Preliminary draft: http://www.eecs.berkeley.edu/~prasad/linsystems.pdf

--I don't understand why anybody would want to use this algorithm for any purpose. It's slower than everything else I can think of and also randomized i.e. can fail. However... perhaps it might be useful if changed to be about finite RINGS not fields. In rings, you cannot divide, hence gaussian elimination is not usable. Hence this algorithm then has less competition. (Even then I'm dubious, but then it'd have at least a hope.)