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Proceedings ArticleDOI

On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms

25 Jul 2007-pp 11-17
TL;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.

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Citations
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Proceedings ArticleDOI
09 Jul 2006
TL;DR: By interpolating the black box evaluated at random primitive roots of unity, this work gives efficient and numerically robust solutions to the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating-point arithmetic.
Abstract: We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating-point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact Ben-Or/Tiwari sparse interpolation algorithm and the classical Prony's method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony's method. We analyze the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications.

85 citations


Additional excerpts

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Journal ArticleDOI
TL;DR: By interpolating the black box evaluated at random primitive roots of unity, this work gives efficient and numerically robust solutions to the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating point arithmetic.

84 citations

Proceedings ArticleDOI
20 Jul 2008
TL;DR: This work generalizes the technique by Peyrl and Parillo to computing lower bound certificates for several well-known factorization problems in hybrid symbolic-numeric computation and certifies accurate rational lower bounds near the irrational global optima.
Abstract: We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several well-known factorization problems in hybrid symbolic-numeric computation. The idea is to transform a numerical sum-of-squares (SOS) representation of a positive polynomial into an exact rational identity. Our algorithms successfully certify accurate rational lower bounds near the irrational global optima for benchmark approximate polynomial greatest common divisors and multivariate polynomial irreducibility radii from the literature, and factor coefficient bounds in the setting of a model problem by Rump (up to n = 14, factor degree = 13.The numeric SOSes produced by the current fixed precision semi-definite programming (SDP) packages (SeDuMi, SOSTOOLS, YALMIP) are usually too coarse to allow successful projection to exact SOSes via Maple 11's exact linear algebra. Therefore, before projection we refine the SOSes by rank-preserving Newton iteration. For smaller problems the starting SOSes for Newton can be guessed without SDP ("SDP-free SOS"), but for larger inputs we additionally appeal to sparsity techniques in our SDP formulation.

72 citations

Proceedings ArticleDOI
29 Jul 2007
TL;DR: Five new algorithms for sparse rational function interpolation algorithm in the hybrid symbolic-numeric setting when the black box for the function returns real and complex values with noise are presented and analyzed.
Abstract: The black box algorithm for separating the numerator from the denominator of a multivariate rational function can be combined with sparse multivariate polynomial interpolation algorithms to interpolate a sparse rational function. domization and early termination strategies are exploited to minimize the number of black box evaluations. In addition, rational number coefficients are recovered from modular images by rational vector recovery. The need for separate numerator and denominator size bounds is avoided via correction, and the modulus is minimized by use of lattice basis reduction, a process that can be applied to sparse rational function vector recovery itself. Finally, one can deploy sparse rational function interpolation algorithm in the hybrid symbolic-numeric setting when the black box for the function returns real and complex values with noise. We present and analyze five new algorithms for the above problems and demonstrate their effectiveness on a mark implementation.

40 citations

Proceedings ArticleDOI
08 Jun 2011
TL;DR: This work improves on the best-known algorithm for interpolation over large finite fields by presenting a Las Vegas randomized algorithm that uses fewer black box evaluations and provides the first provably stable algorithm for this problem, at the cost of modestly more evaluations.
Abstract: We consider the problem of interpolating an unknown multivariate polynomial with coefficients taken from a finite field or as numerical approximations of complex numbers. Building on the recent work of Garg and Schost, we improve on the best-known algorithm for interpolation over large finite fields by presenting a Las Vegas randomized algorithm that uses fewer black box evaluations. Using related techniques, we also address numerical interpolation of sparse polynomials with complex coefficients, and provide the first provably stable algorithm (in the sense of relative error) for this problem, at the cost of modestly more evaluations. A key new technique is a randomization which makes all coefficients of the unknown polynomial distinguishable, producing what we call a diverse polynomial. Another departure from most previous approaches is that our algorithms do not rely on root finding as a subroutine. We show how these improvements affect the practical performance with trial implementations.

38 citations


Cites methods from "On probabilistic analysis of random..."

  • ...Approximate sparse ra­tional function interpolation is considered by Kaltofen and Yang (2007) and Kaltofen et al. (2007), using the Structured Probes Probe degree Computation cost Total cost Type Dense Ben-Or & Tiwari Garg & Schost Randomized G & S Ours d O(t) O (t2 log d) O (t log…...

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References
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Journal ArticleDOI
TL;DR: Vanous fast probabdlsttc algonthms, with probability of correctness guaranteed a prion, are presented for testing polynomial ldentmes and propemes of systems of polynomials and ancdlary fast algorithms for calculating resultants and Sturm sequences are given.
Abstract: The s tar thng success o f the Rabm-S t ra s sen -So lovay p n m a h t y algori thm, together wi th the intr iguing foundat tonal posstbthty that axtoms of randomness may constttute a useful fundamenta l source o f m a t h e m a u c a l truth independent of the standard axmmaUc structure of mathemaUcs, suggests a wgorous search for probabdisuc algonthms In dlustratmn of this observaUon, vanous fast probabdlsttc algonthms, with probability of correctness guaranteed a prion, are presented for testing polynomial ldentmes and propemes of systems of polynomials. Ancdlary fast algorithms for calculating resultants and Sturm sequences are given. Probabilistlc calculatton in real anthmetlc, prewously considered by Davis, is justified ngorously, but only in a special case. Theorems of elementary geometry can be proved much more efficiently by the techmques presented than by any known arttficml-mtelhgence approach

