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5.1.79 ludecomp
Syntax:ludecomp (matrix_expression)Type:- list
Purpose:- Computes the LU-decomposition of an (m x n) matrix.
The matrix, A say, must consist of numbers, only. This means that when the basering represents some
![$K[x_1,x_2,\ldots,x_r]$](sing_150.png)
,then all entries of A must come from the ground field K.
The LU-decomposition of A is a triple of matrices P, L, and U such that
- P * A = L * U,
- P is an (m x m) permutation matrix, i.e., its rows/columns form the standard basis of K^m,
- L is an (m x m) matrix in lower triangular form with all diagonal entries equal to 1, and
- U is an (m x n) matrix in upper row echelon form.
From these conditions, it easily follows that also A = P * L * U holds, since P is self-inverse.list L=ludecomp(A);fills a list L with the three above entries P, L, and U. Example:ring r=0,(x),dp; matrix A[3][4]=1,2,3,4,1,1,1,1,2,2,1,1; list plu = ludecomp(A); print(plu[3]); // the matrix U of the decomposition ==> 1,2, 3, 4, ==> 0,-1,-2,-3, ==> 0,0, -1,-1 print(plu[1]*A-plu[2]*plu[3]); // should be the zero matrix ==> 0,0,0,0, ==> 0,0,0,0, ==> 0,0,0,0