### 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 ,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 `system("rref",A);`. See luinverse; lusolve; system.

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