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In this paper, we revisit the generalized block power methods forapproximating the eigenvector associated with $\lambda_1 = 1$ of a Markov chaintransition matrix. Our analysis of the block power method shows that when $s$linearly independent probability vectors are used as the initial block, theconvergence of the block power method to the stationary distribution depends onthe magnitude of the $(s+1)$th dominant eigenvalue $\lambda_{s+1}$ of $P$instead of that of $\lambda_2$ in the power method. Therefore, the block powermethod with block size $s$ is particularly effective for transition matriceswhere $|\lambda_{s+1}|$ is well separated from $\lambda_1 = 1$ but$|\lambda_2|$ is not. This approach is particularly useful when visiting theelements of a large transition matrix is the main computational bottleneck overmatrix--vector multiplications, where the block power method can effectivelyreduce the total number of times to pass over the matrix. To further reduce theoverall computational cost, we combine the block power method with a slidingwindow scheme, taking advantage of the subsequent vectors of the latest $s$iterations to assemble the block matrix. The sliding window scheme correlatesvectors in the sliding window to quickly remove the influences from theeigenvalues whose magnitudes are smaller than $|\lambda_{s}|$ to reduce theoverall number of matrix--vector multiplications to reach convergence. Finally,we compare the effectiveness of these methods in a Markov chain modelrepresenting a stochastic luminal calcium release site.

A Revisit of Block Power Methods for Finite State Markov Chain Applications