A. Bostan, F. Chyzak, Hadrien Notarantonio, M. S. E. Din
Discrete differential equations of order 1 relate polynomially a power series F(t,u) in t with polynomial coefficients in a ''catalytic'' variable~u and one of its specializations, say F(t,u). Such equations are ubiquitous in combinatorics, notably in the enumeration of maps and walks. When the solution F is unique, a celebrated result by Bousquet-Mélou and Jehanne, reminiscent of Popescu's theorem in commutative algebra, states that F is algebraic. We address algorithmic and complexity questions related to this result. In generic situations, we first revisit and analyze known algorithms, based either on polynomial elimination or on the guess-and-prove paradigm. We then design two new algorithms: the first has a geometric flavor, the second blends elimination and guess-and-prove. In the general case (no genericity assumptions), we prove that the total arithmetic size of the algebraic equations for $F(t,1)$ is bounded polynomially in the size of the input discrete differential equation, and that one can compute such equations in polynomial time.
{"title":"Algorithms for Discrete Differential Equations of Order 1","authors":"A. Bostan, F. Chyzak, Hadrien Notarantonio, M. S. E. Din","doi":"10.1145/3476446.3535471","DOIUrl":"https://doi.org/10.1145/3476446.3535471","url":null,"abstract":"Discrete differential equations of order 1 relate polynomially a power series F(t,u) in t with polynomial coefficients in a ''catalytic'' variable~u and one of its specializations, say F(t,u). Such equations are ubiquitous in combinatorics, notably in the enumeration of maps and walks. When the solution F is unique, a celebrated result by Bousquet-Mélou and Jehanne, reminiscent of Popescu's theorem in commutative algebra, states that F is algebraic. We address algorithmic and complexity questions related to this result. In generic situations, we first revisit and analyze known algorithms, based either on polynomial elimination or on the guess-and-prove paradigm. We then design two new algorithms: the first has a geometric flavor, the second blends elimination and guess-and-prove. In the general case (no genericity assumptions), we prove that the total arithmetic size of the algebraic equations for $F(t,1)$ is bounded polynomially in the size of the input discrete differential equation, and that one can compute such equations in polynomial time.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124541718","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 introduce a new type of reduction in a free difference module over a difference field that uses a generalization of the concept of effective order of a difference polynomial. Then we define the concept of a generalized characteristic set of such a module, establish some properties of these characteristic sets and use them to prove the existence, outline a method of computation and find invariants of a dimension polynomial in two variables associated with a finitely generated difference module. As a consequence of these results, we obtain a new type of bivariate dimension polynomials of finitely generated difference field extensions. We also explain the relationship between these dimension polynomials and the concept of Einstein's strength of a system of difference equations.
{"title":"Reduction with Respect to the Effective Order and a New Type of Dimension Polynomials of Difference Modules","authors":"A. Levin","doi":"10.1145/3476446.3535497","DOIUrl":"https://doi.org/10.1145/3476446.3535497","url":null,"abstract":"We introduce a new type of reduction in a free difference module over a difference field that uses a generalization of the concept of effective order of a difference polynomial. Then we define the concept of a generalized characteristic set of such a module, establish some properties of these characteristic sets and use them to prove the existence, outline a method of computation and find invariants of a dimension polynomial in two variables associated with a finitely generated difference module. As a consequence of these results, we obtain a new type of bivariate dimension polynomials of finitely generated difference field extensions. We also explain the relationship between these dimension polynomials and the concept of Einstein's strength of a system of difference equations.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124818124","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 this paper, we describe a new variation of the interpolation algorithm by Möller, proposed in a way that completely avoids Gröbner bases and does not need a term order, but only a well order on terms. This algorithm takes a set of functionals describing a Macaulay chain, namely, roughly speaking, the functionals are chosen and ordered in such a way that the first functional defines a zero-dimensional ideal and all the sets one gets by adding the functionals one after the other define zero-dimensional ideals as well. Starting from this set, the algorithm describes the zero-dimensional ideals of the Macaulay chain via a basis of the quotient algebra and Auzinger-Stetter matrices. Our algorithm shows how Degroebnerization can give symmetric representations to design ideals, a crucial feature for Algebraic Statistics, showing also that such feature can always be attained without using Gröbner bases and Buchberger reduction. The paper further investigates the potential applications of our new algorithm to describe design ideals into non-commutative algebraic settings.
