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5.1.79 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, 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, 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:
See
ludecomp.
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