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5.1.85 luinverse

qcindex Gauss

Syntax:
luinverse ( matrix_expression )
Type:
matrix
Syntax:
luinverse ( matrix_expression, matrix_expression, matrix_expression )
Type:
matrix
Purpose:
Computes the inverse of a matrix A, if A is invertible.

The matrix A must be given either directly, or by its LU-decomposition. In the latter case, three matrices P, L, and U are expected, in this order, which satisfy
- 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 m) matrix in upper row echelon form.
Then, the inverse of A exists if and only if U is invertible, and one has $A^{-1}=U^{-1}\cdot L^{-1}\cdot P$,since P is self-inverse.
In the case of A being given directly, luinverse first computes its LU-decomposition, and then proceeds as in the case when P, L, and U are provided.

list L=luinverse(A); fills the list L with either one entry = 0 (signaling that A is not invertible), or with the two entries $1, A^{-1}$.Thus, in either case the user may first check the condition L[1]==1 to find out whether A is invertible.

Note:
The method will give a warning for any non-quadratic matrix A.

Example:
 
  ring r=0,(x),dp;
  matrix A[3][3]=1,2,3,1,1,1,2,2,1;
  list L = luinverse(A);
  if (L[1] == 1)
  {
    print(L[2]);
    "----- next should be the (3 x 3)-unit matrix:";
    print(A*L[2]);
  }
==> -1,4, -1,
==> 1, -5,2, 
==> 0, 2, -1 
==> ----- next should be the (3 x 3)-unit matrix:
==> 1,0,0,
==> 0,1,0,
==> 0,0,1
See ludecomp; lusolve.