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Introduction

Improving calculations accuracy is the problem in engineering and scientific calculations in various areas [1]: modelling in physics, chemistry, astronomy etc. Frequent practice problem is a choice between a values representation range and calculations accuracy. It is possible to tell today about correlation problem existence between calculated approximated and true abstract task results. There are two basic approaches solving the given problem:

  1. pointed or approximated approach - approaching by rational numbers is used instead of reals. The IEEE-754 [2] is the standard for floating-point numbers;
  2. interval approach - real number is represented as interval. The interval comprised of this number is represented by its edges in form of rational numbers pair [A, B].

Theme actuality

The question about reliability of computer calculations is repeatedly brought up in the literature. There are researches [7-9] which show that using of modern approaches for reals representing leads to:

Figure 1 (animation) demonstrates severity of these problems.

Actuality demonstration

Figure 1 – Demonstration of the work's actuality

(Animation [56 KB]: 8.56 seconds / 25 frames per second / 214 frames / 2 loops)

Hence, researches within the limits of the correlation problem of calculated approximated result with the true abstract task decision are undoubtedly actual.

The objective and solved problems

The objective of the given work is researching the methods of high precision calculation over the multidigit floating-point numbers. Rational numbers are chosen as the research area.

The problems of this work are:

  1. Mathematical correctness' analysis of the existing rational numbers formats (including IEEE-754);
  2. Research of the methods improving calculations accuracy and corresponding data structures;
  3. Theoretical and experimental researches of the developed methods and formats;
  4. Practical implementation of the developed methods and formats in form of an executable library.

Prospective scientific novelty

The result of the given work's theoretical part will be mathematical fundamentals for the floating-point numbers representation as well as methods improving accuracy and stability of the calculations operations over these numbers.

Expected practical results

The expected practical result is the implementation of the proposed methods in the form of the software library of the rational numbers mathematics.

Review of researches and works of theme

Official standards for rational numbers representing and work with them are [2] and [12]. R. Moore's monography [4] is the fundamental document of the interval calculations.

It is possible to note [1] and [6] from foreign publications, and among the descriptions of the thematic software implementations - [8] and [9]. Also it is possible to note on the CIS territory: on interval calculations – [7], and on pointed calculations – V.M.Jurovitsky's criticising articles [13, 14].

In DonNtU were spent researches neither on the given theme, nor in the area of the rational numbers theory.

Conclusion

On this stage the theme actuality and objective is proved and problems are formulated and the analysis of the chosen subject area is almost finished. Also there has been begun the theoretical analysis of a subject area and article about the theory of rational numbers has been written. This article is prepared for publication in the collection of DonNTU-2010 proceedings.

References

  1. У. Кулиш, Д. Рац, Р. Хаммер, М. Хокс. Достоверные вычисления. Базовые численные методы. Изд-во: «Регулярная и хаотическая динамика», - 1995. – 496 с.
  2. IEEE Standard for Floating-Point Arithmetic (Revision of IEEE Std 754-1985) // IEEE Computer Society. – 2008. – 70 p.
  3. Wilkinson J. H. Modern error analysis // SIAM Rev. — 1971. — Vol. 13, № 4. — P. 548–568.
  4. Moore R. E. Interval analysis. — Englewood Cliffs; Prentice Hall, 1966. — 145 p.
  5. Moore R. E. Methods and applications of interval analysis. — Philadelphia; SIAM, 1979. — xi, 190 p.
  6. Alefeld G., Herzberger J. Introduction to interval computations. — New York etc.; Academic Press, 1983. — XVIII, 333 p.; Рус. перев.; Алефельд Г., Херцбергер Ю. Введение в интервальные вычисления: Пер. с англ. — М.: Мир, 1987. — 356 с.
  7. Добронец Б. С., Шайдуров В. В. Двусторонние численные методы. — Новосибирск; Наука, 1990. — 208 с.
  8. Yohe J. M. Portable software for interval arithmetic // Fundamentals of numerical computation (computer-oriented numerical analysis) / Ed.: G. Alefeld, R. D. Grigorieff. — Wien etc.: Springer-Verlag, 1980. — (Computing; Suppl. 2). — P. 211–229.
  9. Klatte R., Kulisch U., Wiethoff A., Lawo C., Rauch M. C-XSC. A C++ class library for extended scientific computing. — Berlin etc.: Springer Verlag, 1993. — 270 p.
  10. Nickel K. Can we trust the results of our computing? // Mathematics for Computer Science; Proc. Symposium held in Paris, March 16–18, 1982. — S. l.; Association francaise pour la cybernetique et technique (AFCET), 1982. — P. 167–175
  11. Верещагин Н.К., Шень А. Математическая логика и теория алгоритмов: Часть 1. Начала теории множеств. – М.: Изд-во МЦНМО, - 2008. – 128 с.
  12. DRAFT Standard for Floating-Point Arithmetic P754 // IEEE Computer Society. – 2006. – 64 p.
  13. Юровицкий В.М. IEEE754-тика угрожает человечеству [электронный ресурс]: http://www.yur.ru/science/computer/IEEE754.htm
  14. Юровицкий В.М. Компьютерные числа − угроза человечеству // Млечный Путь. Сверхновый литературный журнал [электронный ресурс]: http://milkywaycenter.com
  15. Edmond / HI-TECH. FPU посвящается (часть 1) [электронный ресурс]: wasm.ru/print.php?article=edfpu01