We present an Hermitian version of the classical floating-point LLL reduction algorithm of Nguyen and Stehlé[16]. This new variant works on imaginary quadratic fields which are norm-Euclidean and also for some adequate cyclotomic fields. An optimized C++ implementation has been performed, based on the fpLLL code and results show a significant improvement for Hermitian lattices reduction of dimension N when compared to fpLLL reduction on the corresponding Euclidean lattice of dimension 2N. We demonstrate our implementation in the special case of the Gaussian integers.
{"title":"An extension of the fpLLL library to Hermitian lattices","authors":"P. Elbaz-Vincent, Etienne Marcatel","doi":"10.1145/3493492.3493498","DOIUrl":"https://doi.org/10.1145/3493492.3493498","url":null,"abstract":"We present an Hermitian version of the classical floating-point LLL reduction algorithm of Nguyen and Stehlé[16]. This new variant works on imaginary quadratic fields which are norm-Euclidean and also for some adequate cyclotomic fields. An optimized C++ implementation has been performed, based on the fpLLL code and results show a significant improvement for Hermitian lattices reduction of dimension N when compared to fpLLL reduction on the corresponding Euclidean lattice of dimension 2N. We demonstrate our implementation in the special case of the Gaussian integers.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"54 - 58"},"PeriodicalIF":0.1,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43934073","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}
The 22nd Workshop on Computer Algebra in memory of Professor Vladimir Gerdt was held in Dubna, Russia from May 24-25, 2021. Co-organized by the Dorodnicyn Computing Centre (Federal Research Center "Computer Science and Control") of Russian Academy of Sciences and the Meshcheryakov Laboratory of Information Technologies of the Joint Institute for Nuclear Research, it was devoted to computer algebra and related topics and included a memorial session. The workshop web-site is https://ca2021.jinr.ru.
{"title":"22nd Workshop on Computer Algebra in Memory of Professor Vladimir Gerdt in Dubna","authors":"S. Abramov, A. Bogolubskaya","doi":"10.1145/3493492.3493493","DOIUrl":"https://doi.org/10.1145/3493492.3493493","url":null,"abstract":"The 22nd Workshop on Computer Algebra in memory of Professor Vladimir Gerdt was held in Dubna, Russia from May 24-25, 2021. Co-organized by the Dorodnicyn Computing Centre (Federal Research Center \"Computer Science and Control\") of Russian Academy of Sciences and the Meshcheryakov Laboratory of Information Technologies of the Joint Institute for Nuclear Research, it was devoted to computer algebra and related topics and included a memorial session. The workshop web-site is https://ca2021.jinr.ru.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"21 - 29"},"PeriodicalIF":0.1,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45815285","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}
It is our deepest regret to inform you that Vladimir Petrovich Gerdt, Professor, Head of the Algebraic and Quantum Computing Group of the Scientific Department of Computational Physics of the Laboratory of Information Technologies (LIT) at the Joint Institute of Nuclear Research (JINR) in Dubna, Oblast Moscow, Russia, died on January 5th, 2021 at the age of 73, following complications caused by COVID-19. Vladimir Gerdt was born on January 21, 1947 in the town of Engels, Saratov region of the USSR. He began his scientific activity at JINR in November 1971, after graduating from the Physics Department of Saratov State University, first in the Department of Radiation Safety, and from February 1977 on in the Laboratory of Computer Technology and Automation, which was renamed in the year 2000 to Laboratory of Information Technologies, where he was engaged in the deployment of analytical computing software systems on the computers of the JINR Central Research Center, their development and application for solving physical problems. Starting in 1983, he was the head of the Computer Algebra Research Group (renamed in 2007 to Algebraic and Quantum Computing Group) at LIT. In 1976, Vladimir Gerdt successfully defended his Ph.D. thesis (for Kandidat nauk/Kandidat nauk) in the field Theoretical and Mathematical Physics, and in 1992, his doctoral dissertation (for Doktor nauk/Doktor nauk, D.Sc.) in the field Application of Computer Technology, Mathematical Modeling, and Mathematical Methods for Scientific Research. In 1997, he was awarded the academic title of Professor. Vladimir Gerdt started his career with work on the integrability analysis of nonlinear evolution equations using symmetries, and he never ceased to be interested in symmetry methods for di↵erential equations. Later, the involution analysis of polynomial systems and systems of di↵erential or di↵erence equations became a core theme in