I want to solve a small (20 $\times$ 20 up to 30$\times$30) system which is not symmetric, not diagonally dominant, and not sparse. Each row contains a modified form of the Legendre coefficient of a set of user specified functions. The difference between between each row is that the functions are raised to a different power (row $i$ has $[f]^i$ and row $i+1$ has $[f]^{i+1}$). Currently the system has Cond(A) ~ $10^{19}$ and for a 20$\times$20 system Rank(A) =15.  
Solving a small non-symmetric, non-diagonally dominant, and non-sparse system
linear algebra;computational physics