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,[0],[1],[2]
FROM (
SELECT
*
FROM
wct.LUdecomp('#m','*','',NULL)
) d
PIVOT(sum(Value) for ColNum in([0],[1],[2])) 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
See Also