On optimal condition numbers for markov chains

Numerische Mathematik

Volumn 110, Issue 4, 2008, Pages 521-537

  Kirkland, Stephen J ^a   Neumann, Michael ^b   Sze, Nung Sing ^b  

^a UNIVERSITY OF REGINA   (Canada)

^b UNIVERSITY OF CONNECTICUT   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 52949100515     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-008-0172-8     Document Type: Article

References (33)

1. K. Anstreicher U. Rothblum 1987 Using Gauss-Jordan elimination to compute the index, generalized nullspace, and Drazin inverse Lin. Alg. Appl. 85 221 239
2. Ben-Israel A., Greville T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)
3. Berman A., Plemmons R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM Publications, Philadelphia (1994)
4. Campbell S.L., Meyer C.D.: Generalized Inverses of Linear Transformations. Dover Publications, New York (1991)
5. G.E. Cho C.D. Meyer 2000 Markov chain sensitivity measured by mean first passage times Lin. Alg. Appl. 316 21 28
6. G.E. Cho C.D. Meyer 2001 Comparison of perturbation bounds for the stationary distribution of a Markov chain Lin. Alg. Appl. 335 137 150
7. Dobrushin, R.L.: Central limit theorem for non-stationary Markov chains, I, II. Theory Probab. Appl. 1, 65-80, 329-383 (1956)
8. Feller W.: An Introduction to Probability Theory and its Applications, vol. I. Wiley, New York (1960)
9. R. Funderlic C. Meyer 1986 Sensitivity of the stationary distribution vector for an ergodic Markov chain Lin. Alg. Appl. 76 1 17
10. G.H. Golub C.D. Meyer 1986 Using the QR-factorization and group inversion to compute, differentiate, and estimate the sensitivity of stationary probabilities for Markov chains SIAM J. Alg. Discrete Meth. 7 273 281
11. D.J. Hartfiel C.D. Meyer 1998 On the structure of stochastic matrices with a subdominant eigenvalue near 1 Lin. Alg. Appl. 272 193 203
12. d Math. Japon. 26 37 43
13. M. Haviv L. Vander Heyden 1984 Perturbation bounds for the stationary probabilities of a finite Markov chain Adv. Appl. Prob. 16 804 818
14. N.J. Higham P.A. Night 1993 Componentwise error analysis for stationary iterative methods, in Linear algebra, Markov chains, and queueing models Inst. Math. Appl. Minnieap. 48 29 46
15. I. Ipsen C. Meyer 1994 Uniform stability of Markov chains SIAM J. Matrix Anal. Appl. 15 1061 1074
16. Kemeny J.G., Snell J.L.: Finite Markov Chains. Van Nostrand, Princeton (1960)
17. S.J. Kirkland 2002 On a question concerning condition numbers for Markov chains SIAM J. Matrix Anal. Appl. 23 1109 1119
18. Kirkland, S.J., Neumann, M.: The case of equality in the Dobrushin-Deutsch-Zenger bound. Lin. Alg. Appl. (2008) (submitted)
19. S. Kirkland M. Neumann B. Shader 1998 Applications of Pazs inequality to perturbation bounds for Markov chains Lin. Alg. Appl. 268 183 196
20. S. Kirkland M. Neumann J. Xu 2007 Transition matrices for well-conditioned Markov chains Lin. Alg. Appl. 424 118 131
21. C.D. Meyer 1975 The role of the group generalized inverse in the theory of finite Markov chains SIAM Rev. 17 443 464
22. C.D. Meyer 1980 The condition of a finite Markov chain and perturbation bounds for the limiting probabilities SIAM J. Alg. Discrete Meth. 1 273 283
23. C.D. Meyer 1994 Sensitivity of the stationary distribution of a Markov chain SIAM J. Matrix Anal. Appl. 15 715 728
24. Meyer C.D.: Matrix Analysis and Applied Linear Algebra. SIAM Publications, Philadelphia (2000)
25. C.D. Meyer J.M. Shoaf 1980 Updating finite Markov chains by using techniques of group matrix inversion J. Stat. Comput. Simul. 11 163 181
26. U. Rothblum C.-P. Tan 1985 Upper bounds on the maximum modulus of subdominant eigenvlaues. of nonnegative matrices Lin. Alg. Appl. 66 45 86
27. P. Schweitzer 1968 Perturbation theory and finite Markov chains J. Appl. Probab. 5 410 413
28. Seneta E.: Non-Negative Matrices and Markov Chains. Springer, New York (1981)
29. E. Seneta 1988 Perturbation of the stationary distribution measured by ergodicity coefficients Adv. Appl. Probab. 20 228 230
30. Seneta E.: Sensitivity analysis, ergodicity coefficients, and rank-one updates for finite Markov chains. In: Stewart, W.J.(eds) Numerical Solution of Markov Chains, Marcel Dekker, New York (1991)
31. G.W. Stewart 1993 Gaussian elimination, perturbation theory, and Markov chains, in Linear algebra, Markov chains, and queueing models Inst. Math. Appl. Minneap. 48 59 69
32. Varga R.S.: Matrix Iterative Analysis, Second Revised and Expanded Edition. Springer, Heidelberg (2000)
33. J.H. Wilkinson 1982 Note on the practical significance of the Drazin inverse, in Recent applications of generalized inverses Res. Notes Math. 66 82 99