Updating finite markov chains by using techniques of group matrix inversion

Meyer, Carl D.; Shoaf, James M.

doi:10.1080/00949658008810405

Cited by 46 publications

(24 citation statements)

References 4 publications

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“…The theory of Drazin inverse has numerous applications, such as difference equations, statistics, Markov chains and numerical analysis and so on (see [2][3][4][5][6][7][8][9][10][11]). In 1977, Meyer and Rose gave the computational formula of the Drazin inverse for complex block matrix A B 0 C (A and C are square) (see [2]).…”

Section: Introductionmentioning

confidence: 99%

A note on computational formulas for the Drazin inverse of certain block matrices

Feng

Dong

2011

J. Appl. Math. Comput.

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Section: Introductionmentioning

confidence: 99%

A note on computational formulas for the Drazin inverse of certain block matrices

Feng

Dong

2011

J. Appl. Math. Comput.

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“…We have found that the algorithms suggested in [1,12], both of which are based on the shuffle algorithm, generally do very well in the computation of the group inverse of a singular and irreducible M-matrix. Further perturbation analysis for Markov chains, the use of the group inverse in such analysis, and stability analysis for the computation of the group inverse can be found in Meyer and Shoaf [25], Stewart [31], and Wilkinson [33].…”

Section: Introductionmentioning

confidence: 99%

On optimal condition numbers for Markov chains

2008

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Let T andT = T − E be arbitrary nonnegative, irreducible, stochastic matrices corresponding to two ergodic Markov chains on n states. A function κ is called a condition number for Markov chains with respect to the (α, β)-norm pair if π −π α ≤ κ(T ) E β . Here π andπ are the stationary distribution vectors of the two chains, respectively. Various condition numbers, particularly with respect to the (1, ∞) and (∞, ∞)-norm pairs have been suggested in the literature. They were ranked according to their size by Cho and Meyer in a paper from 2001. In this paper we first of all show that what we call the generalized ergodicity coefficient τ p (A # ) = sup y t e=0 y t A # p y 1 , where e is the n-vector of all 1's and A # is the group generalized inverse of A = I − T , is the smallest condition number of Markov chains with respect to the ( p, ∞)-norm pair. We use this result to identify the smallest condition number of Markov chains among the (∞, ∞) and (1, ∞)-norm pairs. These are, respectively, κ 3 and κ 6 in the Cho-Meyer list of 8 condition numbers. Kirkland has studied κ 3 (T ). He has shown that κ 3 (T ) ≥ n−1 2n and he has characterized transition matrices for which equality holds. We prove here again that 2κ 3 (T ) ≤ κ(6) which appears in the Cho-Meyer paper and we characterize the transition matrices T for which κ 6 (T ) = n−1 n . There is actually only one such matrix: T = (J n − I )/(n − 1), where J n is the n × n matrix of all 1's.

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“…This matrix is often involved in questions concerning Markov chains; see [6,23,29] for some general background and [6,9,11,14,23,25,27,28,31,32,38] for Markov chain applications. The precise formula to perform exact updating is as follows.…”

Section: Faster Converging Statesmentioning

confidence: 99%

“…Since exact updating is not the primary focus of this article, the formal proof of Theorem 3.1 is omitted, but the interested reader can find the details that constitute a proof in [31].…”

Section: Faster Converging Statesmentioning

confidence: 99%

Updating Markov Chains with an Eye on Google's PageRank

Langville¹,

Meyer²

2006

SIAM J. Matrix Anal. & Appl.

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Abstract. An iterative algorithm based on aggregation/disaggregation principles is presented for updating the stationary distribution of a finite homogeneous irreducible Markov chain. The focus is on large-scale problems of the kind that are characterized by Google's PageRank application, but the algorithm is shown to work well in general contexts. The algorithm is flexible in that it allows for changes to the transition probabilities as well as for the creation or deletion of states. In addition to establishing the rate of convergence, it is proven that the algorithm is globally convergent. Results of numerical experiments are presented.

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Updating finite markov chains by using techniques of group matrix inversion

Cited by 46 publications

References 4 publications

A note on computational formulas for the Drazin inverse of certain block matrices

A note on computational formulas for the Drazin inverse of certain block matrices

On optimal condition numbers for Markov chains

Updating Markov Chains with an Eye on Google's PageRank

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