Propagation algorithm for computer simulation of stationary paraxial, self-
focusing laser beam in a medium with saturating nonlinearity is given in Li
e-optic form. Accordingly, a very natural piece-wise continuous Lie transfo
rmation that reduces to a restricted Lorentz group of the beam results. It
gives rise to a matrix method for self-focusing beam propagation that is co
nstructed and implemented. Although the results use plasma nonlinearities o
f saturable type, and a gaussian initial beam, these results are applicable
for other media like linear optical fibers and to more general situations.