5.1.81 lusolve

Syntax:

lusolve ( matrix_expression, matrix_expression, matrix_expression, matrix_expression )

Type:

matrix

Purpose:

Computes all solutions of a linear equation system A*x = b, if solvable
The (m x n matrix A must be given by its LU-decomposition, that is, by three matrices P, L, and U, 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 n) matrix in upper row echelon form.
The fourth argument, b, is expected to be an (m x 1) matrix.

list Q=lusolve(P,L,U,b); fills the list Q with either one entry = 0 (signaling that A*x=b has no solution), or with the three entries 1, x, H, where x is any (n x 1) solution of the given linear system, and H is a matrix the columns of which span the solution space of the homogeneous linear system. (I.e., ncols(H) is the dimension of the solution space.)
If there is exactly one solution, then H is the 1x1 matrix with entry zero.

Note:

The method will give a warning if the matrices violate the above conditions regarding row and column numbers, or if the number of rows of the vector b does not equal m.
The method expects matrices with entries coming from the ground field of the given polynomial ring, only.

Example:

ring r=0,(x),dp; matrix A[4][4]=1,1,1,0,1,2,3,1,1,3,5,2,1,4,7,3; matrix b[4][1]=2,5,8,11; list L=ludecomp(A); list Q=lusolve(L[1],L[2],L[3],b); if (Q[1] == 1) { "one solution:"; print(Q[2]); "check whether result is correct (iff next is zero vector):"; print(A*Q[2]-b); if ((nrows(Q[3])==1) and (ncols(Q[3])==1) and (Q[3][1,1]==0)) { "printed solution is the only solution to given linear system" } else { "homogeneous solution space is spanned by columns of:"; print(Q[3]); } } ==> one solution: ==> -1, ==> 3, ==> 0, ==> 0 ==> check whether result is correct (iff next is zero vector): ==> 0, ==> 0, ==> 0, ==> 0 ==> homogeneous solution space is spanned by columns of: ==> -1,-1, ==> 1, 2, ==> 0, -1, ==> -1,0