Advertisement

Monte Carlo Algorithms for Calculating Eigenvalues

  • I. T. Dimov
  • A. N. Karaivanova
  • P. I. Yordanova
Part of the Lecture Notes in Statistics book series (LNS, volume 127)

Abstract

A new Monte Carlo approach for evaluating eigenvalues of real symmetric matrices is proposed and studied. Two Monte Carlo Almost Optimal (MAO) algorithms are presented. The first one is called Resolvent Monte Carlo algorithm (RMC) and uses Monte Carlo iterations by the resolvent matrix. The second one is called Inverse Monte Carlo Iterative algorithm (IMCI) and uses the presentation of the smallest eigenvalue by inverse Monte Carlo iterations. Estimators for speedup and for parallel efficiency are introduced. Estimates are obtained for both algorithms under consideration. Various typical models of computer architectures are considered.

Numerical tests are performed for a number of test matrices - general symmetric dense matrices, sparse symmetric matrices (including band symmetric matrices) with different behaviors on the vector machine CRAY Y—MP C92A.

Some information about the vectorization efficiency of the algorithms is obtained, showing that the studied algorithms are well-vectorized.

Keywords

Markov Chain Mathematical Expectation Symmetric Matrice Small Eigenvalue Monte Carlo Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Di91]
    Dimov I. Minimization of the Probable Error for Some Monte Carlo methods. Proc. Int. Conf. on Mathematical Modeling and Scientific Computation, Varna, 1991Google Scholar
  2. [DT89]
    I. Dimov, O. Tonev, Monte Carlo methods withoverconvergent probable error, in Numerical Methods and Applications, Proc. of Intern. Conf on Numerical Methods and Appl.,Sofia Publ. House of Bulg. Acad. Sci Sofia, 1989), pp. 116–120.Google Scholar
  3. [DT93]
    I. Dimov, O. Toney, Random walk on distant mesh points Monte Carlo methods, Journal of Statistical Physics, vol. 70(5/6), 1993, pp. 1333–1342.MathSciNetGoogle Scholar
  4. [DT93a]
    I. Dimov, O. Toney, Monte Carlo algorithms: performance analysis for some computer architectures. Journal of omputational and Applied Mathematics, vol. 48 (1993), pp. 253–277.CrossRefGoogle Scholar
  5. [Du56]
    V. Dupach, Stochasticke pocetni metody, Cas. pro pest. mat. 81, No 1 (1956), pp. 55–68.Google Scholar
  6. [Ka50]
    H. Kahn, Random sampling (Monte Carlo) techniques in eutron attenuation problems, Nucleonics, 6 No 5 (1950), pp. 27–33; 6 No 6 (1950), pp. 60–65.MathSciNetGoogle Scholar
  7. [KA64]
    L.V. Kantorovich, G.P. Akilov, Functional analysis, Nauka, Moscow, 1977.Google Scholar
  8. [MAD94]
    G. Megson, V. Aleksandrov, I. Dimov, Systolic Matrix Inversion Using a Monte Carlo Method, Journal of Parallel Algorithms and Applications, vol.1, No 1 (1994).Google Scholar
  9. [Mi70]
    G. A. Mikhailov A new Monte Carlo algorithm for estimating the maximum eigenvalue of an integral operator, Docl. Acad. Nauk SSSR, 191, No 5 (1970), pp. 993–996.Google Scholar
  10. [Mi87]
    G.A. Mikhailov, Optimization of the “weight” Monte Carlo methods Nauka, Moscow, 1987.Google Scholar
  11. [So73]
    I.M. Sobol, Monte Carlo numerical methods, Nauka, Moscow, 1973.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • I. T. Dimov
    • 1
  • A. N. Karaivanova
    • 1
  • P. I. Yordanova
    • 1
  1. 1.Central Laboratory for Parallel ComputingBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations