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Computing Multiple Zeros of Polynomial Systems: Case of Breadth One (Invited Talk)

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Computer Algebra in Scientific Computing (CASC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10490))

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Abstract

Given a polynomial system f with a multiple zero x whose Jacobian matrix at x has corank one, we show how to compute the multiplicity structure of x and the lower bound on the minimal distance between the multiple zero x and other zeros of f. If x is only given with limited accuracy, we give a numerical criterion to guarantee that f has \(\mu \) zeros (counting multiplicities) in a small ball around x. Moreover, we also show how to compute verified and narrow error bounds such that a slightly perturbed system is guaranteed to possess an isolated breadth-one singular solution within computed error bounds. Finally, we present modified Newton iterations and show that they converge quadratically if x is close to an isolated exact singular solution of f. This is joint work with Zhiwei Hao, Wenrong Jiang, Nan Li.

This research was supported in part by the National Key Research Project of China 2016YFB0200504 (Zhi) and the National Natural Science Foundation of China under Grants 11571350 (Zhi).

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Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and System Sciences, Beijing, China

    Lihong Zhi

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China

    Lihong Zhi

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  1. Lihong Zhi
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Correspondence to Lihong Zhi .

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Editors and Affiliations

  1. Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Zhi, L. (2017). Computing Multiple Zeros of Polynomial Systems: Case of Breadth One (Invited Talk). In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2017. Lecture Notes in Computer Science(), vol 10490. Springer, Cham. https://doi.org/10.1007/978-3-319-66320-3_28

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  • DOI: https://doi.org/10.1007/978-3-319-66320-3_28

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