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Massivelly parallel modular algorithms for the image of rational maps
Thesis (MSc)--Stellenbosch University, 2024.
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| Format: | Thesis |
| Language: | en_ZA en_ZA |
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Stellenbosch : Stellenbosch University
2024
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| _version_ | 1867613810714476544 |
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| access_status_str | Open Access |
| author | Rakotoarisoa, Hobihasina Patrick |
| author2 | Basson, Dirk |
| author_browse | Basson, Dirk Rakotoarisoa, Hobihasina Patrick |
| author_facet | Basson, Dirk Rakotoarisoa, Hobihasina Patrick |
| author_sort | Rakotoarisoa, Hobihasina Patrick |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2024. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/130511 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA en_ZA |
| last_indexed | 2026-06-10T12:42:03.173Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2024 |
| publishDateRange | 2024 |
| publishDateSort | 2024 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/130511 Massivelly parallel modular algorithms for the image of rational maps Rakotoarisoa, Hobihasina Patrick Basson, Dirk Bohm, Janko Marais, Magdaleen Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Computer algorithms -- Mathematics Algorithms Parallel algorithms -- Mathematical models Modular methods Data structures (Computer science) Gröbner bases Geometry, Algebraic -- Data processing Rational map Algebraic varieties UCTD Thesis (MSc)--Stellenbosch University, 2024. ENGLISH 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. AFRIKAANSE 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. Masters 2024-03-04T10:31:55Z 2024-04-26T20:16:26Z 2024-03-04T10:31:55Z 2024-04-26T20:16:26Z 2024-03 Thesis https://scholar.sun.ac.za/handle/10019.1/130511 en_ZA en_ZA Stellenbosch University ix, 123 pages : illustrations application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Computer algorithms -- Mathematics Algorithms Parallel algorithms -- Mathematical models Modular methods Data structures (Computer science) Gröbner bases Geometry, Algebraic -- Data processing Rational map Algebraic varieties UCTD Rakotoarisoa, Hobihasina Patrick Massivelly parallel modular algorithms for the image of rational maps |
| title | Massivelly parallel modular algorithms for the image of rational maps |
| title_full | Massivelly parallel modular algorithms for the image of rational maps |
| title_fullStr | Massivelly parallel modular algorithms for the image of rational maps |
| title_full_unstemmed | Massivelly parallel modular algorithms for the image of rational maps |
| title_short | Massivelly parallel modular algorithms for the image of rational maps |
| title_sort | massivelly parallel modular algorithms for the image of rational maps |
| topic | Computer algorithms -- Mathematics Algorithms Parallel algorithms -- Mathematical models Modular methods Data structures (Computer science) Gröbner bases Geometry, Algebraic -- Data processing Rational map Algebraic varieties UCTD |
| url | https://scholar.sun.ac.za/handle/10019.1/130511 |
| work_keys_str_mv | AT rakotoarisoahobihasinapatrick massivellyparallelmodularalgorithmsfortheimageofrationalmaps |