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An algorithm to compute certain euler characteristics and Chern-Schwartz-MacPherson classes 一种计算欧拉特征和陈-施瓦茨-麦克弗森类的算法
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631972
M. Helmer
{"title":"An algorithm to compute certain euler characteristics and Chern-Schwartz-MacPherson classes","authors":"M. Helmer","doi":"10.1145/2631948.2631972","DOIUrl":"https://doi.org/10.1145/2631948.2631972","url":null,"abstract":"Let <i>V</i> be a possibly singular scheme-theoretic global complete intersection subscheme of P<sup><i>n</i></sup> and assume that <i>V</i> can be written as the intersection of <i>j</i> hypersurfaces such that the intersection of <i>j</i> -- 1 of the hypersurfaces is smooth (scheme theoretically). Using a result of Fullwood [5] we develop a probabilistic algorithm to compute the Chern-Schwartz-MacPherson class (<i>c</i><sub><i>SM</i></sub>) and Euler characteristic of <i>V</i>. This algorithm complements existing algorithms by providing performance improvements in the computation of the <i>c</i><sub><i>SM</i></sub> class and Euler characteristic for schemes having the special structure described above.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"79 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":"127946690","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
A recursive decision method for termination of linear programs 线性规划终止的递归判定方法
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631966
Yi Li
{"title":"A recursive decision method for termination of linear programs","authors":"Yi Li","doi":"10.1145/2631948.2631966","DOIUrl":"https://doi.org/10.1145/2631948.2631966","url":null,"abstract":"In their CAV 2004 and 2006 papers, Tiwari and Braverman have proved that, for a class of linear programs over the reals, termination is decidable. In this paper, we propose a new algorithm to decide whether a program of the same class terminates or not. In our approach, a program with an assignment matrix having a single Jordan block or having several Jordan blocks with the same eigenvalue is treated as a basic program to which we reduce a program with arbitrary assignment matrices in a recursive process. Furthermore, if a basic program is non-terminating, our method constructs at least one point on which a given basic program does not terminate. In contrast, for a non-terminating basic program, in most cases, the methods of Tiwari and Braverman provide only a so-called N-nonterminating point. Also, different from their methods, we do not need to guess a dominant term from every loop condition in our recursive procedure.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"17 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":"132635660","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
Isotopic epsilon-meshing of real algebraic space curves 实代数空间曲线的同位素epsilon网格划分
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631970
Kai Jin, Jin-San Cheng
{"title":"Isotopic epsilon-meshing of real algebraic space curves","authors":"Kai Jin, Jin-San Cheng","doi":"10.1145/2631948.2631970","DOIUrl":"https://doi.org/10.1145/2631948.2631970","url":null,"abstract":"Based on an efficient generic position checking method and on a method to solve bivariate polynomial systems, we give a new algorithm to compute the topology of an algebraic space curve. Compared to the method presented by the authors, in a joint work with Lazard, the new algorithm is efficient because of two reasons. One is the bitsize of the coefficients that may appear in projections is improved. The other is that one projection is enough for most general case in the new algorithm. We also give an ε-meshing of the space curve after we obtain its topology. Many nontrivial experiments show the efficiency of the algorithm.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"54 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":"122214632","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: 10
On the interplay between asymptotic and numerical methods to solve differential equations problems 解微分方程问题的渐近方法与数值方法的相互作用
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631974
R. Spigler
{"title":"On the interplay between asymptotic and numerical methods to solve differential equations problems","authors":"R. Spigler","doi":"10.1145/2631948.2631974","DOIUrl":"https://doi.org/10.1145/2631948.2631974","url":null,"abstract":"Asymptotic and numerical methods usually represent two independent approaches to solve applied mathematical problems, in particular for ordinary or partial differential equations. Here we present some cases when both techniques are used simultaneously, in a complementary way, rather than being considered as an alternative to each other.","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":"131066489","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
Numerical and geometric properties of a method for finding points on real solution components 求实解分量上点的一种方法的数值和几何性质
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631969
Wenyuan Wu, G. Reid, Yong Feng
{"title":"Numerical and geometric properties of a method for finding points on real solution components","authors":"Wenyuan Wu, G. Reid, Yong Feng","doi":"10.1145/2631948.2631969","DOIUrl":"https://doi.org/10.1145/2631948.2631969","url":null,"abstract":"We consider a critical point method developed in our earlier work for finding certain solution (witness) points on real solution components of real polynomial systems of equations. The method finds points that are critical points of the distance from a plane to the component with the requirement that certain regularity conditions are satisfied. In this paper we analyze the numerical stability of the method. We aim to find at least one well conditioned witness point on each connected component by using perturbation, path tracking and projection techniques. An optimal-direction strategy and an adaptive step size control strategy for path following on high dimensional components are given.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"18 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":"133645580","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
A multiprecision algorithm for the solution of polynomials and polynomial eigenvalue problems 求解多项式和多项式特征值问题的多精度算法
