Advancement of a long arithmetic technology in the construction of algorithms for studying linear systems

Authors

DOI:

https://doi.org/10.15587/1729-4061.2019.157521

Keywords:

method of basic matrices, precise calculations, ill-conditioned system of linear equations

Abstract

We have advanced the application of algorithms within a method of basic matrices, which are equipped with the technology of long arithmetic to improve the precision of performing the basic operations in the course of studying the ill-conditioned linear systems, specifically, the systems of linear algebraic equations (SLAE). Identification of the fact of ill-conditionality of a system is a rather time-consuming computational procedure. The possibility to control computations entering the state of incorrectness and the impossibility of accumulating calculation errors, which is a desirable property of the methods and algorithms for solving practical problems, were introduced.

Modern computers typically use the standard types of integers whose size does not exceed 64 bytes. This hardware limitation was resolved using software, specifically, by developing a proprietary type of data in the form of a special Longnum library in the C++ language (using the STL (Standard Template Library)). Software implementation was aimed at carrying out computations for methods of basic matrices (MBM) and Gauss matrices, that is, long arithmetic for models with rational elements was used. We have proposed the algorithms and computer realization of the Gauss type methods and methods of artificial basic matrices (a variant of the method of basic matrices) in MatLAB environment and Visual C++ environment using precise computation of the methods' elements, first of all, for the ill-conditioned systems of varying dimensionality. The Longnum library with the types of long integers (longint3) and rational numbers (longrat3) with the numerator and denominator of the longint3 type was developed. Arithmetic operations on long integers were performed based on the modern methods, including the Strassen multiplication method. We give the results from the computational experiment employing the mentioned methods, in which test models of the systems were generated, specifically, based on the Gilbert matrices of different dimensionality

Author Biographies

Volodymyr Kudin, Taras Shevchenko National University of Kyiv Volodymyrska str., 60, Kyiv, Ukraine, 01033

Doctor of Technical Sciences, Professor

Department of Intelligent and Information Systems

Viacheslav Onotskyi, Taras Shevchenko National University of Kyiv Volodymyrska str., 60, Kyiv, Ukraine, 01033

PhD, Assistant

Department of Computational Mathematics

Ali Al-Ammouri, National Transport University Mykhailа Omelianovycha-Pavlenka str., 1, Kyiv, Ukraine, 01010

Doctor of Technical Sciences, Professor

Department of Information Analysis and Information Security

Lyudmyla Shkvarchuk, Lviv Polytechnic National University S. Bandery str., 12, Lviv, Ukraine, 79013

Doctor of Еconomic Sciences, Professor

Department of Finance

References

  1. Kahaner, D., Mouler, K., Nesh, S. (2001). Chislennye metody i programmnoe obespechenie. Moscow: Mir, 575.
  2. Demmel', Dzh. (2001). Vychislitel'naya lineynaya algebra. Teoriya i prilozhenie. Moscow: Mir, 430.
  3. Han, D., Zhang, J. (2007). A comparison of two algorithms for predicting the condition number. Sixth International Conference on Machine Learning and Applications (ICMLA 2007). doi: https://doi.org/10.1109/icmla.2007.8
  4. Ebrahimian, R., Baldick, R. (2001). State Estimator Condition Number Analysis. IEEE Power Engineering Review, 21 (5), 64–64. doi: https://doi.org/10.1109/mper.2001.4311389
  5. Nishi, T., Rump, S., Oishi, S. (2013). A consideration on the condition number of extremely ill-conditioned matrices. 2013 European Conference on Circuit Theory and Design (ECCTD). doi: https://doi.org/10.1109/ecctd.2013.6662260
  6. BLAS (Basic Linear Algebra Subprograms). Available at: http://www.netlib.org/blas/sblat1
  7. Li, H., Yang, H., Shao, H. (2010). A note on the perturbation analysis for the generalized Cholesky factorization. Applied Mathematics and Computation, 215 (11), 4022–4027. doi: https://doi.org/10.1016/j.amc.2009.12.009
  8. The GNU Multiple Precision Arithmetic Library. Available at: https://gmplib.org/
  9. Multiple Precision Integer Library (MPI). Available at: https://github.com/servo/nss/tree/master/lib/freebl/mpi
  10. OpenSSL Cryptographic Toolkit. Available at: http://openssl.org
  11. Large Integer Package. Available at: https://github.com/luckyaibin/BigInt/tree/master/freelip
  12. Denis, T., Rose, G. (2006). BigNum Math. Implementing Cryptographic Multiple Precision Arithmetic. Syngress, 291. doi: https://doi.org/10.1016/b978-1-59749-112-9.x5000-x
  13. Galovic, Ya. (2018). C++17 STL. Standartnaya biblioteka shablonov. Sankt-Peterburg: Piter, 432.
  14. Kudin, V. I., Lyashko, S. I., Hritonenko, N. M., Yacenko, Yu. P. (2007). Analiz svoystv lineynoy sistemy metodom iskusstvennyh bazisnih matric. Kibernetika i sistemniy analiz, 4, 119–127.
  15. Bohaienko, V. O., Kudin, V. I., Skopetskyj, V. V. (2009). Analysis of computational schemes for basic matrix method. Komp'yuternaya matematika, 2, 3–13.
  16. Bogaenko, V. A., Kudin, V. I., Skopeckiy, V. V. (2009). Analiz vychislitel'nyh skhem modelirovaniya processov geogidrodinamiki. Probl. upr. i informatiki, 4, 62–72.
  17. Bogaenko, V. A., Kudin, V. I., Skopeckiy, V. V. (2012). Ob osobennostyah organizacii vychisleniy na osnove metoda bazisnyh matric. Kibernetika i sistemniy analiz, 48 (4), 146–155.
  18. Bogainenko, V., Kudin, V. (2014). Building preconditioners using basis matrix method. International journal Information Content and Processing, 1 (2), 182–187.
  19. Knut, D. (2000). Iskusstvo programmirovaniya. Vol. 2. Moscow: Izdatel'skiy dom «Vil'yams», 788.
  20. Krendall, R., Pomerans, K. (2011). Prostye chisla: Kriptograficheskie i vychislitel'nye aspekty. Moscow: URSS, 664.
  21. Straustrup, B. (2006). Yazyk programmirovaniya C++. Special'noe izdanie. Sankt-Peterburg-Moscow: «Nevskiy dialekt» - «BINOM», 1104.
  22. Kudin, V. I., Onotskyi, V. V. (2011). Rozvynennia tekhnolohiyi dovhoi aryfmetyky pry pobudovi alhorytmiv doslidzhennia zadachi liniynoho prohramuvannia. Zhurnal obchysliuvalnoi ta prykladnoi matematyky, 1, 77–84.

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Published

2019-02-26

How to Cite

Kudin, V., Onotskyi, V., Al-Ammouri, A., & Shkvarchuk, L. (2019). Advancement of a long arithmetic technology in the construction of algorithms for studying linear systems. Eastern-European Journal of Enterprise Technologies, 1(4), 14–22. https://doi.org/10.15587/1729-4061.2019.157521

Issue

Vol. 1 No. 4 (2019): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

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Copyright (c) 2019 Volodymyr Kudin, Viacheslav Onotskyi, Ali Al-Ammouri, Lyudmyla Shkvarchuk

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