We consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We start by providing a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone. It computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. An exact SOS decomposition is obtained thanks to the perturbation terms. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and simply exponential in the number of variables. Next, we apply this algorithm to compute exact Polya and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also compare the implementation of our algorithms with existing methods in computer algebra including cylindrical algebraic decomposition and critical point method.
{"title":"On Exact Polya and Putinar's Representations","authors":"Victor Magron, M. S. E. Din","doi":"10.1145/3208976.3208986","DOIUrl":"https://doi.org/10.1145/3208976.3208986","url":null,"abstract":"We consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We start by providing a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone. It computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. An exact SOS decomposition is obtained thanks to the perturbation terms. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and simply exponential in the number of variables. Next, we apply this algorithm to compute exact Polya and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also compare the implementation of our algorithms with existing methods in computer algebra including cylindrical algebraic decomposition and critical point method.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"303 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121738065","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 novel recursive algorithm for reducing a symmetric matrix to a triangular factorization which reveals the rank profile matrix. That is, the algorithm computes a factorization P TA P = L D L T where P is a permutation matrix, L is lower triangular with a unit diagonal and D is symmetric block diagonal with 1 x 1 and 2 x 2 antidiagonal blocks. This algorithm requires O(n2rømega-2) arithmetic operations, with n the dimension of the matrix, r its rank and ømega an admissible exponent for matrix multiplication. Furthermore, experimental results demonstrate that our algorithm has very good performance: its computational speed matches that of its numerical counterpart and is twice as fast as the unsymmetric exact Gaussian factorization. By adapting the pivoting strategy developed in the unsymmetric case, we show how to recover the rank profile matrix from the permutation matrix and the support of the block-diagonal matrix. We also note that there is an obstruction in characteristic 2 for revealing the rank profile matrix, which requires to relax the shape of the block diagonal by allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient. This relaxed decomposition can then be transformed into a standard PLDLTP T decomposition at a negligible cost.
我们提出了一种新的递归算法，将一个对称矩阵化为一个揭示秩轮廓矩阵的三角形分解。即算法计算一个分解P TA P = L D L T，其中P为置换矩阵，L为具有单位对角的下三角形，D为具有1 × 1和2 × 2反对角块的对称块对角。该算法需要O(n2rømega-2)次算术运算，其中n为矩阵的维数，r为矩阵的秩，而ømega是矩阵乘法的可接受指数。此外，实验结果表明，我们的算法具有非常好的性能，其计算速度与数值对应的计算速度相当，并且是非对称精确高斯分解的两倍。通过采用在非对称情况下开发的旋转策略，我们展示了如何从排列矩阵和块对角矩阵的支持中恢复秩轮廓矩阵。我们还注意到，在特征2中有一个阻碍，用于显示秩轮廓矩阵，这需要通过允许二维块具有非零的右下系数来放松块对角线的形状。然后可以将这种松弛分解转换为标准PLDLTP T分解，成本可以忽略不计。
{"title":"Symmetric Indefinite Triangular Factorization Revealing the Rank Profile Matrix","authors":"J. Dumas, Clément Pernet","doi":"10.1145/3208976.3209019","DOIUrl":"https://doi.org/10.1145/3208976.3209019","url":null,"abstract":"We present a novel recursive algorithm for reducing a symmetric matrix to a triangular factorization which reveals the rank profile matrix. That is, the algorithm computes a factorization P TA P = L D L T where P is a permutation matrix, L is lower triangular with a unit diagonal and D is symmetric block diagonal with 1 x 1 and 2 x 2 antidiagonal blocks. This algorithm requires O(n2rømega-2) arithmetic operations, with n the dimension of the matrix, r its rank and ømega an admissible exponent for matrix multiplication. Furthermore, experimental results demonstrate that our algorithm has very good performance: its computational speed matches that of its numerical counterpart and is twice as fast as the unsymmetric exact Gaussian factorization. By adapting the pivoting strategy developed in the unsymmetric case, we show how to recover the rank profile matrix from the permutation matrix and the support of the block-diagonal matrix. We also note that there is an obstruction in characteristic 2 for revealing the rank profile matrix, which requires to relax the shape of the block diagonal by allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient. This relaxed decomposition can then be transformed into a standard PLDLTP T decomposition at a negligible cost.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128742997","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}
Evangelos Bartzos, I. Emiris, Jan Legerský, Elias P. Tsigaridas
The number of embeddings of minimally rigid graphs in RD is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap between upper and lower bounds is still enormous. Specific values and its asymptotic behavior are major and fascinating open problems in rigidity theory. Our work considers the maximal number of real embeddings of minimally rigid graphs in R3. We modify a commonly used parametric semi-algebraic formulation that exploits the Cayley-Menger determinant to minimize the a priori number of complex embeddings, where the parameters correspond to edge lengths. To cope with the huge dimension of the parameter space and find specializations of the parameters that maximize the number of real embeddings, we introduce a method based on coupler curves that makes the sampling feasible for spatial minimally rigid graphs. Our methodology results in the first full classification of the number of real embeddings of graphs with 7 vertices in R3, which was the smallest open case. Building on this and certain 8-vertex graphs, we improve the previously known general lower bound on the maximum number of real embeddings in R3.
