G.W. Stewart, On the sensitivity of nearly uncoupled Markov chains, in Numerical Methods for Markov chains, W.J. Stewart ed., North Holland, Amsterdam, 1990, pp. 102-120.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....T ; T 1 = b T 1 = 1; where 1 is the vector of all ones. According to the perturbation theory for Markov chains, see e.g. 5, 7] k T Gamma e T k kA # kkFk; 3) where A # is the group inverse of the matrix A = I Gamma P: Equality in (3) can be attained for some F: It is shown in [14] that kA # k O 1 ffl : This means that small perturbations in the transition matrices of nearly uncoupled Markov chains can result in large errors in their stationary distributions. The smaller ffl is, the more sensitive the stationary distributions are to the perturbations. However, ....

G.W. Stewart, On the sensitivity of nearly uncoupled Markov chains, in Numerical Methods for Markov chains, W.J. Stewart ed., North Holland, Amsterdam, 1990, pp. 102-120.

....in various ways. Schweitzer (1968) derives a value for # from the fundamental matrix of Kemeny and Snell (1960) whereas the group inverse A # of A = I P is used by Meyer (1980) Golub and Meyer (1986) Funderlic and Meyer (1986) Meyer (1994) Meyer and Stewart (1988) Barlow (1993) and Stewart (1991). Seneta (1991) suggests using a coe#cient of ergodicity for #. Received by the editors September 24, 1992; accepted for publication (in revised form) July 26, 1993. Computer Science, Yale University, New Haven, Connecticut 06520 (ipsen cs.yale.edu) The work of this author was supported in ....

....# j # n, where the term small is to be interpreted in the context of the underlying application. Su#cient conditions for absolute stability are well known. The results in Barlow (1993) Funderlic and Meyer (1986) Golub and Meyer (1986) Meyer (1980) Meyer (1994) Meyer and Stewart (1988) Stewart (1991), for instance, use the fact that a chain is absolutely stable if the group inverse A # of A = I P has no large entries (relative to 1) The results of 5 will establish that the converse of this statement is also true. 1064 i. c. f. ipsen and c. d. meyer 4. Componentwise analysis. In ....

G. W. Stewart (1991), On the sensitivity of nearly uncoupled Markov chains, in Numerical Solution of Markov Chains, W. J. Stewart, ed., Probability: Pure and Applied, No. 8, Marcel Dekker, New York, pp. 105--119.

....the same structure as A, only they are smaller by a factor proportional to ffl M . At first glance this would seem to be an encouraging result. The perturbation theory for NUMCs shows that the solution is Solution of NUMCs 3 insensitive to perturbations that are relatively small compared to E [10]. However, the theory assumes that the sums of the columns of the perturbation are exactly equal to zero. When this condition is violated, even slightly, the solution becomes quite sensitive to perturbations (see test problem 4 discuessed in [6] To see the failure of Gaussian elimination, we ....

....condition number oe Gamma1 t Gamma1 is not too large compared to ffl Gamma1 , these coupling coefficients are insensitive to the perturbations which are relatively small compared to ffl. It should be noted that these two condition numbers are related to the regularity conditions in [9] [10]. A small means that there is only one eigenvalue of D ii approaching one as ffl 0 while the rest of the eigenvalues remain away from the unity. This is implied in the second regularity condition in [9] 10] The relation between the first regularity condition and oe t Gamma1 can be seen from ....

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G. W. Stewart (1990). "On the Sensitivity of Nearly Uncoupled Markov Chains." Technical Report CS-TR 2406, Department of Computer Science, University of Maryland.

....the analysis of computer systems. Currently, there is a very large research effort being devoted to these methods, by many different research groups. See, for example, 4] 7] 8] 9] 10] 12] 15] 16] 23] 24] 30] 31] 32] 33] 35] 36] 39] 47] 48] 49] 52] 53] [55], 61] 62] 64] and [70] With the advent of parallel and distributed computing systems, their advantages are immediately obvious. Ideally the problem is broken into subproblems that can be solved independently and the global solution obtained by pasting together the subproblem solutions. ....

G.W. Stewart. On the sensitivity of nearly uncoupled Markov chains. First International Workshop on the Numerical Solution of Markov Chains. N. Carolina State University, Raleigh, January 1990. Published by Marcel Dekker, Inc. 1990.

.... 1985; Chatelin and Miranker, 1984; Koury et al. 1984; Schweitzer and Kindle, 1986; Takahashi, 1975; and Vantilborgh, 1985) There is a great number and variety of these methods as evidenced by the different approaches presented at a recent workshop on the numerical solution of Markov chains, (Stewart, 1991). They are not covered in this paper primarily due to space limitations. 1 Iterative and Direct Solution Methods Iterative methods of one type or another are by far the most commonly used methods for obtaining the stationary probability vector from either the stochastic transition probability ....

Stewart, G.W. 1991. On the sensitivity of nearly uncoupled Markov chains. In The Numerical Solution of Markov Chains, William J. Stewart, ed. Marcel Dekker, Inc. New York.

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