his research. Together with some of his students, he developed the theory of involutive bases out of the Janet–Riquier theory. As one application of these results to physics, he was always interested in the study of systems with constraints, in particular in Dirac theory. Another application in recent years consisted of designing structure preserving numerical methods for partial di↵erential equations. In the last years, he also revived the Thomas decomposition for polynomial di↵erential equations and applied it to numerous problems. One of his last significant results was an algorithmic solution of the linearization problem for ordinary di↵erential equations. For many years, Vladimir Gerdt also worked in the field of quantum computing. His group at JINR developed methods and computer algebra programs for studying quantum information processes and for modeling quantum systems. In particular, they applied a combination of computational invariant theory and involutive bases to a qualitative and quantitative study of entangled quantum states. Vladimir Gerdt was the au
{"title":"Vladimir Gerdt dies at 73","authors":"E. Mayr, W. Seiler, E. Vorozhtsov","doi":"10.1145/3466895.3467684","DOIUrl":"https://doi.org/10.1145/3466895.3467684","url":null,"abstract":"It is our deepest regret to inform you that Vladimir Petrovich Gerdt, Professor, Head of the Algebraic and Quantum Computing Group of the Scientific Department of Computational Physics of the Laboratory of Information Technologies (LIT) at the Joint Institute of Nuclear Research (JINR) in Dubna, Oblast Moscow, Russia, died on January 5th, 2021 at the age of 73, following complications caused by COVID-19. Vladimir Gerdt was born on January 21, 1947 in the town of Engels, Saratov region of the USSR. He began his scientific activity at JINR in November 1971, after graduating from the Physics Department of Saratov State University, first in the Department of Radiation Safety, and from February 1977 on in the Laboratory of Computer Technology and Automation, which was renamed in the year 2000 to Laboratory of Information Technologies, where he was engaged in the deployment of analytical computing software systems on the computers of the JINR Central Research Center, their development and application for solving physical problems. Starting in 1983, he was the head of the Computer Algebra Research Group (renamed in 2007 to Algebraic and Quantum Computing Group) at LIT. In 1976, Vladimir Gerdt successfully defended his Ph.D. thesis (for Kandidat nauk/Kandidat nauk) in the field Theoretical and Mathematical Physics, and in 1992, his doctoral dissertation (for Doktor nauk/Doktor nauk, D.Sc.) in the field Application of Computer Technology, Mathematical Modeling, and Mathematical Methods for Scientific Research. In 1997, he was awarded the academic title of Professor. Vladimir Gerdt started his career with work on the integrability analysis of nonlinear evolution equations using symmetries, and he never ceased to be interested in symmetry methods for di↵erential equations. Later, the involution analysis of polynomial systems and systems of di↵erential or di↵erence equations became a core theme in his research. Together with some of his students, he developed the theory of involutive bases out of the Janet–Riquier theory. As one application of these results to physics, he was always interested in the study of systems with constraints, in particular in Dirac theory. Another application in recent years consisted of designing structure preserving numerical methods for partial di↵erential equations. In the last years, he also revived the Thomas decomposition for polynomial di↵erential equations and applied it to numerous problems. One of his last significant results was an algorithmic solution of the linearization problem for ordinary di↵erential equations. For many years, Vladimir Gerdt also worked in the field of quantum computing. His group at JINR developed methods and computer algebra programs for studying quantum information processes and for modeling quantum systems. In particular, they applied a combination of computational invariant theory and involutive bases to a qualitative and quantitative study of entangled quantum states. Vladimir Gerdt was the au","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"1 - 2"},"PeriodicalIF":0.1,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3466895.3467684","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42047218","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}
Shashi Gowda, Yingbo Ma, Alessandro Cheli, Maja Gwóźdź, Viral B. Shah, A. Edelman, Chris Rackauckas
As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. Exploiting this feature, we demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We illustrate how this symbolic system improves numerical computing tasks by showcasing an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.
