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Article #
15_2019
47
p.
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submitted on 21th
Feb 2019 displayed on
website on 30th
Sept
2019
Anastasia
Korzhavina, Vladimir Knyazkov
High-precision computations using residue-interval arithmetic on
FPGAs
The problem of round-off errors arises in a large number of
issues in various fields of knowledge, including computational
mathematics, mathematical physics, biochemistry, quantum mechanics,
mathematical programming. Today, experts place particular emphasis
on accuracy, fault tolerance, stability, and reproducibility of
computation results of numerical models when solving a wide range of
industrial and scientific problems, such as: mathematical modeling
and structural designs of aircrafts, cars, ships; process modeling
and computations for solving large-scale problems in the field of
nuclear physics, aerodynamics, gas, and hydrodynamics; problems on
reliable predictive modeling of climatic processes and forecasting
of global changes in the atmosphere and water environments; faithful
modeling of chemical processes and synthesis of pharmaceuticals,
etc. Floating-point arithmetic is the dominant choice for most
scientific applications. However, there are a lot of unsolvable with
double-precision arithmetic problems. A vast number of
floating-point arithmetic operations would be required to solve such
problems. Each operation carries round-off errors leading to
uncontrolled round-off errors and, consequently, incorrect results.
Many modeling and simulation problems need to increase the accuracy
of number representation to 100-1000 decimal digits or more to
obtain reliable results. In this regard, arbitrary-precision
arithmetic is becoming ever important. With this arithmetic, one can
use numbers, whose arbitrary precision is many times greater than
the word length of the conventional system. The paper proposes a new
way of representing integers and floats for computations in
super-large ranges – hybrid residue-positional interval logarithmic
number representation for performing high-precision and reliable
calculations in super-large numerical ranges.
(in Russian).
Key words: residue arithmetic, hybrid number systems, the
interval logarithmic number evaluation, high-precision computations,
long numbers.
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