Updating $LU$ Factorizations for Computing Stationary Distributions

Abstract

The computation of stationary probability distributions for Markov chains is important in the analysis of many models in the mathematical sciences, such as queueing network models, input-output economic models and compartmental tracer analysis models. These computations often involve the solution of large-scale homogeneous linear equations by Gaussian elimination, where A is a Q-matrix, i.e., $A = ( a_{ij} )$ is irreducible, $a_{ij} \leqq 0$ for all $i \ne j$ and has zero column sums. The stationary distributions are the components of the unique solution vector x of positive components whose sum is one. Stable direct methods for computing x by triangular factorization $A = LU$ have received considerable attention recently and the purpose of this paper is to provide a stable method for updating the factors L and U in $O( n^2 )$ flops in the case where a column of A is modified. Updating formulas are derived here using an approach similar to that for updating the Cholesky factor of a symmetric positive defi...