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:
“After which you led me to water
And bade me drink, which I did, owing to your kindness.
You would not let me out for two days and three nights,
Bringing me books bound in wild thyme and scented wild grasses
As if reading had any interest for me ...”
—John Ashbery (b. 1927)
“Affection, indulgence, and humor alike are powerless against the instinct of children to rebel. It is essential to their minds and their wills as exercise is to their bodies. If they have no reasons, they will invent them, like nations bound on war. It is hard to imagine families limp enough always to be at peace. Wherever there is character there will be conflict. The best that children and parents can hope for is that the wounds of their conflict may not be too deep or too lasting.”
—New York State Division of Youth Newsletter (20th century)