{"title":"High-performance symbolic-numerics via multiple dispatch","authors":"Shashi Gowda, Yingbo Ma, Alessandro Cheli, Maja Gwóźdź, Viral B. Shah, A. Edelman, Chris Rackauckas","doi":"10.1145/3511528.3511535","DOIUrl":"https://doi.org/10.1145/3511528.3511535","url":null,"abstract":"As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. Exploiting this feature, we demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We illustrate how this symbolic system improves numerical computing tasks by showcasing an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"92 - 96"},"PeriodicalIF":0.1,"publicationDate":"2021-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48476713","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}
This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in [4] to global function fields of odd characteristics. First, we present algorithm for checking if a given non-degenerate quadratic form is isotropic or hyperbolic. Next we devise a method for computing the dimension of the anisotropic part of a quadratic form. Finally we present algorithms computing two field invariants: the level and the Pythagoras number.
{"title":"Algorithms for quadratic forms over global function fields of odd characteristic","authors":"Mawunyo Kofi Darkey-Mensah","doi":"10.1145/3511528.3511530","DOIUrl":"https://doi.org/10.1145/3511528.3511530","url":null,"abstract":"This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in [4] to global function fields of odd characteristics. First, we present algorithm for checking if a given non-degenerate quadratic form is isotropic or hyperbolic. Next we devise a method for computing the dimension of the anisotropic part of a quadratic form. Finally we present algorithms computing two field invariants: the level and the Pythagoras number.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"55 1","pages":"68 - 72"},"PeriodicalIF":0.1,"publicationDate":"2021-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46087500","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}
One-block quantifier elimination is comprised of computing a semi-algebraic description of the projection of a semi-algebraic set or of deciding the truth of a semi-algebraic formula with a single quantifier.
一个块量词消去包括计算半代数集的投影的半代数描述，或者用单个量词决定半代数公式的真值。
{"title":"Towards fast one-block quantifier elimination through generalised critical values","authors":"Jérémy Berthomieu, A. Ferguson, M. S. E. Din","doi":"10.1145/3457341.3457348","DOIUrl":"https://doi.org/10.1145/3457341.3457348","url":null,"abstract":"One-block quantifier elimination is comprised of computing a semi-algebraic description of the projection of a semi-algebraic set or of deciding the truth of a semi-algebraic formula with a single quantifier.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"109 - 113"},"PeriodicalIF":0.1,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3457341.3457348","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46627131","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 go from calculus of scalar functions of one variable to multivariate calculus of vector-valued functions is a steep learning curve for many students. It takes a lot of practice to get used to the new concepts such as the directional derivatives, the di.erentiability, the many types of first order di.erential operators, parameterization of surfaces and the fundamental theorems of integrals, e.g. the Divergence Theorem and the Stokes Theorem. Along the learning process of mastering the skills, the students often need to check whether the intermediate steps in the tedious calculations are correct. Unfortunately, this is beyond the capability of an ordinary calculator and the answers provided at the end of the books. This is where computer algebra software, such as Mathematica, can come to students' help. With the developed symbolic computation tools, the students can tweak a given problem, solve a new one by hand and then check the answer against the result obtained by using the computer algebra software.
{"title":"Teaching multivariable calculus and tensor calculus with computer algebra software","authors":"M. Ou","doi":"10.1145/3465002.3465005","DOIUrl":"https://doi.org/10.1145/3465002.3465005","url":null,"abstract":"To go from calculus of scalar functions of one variable to multivariate calculus of vector-valued functions is a steep learning curve for many students. It takes a lot of practice to get used to the new concepts such as the directional derivatives, the di.erentiability, the many types of first order di.erential operators, parameterization of surfaces and the fundamental theorems of integrals, e.g. the Divergence Theorem and the Stokes Theorem. Along the learning process of mastering the skills, the students often need to check whether the intermediate steps in the tedious calculations are correct. Unfortunately, this is beyond the capability of an ordinary calculator and the answers provided at the end of the books. This is where computer algebra software, such as Mathematica, can come to students' help. With the developed symbolic computation tools, the students can tweak a given problem, solve a new one by hand and then check the answer against the result obtained by using the computer algebra software.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"134 - 135"},"PeriodicalIF":0.1,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3465002.3465005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45233984","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}
