 # SQL Server LU decomposition

LUdecomp

Updated: 19 January 2015

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

LU = PA

This means that L has only zeroes above the diagonal and U has only zeroes below the diagonal.

For a 3 x 3 matrix this becomes: Syntax
SELECT * FROM [wct].[LUdecomp](
<@Matrix_TableName, nvarchar(max),>
,<@Matrix_ColumnNames, nvarchar(4000),>
,<@Matrix_GroupedColumnName, nvarchar(4000),>
,<@Matrix_GroupedColumnValue, sql_variant,>)
Arguments
@Matrix_ TableName
the name, as text, of the table or view that contains the values in the square (N x N) array to be used in the LUdecomp calculation.
@Matrix_ ColumnNames
the name, as text, of the columns in the table or view specified by @Matrix_TableName that contains the array values to be used in the LUdecomp calculation. Data returned from the @Matrix_ColumnNames must be of the type float or of a type that implicitly converts to float.
@Matrix_GroupedColumnName
the name, as text, of the column in the table or view specified by @Matrix_TableName which will be used for grouping the results.
@Matrix_GroupedColumnValue
the column value to do the grouping on.
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 LUdecomp_q function for more complicated queries.
·         Use LUdecompN for a table in third-normal form.
·         Use LU for a matrix stored as a string.
·         Type is either 'L', 'U', or 'P'.
·         The function returns an error if the array contains a non-numeric value.
Examples
Since LUdecomp requires a table name as input, we will have to put our matrix into a temporary table before invoking the table-valued function. Make sure that your table structure guarantees that the input matrix will be returned in the correct order.
SELECT
*
INTO
#m
FROM (
SELECT 0.002,1.231,2.471 UNION ALL
SELECT 1.196,3.165,2.54 UNION ALL
SELECT 1.475,4.271,2.142
) n(a,b,c)
GO
SELECT
*
FROM
wct.LUdecomp('#m','a,b,c','',NULL)

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. Also note that we can use * to select all the columns.
SELECT
*
INTO
#m
FROM (
SELECT 0.002,1.231,2.471 UNION ALL
SELECT 1.196,3.165,2.54 UNION ALL
SELECT 1.475,4.271,2.142
) n(a,b,c)
GO
SELECT
Type,,,
FROM (
SELECT
*
FROM
wct.LUdecomp('#m','*','',NULL)
) d
PIVOT(sum(Value) for ColNum in(,,)) as 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 that input matrix using the calculation P'LU.
SELECT
k.*
FROM (
SELECT
Type as MatrixType,
wct.NMATRIX2STRING(RowNum, ColNum, Value) as Matrix
FROM
wct.LUdecomp('#m','*','',NULL)
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

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