Massivelly parallel modular algorithms for the image of rational maps

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dc.contributor.advisorBasson, Dirken_ZA
dc.contributor.advisorBohm, Jankoen_ZA
dc.contributor.advisorMarais, Magdaleenen_ZA
dc.contributor.authorRakotoarisoa, Hobihasina Patricken_ZA
dc.contributor.otherStellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.en_ZA
dc.date.accessioned2024-03-04T10:31:55Z
dc.date.accessioned2024-04-26T20:16:26Z
dc.date.available2024-03-04T10:31:55Z
dc.date.available2024-04-26T20:16:26Z
dc.date.issued2024-03
dc.descriptionThesis (MSc)--Stellenbosch University, 2024. en_ZA
dc.description.abstractENGLISH ABSTRACT: Modular methods are a tool which can be applied in computer algebra to signifi‑ cantly improve the performance of algorithms in characteristic 0 by addressing the problem of intermediate coefficient growth. Computations are done simul‑ taneously over multiple finite fields by reducing the input data, applying the algorithm under consideration in positive characteristic, and then lifting the modular results to the rationals via Chinese remaindering and the Farey map. Even in the existence of bad primes, error tolerance of this process ensures that for a sufficiently large set of good primes the approach terminates with the cor‑ rect answer. The method is clearly parallel and has the potential to scale across multiple computers. It has been applied for various use cases, for example, for the computation of Gröbner bases. In this thesis, we provide a generic modular approach which is applicable to polynomial data structures arising from com‑ mutative algebra and algebraic geometry, such as modules, varieties, and ratio‑ nal maps. Moreover, we develop a massively parallel framework for modular computations, which we model in terms of Petri nets. We give an implementa‑ tion relying on the SINGULAR/GPI‑SPACE framework.en_ZA
dc.description.abstractAFRIKAANSE OPSOMMING: Modulêre metodes is gereedskap wat toegepas kan word in berekeningsalgebra om die prestasie van algoritmes in karakteristiek 0 aansienlik te verbeter deur die probleem van intermediêre koëffisiëntegroei aan te spreek. Berekeninge word gelyktydig oor verskeie eindige liggame uitgevoer deur die invoerdata te beperk, die algoritme onder oorweging in positiewe karakteristiek toe te pas, en dan die modulêre resultate na die rasionele getalle te verhef deur middel van Chinese reste en die Farey‑afbeelding. Selfs indien slegte priemgetalle voor‑ kom, verseker die fouttoleransie van hierdie proses dat vir ’n voldoende groot versameling goeie priemgetalle die benadering met die korrekte antwoord ter‑ mineer. Die metode is duidelik parallel en het die potensiaal om oor verskeie rekenaars te skaal. Dit is aangewend vir verskeie gevalle, byvoorbeeld, vir die berekening van Gröbner‑basisse. In hierdie tesis verskaf ons ’n generiese mo‑ dulêre benadering wat van toepassing is op polinoomdatastrukture wat voort‑ spruit uit kommutatiewe algebra en algebraı̈ese meetkunde, soos modules, va‑ riëteite, en rasionele afbeeldings. Daarbenewens ontwikkel ons ’n grootskaalse parallelle raamwerk vir modulêre berekeninge, wat ons modelleer in terme van Petri‑netwerke. Ons gee ’n implementering wat gebasseer is op die SIN‑ GULAR/GPI‑SPACE raamwerk.af_ZA
dc.description.versionMastersen_ZA
dc.format.extentix, 123 pages : illustrationsen_ZA
dc.identifier.urihttps://scholar.sun.ac.za/handle/10019.1/130511
dc.language.isoen_ZAen_ZA
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : Stellenbosch Universityen_ZA
dc.rights.holderStellenbosch Universityen_ZA
dc.subject.lcshComputer algorithms -- Mathematicsen_ZA
dc.subject.lcshAlgorithmsen_ZA
dc.subject.lcshParallel algorithms -- Mathematical modelsen_ZA
dc.subject.lcshModular methodsen_ZA
dc.subject.lcshData structures (Computer science)en_ZA
dc.subject.lcshGröbner basesen_ZA
dc.subject.lcshGeometry, Algebraic -- Data processingen_ZA
dc.subject.lcshRational mapen_ZA
dc.subject.lcshAlgebraic varietiesen_ZA
dc.subject.nameUCTDen_ZA
dc.titleMassivelly parallel modular algorithms for the image of rational mapsen_ZA
dc.typeThesisen_ZA
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