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5.1.77 lusolve
Syntax:
lusolve ( matrix_expression, matrix_expression,
matrix_expression, matrix_expression )
Type:
- matrix
Purpose:
- Computes a solution 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, i.e., a vector.
list L=lusolve(P,L,U,b); fills the list L with either one entry = 0
(signaling that A*x=b has no solution), or with the three entries 1, x, d,
where x is any (n x 1) solution and d the dimension of the affine solution
space.
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.
Example:
| ring r=0,(x),dp;
matrix A[4][3]=1,1,3,2,1,4,3,0,3,4,0,4;
matrix b[4][1]=5,7,6,8;
list L=ludecomp(A);
list Q=lusolve(L[1],L[2],L[3],b);
if (Q[1] == 1)
{
print(Q[2]);
"----- next should be the zero vector:";
print(A*Q[2]-b);
"solution space has dimension", Q[3];
}
==> 2,
==> 3,
==> 0
==> ----- next should be the zero vector:
==> 0,
==> 0,
==> 0,
==> 0
==> solution space has dimension 1
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See
ludecomp.
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