Pseudo-random number generation algorithms What algorithms are used in modern and good-quality random number generators? 
 A: The Mersenne Twister is one I've come across and used before now.
A: The Xorshift PNG designed by George Marsaglia. Its period (2^128-1) is much shorter than the Mersenne-Twister but the algorithm is very simple to implement and lends itself to parallelization. Performs well on many-core architectures such as DSP chips and Nvidia's Tesla.
A: At http://prng.di.unimi.it/ you can find a shootout of several random number generators tested using TestU01, the modern test suite for pseudorandom number generators that replaced diehard and dieharder. You can pick and choose.
A: In R, the default setting for random number generation are:


*

*For U(0,1), use the Mersenne-Twister algorithm

*For Guassian numbers use  the numerical inversion of the standard normal distribution function. 


You can easily check this, viz.
> RNGkind()
[1] "Mersenne-Twister" "Inversion"

It is possible to change the default generator to other PRNGs, such as Super-Duper,Wichmann-Hill, Marsaglia-Multicarry or even a user-supplied PRNG. See the ?RNGkind for further details. I have never needed to change the default PRNG.
The C GSL library also uses the Mersenne-Twister by default.