The sparsity t≪ deg(f) with respect to the basis Pn has been exploited—since [9] —in interpolation algorithms that reconstruct the degree/coefficient expansion (δj, cj)1≤j≤t from values ai = f(γi) at the arguments x ← γi ∈ K. Current algorithms for standard and Chebyshev bases use i = 1, . . . , N = t + B values when an upper bound B ≥ t is provided on input. The sparsity t can also be computed “on-the-fly” from N = 2t+ 1 values by a randomized algorithm which fails with probability O(ǫ deg(f)), where ǫ≪ 1 can be chosen on input. See [3] for a list of references. This note considers Dickson Polynomials for the basis in which a sparse representation is sought. Wang and Yucas [10, Remark 2.5] define the n-th degree Dickson Polynomials Dn,k(x, a) ∈ K[x] of the (k + 1)’st kind for a parameter a ∈ K, a 6= 0, and k ∈ Z≥0, k 6= 2 recursively as as follows:
{"title":"A note on sparse polynomial interpolation in Dickson polynomial basis","authors":"E. Imamoglu, E. Kaltofen","doi":"10.1145/3465002.3465003","DOIUrl":"https://doi.org/10.1145/3465002.3465003","url":null,"abstract":"The sparsity t≪ deg(f) with respect to the basis Pn has been exploited—since [9] —in interpolation algorithms that reconstruct the degree/coefficient expansion (δj, cj)1≤j≤t from values ai = f(γi) at the arguments x ← γi ∈ K. Current algorithms for standard and Chebyshev bases use i = 1, . . . , N = t + B values when an upper bound B ≥ t is provided on input. The sparsity t can also be computed “on-the-fly” from N = 2t+ 1 values by a randomized algorithm which fails with probability O(ǫ deg(f)), where ǫ≪ 1 can be chosen on input. See [3] for a list of references. This note considers Dickson Polynomials for the basis in which a sparse representation is sought. Wang and Yucas [10, Remark 2.5] define the n-th degree Dickson Polynomials Dn,k(x, a) ∈ K[x] of the (k + 1)’st kind for a parameter a ∈ K, a 6= 0, and k ∈ Z≥0, k 6= 2 recursively as as follows:","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"125 - 128"},"PeriodicalIF":0.1,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3465002.3465003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48164821","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}
In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then [EQUATION] This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If an integer of the form n = 4k + 1, k > 0 is prime, then ([EQUATION]) and (2) If an integer of the form n = 4k + 3, k ≥ 0 is prime, then [EQUATION]. In addition, the author proposes important conjectures based on the converse of the above theorems which aim to establish primality of n. These conjectures are scrutinized by the given combinatorial primality test algorithm which can also distinguish patterns of prime n whether it is of the form 4k + 1 or 4k + 3.
{"title":"Combinatorial primality test","authors":"M. R. Valluri","doi":"10.1145/3465002.3465004","DOIUrl":"https://doi.org/10.1145/3465002.3465004","url":null,"abstract":"In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then [EQUATION] This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If an integer of the form n = 4k + 1, k > 0 is prime, then ([EQUATION]) and (2) If an integer of the form n = 4k + 3, k ≥ 0 is prime, then [EQUATION]. In addition, the author proposes important conjectures based on the converse of the above theorems which aim to establish primality of n. These conjectures are scrutinized by the given combinatorial primality test algorithm which can also distinguish patterns of prime n whether it is of the form 4k + 1 or 4k + 3.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"129 - 133"},"PeriodicalIF":0.1,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3465002.3465004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47725434","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}
We propose solving the power flow equations using monodromy. We prove the variety under consideration decomposes into trivial and nontrivial subvarieties and that the nontrivial subvariety is irreducible. We also show various symmetries in the solutions. We finish by giving numerical results comparing monodromy against polyhedral and total degree homotopy methods and giving an example of a network where we can find all solutions to the power flow equation using monodromy where other homotopy techniques fail. This work gives hope that finding all solutions to the power flow equations for networks of realistic size is possible.
{"title":"Exploiting symmetry in the power flow equations using monodromy","authors":"J. Lindberg, N. Boston, B. Lesieutre","doi":"10.1145/3457341.3457346","DOIUrl":"https://doi.org/10.1145/3457341.3457346","url":null,"abstract":"We propose solving the power flow equations using monodromy. We prove the variety under consideration decomposes into trivial and nontrivial subvarieties and that the nontrivial subvariety is irreducible. We also show various symmetries in the solutions. We finish by giving numerical results comparing monodromy against polyhedral and total degree homotopy methods and giving an example of a network where we can find all solutions to the power flow equation using monodromy where other homotopy techniques fail. This work gives hope that finding all solutions to the power flow equations for networks of realistic size is possible.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"54 1","pages":"100 - 104"},"PeriodicalIF":0.1,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3457341.3457346","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48258669","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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