A. Novocin, David Saunders, Alexander Stachnik, Bryan S. Youse
In the study of strongly regular graphs, ranks of adjacency matrices (Laplacians actually) are extensively used to demonstrate inequivalence of graphs. Constructions have been given for several families of graphs. Formulas for the ranks in these families are an important tool for understanding their properties. The first and computational challenge is to compute rank modulo 3 of some very large matrices. To our advantage is that the ranks are expected to be relatively small. Typically in these families, the matrix dimension is 3k while the rank modulo 3 is in the vicinity of 2k. Here we discuss a high performance parallel solution to the problem. It involves parallelism at three levels: word-level vectorization of field elements, shared-memory multi-core, and a multi-node distributed memory and file-system modulated level. The implementation has been applied to the case k = 16, wherein the matrix contains approximately 1.85 peta-entries. The second challenge is to discern a formula for the sequence of ranks in a given graph family. Our computations provide further evidence for an existing conjecture concerning the Dickson family of strongly regular graphs and provide a starting point towards finding a formula for the Ding-Yuan and Cohen-Ganley families of graphs.
{"title":"3-ranks for strongly regular graphs","authors":"A. Novocin, David Saunders, Alexander Stachnik, Bryan S. Youse","doi":"10.1145/2790282.2790295","DOIUrl":"https://doi.org/10.1145/2790282.2790295","url":null,"abstract":"In the study of strongly regular graphs, ranks of adjacency matrices (Laplacians actually) are extensively used to demonstrate inequivalence of graphs. Constructions have been given for several families of graphs. Formulas for the ranks in these families are an important tool for understanding their properties. The first and computational challenge is to compute rank modulo 3 of some very large matrices. To our advantage is that the ranks are expected to be relatively small. Typically in these families, the matrix dimension is 3k while the rank modulo 3 is in the vicinity of 2k. Here we discuss a high performance parallel solution to the problem. It involves parallelism at three levels: word-level vectorization of field elements, shared-memory multi-core, and a multi-node distributed memory and file-system modulated level. The implementation has been applied to the case k = 16, wherein the matrix contains approximately 1.85 peta-entries. The second challenge is to discern a formula for the sequence of ranks in a given graph family. Our computations provide further evidence for an existing conjecture concerning the Dickson family of strongly regular graphs and provide a starting point towards finding a formula for the Ding-Yuan and Cohen-Ganley families of graphs.","PeriodicalId":384227,"journal":{"name":"Proceedings of the 2015 International Workshop on Parallel Symbolic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122529252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
To interpolate a supersparse polynomial with integer coefficients, two alternative approaches are the Prony-based "big prime" technique, which acts over a single large finite field, or the more recently-proposed "small primes" technique, which reduces the unknown sparse polynomial to many low-degree dense polynomials. While the latter technique has not yet reached the same theoretical efficiency as Prony-based methods, it has an obvious potential for parallelization. We present a heuristic "small primes" interpolation algorithm and report on a low-level C implementation using FLINT and MPI.
{"title":"Parallel sparse interpolation using small primes","authors":"Mohamed Khochtali, Daniel S. Roche, Xisen Tian","doi":"10.1145/2790282.2790290","DOIUrl":"https://doi.org/10.1145/2790282.2790290","url":null,"abstract":"To interpolate a supersparse polynomial with integer coefficients, two alternative approaches are the Prony-based \"big prime\" technique, which acts over a single large finite field, or the more recently-proposed \"small primes\" technique, which reduces the unknown sparse polynomial to many low-degree dense polynomials. While the latter technique has not yet reached the same theoretical efficiency as Prony-based methods, it has an obvious potential for parallelization. We present a heuristic \"small primes\" interpolation algorithm and report on a low-level C implementation using FLINT and MPI.","PeriodicalId":384227,"journal":{"name":"Proceedings of the 2015 International Workshop on Parallel Symbolic Computation","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115951414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dereje Kifle Boku, C. Fieker, W. Decker, Andreas Steenpaß
Although Buchberger's algorithm, in theory, allows us to compute Gröbner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field K = Q(α), a simple extension of Q, where α is an algebraic number, and let f ∈ Q[t] be the minimal polynomial of α. In this paper we present a new efficient method to compute Gröbner bases in polynomial rings over the algebraic number field K. Starting from the ideas of Noro [11], we proceed by joining f to the ideal to be considered, adding t as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2, 3, 10], that is, by inferring information in characteristic zero from information in characteristic p > 0. For suitable primes p, the minimal polynomial f is reducible over Fp. This allows us to apply modular methods once again, on a second level, with respect to the factors of f. The algorithm thus resembles a divide and conquer strategy and is in particular easily parallelizable. At current state, the algorithm is probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithm, which has been implemented in Singular [7], outperforms other known methods by far.
