RLaB is not a contender for fastest speed, but it's definitely the tiniest linear algebra app out there. It weighs in at under 1.5MB.
Makes a wonderful pop-up for that quick calculation, or for high-school students just learning linear algebra.
Very easy to use... and
free.
Welcome to RLaB. New users type `help INTRO'
RLaB version 2.1.05 Copyright (C) 1992-97 Ian Searle
This is free software, and you are welcome to redistribute it under
certain conditions; type `help conditions' for details
> tic(); N=readm("N.dat"); toc()
3.66
> d=readm("d.dat");
> tic(); x=N\d; toc()
28.5
> tic(); x=solve(N,d,"S"); toc()
14.5
>
The second solution method (x=solve(N,d,"S")) tells RLaB that the matrix is symmetric, so it uses LAPACK subroutine DSYTRF (Bunch-Kaufman diagonal pivoting) to solve, which cuts the time in half.