Solving Systems of Nonlinear Equations on Multi-core Processors

  • Lesia MochuradEmail author
  • Nataliya BoykoEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1080)


The paper proposes an approach to solving multidimensional systems of nonlinear equations based on the use of the OpenMP parallelization mechanism and the multicore architecture of modern computers. The software product, which performs the main function - the parallelization of the numerical solution of multidimensional SNE by the Newton method, is developed. The analysis of the speed and efficiency of calculations with different number of processor cores is carried out. As a result, appropriate estimates of the acceleration and efficiency coefficients were obtained. The proposed method is easily scaled to a different number of processor cores. A number of numerical experiments were conducted. The obtained results also indicate the possibility of further optimization of the computational process by developing the multi-core architecture of modern computers.


Newton’s method OpenMP parallel computing technology Multithreading Finite difference 


  1. 1.
    Krste, A., et al.: The landscape of parallel computing research: a view from Berkeley. University of California, Berkeley. Technical report № UCB/EECS-2006-183, 56 p. (2006)Google Scholar
  2. 2.
    Yakovlev, M.F., Gerasymova, T.O., Nesterenko, A.N.: Characteristic feature of the solving both of non-linear systems and systems of ordinary differential equations on parallel computer. In: Proceedings of International Symposium “Optimization Problems of Computations” (OPC - XXXV). V.M. Glushkov Institute of cybernetics of NAS of Ukraine, Kyiv (2009), vol. 2, pp. 435–439 (2009)Google Scholar
  3. 3.
    Yakovlev, M.V.F., Nesterenko, A.N., Brusnikin, V.N.: Problems of the efficient solving of non-linear systems on multi-processor MIMD-architecture computers. Math. Mach. Syst. 4, 12–17 (2014)Google Scholar
  4. 4.
    Khymych, A.N., Molchanov, Y.N., Popov, A.V., et al.: Parallel Algorithms for Solving Problems of Computational Mathematics, 248 p. Scientific Opinion, Kiev (2008)Google Scholar
  5. 5.
    Autar, K.: Nonlinear equations. In: Newton-Raphson Method-More Examples, Civil Engineering, 7 August, 4 p. (2009)Google Scholar
  6. 6.
    Autar, K.: Nonlinear equations. In: Newton-Raphson Method-More Examples, Electrical Engineering, 7 August, 4 p. (2009)Google Scholar
  7. 7.
    Autar, K.: Nonlinear equations. In: Newton-Raphson Method-More Examples, Mechanical Engineering, 7 August, 3 p. (2009)Google Scholar
  8. 8.
    Kakhaner, D., Mouler, K., Nэsh, S.: Numerical Methods and Software, p. 575. Mir publishing house, Moscow (1998)Google Scholar
  9. 9.
    Mochurad, L.I.: Method of reduction model for calculation of electrostatic fields of electronic optics systems. Sci. J. Radioelektronika Inf. Manage. 1(48), 29–40 (2019). (In Ukrainian)Google Scholar
  10. 10.
    Voss, M.: OpenMP Share Memory Parallel Programming, Toronto, Canada (2003)Google Scholar
  11. 11.
    Chapman, B., Jost, G.: Ruud van der Pas: Using OpenMP: Portable Shared Memory Parallel Programming (Scientific and Engineering Computation). The MIT Press, Cambridge (2008)Google Scholar
  12. 12.
    Chandra, R., Menon, R., Dagum, L., Kohr, D., Maydan, D., McDonald, J.: Parallel Programming in OpenMP. Morgan Kaufinann Publishers, San Francisco (2000)Google Scholar
  13. 13.
    Ananth, G., Anshul, G., George, K., Vipin, K.: Introduction to Parallel Computing, 856 p. Addison Wesley (2003). ISBN-0-201-64865-2Google Scholar
  14. 14.
    Boyko, N.: A look trough methods of intellectual data analysis and their applying in informational systems. In: Scientific and Technical Conference Computer Sciences and Information Technologies (CSIT), 2016 XIth International, pp. 183–185. IEEE (2016)Google Scholar
  15. 15.
    Boyko, N.: Advanced technologies of big data research in distributed information systems. In: Radio Electronics, Computer Science, Control, vol. 4, pp. 66–77. Zaporizhzhya National Technical University, Zaporizhzhya (2017)Google Scholar
  16. 16.
    Boyko, N.: Machine learning on data lake. Monograph, p. 189, LAP Lambert Academic Publishing (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Lviv Polytechnic National UniversityLvivUkraine

Personalised recommendations