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Title
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A fast Newton's iteration for M/G/1-type and GI/M/1-type Markov chains
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Author
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Abstract
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| In this article we revisit Newton's iteration as a method to find the G or R matrix in M/G/1-type and GI/M/1-type Markov chains. We start by reconsidering the method proposed in Ref.([15]), which required O(m(6) + Nm(4)) time per iteration, and show that it can be reduced to O(Nm(4)), where m is the block size and N the number of blocks. Moreover, we show how this method is able to further reduce this time complexity to O(Nr(3) + Nm(2)r(2) + m(3)r) when A(0) has rank r < m. In addition, we consider the case where [A(1)A(2) ... A(N)] is of rank r < m and propose a new Newton's iteration method which is proven to converge quadratically and that has a time complexity of O(Nm(3) + Nm(2)r(2) + mr(3)) per iteration. The computational gains in all the cases are illustrated through numerical examples. |
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Language
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English
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Source (journal)
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Stochastic models. - New York, N.Y.
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Publication
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New York, N.Y.
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2012
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ISSN
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1532-6349
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DOI
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10.1080/15326349.2012.726038
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Volume/pages
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28
:4
(2012)
, p. 557-583
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ISI
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000310849700002
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Full text (Publisher's DOI)
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Full text (open access)
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