FREE ESSAY ON MODULAR ARITHMETICS

MODULAR ARITHMETICS

MODULAR ARITHMETIC:
Modular arithmetic can be used to compute
exactly, at low cost, a set of simple computations.
These include most geometric predicates, that
need to be checked exactly, and especially, the
sign of determinants and more general polynomial
expressions. 
Modular arithmetic resides on the Chinese
Remainder Theorem, which states that, when
computing an integer expression, you only have to
compute it modulo several relatively prime integers
called the modulis. The true integer value can then
be deduced, but also only its sign, in a simple and
efficient maner. 
The main drawback with modular arithmetic is its
static nature, because we need to have a bound on
the result to be sure that we preserve ourselves
from overflows (that can't be detected easily
while computing). The smaller this known bound is,
the less computations we have to do. 
We have developped a set of efficient tools to deal
with these problems, and we propose a filtered
approach, that is, an approximate computation
using floating point arithmetic, followed, in the bad
case, by a modular computation of the expression
of which we know a bound, thanks to the floating
point computation we have just done. Theoretical
work has been done in common with , , Victor Pan
and. See the bibliography for details. 
At the moment, only the tools to compute without
filters are available. The aim is now to build a
compiler, that produces exact geometric
predicates with the following scheme: filter +
modular computation. This approach is not
compulsory optimal in all cases, but it has the
advantage of simpleness in most geometric tests,
because it's general enough. 
Concerning the implementation, the Modular
Package contains routines to compute sign of
determinants and polynomial expressions, using
modular arithmetic. It is already usable, to
compute signs of determinants, in any dimension,
with integer entries of less than 53 bits. In the
near future, we plan to add a floating point filter
before the modular computation. 
Bibliography
Explains basically the definition of modular arithmetic, and contents of it.