The Covariation Matrix of Solution of a Linear Algebraic System by the Monte Carlo Method

  • Tatiana M. Tovstik
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 231)


A linear algebraic system is solved by the Monte Carlo method generating a vector stochastic series. The expectation of a stochastic series coincides with the Neumann series presenting the solution of a linear algebraic system. An analytical form of the covariation matrix of this series is obtained, and this matrix is used to estimate the exactness of the system solution. The sufficient conditions for the boundedness of the covariation matrix are found. From these conditions, it follows the stochastic stability of the algorithm using the Monte Carlo method. The number of iterations is found, which provides for the given exactness of solution with the large enough probability. The numerical examples for systems of the order 3 and of the order 100 are presented.


Linear algebraic system Monte Carlo method Covariation matrix of solution 



The work is supported by Russian Foundation of Basic Researches, grant 14.01.00271a.


  1. 1.
    Tovstik, T.M.: On the solution of systems of linear algebraic equations by Gibbs’s method. Vestn St.Peterburg Univ.: Math. 44(4), 317–323 (2011)CrossRefGoogle Scholar
  2. 2.
    Winkler, G.: Image analysis, Random Fields and Dynamic Monte Carlo Methods: A Mathematical Introduction. Springer, Berlin (1995)CrossRefGoogle Scholar
  3. 3.
    Ermakov, S.M.: Monte Carlo Method and Close Problems, 327 p. Nauka, Moscow (1971). [In Russian]Google Scholar
  4. 4.
    Belyaeva, A.A., Ermakov, S.M.: On the Monte Carlo method with remembering of the intermediate results. Vestn. St.Petersburg Univ. Ser. 1 3, 8–11 (1996)zbMATHGoogle Scholar
  5. 5.
    Dmitriev, A.V., Ermakov, S.M.: Monte Carlo and method asynchronic iterations. Vestn. St.Petersburg Univ. 44(4), 517–528 (2011)Google Scholar
  6. 6.
    Tovstik T.M., Volosenko K.S.: Monte Carlo algorithm for a solution of a system linear algebraic eqetions by the Zeidel method. In: Proceedings of the Conference on Actual Problems of Computational and Applied Mathematics, Novosibirsk (2015). ISBN 978-5-9905347-2-8. [In Russian]Google Scholar
  7. 7.
    Demidovich, B.P., Maron, I.A.: Foundations of Computation Mathematic, 664 p. Nauka, Moscow (1970). [In Russian]Google Scholar
  8. 8.
    Feller, W.: An Introduction to Prabability Theory and its Applications. Willey, Chapman & Hall, Limited, New York, London (1957)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations