There's an neat technique for fitting RBF representations to dense point clouds. Instead of the impractical matrix inversion, the trick is to solve the linear system with a fast multipole method, similar to the Barnes-Hut algorithm for the n-body problem.

Carr, J.C. et al, "Reconstruction and Representation of 3D Objects with Radial Basis Functions", Siggraph 2001

http://dl.acm.org/citation.cfm?id=383266

Another relevant difference is that precision is not nearly as critical in games as in most numerical applications. Errors of a few percent will likely go unnoticed. Graphics implementations therefore usually only use single precision.

Ah, that's the context I was missing, thanks!

In games we can usually just transpose the inner 3x3 R+S, invert scale and handle translation similarly to get a semi-quick inverse.

This is the difference between matrices that are SO(3) (which have trivial inverses) and matrices that are just GL(n) (which: don't).

This. Or you know how to calculate the closed-form inverse a priori like with projection matrices (e.g. gluUnProject).