{"title":"On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs","authors":"Evangelos Bartzos, I. Emiris, Jan Legerský, Elias P. Tsigaridas","doi":"10.1145/3208976.3208994","DOIUrl":"https://doi.org/10.1145/3208976.3208994","url":null,"abstract":"The number of embeddings of minimally rigid graphs in RD is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap between upper and lower bounds is still enormous. Specific values and its asymptotic behavior are major and fascinating open problems in rigidity theory. Our work considers the maximal number of real embeddings of minimally rigid graphs in R3. We modify a commonly used parametric semi-algebraic formulation that exploits the Cayley-Menger determinant to minimize the a priori number of complex embeddings, where the parameters correspond to edge lengths. To cope with the huge dimension of the parameter space and find specializations of the parameters that maximize the number of real embeddings, we introduce a method based on coupler curves that makes the sampling feasible for spatial minimally rigid graphs. Our methodology results in the first full classification of the number of real embeddings of graphs with 7 vertices in R3, which was the smallest open case. Building on this and certain 8-vertex graphs, we improve the previously known general lower bound on the maximum number of real embeddings in R3.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128077843","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}
Wen-Ding Li, Ming-Shing Chen, Po-Chun Kuo, Chen-Mou Cheng, Bo-Yin Yang
In ISSAC 2017, van der Hoeven and Larrieu showed that evaluating a polynomial P ın Fq [x] of degree
在ISSAC 2017中，van der Hoeven和Larrieu表明，在Fqd的所有n个单位根上计算次数
{"title":"Frobenius Additive Fast Fourier Transform","authors":"Wen-Ding Li, Ming-Shing Chen, Po-Chun Kuo, Chen-Mou Cheng, Bo-Yin Yang","doi":"10.1145/3208976.3208998","DOIUrl":"https://doi.org/10.1145/3208976.3208998","url":null,"abstract":"In ISSAC 2017, van der Hoeven and Larrieu showed that evaluating a polynomial P ın Fq [x] of degree <n at all n -th roots of unity in Fqd can essentially be computed d times faster than evaluating Q ın Fqd x at all these roots, assuming Fqd contains a primitive n -th root of unity. Termed the Frobenius FFT, this discovery has a profound impact on polynomial multiplication, especially for multiplying binary polynomials, which finds ample application in coding theory and cryptography. In this paper, we show that the theory of Frobenius FFT beautifully generalizes to a class of additive FFT developed by Cantor and Gao-Mateer. Furthermore, we demonstrate the power of Frobenius additive FFT for q=2: to multiply two binary polynomials whose product is of degree <256, the new technique requires only 29,005 bit operations, while the best result previously reported was 33,397. To the best of our knowledge, this is the first time that FFT-based multiplication outperforms Karatsuba and the like at such a low degree in terms of bit-operation count.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122707919","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}
Let A0, ..., An be m x m symmetric matrices with entries in Q, and let A(x) be the linear pencil A0+x1 A1 + ··· + xn An, where x=(x1,...,xn) are unknowns. The linear matrix inequality (LMI) A(x) ≥ 0 defines the subset of Rn, called spectrahedron, containing all points x such that A(x) has non-negative eigenvalues. The minimization of linear functions over spectrahedra is called semidefinite programming (SDP). Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for solving SDP are mostly based on the interior point method, assuming some non-degeneracy properties such as the existence of interior points in the admissible set. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice but we prove that solving such problems can be done in polynomial time if either n or m is fixed.
