dc.contributor.author Birmpilis, Stavros dc.date.accessioned 2021-09-24 15:06:59 (GMT) dc.date.available 2021-09-24 15:06:59 (GMT) dc.date.issued 2021-09-24 dc.date.submitted 2021-09-20 dc.identifier.uri http://hdl.handle.net/10012/17514 dc.description.abstract The focus of this thesis is on fundamental computational problems in exact integer linear algebra. Specifically, for a nonsingular integer input matrix A of dimension n, we consider problems such as linear system solving and computing integer matrix normal forms. en Our goal is to design algorithms that have complexity about the same as the cost of multiplying together two integer matrices of the same dimension and size of entries as the input matrix A. If 2 ≤ ω ≤ 3 is a valid exponent for matrix multiplication, that is, if two n × n matrices can be multiplied in O(n^ω) basic operations from the domain of entries, then our target complexity is O(n^ω log ||A||) bit operations, up to some missing log n and loglog ||A|| factors. Here ||A|| denotes the largest entry in A in absolute value. The first contribution is solving the problem of computing the Smith normal form S of a nonsingular matrix A along with computing unimodular matrices U, V such that AV = US within our target cost. The algorithm we give is a Las Vegas probabilistic algorithm which means that we are able to verify the correctness of its output. The second contribution of the thesis is with respect to linear system solving. We present a deterministic reduction to matrix multiplication for the problem of linear system solving: given as input a nonsingular A and a vector b, solve the system Ax = b. The system solution x is computed within our target complexity. dc.language.iso en en dc.publisher University of Waterloo en dc.subject Smith normal form en dc.subject Integer matrices en dc.subject Linear system solving en dc.subject Exact linear algebra en dc.title Fast algorithms for computing with integer matrices: normal forms and applications en dc.type Doctoral Thesis en dc.pending false uws-etd.degree.department David R. Cheriton School of Computer Science en uws-etd.degree.discipline Computer Science en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Doctor of Philosophy en uws-etd.embargo.terms 0 en uws.contributor.advisor Labahn, George uws.contributor.advisor Storjohann, Arne uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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