NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2001; 8:287–295 (DOI: 10.1002/nla.242) A divide and conquer approach to computing the mean rst passage matrix for Markov chains via Perron complement reductions Stephen J. Kirkland 1 , Michael Neumann 2; *;† and Jianhong Xu 2 1 Department of Mathematics and Statistics; University of Regina; Regina; Saskatchewan; Canada S4S 0A2 2 Department of Mathematics; University of Connecticut; Storrs; Connecticut 06269-3009; U.S.A. SUMMARY Let M T be the mean rst passage matrix for an n-state ergodic Markov chain with a transition matrix T . We partition T as a 2 × 2 block matrix and show how to reconstruct M T eciently by using the blocks of T and the mean rst passage matrices associated with the non-overlapping Perron complements of T . We present a schematic diagram showing how this method for computing M T can be implemented in parallel. We analyse the asymptotic number of multiplication operations necessary to compute M T by our method and show that, for large size problems, the number of multiplications is reduced by about 1= 8, even if the algorithm is implemented in serial. We present ve examples of moderate sizes (of orders 20–200) and give the reduction in the total number of ops (as opposed to multiplications) in the computation of M T . The examples show that when the diagonal blocks in the partitioning of T are of equal size, the reduction in the number of ops can be much better than 1= 8. Copyright ? 2001 John Wiley & Sons, Ltd. KEY WORDS: Markov chain; mean rst passage; Perron complements 1. INTRODUCTION Suppose that we have a homogeneous ergodic Markov chain {X m | m =0; 1;:::} with states S 1 ;:::; S n . If T is the corresponding transition matrix, then the unique positive vector w ∈ R n satisfying w t T = w t and ‖w‖ 1 = 1 is called the stationary distribution vector for the chain. It is well known (see Reference [1], for example) that the stationary distribution * Correspondence to: Michael Neumann, Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009, U.S.A. † E-mail: neumann@math.uconn.edu Contract=grant sponsor: NSERC Grant; contract=grant number: OGP0138251 Contract=grant sponsor: Research supported in part by NSF Grant; contract=grant number: DMS9973247 Received 16 June 2000 Copyright ? 2001 John Wiley & Sons, Ltd. Revised 19 January 2001