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631952
Dario Bini, L. Robol
{"title":"A multiprecision algorithm for the solution of polynomials and polynomial eigenvalue problems","authors":"Dario Bini, L. Robol","doi":"10.1145/2631948.2631952","DOIUrl":"https://doi.org/10.1145/2631948.2631952","url":null,"abstract":"Many applications of the real world are modelled by matrix polynomials [EQUATION] where <i>A</i><sub><i>i</i></sub> are <i>m</i> x <i>m</i> matrices, see for instance [2], [6]. A computational task encountered in this framework is computing the eigenvalues of <i>P</i>(<i>x</i>), that is, the solutions of the polynomial equation det <i>P</i>(<i>x</i>) = 0. This task is generally accomplished by reducing <i>P</i>(<i>x</i>) to a linear pencil of the kind <i>xL</i> -- <i>K</i> for suitable matrices <i>K, L</i> of size <i>mn</i>, and to solving the eigenvalue problem (λ<i>L</i> -- <i>K</i>)<i>v</i> = 0 by means of standard numerical algorithms. A wide literature exists on this approach, we refer the reader to [7] for an example.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"37 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":"128538349","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
Prony systems via decimation and homotopy continuation 通过抽取和同伦延拓的proony系统
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631961
Dmitry Batenkov
{"title":"Prony systems via decimation and homotopy continuation","authors":"Dmitry Batenkov","doi":"10.1145/2631948.2631961","DOIUrl":"https://doi.org/10.1145/2631948.2631961","url":null,"abstract":"We consider polynomial systems of Prony type, appearing in many areas of mathematics. Their robust numerical solution is considered to be difficult, especially in \"near-colliding\" situations. We transform the nonlinear part of the Prony system into a Hankel-type polynomial system. Combining this representation with a recently discovered \"decimation\" technique, we present an algorithm which applies homotopy continuation on a sequence of modified Hankel-type systems as above. In this way, we are able to solve for the nonlinear variables of the original system with high accuracy when the data is perturbed.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"104 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":"122697201","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: 3
Exploring univariate mixed polynomials 探索单变量混合多项式
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631960
M. Elkadi, A. Galligo
{"title":"Exploring univariate mixed polynomials","authors":"M. Elkadi, A. Galligo","doi":"10.1145/2631948.2631960","DOIUrl":"https://doi.org/10.1145/2631948.2631960","url":null,"abstract":"We consider mixed polynomials <i>P</i>(<i>z, z</i>) of the single complex variable <i>z</i> with complex (or real coefficients, of degree <i>n</i> in <i>z</i> and <i>m</i> in <i>z</i>. This data is equivalent to a pair of real bivariate polynomials <i>f</i>(<i>x, y</i>) and <i>g</i>(<i>x, y</i>) obtained by separating real and imaginary parts of <i>P</i>. However specifying the degrees, here we focus on the case where <i>m</i> is small allows to investigate interesting roots structures and roots counting; intermediate between complex and real algebra. Mixed polynomials naturally appear in the study of complex polynomial matrices and complex moment problems, harmonic maps, and in recent papers dealing with Milnor fibrations.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"4 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":"121318724","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: 3
Some ideas for the computation of matrix solvents 计算基质溶剂的一些思路
Symbolic-Numeric Computation Pub Date : 2014-07-28 DOI: 10.1145/2631948.2631953
M. Barkatou, P. Boito, E. Ugalde
{"title":"Some ideas for the computation of matrix solvents","authors":"M. Barkatou, P. Boito, E. Ugalde","doi":"10.1145/2631948.2631953","DOIUrl":"https://doi.org/10.1145/2631948.2631953","url":null,"abstract":"We consider the matrix polynomial [EQUATION], with given coefficients [EQUATION]. A matrix [EQUATION] is called a solvent if P(S) = 0. We explore some approaches to the symbolic and numeric computation of solvents. In particular, we compute formulas for the condition number and backward error of the problem which rely on the contour integral based representation of P(S). Finally, we describe a possible approach for computing exact solvents symbolically.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"146 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":"121999003","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
Computing GCRDs of approximate differential polynomials 近似微分多项式的gcrd计算
Symbolic-Numeric Computation Pub Date : 2014-06-03 DOI: 10.1145/2631948.2631964
M. Giesbrecht, Joseph Haraldson
{"title":"Computing GCRDs of approximate differential polynomials","authors":"M. Giesbrecht, Joseph Haraldson","doi":"10.1145/2631948.2631964","DOIUrl":"https://doi.org/10.1145/2631948.2631964","url":null,"abstract":"Differential (Ore) type polynomials with approximate polynomial coefficients are introduced. These provide a useful representation of approximate differential operators with a strong algebraic structure, which has been used successfully in the exact, symbolic, setting. We then present an algorithm for the approximate Greatest Common Right Divisor (GCRD) of two approximate differential polynomials, which intuitively is the differential operator whose solutions are those common to the two inputs operators. More formally, given approximate differential polynomials f and g, we show how to find \"nearby\" polynomials f and g which have a non-trivial GCRD. Here \"nearby\" is under a suitably defined norm. The algorithm is a generalization of the SVD-based method of Corless et al. (1995) for the approximate GCD of regular polynomials. We work on an appropriately \"linearized\" differential Sylvester matrix, to which we apply a block SVD. The algorithm has been implemented in Maple and a demonstration of its robustness is presented.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124805680","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
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