Accurate Computation of the Fundamental Matrix of a Markov Chain

Associated with every stochastic matrix is another matrix called the fundamental matrix. The fundamental matrix can be used to obtain mean first-passage-times and other interesting operating characteristics. The fundamental matrix is defined as a matrix inverse, and computing it from the definition can be fraught with numerical errors. We establish a new representation of the fundamental matrix where matrix inversion is replaced by multiplying and then adding a pair of matrices. The representation requires the solution of a system of linear equations, and we show that that can be done via back and forward substitution from numbers that have already been calculated when the GTH algorithm is used to compute the steady-state probabilities. An algorithm based on this representation is given. The time complexity of the faster implementation is 75% of the time complexity of using Gaussian elimination.