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5.1.84 ludecomp

qcindex Gauss

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 $K[x_1,x_2,\ldots,x_r]$,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
See luinverse; lusolve.