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Root-Finding with Eigen-Solving

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Symbolic-Numeric Computation

Part of the book series: Trends in Mathematics ((TM))

Abstract

We survey and extend the recent progress in polynomial root-finding via eigen-solving for highly structured generalized companion matrices. We cover the selection of eigen-solvers and matrices and show the benefits of exploiting matrix structure. No good estimates for the rate of global convergence of the eigen-solvers are known, but according to ample empirical evidence it is sufficient to use a constant number of iteration steps per eigenvalue. If so, the resulting root-finders are optimal up to a constant factor because they use linear arithmetic time per step and perform with a constant (double) precision. Some by-products of our study are of independent interest. The algorithms can be extended to solving secular equations

Supported by PSC CUNY Awards 66437-0035 and 67297-0036.

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Author notes

  1. Xinmao Wang

    Present address: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Lehman College of the City University of New York, Bronx, NY, 10468, USA

    Victor Y. Pan, Brian Murphy & Rhys Eric Rosholt

  2. Ph.D. Program in Computer Science, Graduate Center of the City University of New York, New York, NY, 10036, USA

    Dmitriy Ivolgin, Yuqing Tang & Xiaodong Yan

  3. Ph.D. Program in Mathematics, Graduate Center of the City University of New York, New York, NY, 10036, USA

    Xinmao Wang

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  1. Victor Y. Pan
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Editors and Affiliations

  1. School of Science, Beihang University, 37 Xueyuan Road, Beijing, 100083, China

    Dongming Wang

  2. Laboratoire d’Informatique de Paris 6, Université Pierre et Marie Curie - CNRS, 8, rue due Capitaine Scott, 75015, Paris, France

    Dongming Wang

  3. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and System Sciences, Beijing, 100080, China

    Lihong Zhi

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Pan, V.Y. et al. (2007). Root-Finding with Eigen-Solving. In: Wang, D., Zhi, L. (eds) Symbolic-Numeric Computation. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7984-1_12

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