Black Box Linear Algebra with the LinBox Library

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dc.contributor.advisor Erich Kaltofen, Committee Chair en_US
dc.contributor.advisor Carl D. Meyer, Committee Member en_US
dc.contributor.advisor Ralph C. Smith, Committee Member en_US
dc.contributor.advisor B. David Saunders, Committee Member en_US
dc.contributor.advisor Hoon Hong, Committee Member en_US Turner, William J. en_US 2010-04-02T18:26:01Z 2010-04-02T18:26:01Z 2002-07-02 en_US
dc.identifier.other etd-06122002-095342 en_US
dc.description.abstract Black box algorithms for exact linear algebra view a matrix as a linear operator on a vector space, gathering information about the matrix only though matrix-vector products and not by directly accessing the matrix elements. Wiedemann's approach to black box linear algebra uses the fact that the minimal polynomial of a matrix generates the Krylov sequences of the matrix and their projections. By preconditioning the matrix, this approach can be used to solve a linear system, find the determinant of the matrix, or to find the matrix's rank. This dissertation discusses preconditioners based on Benes networks to localize the linear independence of a black box matrix and introduces a technique to use determinantal divisors to find preconditioners that ensure the cyclicity of nonzero eigenvalues. This technique, in turn, introduces a new determinant-preserving preconditioner for a dense integer matrix determinant algorithm based on the Wiedemann approach to black box linear algebra and relaxes a condition on the preconditioner for the Kaltofen-Saunders black box rank algorithm. The dissertation also investigates the minimal generating matrix polynomial of Coppersmith's block Wiedemann algorithm, how to compute it using Beckermann and Labahn's Fast Power Hermite-Pade Solver, and a block algorithm for computing the rank of a black box matrix. Finally, it discusses the design of the LinBox library for symbolic linear algebra. en_US
dc.rights I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to NC State University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. en_US
dc.subject black box linear algebra en_US
dc.subject Wiedemann method en_US
dc.subject block Wiedemann method en_US
dc.subject linear algebra en_US
dc.subject randomized algorithm en_US
dc.subject LinBox library en_US
dc.title Black Box Linear Algebra with the LinBox Library en_US PhD en_US dissertation en_US Computational Mathematics en_US

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