By Tugrul Dayar

ISBN-10: 1461441900

ISBN-13: 9781461441908

Advent -- Preliminaries -- Iterative tools -- Decompositional tools -- Matrix-Analytic tools -- Conclusion.653Computer technology

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Extra info for Analyzing markov chains using kronecker products : theory and applications

Example text

H/ Qk : kDH C1 hD1 T. h/ Recall that this enumeration necessarily implies that Qk D Inh for h D 1; : : : ; H and k ยค h due to the definition of local transitions. h/ matrices Qh for h D 1; : : : ; H . 7). h/ 2 Rn 0nh [cf. h/ W S ! h/ represents the aggregation of all dimensions except the hth. h/ D h 1 O ! In l e O In h O lD1 H O ! m/ 2 Rnh n [cf. h/ . Then the decompositional iterative method can be stated [4] for a user-specified stopping tolerance, tol, as in Algorithm 3. m/ , to the uniform distribution.

H/ represents the aggregation of all dimensions except the hth. h/ D h 1 O ! In l e O In h O lD1 H O ! m/ 2 Rnh n [cf. h/ . Then the decompositional iterative method can be stated [4] for a user-specified stopping tolerance, tol, as in Algorithm 3. m/ , to the uniform distribution. h/ equations is solved subject to a normalization condition. mC1/ > 0 can be computed. h/ transition rate matrices, Qk for k D H C 1; : : : ; K, are specified. m/ / is used. m/ > 0. h/ also guaranteed in this case. m/ /, can be handled systematically.

The second best solver is ML, which takes 232 MB and 24 iterations to converge to the solution in 99 s. It improves only slightly as the synchronized transition rates become smaller. The only other competitive solver is BICGSTAB, which takes 143 iterations and 127 s, requiring 244 MB. In particular, GMRES(20), BICGSTAB, and TFQMR do not benefit from BGS preconditioning. The performances of the Jacobi, GS, and BGS solvers are not affected by a change in the rates of synchronized transitions. BGS performs very poorly due to the large time per iteration.

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Analyzing markov chains using kronecker products : theory and applications by Tugrul Dayar


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