{"title":"Exact Algorithms for Semidefinite Programs with Degenerate Feasible Set","authors":"D. Henrion, Simone Naldi, M. S. E. Din","doi":"10.1145/3208976.3209022","DOIUrl":"https://doi.org/10.1145/3208976.3209022","url":null,"abstract":"Let A0, ..., An be m x m symmetric matrices with entries in Q, and let A(x) be the linear pencil A0+x1 A1 + ··· + xn An, where x=(x1,...,xn) are unknowns. The linear matrix inequality (LMI) A(x) ≥ 0 defines the subset of Rn, called spectrahedron, containing all points x such that A(x) has non-negative eigenvalues. The minimization of linear functions over spectrahedra is called semidefinite programming (SDP). Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for solving SDP are mostly based on the interior point method, assuming some non-degeneracy properties such as the existence of interior points in the admissible set. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice but we prove that solving such problems can be done in polynomial time if either n or m is fixed.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114381443","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 new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs and the erroneous output. Their running time is softly linear in the number of nonzero entries in these matrices when the number of errors is sufficiently small, and they also incorporate fast matrix multiplication so that the cost scales well when the number of errors is large. These algorithms build on the recent result of Gasieniec et al (2017) on correcting matrix products, as well as existing work on verification algorithms, sparse low-rank linear algebra, and sparse polynomial interpolation.
{"title":"Error Correction in Fast Matrix Multiplication and Inverse","authors":"Daniel S. Roche","doi":"10.1145/3208976.3209001","DOIUrl":"https://doi.org/10.1145/3208976.3209001","url":null,"abstract":"We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs and the erroneous output. Their running time is softly linear in the number of nonzero entries in these matrices when the number of errors is sufficiently small, and they also incorporate fast matrix multiplication so that the cost scales well when the number of errors is large. These algorithms build on the recent result of Gasieniec et al (2017) on correcting matrix products, as well as existing work on verification algorithms, sparse low-rank linear algebra, and sparse polynomial interpolation.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126424732","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 extends the classical Hermite-Ostrogradsky reduction for rational functions to more general functions in primitive extensions of certain types. For an element f in such an extension K , the extended reduction decomposes f as the sum of a derivative in K and another element r such that f has an antiderivative in K if and only if r=0; and f has an elementary antiderivative over K if and only if r is a linear combination of logarithmic derivatives over the constants when K is a logarithmic extension. Moreover, r is minimal in some sense. Additive decompositions may lead to reduction-based creative-telescoping methods for nested logarithmic functions, which are not necessarily D -finite.
{"title":"Additive Decompositions in Primitive Extensions","authors":"Shaoshi Chen, Hao Du, Ziming Li","doi":"10.1145/3208976.3208987","DOIUrl":"https://doi.org/10.1145/3208976.3208987","url":null,"abstract":"This paper extends the classical Hermite-Ostrogradsky reduction for rational functions to more general functions in primitive extensions of certain types. For an element f in such an extension K , the extended reduction decomposes f as the sum of a derivative in K and another element r such that f has an antiderivative in K if and only if r=0; and f has an elementary antiderivative over K if and only if r is a linear combination of logarithmic derivatives over the constants when K is a logarithmic extension. Moreover, r is minimal in some sense. Additive decompositions may lead to reduction-based creative-telescoping methods for nested logarithmic functions, which are not necessarily D -finite.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125118346","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 the chordal graph structures of polynomial sets appearing in triangular decomposition in top-down style are studied when the input polynomial set has a chordal associated graph. We prove that the associated graph of one specific triangular set computed in any algorithm for triangular decomposition in top-down style is a subgraph of the chordal graph of the input polynomial set and that all the polynomial sets, including all the computed triangular sets, appearing in one specific algorithm for triangular decomposition in top-down style (Wang's method) have associated graphs which are subgraphs of the chordal graph of the input polynomial set.
