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GCD of multivariate approximate polynomials using beautification with the subtractive algorithm

Authors:

Robert M. Corless,

Erik Postma,

David R. StoutemyerAuthors Info & Claims

SNC '11: Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

Pages 153 - 154

https://doi.org/10.1145/2331684.2331709

Published: 07 June 2012 Publication History

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Abstract

The GCD problem for approximate polynomials, by which we mean polynomials expressed in some fixed basis but having approximately-known coefficients, has been well-studied at least since the paper of [6]. Important papers include those listed in [4, 2.12.3], and more recently includes [5], [8] and [9]. What is new about the present paper is that we hope to take advantage of some new technology, in order to improve our understanding of the GCD problem and not necessarily to try to improve on existing algorithms.

References

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R. M. Corless, P. M. Gianni, B. M. Trager, and S. M. Watt. The Singular Value Decomposition for polynomial systems. In International Symposium on Symbolic and Algebraic Computation, pages 195--207, Montréal, Canada, 1995.

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R. M. Corless, E. Postma, and D. R. Stoutemyer. Rounding coefficients and artificially underflowing terms in non-numeric expressions. Communications in Computer Algebra, to appear, 2011.

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J. Grabmeier, E. Kaltofen, and V. Weispfenning. Computer algebra handbook: foundations, applications, systems. Berlin; New York: Springer, 2003.

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E. Kaltofen, Z. Yang, and L. Zhi. Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials. In International Symposium on Symbolic and Algebraic Computation, pages 169--176, New York, NY, USA, 2006.

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A. Schönhage. Quasi-gcd computations. Journal of Complexity, 1:118--137, 1985.

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K. Shirayanagi and H. Sekigawa. A new Gröbner basis conversion method based on stabilization techniques. Theoretical Computer Science, 409(2):311--317, 2008.

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A. Terui. An iterative method for calculating approximate GCD of univariate polynomials. In International Symposium on Symbolic and Algebraic Computation, pages 351--358, New York, NY, USA, 2009.

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Z. Zeng. Algorithm 835: Multroot---a Matlab package for computing polynomial roots and multiplicities. ACM Transactions on Mathematical Software, 30(2):218--236, 2004.

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Index Terms

GCD of multivariate approximate polynomials using beautification with the subtractive algorithm
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms

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SNC '11: Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

June 2012

194 pages

ISBN:9781450305150

DOI:10.1145/2331684

General Chair:
Ilias Kotsireas
Wilfrid Laurier University, Canada

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Association for Computing Machinery

New York, NY, United States

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Published: 07 June 2012

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ISSAC '11: International Symposium on Symbolic and Algebraic Computation

June 7, 2011 - June 9, 2012

California, San Jose

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Computing GCRDs of approximate differential polynomials

Approximate polynomial gcd: Small degree and small height perturbations

Approximate polynomial GCD over integers

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