  In this paper, we provide a review on the GTH algorithm, which is a
numerically stable algorithm for computing stationary probabilities of a Markov
chain. Mathematically the GTH algorithm is an rearrangement of Gaussian
elimination, and therefore they are mathematically equivalent. All components
in the GTH algorithm can be interpreted probabilistically based on the
censoring concept and each elimination in the GTH algorithm leads to a censored
Markov chain. The $RG$-factorization is a counterpart to the LU-decomposition
for Gaussian elimination. The censored Markov chain can also be treated as an
extended version of the GTH algorithm for a system consisting of infinitely
many linear equations. The censored Markov chain produces a minimal error for
approximating the original chain under the $l_1$-norm.

Probability