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Symbolic-Numeric Computation最新文献

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The nearest polynomial to multiple given polynomials with a given zero 与多个给定多项式最接近的多项式，且多项式为给定零
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631975
Hiroshi Sekigawa
{"title":"The nearest polynomial to multiple given polynomials with a given zero","authors":"Hiroshi Sekigawa","doi":"10.1145/2631948.2631975","DOIUrl":"https://doi.org/10.1145/2631948.2631975","url":null,"abstract":"The following type of problems have been well-studied in the area of symbolic-numeric computation for about twenty years: Given a polynomial f ∈ C[x] and a point z ∈ C, find the nearest polynomial f̃ ∈ C[x] to f with f̃(z) = 0. A common framework for such problems is described in [7]. In previous works, for example [3, 4, 7, 6], problems for one given polynomial were considered. Here, we consider a problem for multiple given polynomials. Through observation or by using different numerical algorithms for a given input data, we may obtain multiple polynomials being equal in theory but being slightly different each other. Thus, it is worth considering the problem for multiple polynomials. In this abstract, after the preliminaries, we define the nearest polynomial to multiple given polynomials. In the definition, we use a pair of norms to measure the nearness between polynomials. We remark the difficulty of the problem of finding the nearest polynomial depends on the norm pair. Finally, we describe an algorithm for the problem when both of the norms are the ∞-norm.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130812098","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}
引用次数: 4
Two variants of HJLS-PSLQ with applications 带应用程序的HJLS-PSLQ的两个变体
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631965
Yong Feng, Jingwei Chen, Wenyuan Wu
{"title":"Two variants of HJLS-PSLQ with applications","authors":"Yong Feng, Jingwei Chen, Wenyuan Wu","doi":"10.1145/2631948.2631965","DOIUrl":"https://doi.org/10.1145/2631948.2631965","url":null,"abstract":"The HJLS and PSLQ algorithms are the most popular algorithms for finding nontrivial integer relations for several real numbers. It has been already shown that PSLQ is essentially equivalent to HJLS under certain settings. We here call them HJLS-PSLQ.\u0000 In the present work, we provide two variants of HJLS-PSLQ. The first one is a new modification of Bailey and Broadhurst's multi-pair version. We prove the termination of our modification, while the original multi-pair version may not terminate. The second one is an incremental version of HJLS-PSLQ. For those applications requiring to call HJLS-PSLQ many times, such as finding the minimal polynomial of an algebraic number without knowing the degree, we show the incremental version is more efficient than HJLS-PSLQ, both theoretically and practically.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122140289","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}
引用次数: 5
Optimizing a linear function over a noncompact real algebraic variety 非紧实代数变量上线性函数的优化
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631957
Feng Guo, Chu Wang, L. Zhi
{"title":"Optimizing a linear function over a noncompact real algebraic variety","authors":"Feng Guo, Chu Wang, L. Zhi","doi":"10.1145/2631948.2631957","DOIUrl":"https://doi.org/10.1145/2631948.2631957","url":null,"abstract":"Our aim is to compute such a polynomial Φ of the least possible degree. In [3, 4], Rostalski and Sturmfels explored dualities and their interconnections in the context of polynomial optimization (1.1). Assuming that the feasible regionX is irreducible, compact and smooth, they showed that the optimal value function Φ is represented by the defining equation of the hypersurface dual to the projective closure of X [4, Theorem 5.23]. In the present paper, we prove this conclusion is still true for a noncompact real algebraic variety X, when X is irreducible, smooth and the recession cone of the closure of the convex hull co (X) of X is pointed.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130589088","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}
引用次数: 2
Finding a sparse solution of a class of linear differential equations by solving a nonlinear system 通过求解一个非线性系统求一类线性微分方程的稀疏解
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631963
Libin Jiao, Bo Yu
