Lower Bound
The discrepancy of arbitrary points satisfies
- ≥
for any nonzero lattice point, where denotes the number of nonzero coordinates of .
These two theorems show that the CIG is not perfect because the discrepancy is greater strictly than a positive value but also the CIG is not the worst generator as the discrepancy is lower than a value less than 1.
There exist also theorems which bound the average value of the discrepancy for Compound Inversive Generators and also ones which take values such that the discrepancy is bounded by some value depending on the parameters. For more details see the original paper.
Read more about this topic: Inversive Congruential Generator
Famous quotes containing the word bound:
“I wish either my father or my mother, or indeed both of them, as they were in duty both equally bound to it, had minded what they were about when they begot me.”
—Laurence Sterne (1713–1768)
“The roaring alongside he takes for granted,
and that every so often the world is bound to shake.
He runs, he runs to the south, finical, awkward,
in a state of controlled panic, a student of Blake.”
—Elizabeth Bishop (1911–1979)