SQL Server LU Function

Updated 2023-10-19 16:12:06.687000

Description

Use the scalar function LU to calculate the LU factorization of an N x N matrix A using partial pivoting. LU returns a lower triangular matrix L, an upper triangular matrix U, or 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.

LU expects a string representation of the matrix, with columns separated by commas and rows separated by semi-colons.

Syntax

SELECT [westclintech].[wct].[LU](
  <@Matrix, nvarchar(max),>
 ,<@MatrixName, nvarchar(4000),>)

Arguments

@Matrix

a string representation of the matrix.

@MatrixName

a string identifying the name of matrix to be returned; 'L' for the lower triangular matrix, 'U' for the upper triangular matrix, and 'LU' for the combined matrix.

Return Type

nvarchar(max)

Remarks

The number of columns in @Matrix must be equal to the number of rows or an error will be returned.

The string representations of @Matrix must only contain numbers, commas (to separate the columns), and semi-colons to separate the rows.

Consecutive commas will generate an error.

Consecutive semi-colons will generate an error.

Non-numeric data between commas will generate an error

Non-number data between semi-colons will generate an error

To convert non-normalized data to a string format, use the MATRIX2STRING or the MATRIX2STRING_q function.

To convert normalized data to a string format, us the NMATRIX2STRING or the NMATRIX2STRING_q function.

To convert the string result to a table, us the table-valued function MATRIX.

@MatrixName must be either 'L', 'U', or 'P' or an error message will be generated.

Examples

In this example, we return the L matrix from the LU decomposition.

DECLARE @A as varchar(max) = '0.002,1.231,2.471;1.196,3.165,2.54;1.475,4.271,2.142';
SELECT wct.LU(@A, 'L') as L;

This produces the following result.

To get the U matrix, we would have entered:

SELECT wct.LU(@A, 'U') as U;

This produces the following result.

And, to get the P matrix we enter the following:

SELECT
   wct.LU(@A,'P') as P;

returning

In this example we calculate P'LU returning the original matrix.

SELECT wct.MATMULT(wct.MATMULT(wct.TRANSPOSE(wct.LU(@A, 'P')), wct.LU(@A, 'L')), 
          wct.LU(@A, 'U')) as A;

This produces the following result.

We can use the table-valued function MATRIX to convert the string into third-normal form.

SELECT *
FROM wct.MATRIX(wct.MATMULT(wct.MATMULT(wct.TRANSPOSE(wct.LU(@A, 'P')), wct.LU(@A,
          'L')), wct.LU(@A, 'U')));

This produces the following result.

See Also

LUDECOMP - Calculate the LU factorization of an N x N matrix using partial pivoting.

LUDECOMP_Q - Calculate the LU factorization of an N x N matrix using partial pivoting.

LUDECOMPN - Calculate the LU factorization of an N x N matrix using partial pivoting.

LUDECOMPN_Q - Calculate the LU factorization of an N x N matrix using partial pivoting.

QR - QR decomposition