Symbolic-Numeric Computation最新文献

{"title":"Roots of the derivatives of some random polynomials","authors":"A. Galligo","doi":"10.1145/2331684.2331703","DOIUrl":"https://doi.org/10.1145/2331684.2331703","url":null,"abstract":"Our observations show that the sets of real (respectively complex) roots of the derivatives of some classical families of random polynomials admit a rich variety of patterns looking like discretized curves. To bring out the shapes of the suggested curves, we introduce an original use of fractional derivatives. Then we present several conjectures and outline a strategy to explain the presented phenomena. This strategy is based on asymptotic geometric properties of the corresponding complex critical points sets.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128503910","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}

{"title":"Computing clustered close-roots of univariate polynomials","authors":"Tateaki Sasaki, Akira Terui","doi":"10.1145/1577190.1577217","DOIUrl":"https://doi.org/10.1145/1577190.1577217","url":null,"abstract":"Given a univariate polynomial having well-separated clusters of close roots, we give a method of computing close roots in a cluster simultaneously, without computing other roots. We first determine the position and the size of the cluster, as well as the number of close roots contained. Then, we move the origin to a near center of the cluster and perform the scale transformation so that the cluster is enlarged to be of size O(1). These operations transform the polynomial to a very characteristic one. We modify Durand-Kerner's method so as to compute only the close roots in the cluster. The method is very efficient because we can discard most terms of the transformed polynomial. We also give a formula of quite tight error bound. We show high efficiency of our method by empirical experiments.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"369 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114366579","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}

{"title":"Approximate factorization of polynomials over Z","authors":"Tateaki Sasaki, Yasutaka Ookura","doi":"10.1145/1577190.1577216","DOIUrl":"https://doi.org/10.1145/1577190.1577216","url":null,"abstract":"We propose three algorithms for approximate factorization of univariate polynomials over Z; the first one uses sums of powers of roots (SPR method), the second one utilizes factor-differentiated polynomials (FD method), and the third one is a robust but slow method. The SPR method is applicable to monic polynomials well but it is almost useless for non-monic polynomials unless their leading coefficients are sufficiently small. The FD method is applicable to both monic and non-monic polynomials, but it also becomes useless if both the leading and the tail coefficients increase. The third one is applicable to any polynomial factorizable approximately over Z, but it is slow. We discuss two types of polynomials which are ill-conditioned for rootfinding, Wilkinson-type polynomials and polynomials with close roots. Furthermore, we consider briefly approximate factorization of multivariate polynomials over Z.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130311048","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}

{"title":"Experimental evaluation and cross-benchmarking of univariate real solvers","authors":"M. Hemmer, Elias P. Tsigaridas, Zafeirakis Zafeirakopoulos, I. Emiris, M. Karavelas, B. Mourrain","doi":"10.1145/1577190.1577202","DOIUrl":"https://doi.org/10.1145/1577190.1577202","url":null,"abstract":"Real solving of univariate polynomials is a fundamental problem with several important applications. This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials over the integers. We have tested 9 different implementations based on symbolic-numeric methods, Sturm sequences, Continued Fractions and Descartes' rule of sign. The methods under consideration were developed at the GALAAD group at INRIA,the VEGAS group at LORIA and the MPI Saarbrücken. We compared their sensitivity with respect to various aspects such as degree, bitsize or root separation of the input polynomials. Our datasets consist of 5,000 polynomials from many different settings, which have maximum coefficient bitsize up to bits 8,000, and the total running time of the experiments was about 50 hours. Thereby, all implementations of the theoretically exact methods always provided correct results throughout this extensive study. For each scenario we identify the currently most adequate method, and we point to weaknesses in each approach, which should lead to further improvements. Our results indicate that there is no \"best method\" overall, but one can say that for most instances the solvers based on Continued Fractions are among the best methods. To the best of our knowledge, this is the largest number of tests for univariate real solving up to date.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121691148","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}

{"title":"Finding positively invariant sets of a class of nonlinear loops via curve fitting","authors":"L. Shen, Min Wu, Zhengfeng Yang, Zhenbing Zeng","doi":"10.1145/1577190.1577218","DOIUrl":"https://doi.org/10.1145/1577190.1577218","url":null,"abstract":"In this paper, we study positively invariant sets of a class of nonlinear loops and discuss the relation between these sets and the attractors of the loops. For the canonical Hénon map, a numerical method based on curve fitting is proposed to find a positively invariant set containing the strange attractor. This work can be generalized to find inequality termination conditions for loops with nonlinear assignments.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123678161","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}

{"title":"Application of filter diagonalization method to numerical solution of algebraic equations","authors":"H. Murakami","doi":"10.1145/1577190.1577208","DOIUrl":"https://doi.org/10.1145/1577190.1577208","url":null,"abstract":"By the use of symbolic computation, a problem given by a set of multivariate algebraic relations is often reduced to a univariate algebraic equation which is quite high degree. And, if the roots are required in numbers we generally have to solve the higher degree algebraic equation by some iterative method. In this paper, an application of the filter diagonalization method (FDM) [5] is studied to solve the higher degree univariate algebraic equation of numerical coefficients when only a small portion of roots are required which are near the specified location in the complex plane or near the specified interval. Recently, FDM has been developing as the technique to solve a small portion of eigenpairs of a matrix selectively depending on their eigenvalues.\u0000 By the companion method, roots of an algebraic equation of higher degree N are solved as eigenvalues of companion matrix A after the balancing is made. Usually all eigenvalues can be solved by the method of shifted QR iteration. The amount of computation of the ordinal shifted QR iteration which does not use the special non-zero structure of the Frobenius companion matrix of degree N is O(N3). In the paper [1], it was shown that the amount of computation to solve all eigenvalues of size N Frobenius companion matrix by the special QR iteration which uses the structure is O(N2).\u0000 In this paper, we assume that not all but only a small portion of roots are required. To reduce the elapsed time, the inverse iteration (Rayleigh-quotient iteration) will be used in parallel. In the inverse iteration, the linear equation of shifted matrix A-ρI is solved. (For the case of a univariate algebraic equation of N-th degree, A is the Frobenius companion (or its balanced matrix). By the use of the special sparse structure of the shifted matrix in LU-decomposition or QR-decomposition, the complexity of the solution of the linear equation is O(N) in both arithmetics and space.) By the use of a well-tuned filter which is a linear combination of resolvents, FDM gives well approximated eigenpairs whose eigenvalues are near the specified location in the complex plane. From the approximated eigenpairs as initial values, inverse-iteration method quickly improves eigenpairs in a few iterations.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130431409","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}

