By Tugrul Dayar
Creation -- Preliminaries -- Iterative equipment -- Decompositional tools -- Matrix-Analytic equipment -- Conclusion.653Computer technology
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Extra info for Analyzing markov chains using kronecker products : theory and applications
MC1;lC1/ e D 1. The level to end recursion depends on available memory since there must be space to store and factorize the aggregated CTMC at that level. m;l/ , at each level. 13) changes from iteration to iteration, and hence, the method is nonstationary. l/ . l/ . l/ were ordered antilexicographically. l/ . l/ j since it has one nonzero per column by definition. These PH 1 QH vectors amount to a total storage of lD0 hDlC1 nh floating-point values if the recursion terminates at level H . l C2/ through H .
Therefore, there tends to be more common structure among the diagonal blocks of a MC expressed as a sum of Kronecker products. Diagonal blocks having identical off-diagonal parts and diagonals that differ by a multiple of the identity are exploited in . There, it is shown that such diagonal blocks can share and work with the factorization of only one diagonal block. This approach saves not only time spent for factorization of diagonal blocks at the outset but also space. The same paper also discusses a three-level version of BSOR for MCs based on Kronecker products in which diagonal blocks that are too large to be factorized are solved using BSOR.
1). So far, various preconditioners have been proposed for Kronecker structured representations such as those based on truncated Neumann series [139, 141], the cheap and separable preconditioner , circulant preconditioners for a specific class of problem , and the Kronecker sum preconditioner , which has been shown to work effectively on some small problems. The Kronecker product approximate preconditioner for MCs based on Kronecker products developed in a sequence of papers [98–100], although encouraging, is in the form of a prototype implementation.
Analyzing markov chains using kronecker products : theory and applications by Tugrul Dayar