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The Algorithmic Numbers in Non-Archimedean Numerical Computing Environments
There are many natural phenomena that can best be described by the
use of infinitesimal and infinite numbers (see e.g. [1, 5, 13, 23]. However,
until now, the Non-standard techniques have been applied to theoretical
models. In this paper we investigate the possibility to implement such
models in numerical simulations. First we define the field of Euclidean
numbers which is a particular eld of hyperreal numbers. Then, we introduce
a set of families of Euclidean numbers, that we have called altogether
algorithmic numbers, some of which are inspired by the IEEE 754 standard
for floating point numbers. In particular, we suggest three formats which
are relevant from the hardware implementation point of view:
the Polynomial Algorithmic Numbers, the Bounded Algorithmic Numbers and the
Truncated Algorithmic Numbers. In the second part of the paper, we show
a few applications of such numbers
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