{"title":"A Degroebnerization Approach to Algebraic Statistics","authors":"M. Ceria, Ferdinando Mora","doi":"10.1145/3476446.3536195","DOIUrl":"https://doi.org/10.1145/3476446.3536195","url":null,"abstract":"In this paper, we describe a new variation of the interpolation algorithm by Möller, proposed in a way that completely avoids Gröbner bases and does not need a term order, but only a well order on terms. This algorithm takes a set of functionals describing a Macaulay chain, namely, roughly speaking, the functionals are chosen and ordered in such a way that the first functional defines a zero-dimensional ideal and all the sets one gets by adding the functionals one after the other define zero-dimensional ideals as well. Starting from this set, the algorithm describes the zero-dimensional ideals of the Macaulay chain via a basis of the quotient algebra and Auzinger-Stetter matrices. Our algorithm shows how Degroebnerization can give symmetric representations to design ideals, a crucial feature for Algebraic Statistics, showing also that such feature can always be attained without using Gröbner bases and Buchberger reduction. The paper further investigates the potential applications of our new algorithm to describe design ideals into non-commutative algebraic settings.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"20 11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128451548","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}
There are two types of integer gcd algorithms: those which compute the sequence of remainders of Euclid's algorithm and those which build different sequences. The former are more difficult to validate and analyse, whereas the latter are simpler and more efficient. When one wants the euclidean remainders (for instance if one wants to compute continued fractions), only the former can be used. Our main focus is the subquadratic time Thull-Yap GCD algorithm, and in fact on its core computing a half gcd (TYHGCD). This algorithm is tricky due to the difficulty in correcting the remainder sequence that comes back from a recursive call. The aim of this work is to revise TYHGCD in order to implement it using GMP. We clarify some points of the algorithm, in particular the stopping conditions that are always difficult to set correctly. We add a base case to speed up the whole algorithm, using Jebelean's quadratic algorithm with a stopping condition. We give our own modified version and add the proofs where needed. We insist on the test phase for this algorithm, giving families of hard cases for all branches, some of which are rarely activated. We give some details on our implementation in GMP using low-level functions, adding some remarks on the use of fast multiplications techniques. We pay attention to the data structure needed to store partial quotients, enabling to navigate rapidly back and forth in the sequence of Euclidean remainders. Benchmarks are provided. Some comments are made on Lichtblau's algorithm, which is close in spirit to the Thull-Yap algorithm.
{"title":"Implementing the Thull-Yap Algorithm for Computing Euclidean Remainder Sequences","authors":"F. Morain","doi":"10.1145/3476446.3536188","DOIUrl":"https://doi.org/10.1145/3476446.3536188","url":null,"abstract":"There are two types of integer gcd algorithms: those which compute the sequence of remainders of Euclid's algorithm and those which build different sequences. The former are more difficult to validate and analyse, whereas the latter are simpler and more efficient. When one wants the euclidean remainders (for instance if one wants to compute continued fractions), only the former can be used. Our main focus is the subquadratic time Thull-Yap GCD algorithm, and in fact on its core computing a half gcd (TYHGCD). This algorithm is tricky due to the difficulty in correcting the remainder sequence that comes back from a recursive call. The aim of this work is to revise TYHGCD in order to implement it using GMP. We clarify some points of the algorithm, in particular the stopping conditions that are always difficult to set correctly. We add a base case to speed up the whole algorithm, using Jebelean's quadratic algorithm with a stopping condition. We give our own modified version and add the proofs where needed. We insist on the test phase for this algorithm, giving families of hard cases for all branches, some of which are rarely activated. We give some details on our implementation in GMP using low-level functions, adding some remarks on the use of fast multiplications techniques. We pay attention to the data structure needed to store partial quotients, enabling to navigate rapidly back and forth in the sequence of Euclidean remainders. Benchmarks are provided. Some comments are made on Lichtblau's algorithm, which is close in spirit to the Thull-Yap algorithm.