How Computers Generate Random Numbers - Chapter 6
As stated earlier, for a linear congruential random number generator,
a 2-D plot will show that all the points lie on parallel lines, and a
3-D plot will show that all the points lie on parallel planes. In
general, an n-D plot would show that all the points lie on (n-1)
One test of a linear congruential random number generator is to see how
far apart these parallel planes are for each dimension. The spectral
test calculates the distances between these parallel planes for a
linear congruential random number generator given the modulus M and the
multiplier A of the generator. A good random number generator would
need to have the planes closely spaced for each dimension so that the
individual planes do not become apparent early on. RANDU, having only
16 widely spaced parallel planes in the 3-D unit cube, fails the
spectral test quite horribly.
We will not discuss further the spectral test here. Further information
on the spectral test can be found in the book by Knuth.