On the stability of the computation of the stationary probabilities of Markov chains using Perron complements.

*(English)*Zbl 1071.65504Summary: For an \(n\)-state ergodic homogeneous Markov chain whose transition matrix is \(T \in \mathbb R^{n,n}\), it has been shown by C. D. Meyer on the one hand and by S. J. Kirkland, M. Neumann and J. Xu on the other hand that the stationary distribution vector and that the mean first passage matrix, respectively, can be computed by a divide and conquer parallel method from the Perron complements of \(T\). This is possible due to the facts, that the Perron complements of \(T\) are themselves transition matrices for finite ergodic homogeneous Markov chains with fewer states and that their stationary distribution vectors are multiples of the corresponding subvectors of the stationary distribution vector of the entire chain.

Here we examine various questions concerning the stability of computing the stationary distribution vectors of the Perron complements and compare them with the stability of computing the stationary distribution vector for the entire chain. In particular, we obtain that condition numbers which are related to the coefficients of ergodicity improve as we pass from the entire chain to the chains associated with its Perron complements.

Here we examine various questions concerning the stability of computing the stationary distribution vectors of the Perron complements and compare them with the stability of computing the stationary distribution vector for the entire chain. In particular, we obtain that condition numbers which are related to the coefficients of ergodicity improve as we pass from the entire chain to the chains associated with its Perron complements.

##### MSC:

65C40 | Numerical analysis or methods applied to Markov chains |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

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\textit{M. Neumann} and \textit{J. Xu}, Numer. Linear Algebra Appl. 10, No. 7, 603--618 (2003; Zbl 1071.65504)

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