1,904 citations


"On probabilistic analysis of random..." refers methods in this paper

  • ...On Probabilistic Analysis of Randomization in Hybrid Symbolic-numeric Algorithms* Erich Kaltofen, Zhengfeng Yang Department of Mathematics North Carolina State University Raleigh, North Carolina 27695-8205, USA kaltofen,zyang4@math.ncsu.edu http://www.kaltofen.us Lihong Zhi Key Laboratory of…...

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  • ...On Probabilistic Analysis of Randomization in Hybrid Symbolic-numeric Algorithms* Erich Kaltofen, Zhengfeng Yang Department of Mathematics North Carolina State University Raleigh, North Carolina 27695-8205, USA kaltofen,zyang4@math.ncsu.edu http://www.kaltofen.us Lihong Zhi Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science Beijing 100080, China lzhi@mmrc.iss.ac.cn http://www.mmrc.iss.ac.cn/~lzhi/ ABSTRACT Algebraic randomization techniques can be applied to hy­brid symbolic-numeric algorithms....

    [...]

  • ...Hybrid symbolic-numeric algorithms permit errors in the input scalars due to .oating point round-o. or through phys­ical measurement....

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Book ChapterDOI
01 Jun 1979
TL;DR: This work has tried to demonstrate how sparse techniques can be used to increase the effectiveness of the modular algorithms of Brown and Collins and believes this work has finally laid to rest the bad zero problem.
Abstract: In this paper we have tried to demonstrate how sparse techniques can be used to increase the effectiveness of the modular algorithms of Brown and Collins. These techniques can be used for an extremely wide class of problems and can applied to a number of different algorithms including Hensel's lemma. We believe this work has finally laid to rest the bad zero problem.

1,297 citations


"On probabilistic analysis of random..." refers methods in this paper

  • ...On Probabilistic Analysis of Randomization in Hybrid Symbolic-numeric Algorithms* Erich Kaltofen, Zhengfeng Yang Department of Mathematics North Carolina State University Raleigh, North Carolina 27695-8205, USA kaltofen,zyang4@math.ncsu.edu http://www.kaltofen.us Lihong Zhi Key Laboratory of…...

    [...]

  • ...On Probabilistic Analysis of Randomization in Hybrid Symbolic-numeric Algorithms* Erich Kaltofen, Zhengfeng Yang Department of Mathematics North Carolina State University Raleigh, North Carolina 27695-8205, USA kaltofen,zyang4@math.ncsu.edu http://www.kaltofen.us Lihong Zhi Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science Beijing 100080, China lzhi@mmrc.iss.ac.cn http://www.mmrc.iss.ac.cn/~lzhi/ ABSTRACT Algebraic randomization techniques can be applied to hy­brid symbolic-numeric algorithms....

    [...]

Journal ArticleDOI
TL;DR: A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described and a probabilistic algorithm is shown to exist for finding the determinant of a square matrix.
Abstract: A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described. The algorithms discussed all require O(n_{1}(\omega + n_{1})\log^{k}n_{1}) field operations, where n_{1} is the maximum dimension of the coefficient matrix, \omega is approximately the number of field operations required to apply the matrix to a test vector, and the value of k depends on the algorithm. A probabilistic algorithm is shown to exist for finding the determinant of a square matrix. Also, probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error.

617 citations


"On probabilistic analysis of random..." refers methods in this paper

  • ...We show experimentally that our algorithm can han­dle sparse polynomials with large degrees....

    [...]

Journal ArticleDOI
TL;DR: A probabilistic solution is presented which achieves small probability of error on 30 points for m-ary multinomials in Howden's method for algebraic program testing.

525 citations


"On probabilistic analysis of random..." refers methods in this paper

  • ...On Probabilistic Analysis of Randomization in Hybrid Symbolic-numeric Algorithms* Erich Kaltofen, Zhengfeng Yang Department of Mathematics North Carolina State University Raleigh, North Carolina 27695-8205, USA kaltofen,zyang4@math.ncsu.edu http://www.kaltofen.us Lihong Zhi Key Laboratory of…...

    [...]

  • ...On Probabilistic Analysis of Randomization in Hybrid Symbolic-numeric Algorithms* Erich Kaltofen, Zhengfeng Yang Department of Mathematics North Carolina State University Raleigh, North Carolina 27695-8205, USA kaltofen,zyang4@math.ncsu.edu http://www.kaltofen.us Lihong Zhi Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science Beijing 100080, China lzhi@mmrc.iss.ac.cn http://www.mmrc.iss.ac.cn/~lzhi/ ABSTRACT Algebraic randomization techniques can be applied to hy­brid symbolic-numeric algorithms....

    [...]

  • ...Hybrid symbolic-numeric algorithms permit errors in the input scalars due to .oating point round-o. or through phys­ical measurement....

    [...]