PROGRAM SYSTEMS: THEORY AND APPLICATIONS

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Supercomputing Software and Hardware
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Mathematical Foundations of Programming
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• Содержание выпуска •
• Supercomputing Software and Hardware •
• Software and Hardware for Distributed Systems and Supercomputers •
• Mathematical Foundations of Programming •
• Artificial Intelligence, Intelligence Systems, Neural Networks •

Mathematical Foundations of Programming

Responsible for the Section: doctor of physico-mathematical Sciences Nikolay Nepeivoda

On the left: assigned number of the paper, submission date, the number of A5 pages contained in the paper, and the reference to the full-text PDF .

 

Article # 15_2019

47 p.

PDF

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.

article citation

http://psta.psiras.ru/read/psta2019_3_81-127.pdf

 DOI

https://doi.org/10.25209/2079-3316-2019-10-3-81-127

• Содержание выпуска •
• Supercomputing Software and Hardware •
• Software and Hardware for Distributed Systems and Supercomputers •
• Mathematical Foundations of Programming •
• Artificial Intelligence, Intelligence Systems, Neural Networks •

 

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