{"title":"Finding a sparse solution of a class of linear differential equations by solving a nonlinear system","authors":"Libin Jiao, Bo Yu","doi":"10.1145/2631948.2631963","DOIUrl":"https://doi.org/10.1145/2631948.2631963","url":null,"abstract":"In this paper, a new numerical method is proposed for finding a sparse solution of a class of linear differential equations with highly oscillatory coefficients. In contrast to spectral methods and finite element methods: 1) The numerical solution is represented by a linear combination of undetermined basis functions instead of a linear combination of predetermined basis functions; 2) A small nonlinear system is obtained rather than a large linear one and, the nonlinear system can be efficiently solved by Prony method; 3) The amount of computational work of our new method is independent of the parameter in the oscillatory coefficient. Some numerical examples are given to show that our new method is promising.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123762931","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}
引用次数: 0
Automated theorem proving for special functions: the next phase 特殊函数的自动定理证明:下一阶段
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631950
Lawrence Charles Paulson
{"title":"Automated theorem proving for special functions: the next phase","authors":"Lawrence Charles Paulson","doi":"10.1145/2631948.2631950","DOIUrl":"https://doi.org/10.1145/2631948.2631950","url":null,"abstract":"Automated theorem proving, in a nutshell, is the combination of symbolic logic with syntactic algorithms. A formal proof calculus is chosen with two criteria in mind: expressiveness and ease of automation. These desiderata pull in opposite directions: Boolean logic and linear arithmetic are decidable, so the answers to all questions can simply be calculated, but these theories are not very expressive. At the other extreme, a dependent type theory such as the calculus of constructions used in Coq [6] is highly expressive and flexible, but complicates automation; even basic rewriting is difficult. Higher-order logic is often seen as a suitable compromise, expressive enough to reason directly about sets and functions, while still admitting substantial automation (especially in the case of Isabelle [18]).","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124875772","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}
引用次数: 1
Safety verification of nonlinear systems based on rational invariants 基于有理不变量的非线性系统安全性验证
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631967
Wang Lin, Min Wu, Zhengfeng Yang, Zhenbing Zeng
{"title":"Safety verification of nonlinear systems based on rational invariants","authors":"Wang Lin, Min Wu, Zhengfeng Yang, Zhenbing Zeng","doi":"10.1145/2631948.2631967","DOIUrl":"https://doi.org/10.1145/2631948.2631967","url":null,"abstract":"where x ∈ R is the state variable, and f(x) is a vector of rational functions in x over Q. We consider the dynamics of (1) in a bounded domain of the state space R, given by Ψ , {x ∈ R|ψ1(x) ≥ 0 ∧ · · · ∧ ψr(x) ≥ 0}, with ψi(x) ∈ Q[x] for 1 ≤ i ≤ r. We say that the system (1) is safe if all trajectories of (1) starting from any state in the initial set, can not evolve to the unsafe states. We are interested in the problem of safety verification of nonlinear system (1), described as follows.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122812357","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}
引用次数: 0
The euclidean distance degree 欧氏距离度
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631951
J. Draisma, Emil Horobet, G. Ottaviani, B. Sturmfels, Rekha R. Thomas
{"title":"The euclidean distance degree","authors":"J. Draisma, Emil Horobet, G. Ottaviani, B. Sturmfels, Rekha R. Thomas","doi":"10.1145/2631948.2631951","DOIUrl":"https://doi.org/10.1145/2631948.2631951","url":null,"abstract":"The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for computation.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124994587","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}
引用次数: 19
Numerical linear system solving with parametric entries by error correction 采用误差校正方法求解参数项线性系统
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631956
Brice Boyer, E. Kaltofen
