[Abstract] This thesis is divided into two parts, both containing algorithms for dealing with matrices and matrix groups. Part I is concerned with individual matrices over an arbitrary field. Our algorithms make use of a sequence called the rank profile which is related to the linear dependence relations between the columns of a matrix. First we look at LSP decompositions of matrices as defined by Ibarra et al. in 1982. This decomposition is related to, and a little more general than, the LUP decomposition. The algorithm given by Ibarra et al. to compute an LSP decomposition was only defined for m?n matrices where m ≤ n and is claimed to have the same asymptotic cost as matrix multiplication. We prove that their cost analysis overlooked some aspects of the computation and present a new version of the algorithm which finds both an LSP decomposition and the rank profile of any matrix. The cost of our algorithm is the same as that claimed by Ibarra et al. when m ≤ n and has a similar cost when m > n. One of the steps in the Ibarra et al. algorithm is not completely explicit, so that any one of several choices can be made. Our algorithm is designed so that the particular choice made at this point allows for the simultaneous calculation of the rank profile. Next we study algorithms to find the characteristic polynomial of a square matrix. The current fastest algorithm to find the characteristic polynomial of a square matrix was developed by Keller-Gehrig in 1985. We present a new, simpler version of this algorithm with the same cost which makes the algorithm?s reliance on the rank profile explicit. In Part II we present generalised sifting, a scheme for creating Monte Carlo black box constructive group recognition algorithms. Generalised sifting is designed to facilitate computation in a known group, specifically re-writing arbitrary elements as words or straight-line programs in a standard generating set. It can also be used to create membership tests in black-box groups. Generalised sifting was inspired by the subgroup sifting techniques originally introduced by Sims in 1970 but uses a chain of subsets rather than subgroups. We break the problem down into a sequence of separately analysed and proven steps which sift down into each subset in turn ... All of the algorithms in Parts I and II are given with a theoretical proof and (where appropriate) complexity analysis. The LSP decomposition, characteristic polynomial and generalised sifting algorithms have all been implemented and tested in the computer algebra package GAP.
| Identifer | oai:union.ndltd.org:ADTP/221274 |
| Date | January 2006 |
| Creators | Ambrose, Sophie |
| Publisher | University of Western Australia. School of Mathematics and Statistics |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | Copyright Sophie Ambrose, http://www.itpo.uwa.edu.au/UWA-Computer-And-Software-Use-Regulations.html |
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