LU decomposition is the most common choice, which is a similar operation to matrix inversion in that you're iterating row (or column) operations on both sides of an equation.  Numerical Recipes has a great section on this, IIRC.<p>But that kind of obscures the fact that, while algorithms like LU decomposition are faster than a pure matrix inversion, they're faster by a constant factor of about 2 or so.  It's not a fundamentally different algorithm, just a different set of tricks to solve a bunch of simultaneous equations.  If you aren't hugely performance constrained and already have a matrix inversion routine, you'd never be bothered to re-write your equation solver just for this speedup.