We explore the problem of computing a nearest singular matrix to a given regular Hankel matrix while preserving the structure of the matrix. Nearness is measured in a matrix norm, or a componentwise norm. A recent result for structured condition numbers leads to an efficient algorithm in the spectral norm. We devise a parametrization of singular Hankel matrices, to discuss other norms.
{"title":"On computing nearest singular hankel matrices","authors":"M. Hitz","doi":"10.1145/1073884.1073909","DOIUrl":"https://doi.org/10.1145/1073884.1073909","url":null,"abstract":"We explore the problem of computing a nearest singular matrix to a given regular Hankel matrix while preserving the structure of the matrix. Nearness is measured in a matrix norm, or a componentwise norm. A recent result for structured condition numbers leads to an efficient algorithm in the spectral norm. We devise a parametrization of singular Hankel matrices, to discuss other norms.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122036434","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}
Extremalization and optimization problems are considered as of utmost importance both in the past and in the present days. Thus, in the early beginning of infinitesimal calculus the determination of maxima and minima is one of the stimulating problems causing the creation of calculus and one of the successful applications which stimulates its rapid further development. However, the maxima and minima involved are all of local character leading to equations difficult to be solved, not to say the inherent logical difficulties involving necessary and/or sufficient conditions to be satisfied. In recent years owing to creation of computers various kinds of numerical methods have been developed which involve usually some converging processes. These methods, besides such problems as stability or error-control, can hardly give the greatest or least value, or global-optimal value for short over the whole domain, supposed to exist in advance. However, the problem becomes very agreeable if we limit ourselves to the polynomial-type case. In fact, based on the classical treatment of polynomial equations-solving in ancient China and its modernization due to J.F. Ritt, we have discovered a Finite Kernel Theorem to the effect that a finite set of real values, to be called the finite kernel set of the given problem, may be so determined that all possible extremal values will be found among this finite set and the corresponding extremal zeros are then trivially determined. Clearly it will give the global optimal value over the whole domain in consideration, if it is already known to exist in some way. Associated packages wsolve and e_val have been given by D. K. Wang and had been applied with success in various kinds of problems, polynomial-definiteness, non-linear programming, etc., particularly problems involving inequalities.
{"title":"On a finite kernel theorem for polynomial-type optimization problems and some of its applications","authors":"Wu Wen-tsun","doi":"10.1145/1073884.1073887","DOIUrl":"https://doi.org/10.1145/1073884.1073887","url":null,"abstract":"Extremalization and optimization problems are considered as of utmost importance both in the past and in the present days. Thus, in the early beginning of infinitesimal calculus the determination of maxima and minima is one of the stimulating problems causing the creation of calculus and one of the successful applications which stimulates its rapid further development. However, the maxima and minima involved are all of local character leading to equations difficult to be solved, not to say the inherent logical difficulties involving necessary and/or sufficient conditions to be satisfied. In recent years owing to creation of computers various kinds of numerical methods have been developed which involve usually some converging processes. These methods, besides such problems as stability or error-control, can hardly give the greatest or least value, or global-optimal value for short over the whole domain, supposed to exist in advance. However, the problem becomes very agreeable if we limit ourselves to the polynomial-type case. In fact, based on the classical treatment of polynomial equations-solving in ancient China and its modernization due to J.F. Ritt, we have discovered a Finite Kernel Theorem to the effect that a finite set of real values, to be called the finite kernel set of the given problem, may be so determined that all possible extremal values will be found among this finite set and the corresponding extremal zeros are then trivially determined. Clearly it will give the global optimal value over the whole domain in consideration, if it is already known to exist in some way. Associated packages wsolve and e_val have been given by D. K. Wang and had been applied with success in various kinds of problems, polynomial-definiteness, non-linear programming, etc., particularly problems involving inequalities.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125871934","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 introduces an improved method for constructing cylindrical algebraic decompositions (CADs) for formulas with two polynomial equations as implied constraints. The fundamental idea is that neither of the varieties of the two polynomials is actually represented by the CAD the method produces, only the variety defined by their common zeros is represented. This allows for a substantially smaller projection factor set, and for a CAD with many fewer cells.In the current theory of CADs, the fundamental object is to decompose n-space into regions in which a polynomial equation is either identically true or identically false. With many polynomials, one seeks a decomposition into regions in which each polynomial equation is identically true or false independently. The results presented here are intended to be the first step in establishing a theory of CADs in which systems of equations are fundamental objects, so that given a system we seek a decomposition into regions in which the system is identically true or false --- which means each equation is no longer considered independently. Quantifier elimination problems of this form (systems of equations with side conditions) are quite common, and this approach has the potential to bring large problems of this type into the scope of what can be solved in practice. The special case of formulas containing two polynomial equations as constraints is an important one, but this work is also intended to be extended in the future to the more general case.
