{"title":"Rigorous global search using taylor models","authors":"M. Berz, K. Makino","doi":"10.1145/1577190.1577198","DOIUrl":"https://doi.org/10.1145/1577190.1577198","url":null,"abstract":"A Taylor model of a smooth function f over a sufficiently small domain D is a pair (P,I) where P is the Taylor polynomial of f at a point d in D, and I is an interval such that f differs from P by not more than I over D. As such, they represent a hybrid between numerical techniques for the interval and the coefficients of P and algebraic techniques for the manipulation of polynomials. A calculus including addition, multiplication and differentiation/integration is developed to compute Taylor models for code lists, resulting in a method to compute rigorous enclosures of arbitrary computer functions in terms of Taylor models. The methods combine the advantages of numeric methods, namely finite size of representation, speed, and no limitations on the objects on which operations can be carried out with those of symbolic methods, namely the ability to treat functions instead of points and making rigorous statements.\u0000 We show how the methods can be used for the problem of rigorous global search based on a branch and bound approach, where Taylor models are used to prune the search space and resolve constraints to high order. Compared to other rigorous global optimizers based on intervals and linearizations, the methods allow the treatment of complicated functions with long code lists and with large amounts of dependency. Furthermore, the underlying polynomial form allows the use of other efficient bounding and pruning techniques, including the linear dominated bounder (LDB) and the quadratic fast bounder (QFB).","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"20 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":"128277341","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":"Pseudospectra for exponential polynomial matrices","authors":"Robert M Corless","doi":"10.1145/1577190.1577192","DOIUrl":"https://doi.org/10.1145/1577190.1577192","url":null,"abstract":"ẋ(t) = Ax(t) (1) where x ∈ C and A ∈ C(n×n), together with (say) initial conditions x(0) = x0, occurs often as a simple model of many applied dynamical phenomena, for instance in theoretical evolution or in the physics of lasers, to name only two out of many possibilities. The so-called characteristic equation det(λI − A) = 0 occurs when looking for solutions of the form x(t) = ev for some vector v ∈ C. Understanding the exact solution x(t) = exp(At)x0 comes from the eigenvalues (spectrum) of A, and more recently from the pseudospectrum of A, by which is meant the set [5, 17]","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"37 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":"121458310","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}
T. Sakurai, Junko Asakura, Hiroto Tadano, T. Ikegami, Kinji Kimura
{"title":"A method for finding zeros of polynomial equations using a contour integral based eigensolver","authors":"T. Sakurai, Junko Asakura, Hiroto Tadano, T. Ikegami, Kinji Kimura","doi":"10.1145/1577190.1577213","DOIUrl":"https://doi.org/10.1145/1577190.1577213","url":null,"abstract":"In this paper, we present a method for finding zeros of polynomial equations in a given domain. We apply a numerical eigensolver using contour integral for a polynomial eigenvalue problem that is derived from polynomial equations. The Dixon resultant is used to derive the matrix polynomial of which eigenvalues involve roots of the polynomial equations with respect to one variable. The matrix polynomial obtained by the Dixon resultant is sometimes singular. By applying the singular value decomposition for a matrix which appears in the eigensolver, we can obtain the roots of given polynomial systems. Experimental results demonstrate the efficiency of the proposed method.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"21 2 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":"123553873","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":"De-Sinc numerical methods","authors":"M. Sugihara","doi":"10.1145/1577190.1577196","DOIUrl":"https://doi.org/10.1145/1577190.1577196","url":null,"abstract":"The present talk gives a survey of the DE-Sinc numerical methods (= the Sinc numerical methods, which have been developed by Stenger and his school, incorporated with double-exponential transformations). The DE-Sinc numerical methods have a feature that they enjoys the convergence rate O(exp(-κ'n/log n)) with some κ'>0 even if the function, or the solution to be approximated has end-point singularity, where n is the number of nodes or bases used in the methods.","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":"124177778","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":"Preconditioning, randomization, solving linear systems, eigen-solving, and root-finding","authors":"V. Pan, G. Qian, Ailong Zheng","doi":"10.1145/1577190.1577194","DOIUrl":"https://doi.org/10.1145/1577190.1577194","url":null,"abstract":"We propose novel randomized preprocessing techniques for solving linear systems of equations and eigen-solving with extensions to the solution of polynomial and secular equations. According to our formal study and extensive experiments, the approach turns out to be effective, particularly in the case of structured input matrices.