Summary
The standard perturbation theory for linear equations states that nearly uncoupled Markov chains (NUMCs) are very sensitive to small changes in the elements. Indeed, some algorithms, such as standard Gaussian elimination, will obtain poor results for such problems. A structured perturbation theory is given that shows that NUMCs usually lead to well conditioned problems. It is shown that with appropriate stopping, criteria, iterative aggregation/disaggregation algorithms will achieve these structured error bounds. A variant of Gaussian elimination due to Grassman, Taksar and Heyman was recently shown by O'Cinneide to achieve such bounds.
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References
Barlow, J.L. (1986): On the smallest positive singular value of a singular M-matrix with applications to ergodic Markov chains. SIAM J. Alg. Dis. Methods7, 414–424
Barlow, J.L. (1991): Error bounds and condition estimates for the computation of null vectors with applications to Markov chains. Technical Report, CS-91-20, The Pennsylvania State University, Department of Computer Science, University Park, PA. SIAM J. Matrix Anal. Appl.
Funderlic, R.E., Meyer, Jr. C.D. (1986): Sensitivity of the stationary distribution for an ergodic Markov chain. Linear Algebra Appl.76, 1–17
Geurts, A.J. (1982): A contribution to the theory of condition. Numer. Math.39, 85–96
Grassman, W.K., Taksar, M.I., Heyman, D.P. (1985): Regenerative analysis and steady state distribution for Markov chains. Oper. Res.33, 1107–1116
McAllister, D.F., Stewart, G.W., Stewart, W.J. (1984): On a Rayleigh-Ritz refinement technique for nearly uncoupled stochastic matrices. Linear Algebra Appl.60, 1–25
Meyer, Jr. C.D. (1975): The role of the group inverse in the theory of finite Markov chains. SIAM Review17, 443–464
Meyer, Jr. C.D., Stewart, G.W. (1988): Derivatives and perturbations of eigenvectors. SIAM J. Numer. Anal.25, 679–691
O'Cinneide, C.A. (1992): Error analysis of a variant of Gaussian elimination for steady-state distributions of Markov chains. Technical report, Purdue University, West Lafayette
Schweitzer, P.J. (1988): Perturbation theory and finite Markov chains. J. Appl. Probab.,5, 401–413
Stewart, G.W. (1971): Error bounds for approximate invariant subspaces for close linear operators. SIAM J. Numer. Anal.8, 796–808
Stewart, G.W., Stewart, W.J., McAllister, D.F. (1984): A two-stage iteration for solving nearly uncoupled Markov chains. Technical Report CSC TR-138, Department of Computer Science, University of Maryland, College Park
Stewart, G.W., Zhang, G. (1991): On a direct method for the solution of nearly uncoupled Markov chains. Numer. Math.59, 1–11
Zhang, G. (1991): On the sensitivity of the solution of nearly uncoupled Markov chains. Technical Report UMIACS TR 91-18, Institute for Advanced Computer Studies, University of Maryland
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Supported by the National Science Foundation under grant CCR-9000526 and its renewal, grant CCR-9201692. This research was done in part, during the author's visit to the Institute for Mathematics and its Applications, 514 Vincent Hall, 206 Church St. S.E., University of Minnesota, Minneapolis, MN 55455, USA
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Barlow, J.L. Perturbation results for nearly uncoupled Markov chains with applications to iterative methods. Numer. Math. 65, 51–62 (1993). https://doi.org/10.1007/BF01385739
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DOI: https://doi.org/10.1007/BF01385739