{"title":"Numerical linear system solving with parametric entries by error correction","authors":"Brice Boyer, E. Kaltofen","doi":"10.1145/2631948.2631956","DOIUrl":"https://doi.org/10.1145/2631948.2631956","url":null,"abstract":"We consider the problem of solving a full rank consistent linear system <i>A</i>(<i>u</i>)<b>x</b> = <i>b</i>(<i>u</i>) where the <i>m</i> x <i>n</i> matrix <i>A</i> and the <i>m</i>-dimensional vector <i>b</i> has entries that are polynomials in <i>u</i> over a field. We give an algorithm that computes the unique solution <b>x</b> = <b>f</b>(<i>u</i>)/<i>g</i>(<i>u</i>), which is a vector of rational functions, by evaluating the parameter <i>u</i> at distinct points. Those points ξ<sub>λ</sub> where the matrix <i>A</i> evaluates to a matrix <i>A</i>(ξ<sub>λ</sub>), with entries over the scalar field, of lower rank, or in the numeric setting to an ill-conditioned matrix, are not identified but accounted for by error-correcting code techniques. We also correct true errors where the evaluation at some <i>u</i> = ξ<sub>λ</sub> results in an erroneous, possibly full rank consistent and well-conditioned scalar linear system. Our algorithm generalizes Welch/Berlekamp decoding of Reed/Solomon error correcting codes and their numeric floating point counterparts.\u0000 We have implemented our algorithms with floating point arithmetic. For the determination of the exact numerator and denominator degrees and number of errors we use singular values based numeric rank computations. The arising linear systems for the error-corrected parametric solution are demonstrated to be well-conditioned even when the input scalars have noise. In several initial experiments we have shown that our approach is numerically stable even for larger systems <i>m</i> = <i>n</i> = 100, provided the degrees in the solution are small (≤ 2). For smaller systems <i>m</i> = <i>n</i> = 10 with higher degrees (≤ 20) the algorithm works similarly to rational function recovery. Our implementation can correct 13 true errors in both settings.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"76 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120839881","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}
引用次数: 7
Cleaning-up data for sparse model synthesis: when symbolic-numeric computation meets error-correcting codes 稀疏模型综合的数据清理:当符号-数值计算遇到纠错码时
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631949
E. Kaltofen
{"title":"Cleaning-up data for sparse model synthesis: when symbolic-numeric computation meets error-correcting codes","authors":"E. Kaltofen","doi":"10.1145/2631948.2631949","DOIUrl":"https://doi.org/10.1145/2631948.2631949","url":null,"abstract":"The discipline of symbolic computation contributes to mathematical model synthesis in several ways. One is the pioneering creation of interpolation algorithms that can account for sparsity in the resulting multi-dimensional models, for example, by Zippel [12], Ben-Or and Tiwari [1], and in their recent numerical counterparts by Giesbrecht-Labahn-Lee [5] and Kaltofen-Yang-Zhi [9].","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122381173","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}
引用次数: 1
Structure of symmetry of PDE: exploiting partially integrated systems PDE的对称结构:利用部分集成系统
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631962
I. Lisle, Tracy Shih-lung Huang, G. Reid
{"title":"Structure of symmetry of PDE: exploiting partially integrated systems","authors":"I. Lisle, Tracy Shih-lung Huang, G. Reid","doi":"10.1145/2631948.2631962","DOIUrl":"https://doi.org/10.1145/2631948.2631962","url":null,"abstract":"This work is part of a sequence in which we develop and refine algorithms for computer symmetry analysis of differential equations. We show how to exploit partially integrated forms of symmetry defining systems to assist the differential elimination algorithms that uncover structure of the Lie symmetry algebras. We thus incorporate a key advantage of heuristic integration methods, that of exploiting easy integrals of simple (e.g. one term) PDE that frequently occur in such analyses. A single unified method is given that computes structure constants whether the defining system is unsolved, or has been partially or completely integrated.\u0000 We also give a symbolic-numeric algorithm which for the first time can determine the structure of Lie symmetry algebras specified by defining systems that contain floating point coefficients. This algorithm incorporates a numerical version of the Cartan-Kuranishi prolongation projection algorithm from the geometry of differential equations.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121923119","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}
引用次数: 6
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