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"157 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124393411","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 addition to the traditional theory and experimental pillars of science, we are witnessing the emergence of three more recent pillars, which are simulation, data analysis, and machine learning. All three recent pillars of science rely on computing but in different ways. Matrices, and sparse matrices in particular, play an outsized role in all three computing related pillars of science, which will be the topic of my talk. Solving systems of linear equations have traditionally driven research in sparse matrix computation for decades. Direct and iterative solvers, together with finite element computations, still account for the primary use case for sparse matrix data structures and algorithms. These solvers are the workhorses of scientific simulations. Modern methods for data analysis, such as matrix decompositions and graph analytics, often use the same underlying sparse matrix technology. The same can be said for various machine learning methods, where the data and/or the models are often sparse. I highlight some of the emerging use cases of sparse matrices outside the domain of solvers. These include graph computations, computational biology and emerging techniques in machine learning. A recurring theme in all these novel use cases is the concept of a semiring on which the sparse matrix computations are carried out. By overloading scalar addition and multiplication operators of a semiring, we can attack a much richer set of computational problems using the same sparse data structures and algorithms. This approach has been formalized by the GraphBLAS effort. I will illustrate one example application from each problem domain, together with the most computationally demanding sparse matrix primitive required for its efficient execution. I will also cover available software, such as various implementations of the GraphBLAS standard, that implement these sparse matrix primitives efficiently on various architectures.
{"title":"Sparse Matrices Powering Three Pillars of Science: Simulation, Data, and Learning","authors":"A. Buluç","doi":"10.1145/3476446.3535507","DOIUrl":"https://doi.org/10.1145/3476446.3535507","url":null,"abstract":"In addition to the traditional theory and experimental pillars of science, we are witnessing the emergence of three more recent pillars, which are simulation, data analysis, and machine learning. All three recent pillars of science rely on computing but in different ways. Matrices, and sparse matrices in particular, play an outsized role in all three computing related pillars of science, which will be the topic of my talk. Solving systems of linear equations have traditionally driven research in sparse matrix computation for decades. Direct and iterative solvers, together with finite element computations, still account for the primary use case for sparse matrix data structures and algorithms. These solvers are the workhorses of scientific simulations. Modern methods for data analysis, such as matrix decompositions and graph analytics, often use the same underlying sparse matrix technology. The same can be said for various machine learning methods, where the data and/or the models are often sparse. I highlight some of the emerging use cases of sparse matrices outside the domain of solvers. These include graph computations, computational biology and emerging techniques in machine learning. A recurring theme in all these novel use cases is the concept of a semiring on which the sparse matrix computations are carried out. By overloading scalar addition and multiplication operators of a semiring, we can attack a much richer set of computational problems using the same sparse data structures and algorithms. This approach has been formalized by the GraphBLAS effort. I will illustrate one example application from each problem domain, together with the most computationally demanding sparse matrix primitive required for its efficient execution. I will also cover available software, such as various implementations of the GraphBLAS standard, that implement these sparse matrix primitives efficiently on various architectures.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133232217","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 purpose of this article is to present some recent applications of computer algebra to answer structural and numerical questions in applied sciences. A first example concerns identifiability which is a pre-condition for safely running parameter estimation algorithms and obtaining reliable results. Identifiability addresses the question whether it is possible to uniquely estimate the model parameters for a given choice of measurement data and experimental input. As discussed in this paper, symbolic computation offers an efficient way to do this identifiability study and to extract more information on the parameter properties. A second example addressed hereafter is the diagnosability in nonlinear dynamical systems. The diagnosability is a prior study before considering diagnosis. The diagnosis of a system is defined as the detection and the isolation of faults (or localization and identification) acting on the system. The diagnosability study determines whether faults can be discriminated by the mathematical model from observations. These last years, the diagnosability and diagnosis have been enhanced by exploitting new analytical redundancy relations obtained from differential algebra algorithms and by the exploitation of their properties through computer algebra techniques.
