Essay, Research Paper: Modular Arithmetic

Mathematics

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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.
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