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On the complexity of matrix reduction over finite fields
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
2007 (English)In: Advances in Applied Mathematics, ISSN 0196-8858, Vol. 39, no 4, 428-452 p.Article in journal (Refereed) Published
Abstract [en]

We study matrix elimination over finite fields, and present an algorithm which is asymptotically faster than the traditional Gauss--Jordan elimination. We also bound the average and worst-case complexity for the problem, proving that our algorithm is close to being optimal, and show related concentration results for random matrices.

Next we present the results of a large computational study of the complexities for small matrices and fields. Here we determine the exact distribution of the complexity for matrices from $\mathrm{GL}_{n}(\mathbb{F}_{q})$, with $n$ an $q$ small

Finally we consider an extension of the problems studied for finite fields to finite semifields. We give a conjecture on the behaviour of a natural analogue of $\mathrm{GL}_{n}$ for semifields and prove this for a certain class of semifields.

Place, publisher, year, edition, pages
2007. Vol. 39, no 4, 428-452 p.
Keyword [en]
Matrix reduction, Complexity, Finite fields, Semifields
Identifiers
OAI: oai:DiVA.org:umu-7595DiVA: diva2:147266
Available from: 2008-01-11 Created: 2008-01-11 Last updated: 2009-08-24Bibliographically approved
In thesis
1. On the Ising problem and some matrix operations
Open this publication in new window or tab >>On the Ising problem and some matrix operations
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The first part of the dissertation concerns the Ising problem proposed to Ernst Ising by his supervisor Wilhelm Lenz in the early 20s. The Ising model, or perhaps more correctly the Lenz-Ising model, tries to capture the behaviour of phase transitions, i.e. how local rules of engagement can produce large scale behaviour.

Two decades later Lars Onsager solved the Ising problem for the quadratic lattice without an outer field. Using his ideas solutions for other lattices in two dimensions have been constructed. We describe a method for calculating the Ising partition function for immense square grids, up to linear order 320 (i.e. 102400 vertices).

In three dimensions however only a few results are known. One of the most important unanswered questions is at which temperature the Ising model has its phase transition. In this dissertation it is shown that an upper bound for the critical coupling Kc, the inverse absolute temperature, is 0.29 for the tree dimensional cubic lattice.

To be able to get more information one has to use different statistical methods. We describe one sampling method that can use simple state generation like the Metropolis algorithm for large lattices. We also discuss how to reconstruct the entropy from the model, in order to obtain parameters as the free energy.

The Ising model gives a partition function associated with all finite graphs. In this dissertation we show that a number of interesting graph invariants can be calculated from the coefficients of the Ising partition function. We also give some interesting observations about the partition function in general and show that there are, for any N, N non-isomorphic graphs with the same Ising partition function.

The second part of the dissertation is about matrix operations. We consider the problem of multiplying them when the entries are elements in a finite semiring or in an additively finitely generated semiring. We describe a method that uses O(n3 / log n) arithmetic operations.

We also consider the problem of reducing n x n matrices over a finite field of size q using O(n2 / logq n) row operations in the worst case.

Place, publisher, year, edition, pages
Umeå: Matematik och matematisk statistik, 2007. 7 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 37
Keyword
Ising problem, phase tansition, matrix multiplicatoin, matrix inversion
National Category
Discrete Mathematics
Identifiers
urn:nbn:se:umu:diva-1129 (URN)987-91-7264-323-9 (ISBN)
Public defence
2007-05-31, MA 121, MIT, Umeå, 13:15 (English)
Supervisors
Available from: 2007-05-10 Created: 2007-05-10 Last updated: 2011-04-21Bibliographically approved

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