{"title":"Applications of Computer Algebra to Parameter Analysis of Dynamical Systems","authors":"N. Verdière, S. Orange","doi":"10.1145/3476446.3535473","DOIUrl":"https://doi.org/10.1145/3476446.3535473","url":null,"abstract":"The purpose of this article is to present some recent applications of computer algebra to answer structural and numerical questions in applied sciences. A first example concerns identifiability which is a pre-condition for safely running parameter estimation algorithms and obtaining reliable results. Identifiability addresses the question whether it is possible to uniquely estimate the model parameters for a given choice of measurement data and experimental input. As discussed in this paper, symbolic computation offers an efficient way to do this identifiability study and to extract more information on the parameter properties. A second example addressed hereafter is the diagnosability in nonlinear dynamical systems. The diagnosability is a prior study before considering diagnosis. The diagnosis of a system is defined as the detection and the isolation of faults (or localization and identification) acting on the system. The diagnosability study determines whether faults can be discriminated by the mathematical model from observations. These last years, the diagnosability and diagnosis have been enhanced by exploitting new analytical redundancy relations obtained from differential algebra algorithms and by the exploitation of their properties through computer algebra techniques.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"100 7-8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130513135","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}
Some well-known correspondences between sets of linearly independent rows and columns of matrices over fields carry over to matrices over non-commutative rings without nontrivial zero divisors.
{"title":"On Linear Dependence of Rows and Columns in Matrices over Non-commutative Domains","authors":"S. Abramov, M. Petkovšek, A. Ryabenko","doi":"10.1145/3476446.3535490","DOIUrl":"https://doi.org/10.1145/3476446.3535490","url":null,"abstract":"Some well-known correspondences between sets of linearly independent rows and columns of matrices over fields carry over to matrices over non-commutative rings without nontrivial zero divisors.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125542450","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}
Solving systems of polynomial equations is a central problem in nonlinear and computational algebra. Since Buchberger's algorithm for computing Gröbner bases in the 60s, there has been a lot of progress in this domain. Moreover, these equations have been employed to model and solve problems from diverse disciplines such as biology, cryptography, and robotics. Currently, we have a good understanding of how to solve generic systems from a theoretical and algorithmic point of view. However, polynomial equations encountered in practice are usually structured, and so many properties and results about generic systems do not apply to them. For this reason, a common trend in the last decades has been to develop mathematical and algorithmic frameworks to exploit specific structures of systems of polynomials. Arguably, the most common structure is sparsity; that is, the polynomials of the systems only involve a few monomials. Since Bernstein, Khovanskii, and Kushnirenko's work on the expected number of solutions of sparse systems, toric geometry has been the default mathematical framework to employ sparsity. In particular, it is the crux of the matter behind the extension of classical tools to systems, such as resultant computations, homotopy continuation methods, and most recently, Gröbner bases. In this work, we will review these classical tools, their extensions, and recent progress in exploiting sparsity for solving polynomial systems.
