dc.contributor.author Lyu, Saiyue dc.date.accessioned 2022-05-09 19:01:10 (GMT) dc.date.available 2022-05-09 19:01:10 (GMT) dc.date.issued 2022-05-09 dc.date.submitted 2022-04-29 dc.identifier.uri http://hdl.handle.net/10012/18243 dc.description.abstract Sparse polynomials are those polynomials with only a few non-zero coefficients relative to their degree. They can appear in practice in polynomial systems as inputs, where the degree of the input sparse polynomial can be exponentially larger than the bit length of the representation of it. This leads to the difficulties when computing with sparse polynomials, as many efficient algorithms for dense polynomials take polynomial-time in the degree, and hence an exponential number of operations in a natural representation of the sparse polynomial. en In this thesis, we explore new and faster methods for sparse polynomials and power series. We reconsider algorithms for the sparse perfect power problem and derive a faster sparsity-sensitive algorithm. We then show a fast new algorithm for sparse polynomial decomposition, again sensitive to the sparsity of the input and output. Finally, our algorithms to solve the sparse perfect power and decomposition problems lead us to explore a generalization to solving the linear differential equation with sparse polynomial coefficients using a Newton-like method. We demonstrate an algorithm which will find sparse solutions if they exist, in time polynomial in the input and the output. dc.language.iso en en dc.publisher University of Waterloo en dc.subject sparse polynomial en dc.subject polynomial decomposition en dc.subject differential equation en dc.subject complexity en dc.subject algorithm en dc.title Faster Algorithms for Sparse Decomposition and Sparse Series Solutions to Differential Equations en dc.type Master 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 Master of Mathematics en uws-etd.embargo.terms 0 en uws.contributor.advisor Giesbrecht, Mark 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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