('
SELECT
*
FROM (VALUES
(0,0,0.002),
(0,1,1.231),
(0,2,2.471),
(1,0,1.196),
(1,1,3.165),
(1,2,2.54),
(2,0,1.475),
(2,1,4.271),
(2,2,2.142)
) m(r,c,x)'
)
GROUP BY
Type
) p PIVOT(MAX(Matrix) FOR MatrixType IN(L,P,U))d
CROSS APPLY
wct.MATRIX(wct.MATMULT(wct.TRANSPOSE(P),wct.MATMULT(L,U)))K

This produces the following result.
RowNum      ColNum      ItemValue
----------- ----------- ----------------------
0           0           0.00200000000000001
0           1           1.231
0           2           2.471
1           0           1.196
1           1           3.165
1           2           2.54
2           0           1.475
2           1           4.27099999999998
2           2           2.14199999999999

This example demonstrates how to use the function by selecting data from a table.
SELECT
*
INTO
#A
FROM (VALUES
(0,0,0.002),
(0,1,1.231),
(0,2,2.471),
(1,0,1.196),
(1,1,3.165),
(1,2,2.54),
(2,0,1.475),
(2,1,4.271),
(2,2,2.142)
) m(r,c,x)

SELECT
*
FROM
wct.LUdecompN_q('SELECT r,c,x FROM #A')