{"title":"On using bi-equational constraints in CAD construction","authors":"Christopher W. Brown, S. McCallum","doi":"10.1145/1073884.1073897","DOIUrl":"https://doi.org/10.1145/1073884.1073897","url":null,"abstract":"This paper introduces an improved method for constructing cylindrical algebraic decompositions (CADs) for formulas with two polynomial equations as implied constraints. The fundamental idea is that neither of the varieties of the two polynomials is actually represented by the CAD the method produces, only the variety defined by their common zeros is represented. This allows for a substantially smaller projection factor set, and for a CAD with many fewer cells.In the current theory of CADs, the fundamental object is to decompose n-space into regions in which a polynomial equation is either identically true or identically false. With many polynomials, one seeks a decomposition into regions in which each polynomial equation is identically true or false independently. The results presented here are intended to be the first step in establishing a theory of CADs in which systems of equations are fundamental objects, so that given a system we seek a decomposition into regions in which the system is identically true or false --- which means each equation is no longer considered independently. Quantifier elimination problems of this form (systems of equations with side conditions) are quite common, and this approach has the potential to bring large problems of this type into the scope of what can be solved in practice. The special case of formulas containing two polynomial equations as constraints is an important one, but this work is also intended to be extended in the future to the more general case.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129434034","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 present formulae for the computation of the partial degrees w.r.t. each variable of the implicit equation of a rational surface given by means of a proper parametrization. Moreover, when the parametrization is not proper we give upper bounds. These formulae generalize the results in [17] to the surface case, and they are based on the computation of the degree of the rational maps induced by the projections, onto the coordinate planes of the three dimensional space, of the input surface parametrization. In addition, using the results presented in [9] and [10], the formulae simply involve the computation of the degree of univariate polynomials directed determined from the parametrization by means of some univariate resultants and some polynomial gcds.
{"title":"Partial degree formulae for rational algebraic surfaces","authors":"S. Pérez-Díaz, J. Sendra","doi":"10.1145/1073884.1073926","DOIUrl":"https://doi.org/10.1145/1073884.1073926","url":null,"abstract":"In this paper, we present formulae for the computation of the partial degrees w.r.t. each variable of the implicit equation of a rational surface given by means of a proper parametrization. Moreover, when the parametrization is not proper we give upper bounds. These formulae generalize the results in [17] to the surface case, and they are based on the computation of the degree of the rational maps induced by the projections, onto the coordinate planes of the three dimensional space, of the input surface parametrization. In addition, using the results presented in [9] and [10], the formulae simply involve the computation of the degree of univariate polynomials directed determined from the parametrization by means of some univariate resultants and some polynomial gcds.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114495411","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}
A key step in the computation of the unitary dual of a Lie group is the determination if certain rational symmetric matrices are positive semi-definite. The size of some of the computations dictates that high performance integer matrix computations be used. We explore the feasibility of this approach by developing three algorithms for integer symmetric matrix signature and studying their performance both asymptotically and experimentally on a particular matrix family constructed from the exceptional Weyl group E8. We conclude that the computation is doable, with a parallel implementation needed for the largest representations.
{"title":"Signature of symmetric rational matrices and the unitary dual of lie groups","authors":"Jeffrey Adams, B. D. Saunders, Z. Wan","doi":"10.1145/1073884.1073889","DOIUrl":"https://doi.org/10.1145/1073884.1073889","url":null,"abstract":"A key step in the computation of the unitary dual of a Lie group is the determination if certain rational symmetric matrices are positive semi-definite. The size of some of the computations dictates that high performance integer matrix computations be used. We explore the feasibility of this approach by developing three algorithms for integer symmetric matrix signature and studying their performance both asymptotically and experimentally on a particular matrix family constructed from the exceptional Weyl group E8. We conclude that the computation is doable, with a parallel implementation needed for the largest representations.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"128 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125271112","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}
Starting from the well-known factorization of linear ordinary differential equations, we define the generalized Loewy decomposition for a D-module. To this end, for any module I, overmodules J ⊇ I are constructed. They subsume the conventional factorization as special cases. Furthermore, the new concept of the module of relative syzygies Syz(I,J) is introduced. The invariance of this module and its solution space w.r.t. the set of generators is shown. We design an algorithm which constructs the Loewy-decomposition for finite-dimensional and some kinds of general D modules. These results are applied for solving various second- and third-order linear partial differential equations.
