Computation of the stationary distribution of a markov chain

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Abstract
Eight algorithms are considered for the computation of the stationary distribution l´ of a finite Markov chain with associated probability transition matrix P. The recommended algorithm is based on solving l´(I—P+eú)=ú, where e is the column vector of ones and u´ is a row vector satisfying u´e ≠0.An error analysis is presented for any such u including the choices ú= ejP and ú=e´j where éj is the jth row of the identity matrix. Computationalcomparisons between five of the algorithms are made based on twenty 8 x 8, twenty 20 x 20, and twenty 40 x 40 transition matrices. The matrix (I—P+eú)−1 is shown to be a non-singular generalized inverse of I—P when the unit root of P is simple and úe ≠ 0. A simple closed form expression is obtained for the Moore-Penrose inverse of I—P whenI—P has nullity one

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