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 publication(s) have been published by the conference receiving 980 citation(s).

Topics: Polynomial, Symbolic computation, Matrix (mathematics), Matrix polynomial, Square-free polynomial

##### Papers

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01 Jan 2007TL;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.

76 citations

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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

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28 Jul 2014Abstract: 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

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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.

48 citations

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25 Jul 2007TL;DR: A new hybrid algorithm based on Zippel's original sparse polynomial interpolation technique is developed, and it is shown that the random Fourier-like matrices arising in the algorithm, have the desired rank property in the exact case, and appear usable numerically.

Abstract: Algebraic randomization techniques can be applied to hybrid symbolic-numeric algorithms. Here we consider the problem of interpolating a sparse rational function from noisy values. We develop a new hybrid algorithm based on Zippel's original sparse polynomial interpolation technique. We show experimentally that our algorithm can handle sparse polynomials with large degrees. We also give a (partial) mathematical justification why the Zippel's algebraic randomization technique can be used with our approximate data: the randomly generated non-zero values are expected to be bounded away from zero. We show that the random Fourier-like matrices arising in our algorithm, have the desired rank property in the exact case, and appear usable numerically.Algebraic randomization techniques can be applied to hybrid symbolic-numeric algorithms. Here we consider the problem of interpolating a sparse rational function from noisy values. We develop a new hybrid algorithm based on Zippel's original sparse polynomial interpolation technique. We show experimentally that our algorithm can handle sparse polynomials with large degrees. We also give a (partial) mathematical justification why the Zippel's algebraic randomization technique can be used with our approximate data: the randomly generated non-zero values are expected to be bounded away from zero. We show that the random Fourier-like matrices arising in our algorithm, have the desired rank property in the exact case, and appear usable numerically.

31 citations