Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
Winkler, G.: Image analysis, Random Fields and Dynamic Monte Carlo Methods: A Mathematical Introduction. Springer, Berlin (1995)
Ermakov, S.M.: Monte Carlo Method and Close Problems, 327 p. Nauka, Moscow (1971). [In Russian]
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)
Dmitriev, A.V., Ermakov, S.M.: Monte Carlo and method asynchronic iterations. Vestn. St.Petersburg Univ. 44(4), 517–528 (2011)
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]
Demidovich, B.P., Maron, I.A.: Foundations of Computation Mathematic, 664 p. Nauka, Moscow (1970). [In Russian]
Feller, W.: An Introduction to Prabability Theory and its Applications. Willey, Chapman & Hall, Limited, New York, London (1957)
Acknowledgements
The work is supported by Russian Foundation of Basic Researches, grant 14.01.00271a.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Tovstik, T.M. (2018). The Covariation Matrix of Solution of a Linear Algebraic System by the Monte Carlo Method. In: Pilz, J., Rasch, D., Melas, V., Moder, K. (eds) Statistics and Simulation. IWS 2015. Springer Proceedings in Mathematics & Statistics, vol 231. Springer, Cham. https://doi.org/10.1007/978-3-319-76035-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-76035-3_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-76034-6
Online ISBN: 978-3-319-76035-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)