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"455 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":"124320284","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":"Rigorous integration of flows and ODEs using taylor models","authors":"K. Makino, M. Berz","doi":"10.1145/1577190.1577206","DOIUrl":"https://doi.org/10.1145/1577190.1577206","url":null,"abstract":"Taylor models combine the advantages of numerical methods and algebraic approaches of efficiency, tightly controlled recourses, and the ability to handle very complex problems with the advantages of symbolic approaches, in particularly the ability to be rigorous and to allow the treatment of functional dependencies instead of merely points. The resulting differential algebraic calculus involving an algebra with differentiation and integration is particularly amenable for the study of ODEs and PDEs based on fixed point problems from functional analysis. We describe the development of rigorous tools to determine enclosures of flows of general nonlinear differential equations based on Picard iterations. Particular emphasis is placed on the development of methods that have favorable long term stability, which is achieved using suitable preconditioning and other methods. Applications of the methods are presented, including determinations of rigorous enclosures of flows of ODEs in the theory of chaotic dynamical systems.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"30 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":"126464749","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":"Optimization and NP_R-completeness of certain fewnomials","authors":"P. Pébay, J. Rojas, David C. Thompson","doi":"10.1145/1577190.1577212","DOIUrl":"https://doi.org/10.1145/1577190.1577212","url":null,"abstract":"We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n+2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are polynomial in n and the logarithm of a certain condition number. For the special case of polynomials (i.e., integer exponents), the log of our condition number is sub-quadratic in the sparse size. The best previous complexity bounds were exponential in the size, even for n fixed. Along the way, we partially extend the theory of A-discriminants to real exponents and exponential sums, and find new and natural NPR-complete problems.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131367530","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":"The nearest polynomial of lower degree","authors":"Robert M Corless, N. Rezvani","doi":"10.1145/1277500.1277530","DOIUrl":"https://doi.org/10.1145/1277500.1277530","url":null,"abstract":"Suppose one is working with the polynomial p(x) = −0.99B3 0(x)−1.03B 1(x)− 0.33B 2(x) + B 3 3(x) expressed in terms of degree 3 Bernstein polynomials and suppose one has reason to believe that p(x) is really of degree 2. Applying the degree-reducing procedure [6] one gets −0.99B2 0(x)− 1.05B 1(x) + B 2(x). But is this correct, or have we treated p(x) in a Procrustean 1 fashion? Checking by converting p(x) to the power basis, we find that the coefficient of x is 0.11. Is this zero or not?","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123971939","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":"A numerical study of extended Hensel series","authors":"D. Inaba, Tateaki Sasaki","doi":"10.1145/1277500.1277517","DOIUrl":"https://doi.org/10.1145/1277500.1277517","url":null,"abstract":"The extended Hensel construction is a Hensel construction at a singular point of the multivariate polynomial, and it allows us to expand the roots of a given multivariate poly-nomial into a kind of series which we call an extended Hensel series. This paper investigates the behavior of the extended Hensel series numerically, and clarifies the following four points. 1) The convergence domain of the extended Hensel series is very different from those of the Taylor series; both convergence and divergence domains coexist in the neighborhood of the expansion point. 2) The extended Hensel series truncated at 7 ~ 8 order coincides very well with the corresponding algebraic function in the convergence domain, while it behaves very wildly in the divergence domain. 3) In the case of non-monic polynomial, the factors of leading co-efficient are distributed among the extended Hensel series, and the singular behaviors of the roots at the zero-points of the leading coefficient are expressed nicely by the Hensel series. 4) Although many-valuedness of extended Hensel series is usually different from that of the corresponding exact roots, the Hensel series reproduce the behaviors of the exact roots by jumping from one branch to another occasionally.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"10 15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128519795","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":"Real implicitization of curves and geometric extraneous components","authors":"M. Fioravanti, L. González-Vega, A. Seidl","doi":"10.1145/1277500.1277515","DOIUrl":"https://doi.org/10.1145/1277500.1277515","url":null,"abstract":"Real implicitization of parametric curves has important applications in computer aided geometric design. Implicitization of parametric curves by resultant computations may lead to super uous isolated points. Hence, an exact implicit description should consist of equations and further conditions excluding these geometric extraneous components. Although a real implicit description of this kind can be obtained by real quantifier elimination, we give a direct way to find the conditions to add for an exact description. This results in a more effective algorithm and nicer formulas.","PeriodicalId":308716,"journal":{"name":"Symbolic-Numeric Computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128238176","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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