{"title":"Symbolic-numeric problems in the automatic analysis and verification of cyber-physical systems","authors":"Stefan Ratschan","doi":"10.1145/1577190.1577195","DOIUrl":"https://doi.org/10.1145/1577190.1577195","url":null,"abstract":"Cyber-Physical Systems (CPS) are integrations of computation and physical processes. Already now, more or less no new consumer device or industrial machinery does not have some form of integrated computation. Since such systems not only interact with each other, but also with humans, their malfunction can endanger human life, and hence it is essential for them to work correctly. Important examples of properties that are used for specifying system correctness are:\u0000 \" Safety: The system state always stays in a certain set considered to be safe.\u0000 \" Progress: The system state will eventually reach some set considered to be desirable.\u0000 It is important to notice that here we deal with nondeterministic systems: They do not possess a single initial state, but an uncountable set of initial states, and for a given state, further evolution of a system is not fixed but, in general, there are uncountably many further evolutions.\u0000 So, when we want to automatically verify the correctness of such systems, due to this non-determinism, we need some form of global reasoning and a form of representing the above uncountable sets. Or, in other words, we need symbolic computation.\u0000 Considering the two aspects of CPS, computation and physical processes, the first aspect is based on computer programs, which are fixed abstract objects. Hence, for analyzing pure software systems, classical symbolic computation is the natural candidate. However, the second aspect, physical processes, is prone to perturbations, whose analysis is one of the main tasks of numerical analysis.\u0000 As a consequence, for analyzing cyber-physical systems, we need global reasoning in the presence of perturbations, or in other words, symbolic-numeric computation. In the talk we will discuss the problem of computing with the resulting symbolic objects, and their usage in algorithms for the automatic analysis and verification of cyber-physical systems.\u0000 The talk will draw on joint work with Zhikun She, Tomáš Dzetkulič and many others.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127840681","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}

{"title":"Exact polynomial factorization by approximate high degree algebraic numbers","authors":"Jingwei Chen, Yong Feng, Xiaolin Qin, Jingzhong Zhang","doi":"10.1145/1577190.1577199","DOIUrl":"https://doi.org/10.1145/1577190.1577199","url":null,"abstract":"For factoring polynomials in two variables with rational coefficients, an algorithm using transcendental evaluation was presented by Hulst and Lenstra. In their algorithm, transcendence measure was computed. However, a constant c is necessary to compute the transcendence measure. The size of c involved the transcendence measure can influence the efficiency of the algorithm greatly.\u0000 In this paper, we overcome the problem arising in Hulst and Lenstra's algorithm and propose a new polynomial time algorithm for factoring bivariate polynomials with rational coefficients. Using an approximate algebraic number of high degree instead of a variable of a bivariate polynomial, we can get a univariate one. A factor of the resulting univariate polynomial can then be obtained by a numerical root finder and the purely numerical LLL algorithm. The high degree of the algebraic number guarantees that this factor corresponds to a factor of the original bivariate polynomial. We prove that our algorithm saves a (log2(mn))2+ε factor in bit-complexity comparing with the algorithm presented by Hulst and Lenstra, where (n, m) represents the bi-degree of the polynomial to be factored. We also demonstrate on many significant experiments that our algorithm is practical. Moreover our algorithm can be generalized to polynomials with variables more than two.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132740099","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}

{"title":"Error free transformations of floating point numbers and its applications to constructing efficient error free numerical algorithms","authors":"S. Oishi","doi":"10.1145/1577190.1577193","DOIUrl":"https://doi.org/10.1145/1577190.1577193","url":null,"abstract":"We have proposed error free transformations for floating point numbers[1]-[3]. In the first place, this talk will briely survey this result. Then, the suthor will clarify that this new methodology is very usefull to make efficient error free numerical algorithms including error free fast computational geometric algorithms[4], [5].","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129309701","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}

{"title":"Extracting numerical factors of multivariate polynomials from taylor expansions","authors":"A. Cuyt, Wen-shin Lee","doi":"10.1145/1577190.1577201","DOIUrl":"https://doi.org/10.1145/1577190.1577201","url":null,"abstract":"We present a method to extract factors of multivariate polynomials with complex coefficients in floating point arithmetic. We establish the connection between the reciprocal of a multivariate polynomial and its Taylor expansion. Since the multivariate Taylor coefficients are determined by the irreducible factors of the given polynomial, we reconstruct the factors from the Taylor expansion. As each irreducible factor, regardless of its multiplicity, can be separately extracted, our method can lead toward the complete numerical factorization of multivariate polynomials.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126971262","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}