I could notice about an ?iterative method? the author gives in order to get samples from a given cdf F(x) (cumulative distribution function) to which an algebraic inversion is impossible to be performed. A method I ordinary use is based on loops (say 3) in each one the steps are got narrower and narrower, for example st0 = .5, st1=0.01*st0, st2=0.01*st1. The procedure goes as: __1__Given a RND=y0 we start from the origin (relative to the loop) and proceed calculating, with step= stk as long that F(x) is less than y0. This value just exceeded (x´´) one pass to the following loop. Except for the first loop the origin is set at x´´- 1.05*stk. __2__The final x´´ is a very close to that make F(x´´)= y0 as I could state by an F-invertible example.
Example A Gambel distribution has _____F(x) = Exp (-Exp (1-x)/2)) From 100000 random numbers we find that circa 0.7 % values the difference from the exact values is less than 0.000010 (!), the exact being obtained from the inversion : x = 1 - 2* Log(-Log(y0)). Note that using a more elaborated language than Basic this exactness could be much better