Journal of the Korean Operations Research and Management Science Society (한국경영과학회지)

The Korean Operations Research and Management Science Society (한국경영과학회)
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An Algorithm for Computing the Fundamental Matrix of a Markov Chain

  • Park, Jeong-Soo (Department of Statistics, Chonnam N. University) ;
  • Gho, Geon (Department of Business administration, Chonnam N. University)
  • Published : 1997.03.01
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Abstract

A stable algorithm for computing the fundamental matrix (I-Q)$^{-1}$ of a Markov chain is proposed, where Q is a substochastic matrix. The proposed algorithm utilizes the GTH algorithm (Grassmann, Taskar and Heyman, 1985) which is turned out to be stable for finding the steady state distribution of a finite Markov chain. Our algorithm involves no subtractions and therefore loss of significant digits due to concellation is ruled out completely while Gaussian elimination involves subtractions and thus may lead to loss of accuracy due to cancellation. We present numerical evidence to show that our algorithm achieves higher accuracy than the ordinagy Gaussian elimination.

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References

  1. Matrix Computation Golub, G.H.;Van Loan, C.F.
  2. Technical Report 73, Dept. of Operations Research Rounding Errors in Some Recursive Methods Used in Computational Probability Grassmann, W.K.
  3. Operations Research v.33 Regenerative Analysis and Steady State Distributions for Markov Chains Grassmann, W.K.;Taksar, M.I.;Heyman, D.P.
  4. SIAM Journal of Scientific Computing v.5 Comparison of Some Direct Methods for Computing Stationary Distributions of Markov Chains Harrod, W.J.;Plemmons, R.J.
  5. SIAM Journal of Algebraic and Discrete Mathematics v.8 Further Comparisons of Direct Methods for Computing Stationary Distributions of Markov Chains Heyman, D.P.
  6. Finite Markov Chains Kemeny, J.G.;Snell, J.L.
  7. User's Manual(Version 1.0) IMSL, Inc.
  8. Numerische Mathematik v.66 Entrywise Perturbation Theory and Error Analysis for Markov Chains O'Cinneide, C.A.
  9. Journal of Statistical Computation and Simulation v.4 Computation of The Stationary Distribution of A Markov Chain Paige, C.C.;Styan, P.H.;Wachter, P.G.
  10. Non-negative Matrices Seneta, E.