{"title":"Solving Sparse Polynomial Systems using Gröbner Bases and Resultants","authors":"M. Bender","doi":"10.1145/3476446.3535498","DOIUrl":"https://doi.org/10.1145/3476446.3535498","url":null,"abstract":"Solving systems of polynomial equations is a central problem in nonlinear and computational algebra. Since Buchberger's algorithm for computing Gröbner bases in the 60s, there has been a lot of progress in this domain. Moreover, these equations have been employed to model and solve problems from diverse disciplines such as biology, cryptography, and robotics. Currently, we have a good understanding of how to solve generic systems from a theoretical and algorithmic point of view. However, polynomial equations encountered in practice are usually structured, and so many properties and results about generic systems do not apply to them. For this reason, a common trend in the last decades has been to develop mathematical and algorithmic frameworks to exploit specific structures of systems of polynomials. Arguably, the most common structure is sparsity; that is, the polynomials of the systems only involve a few monomials. Since Bernstein, Khovanskii, and Kushnirenko's work on the expected number of solutions of sparse systems, toric geometry has been the default mathematical framework to employ sparsity. In particular, it is the crux of the matter behind the extension of classical tools to systems, such as resultant computations, homotopy continuation methods, and most recently, Gröbner bases. In this work, we will review these classical tools, their extensions, and recent progress in exploiting sparsity for solving polynomial systems.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125190901","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 this paper, we consider nonlocal, nonlinear partial differential equations to model anisotropic dynamics of complex root sets of random polynomials under differentiation. These equations aim to generalise the recent PDE obtained by Stefan Steinerberger (2019) in the real case, and the PDE obtained by Sean O'Rourke and Stefan Steinerberger (2020) in the radial case, which amounts to work in 1D. These PDEs approximate dynamics of the complex roots for random polynomials of sufficiently high degree n. The unit of the time t corresponds to n differentiations, and the increment Δt corresponds to 1/n. The general situation in 2D, in particular for complex roots of real polynomials, was not yet addressed. The purpose of this paper is to present a first attempt in that direction. We assume that the roots are distributed according to a regular distribution with a local homogeneity property (defined in the text), and that this property is maintained under differentiation. This allows us to derive a system of two coupled equations to model the motion. Our system could be interesting for other applications. The paper is illustrated with examples computed with the Maple system.
{"title":"Modeling Complex Root Motion of Real Random Polynomials under Differentiation","authors":"A. Galligo","doi":"10.1145/3476446.3536194","DOIUrl":"https://doi.org/10.1145/3476446.3536194","url":null,"abstract":"In this paper, we consider nonlocal, nonlinear partial differential equations to model anisotropic dynamics of complex root sets of random polynomials under differentiation. These equations aim to generalise the recent PDE obtained by Stefan Steinerberger (2019) in the real case, and the PDE obtained by Sean O'Rourke and Stefan Steinerberger (2020) in the radial case, which amounts to work in 1D. These PDEs approximate dynamics of the complex roots for random polynomials of sufficiently high degree n. The unit of the time t corresponds to n differentiations, and the increment Δt corresponds to 1/n. The general situation in 2D, in particular for complex roots of real polynomials, was not yet addressed. The purpose of this paper is to present a first attempt in that direction. We assume that the roots are distributed according to a regular distribution with a local homogeneity property (defined in the text), and that this property is maintained under differentiation. This allows us to derive a system of two coupled equations to model the motion. Our system could be interesting for other applications. The paper is illustrated with examples computed with the Maple system.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127063073","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 present a symbolic-numeric Las Vegas algorithm for factoring Fuchsian ordinary differential operators with rational function coefficients. The new algorithm combines ideas of van Hoeij's "local-to-global" method and of the "analytic" approach proposed by van der Hoeven. It essentially reduces to the former in "easy" cases where the local-to-global method succeeds, and to an optimized variant of the latter in the "hardest" cases, while handling intermediate cases more efficiently than both.
提出了一种具有有理函数系数的Fuchsian常微分算子的符号-数值Las Vegas算法。新算法结合了van Hoeij的“局部到全局”方法和van der Hoeven提出的“解析”方法的思想。在局部到全局方法成功的“简单”情况下，它本质上简化为前者，在“最难”情况下，它简化为后者的优化变体，同时比两者更有效地处理中间情况。
{"title":"Symbolic-Numeric Factorization of Differential Operators","authors":"F. Chyzak, Alexandre Goyer, M. Mezzarobba","doi":"10.1145/3476446.3535503","DOIUrl":"https://doi.org/10.1145/3476446.3535503","url":null,"abstract":"We present a symbolic-numeric Las Vegas algorithm for factoring Fuchsian ordinary differential operators with rational function coefficients. The new algorithm combines ideas of van Hoeij's \"local-to-global\" method and of the \"analytic\" approach proposed by van der Hoeven. It essentially reduces to the former in \"easy\" cases where the local-to-global method succeeds, and to an optimized variant of the latter in the \"hardest\" cases, while handling intermediate cases more efficiently than both.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116857547","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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