Hybrid Monte Carlo Methods for Matrix Computation

  • Vassil Alexandrov
  • Bo Liu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2542)


In this paper we consider hybrid (fast stochastic approximation and deterministic refinement) algorithms for Matrix Inversion (MI) and Solving Systems of Linear Equations (SLAE). Monte Carlo methods are used for the stochastic approximation, since it is known that they are very efficient in finding a quick rough approximation of the element or a row of the inverse matrix or finding a component of the solution vector. We show how the stochastic approximation of the MI can be combined with a deterministic refinement procedure to obtain MI with the required precision and further solve the SLAE using MI. We employ a splitting A = D - C of a given non-singular matrix A, where D is a diagonal dominant matrix and matrix C is a diagonal matrix. In our algorithm for solving SLAE and MI different choices of D can be considered in order to control the norm of matrix T = D-1C, of the resulting SLAE and to minimize the number of the Markov Chains required to reach given precision. Experimental results with dense and sparse matrices are presented.


Monte Carlo Method Markov Chain Matrix Inversion Solution of sytem of Linear Equations Matrix Decomposition Diagonal Dominant Matrices SPAI 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fathi Vajargah, B., Liu, B., and Alexandrov, V.: On the preconditioned Monte Carlo methods for solving linear systems. MCM 2001, Salzburg, Austria (presented).Google Scholar
  2. 2.
    Ortega, J.: Numerical Analysis. SIAM edition, USA (1990)Google Scholar
  3. 3.
    Alexandrov V.N.: Efficient parallel Monte Carlo Methods for Matrix Computation. Mathematics and computers in Simulation, Elsevier, Netherlands, Vol. 47 (1998) 113–122.CrossRefMathSciNetGoogle Scholar
  4. 4.
    Golub, G.H., Ch., F., Van Loan, Matrix Computations, The Johns Hopkins Univ. Press, Baltimore and London, (1996)Google Scholar
  5. 5.
    Taft, K. and Fathi Vajargah, B.: Monte Carlo Method for Solving Systems of Linear lgebraic Equations with Minimum Markov Chains. International Conference PDPTA’2000 Las Vegas (2000)Google Scholar
  6. 6.
    Sobol, I.M.: Monte Carlo Numerical Methods. Moscow, Nauka (1973) (in Russian)Google Scholar
  7. 7.
    Dimov, I., Alexandrov, V.N., and Karaivanova, A.: ResolventMonte Carlo Methods for Linear Algebra Problems. Mathematics and Computers in Simulation, Vol. 155 (2001) 25–36.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Fathi Vajargah, B. and Alexandrov, V.N.: Coarse Grained Parallel Monte Carlo Algorithms for Solving Systems of Linear Equations with Minimum Communication. In: Proc. of PDPTA, June 2001, Las Vegas (2001) 2240–2245.Google Scholar
  9. 9.
    Alexandrov, V.N. and Karaivanova, A.: Parallel Monte Carlo Algorithms for Sparse SLAE using MPI. Lecture Notes in Computer Science, Vol. 1697, Springer-Verlag (1999) 283–290.Google Scholar
  10. 10.
    Alexandrov, V.N., Rau-Chaplin, A., Dehne, F., and Taft, K.: Efficient Coarse Grain Monte Carlo Algorithms for matrix computation using PVM. Lecture Notes in Computer Science, Vol. 1497, Springer-Verlag (1998) 323–330.Google Scholar
  11. 11.
    Dimov, I.T., Dimov, T.T., Gurov, T.V.: A new iterative Monte Carlo Approach for Inverse Matrix Problem. J. of Computational and Applied Mathematics, 92 (1998) 15–35.zbMATHMathSciNetGoogle Scholar
  12. 12.
    Fathi Vajargah, B., Bo Liu, and Alexandrov, V.N.: Mixed Monte Carlo Parallel Algorithms for Matrix Computation. Lecture Notes in Computer Science, Vol. 2330, Springer-Verlag (2002) 609–618.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Vassil Alexandrov
    • 1
  • Bo Liu
    • 1
  1. 1.Department of Computer ScienceThe University of ReadingWhiteknightsUK

Personalised recommendations