Form: $x_i+x_j-x_k-C_i=0$
To clarify, it might have been better to write this here as: $x_i+x_j-y_k-C_i=0$, since $x_i$ and $x_j$ come from a single set of unknowns. $y_k$ are considered different.
All are constants (but dependent mostly) but only $C_i$ are known. The problem is, it's not in matrix form. I have a set of equations, n in number, with all the different $x_k$ being exactly n in number, while $x_i$ and $x_j$ added together are fewer than n in number. I want to elucidate the dependent variables, but doing it by hand takes too long. How to express this as input to a (preferably free) pc program, to get solution e.g., as an upper triangular matrix?
Update: This seems easy at the moment (although I haven't tested yet whether this is correct), but the format of the problem made getting here difficult: I need to make a mostly sparse matrix with +/-1's for the three types of unknowns. One side would be as long as the sum of their indices, an additional column for $C_i$, and the number of rows equal to the number of equations. If this is correct, then what remains is either to find a text parser to convert plain text equations to matrix form, or to just fill in the matrix by hand.
$A$, in sparse format (row, col, value):
1 1 1 2 1 1 3 2 1 4 3 1 5 3 1 6 4 1 7 4 1 8 4 1 9 5 1 10 5 1 11 6 1 12 6 1 13 7 1 14 9 1 15 9 1 16 10 1 17 11 1 18 12 1 19 12 1 20 13 1 21 13 1 22 14 1 23 14 1 24 14 1 25 15 1 26 15 1 27 16 1 28 17 1 29 17 1 30 17 1 31 18 1 32 19 1 33 20 1 34 20 1
Coefficients:
30 27 26 26 24 25 25 20 17 21 13 14 17 18 17 13 14 13 12 12 11 6 2 3 3 2 4 2 3 0 2 1 4 4