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Conference

Symbolic Numeric Computation 

About: Symbolic Numeric Computation is an academic conference. The conference publishes majorly in the area(s): Polynomial & Symbolic computation. Over the lifetime, 119 publications have been published by the conference receiving 1014 citations.

Papers published on a yearly basis

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Book ChapterDOI
01 Jan 2007
TL;DR: This paper presents two results on the complexity of root isolation via Sturm sequences, both of which exploit amortization arguments.
Abstract: This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments.

80 citations

Proceedings ArticleDOI
03 Aug 2009
TL;DR: An effective symbolic-numeric cylindrical algebraic decomposition (SNCAD) algorithm and its variant specially designed for QE are proposed based on the authors' previous work and the implementation of those is reported.
Abstract: Recently quantifier elimination (QE) has been of great interest in many fields of science and engineering. In this paper an effective symbolic-numeric cylindrical algebraic decomposition (SNCAD) algorithm and its variant specially designed for QE are proposed based on the authors' previous work and our implementation of those is reported. Based on analysing experimental performances, we are improving our design/synthesis of the SNCAD for its practical realization with existing efficient computational techniques and several newly introduced ones. The practicality of the SNCAD is now examined by a number of experimental results including practical engineering problems, which also reveals the quality of the implementation.

54 citations

Proceedings ArticleDOI
28 Jul 2014
TL;DR: In this paper, the complexity of solving nonsingular linear systems of equations with structured matrices has been studied and the authors present a Boolean complexity analysis for the problem of polynomial multiplication and division.
Abstract: We estimate the Boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices. We study four basic and most popular classes, that is, Toeplitz, Hankel, Cauchy and Vandermonde matrices, for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation. The Boolean cost estimates for the latter problems have been obtained by Kirrinnis in [10], except for rational interpolation, and we supply them now. All known Boolean cost estimates from [10] for these problems rely on using Kronecker product. This implies the d-fold precision increase for the d-th degree output, but we avoid such an increase by relying on distinct techniques based on employing FFT. Furthermore we simplify the analysis and make it more transparent by combining the representations of our tasks and algorithms both via structured matrices and via polynomials and rational functions. This also enables further extensions of our estimates to cover Trummer's important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes.

51 citations

Proceedings ArticleDOI
03 Aug 2009
TL;DR: 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 and indicates that for most instances the solvers based on Continued Fractions are among the best methods.
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 Saarbrucken. 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.

50 citations

Proceedings ArticleDOI
03 Aug 2009
TL;DR: This work describes the development of rigorous tools to determine enclosures of flows of general nonlinear differential equations based on Picard iterations, with particular emphasis on methods that have favorable long term stability, which is achieved using suitable preconditioning and other methods.
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.

36 citations

Performance
Metrics
No. of papers from the Conference in previous years
YearPapers
201428
201229
200929
200732
19851