{"title":"On the Chordality of Polynomial Sets in Triangular Decomposition in Top-Down Style","authors":"Chenqi Mou, Yang Bai","doi":"10.1145/3208976.3208997","DOIUrl":"https://doi.org/10.1145/3208976.3208997","url":null,"abstract":"In this paper the chordal graph structures of polynomial sets appearing in triangular decomposition in top-down style are studied when the input polynomial set has a chordal associated graph. We prove that the associated graph of one specific triangular set computed in any algorithm for triangular decomposition in top-down style is a subgraph of the chordal graph of the input polynomial set and that all the polynomial sets, including all the computed triangular sets, appearing in one specific algorithm for triangular decomposition in top-down style (Wang's method) have associated graphs which are subgraphs of the chordal graph of the input polynomial set.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127191070","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}
For a given computational problem, a certificate is a piece of data that one (the prover) attaches to the output with the aim of allowing efficient verification (by the verifier) that this output is correct. Here, we consider the minimal approximant basis problem, for which the fastest known algorithms output a polynomial matrix of dimensions m x m and average degree D/m using O~(mømega D/m) field operations. We propose a certificate which, for typical instances of the problem, is computed by the prover using O(mømega D/m) additional field operations and allows verification of the approximant basis by a Monte Carlo algorithm with cost bound O(mømega + m D). Besides theoretical interest, our motivation also comes from the fact that approximant bases arise in most of the fastest known algorithms for linear algebra over the univariate polynomials; thus, this work may help in designing certificates for other polynomial matrix computations. Furthermore, cryptographic challenges such as breaking records for discrete logarithm computations or for integer factorization rely in particular on computing minimal approximant bases for large instances: certificates can then be used to provide reliable computation on outsourced and error-prone clusters.
对于给定的计算问题，证书是一个人(证明者)附加到输出的一段数据，目的是允许(由验证者)有效地验证该输出是正确的。在这里，我们考虑最小近似基问题，对于该问题，已知最快的算法使用O~(mømega D/m)现场操作输出维数为m x m，平均度为D/m的多项式矩阵。对于该问题的典型实例，我们提出了一个证书，该证书由证明者使用O(mømega D/m)额外的现场操作计算，并允许通过成本限为O(mømega + m D)的蒙特卡罗算法验证近似基。除了理论兴趣之外，我们的动机还来自于这样一个事实，即近似基出现在大多数已知最快的线性代数单变量多项式算法中;因此，这项工作可能有助于为其他多项式矩阵计算设计证书。此外，诸如打破离散对数计算或整数分解记录之类的加密挑战特别依赖于计算大型实例的最小近似基:然后可以使用证书在外包和易出错的集群上提供可靠的计算。
{"title":"Certification of Minimal Approximant Bases","authors":"Pascal Giorgi, Vincent Neiger","doi":"10.1145/3208976.3208991","DOIUrl":"https://doi.org/10.1145/3208976.3208991","url":null,"abstract":"For a given computational problem, a certificate is a piece of data that one (the prover) attaches to the output with the aim of allowing efficient verification (by the verifier) that this output is correct. Here, we consider the minimal approximant basis problem, for which the fastest known algorithms output a polynomial matrix of dimensions m x m and average degree D/m using O~(mømega D/m) field operations. We propose a certificate which, for typical instances of the problem, is computed by the prover using O(mømega D/m) additional field operations and allows verification of the approximant basis by a Monte Carlo algorithm with cost bound O(mømega + m D). Besides theoretical interest, our motivation also comes from the fact that approximant bases arise in most of the fastest known algorithms for linear algebra over the univariate polynomials; thus, this work may help in designing certificates for other polynomial matrix computations. Furthermore, cryptographic challenges such as breaking records for discrete logarithm computations or for integer factorization rely in particular on computing minimal approximant bases for large instances: certificates can then be used to provide reliable computation on outsourced and error-prone clusters.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"128 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115253917","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 consider the computation of two normal forms for matrices over the univariate polynomials: the Popov form and the Hermite form. For matrices which are square and nonsingular, deterministic algorithms with satisfactory cost bounds are known. Here, we present deterministic, fast algorithms for rectangular input matrices. The obtained cost bound for the Popov form matches the previous best known randomized algorithm, while the cost bound for the Hermite form improves on the previous best known ones by a factor which is at least the largest dimension of the input matrix.
{"title":"Computing Popov and Hermite Forms of Rectangular Polynomial Matrices","authors":"Vincent Neiger, J. Rosenkilde, Grigory Solomatov","doi":"10.1145/3208976.3208988","DOIUrl":"https://doi.org/10.1145/3208976.3208988","url":null,"abstract":"We consider the computation of two normal forms for matrices over the univariate polynomials: the Popov form and the Hermite form. For matrices which are square and nonsingular, deterministic algorithms with satisfactory cost bounds are known. Here, we present deterministic, fast algorithms for rectangular input matrices. The obtained cost bound for the Popov form matches the previous best known randomized algorithm, while the cost bound for the Hermite form improves on the previous best known ones by a factor which is at least the largest dimension of the input matrix.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130369842","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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