New algorithms for finding the limiting and differential matrices in Markov chains

Alexandru Lazari ; Dmitrii Lozovanu

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2020), pp. 75-88

Résumé

New algorithms for determining the limiting and differential matrices in Markov chains, using fast matrix multiplication methods, new computation procedure of the characteristic polynomial and algorithms of resuming matrix polynomials, are proposed. We show that the complexity of finding the limiting matrix is $O(n^3)$ and the complexity of calculating differential matrices is $O(n^{\omega+1})$, where $n$ is the number of the states of the Markov chain and $O(n^\omega)$ is the complexity of the used matrix multiplication algorithm. The theoretical computational complexity estimation of the algorithm is governed by the fastest known matrix multiplication algorithm, for which $\omega2.372864$.