{"title":"Gröbner bases over algebraic number fields","authors":"Dereje Kifle Boku, C. Fieker, W. Decker, Andreas Steenpaß","doi":"10.1145/2790282.2790284","DOIUrl":"https://doi.org/10.1145/2790282.2790284","url":null,"abstract":"Although Buchberger's algorithm, in theory, allows us to compute Gröbner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field K = Q(α), a simple extension of Q, where α is an algebraic number, and let f ∈ Q[t] be the minimal polynomial of α. In this paper we present a new efficient method to compute Gröbner bases in polynomial rings over the algebraic number field K. Starting from the ideas of Noro [11], we proceed by joining f to the ideal to be considered, adding t as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2, 3, 10], that is, by inferring information in characteristic zero from information in characteristic p > 0. For suitable primes p, the minimal polynomial f is reducible over Fp. This allows us to apply modular methods once again, on a second level, with respect to the factors of f. The algorithm thus resembles a divide and conquer strategy and is in particular easily parallelizable. At current state, the algorithm is probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithm, which has been implemented in Singular [7], outperforms other known methods by far.","PeriodicalId":384227,"journal":{"name":"Proceedings of the 2015 International Workshop on Parallel Symbolic Computation","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116844111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Numerical continuation methods track a solution path defined by a homotopy. The systems we consider are defined by polynomials in several variables with complex coefficients. For larger dimensions and degrees, the numerical conditioning worsens and hardware double precision becomes often insufficient to reach the end of the solution path. With double double and quad double arithmetic, we can solve larger problems that we could not solve with hardware double arithmetic, but at a higher computational cost. This cost overhead can be compensated by acceleration on a Graphics Processing Unit (GPU). We describe our implementation and report on computational results on benchmark polynomial systems.
{"title":"Accelerating polynomial homotopy continuation on a graphics processing unit with double double and quad double arithmetic","authors":"J. Verschelde, Xiangcheng Yu","doi":"10.1145/2790282.2790294","DOIUrl":"https://doi.org/10.1145/2790282.2790294","url":null,"abstract":"Numerical continuation methods track a solution path defined by a homotopy. The systems we consider are defined by polynomials in several variables with complex coefficients. For larger dimensions and degrees, the numerical conditioning worsens and hardware double precision becomes often insufficient to reach the end of the solution path. With double double and quad double arithmetic, we can solve larger problems that we could not solve with hardware double arithmetic, but at a higher computational cost. This cost overhead can be compensated by acceleration on a Graphics Processing Unit (GPU). We describe our implementation and report on computational results on benchmark polynomial systems.","PeriodicalId":384227,"journal":{"name":"Proceedings of the 2015 International Workshop on Parallel Symbolic Computation","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122992724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Proceedings of the 2015 International Workshop on Parallel Symbolic Computation","authors":"","doi":"10.1145/2790282","DOIUrl":"https://doi.org/10.1145/2790282","url":null,"abstract":"","PeriodicalId":384227,"journal":{"name":"Proceedings of the 2015 International Workshop on Parallel Symbolic Computation","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115639614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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