{"title":"Generalized Loewy-decomposition of d-modules","authors":"D. Grigoriev, F. Schwarz","doi":"10.1145/1073884.1073908","DOIUrl":"https://doi.org/10.1145/1073884.1073908","url":null,"abstract":"Starting from the well-known factorization of linear ordinary differential equations, we define the generalized Loewy decomposition for a D-module. To this end, for any module I, overmodules J ⊇ I are constructed. They subsume the conventional factorization as special cases. Furthermore, the new concept of the module of relative syzygies Syz(I,J) is introduced. The invariance of this module and its solution space w.r.t. the set of generators is shown. We design an algorithm which constructs the Loewy-decomposition for finite-dimensional and some kinds of general D modules. These results are applied for solving various second- and third-order linear partial differential equations.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124847737","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}
Blocked versions of the Lanczos procedure have been successfully applied to sample nullspace elements of very large sparse matrices over small finite fields. The heuristic currently in use, namely, Montgomery's method [10], is unreliable for certain input matrices. This paper introduces a new biconditional block Lanczos approach based on lookahead, a technique designed to improve the reliability of the scalar Lanczos algorithm. Empirical data show that the performance of the lookahead-based algorithm is competitive with that of Montgomery's heuristic when their relative reliability is taken into account. The reliability of this new algorithm for arbitrary matrices over small finite fields is then established. In the process, some results on the ranks of certain submatrices of a randomly determined block Hankel matrix are established. These results may be applicable in other contexts, such as Coppersmith's block Wiedemann algorithm [3].
{"title":"A reliable block Lanczos algorithm over small finite fields","authors":"B. Hovinen, W. Eberly","doi":"10.1145/1073884.1073910","DOIUrl":"https://doi.org/10.1145/1073884.1073910","url":null,"abstract":"Blocked versions of the Lanczos procedure have been successfully applied to sample nullspace elements of very large sparse matrices over small finite fields. The heuristic currently in use, namely, Montgomery's method [10], is unreliable for certain input matrices. This paper introduces a new biconditional block Lanczos approach based on lookahead, a technique designed to improve the reliability of the scalar Lanczos algorithm. Empirical data show that the performance of the lookahead-based algorithm is competitive with that of Montgomery's heuristic when their relative reliability is taken into account. The reliability of this new algorithm for arbitrary matrices over small finite fields is then established. In the process, some results on the ranks of certain submatrices of a randomly determined block Hankel matrix are established. These results may be applicable in other contexts, such as Coppersmith's block Wiedemann algorithm [3].","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124905321","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 study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal M is bounded by the product of the regularity of M by the degree of M. In some special cases, such as partial degree truncation, this estimate carries over to total complexity. This leads to complexity improvements for some basic algorithms with algebraic numbers, and some polynomial system solving algorithms.
{"title":"Multivariate power series multiplication","authors":"É. Schost","doi":"10.1145/1073884.1073925","DOIUrl":"https://doi.org/10.1145/1073884.1073925","url":null,"abstract":"We study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal M is bounded by the product of the regularity of M by the degree of M. In some special cases, such as partial degree truncation, this estimate carries over to total complexity. This leads to complexity improvements for some basic algorithms with algebraic numbers, and some polynomial system solving algorithms.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130314654","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}
Given a second order linear differential equations with coefficients in a field k=C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein's theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semi-invariants.
{"title":"Solving second order linear differential equations with Klein's theorem","authors":"M. V. Hoeij, Jacques-Arthur Weil","doi":"10.1145/1073884.1073931","DOIUrl":"https://doi.org/10.1145/1073884.1073931","url":null,"abstract":"Given a second order linear differential equations with coefficients in a field k=C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein's theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semi-invariants.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113990342","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 provide algorithms that find, in case of existence, indefinite nested sum extensions in which a (creative) telescoping solution can be expressed with minimal nested depth.
{"title":"Finding telescopers with minimal depth for indefinite nested sum and product expressions","authors":"Carsten Schneider","doi":"10.1145/1073884.1073924","DOIUrl":"https://doi.org/10.1145/1073884.1073924","url":null,"abstract":"We provide algorithms that find, in case of existence, indefinite nested sum extensions in which a (creative) telescoping solution can be expressed with minimal nested depth.","PeriodicalId":311546,"journal":{"name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126822554","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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