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Fast Approximate Computations with Cauchy Matrices, Polynomials and Rational Functions

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Computer Science - Theory and Applications (CSR 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8476))

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Abstract

The papers [18], [9], [29], and [28] combine the techniques of the Fast Multipole Method of [15], [8] with the transformations of matrix structures, traced back to [19]. The resulting numerically stable algorithms approximate the solutions of Toeplitz, Hankel, Toeplitz-like, and Hankel-like linear systems of equations in nearly linear arithmetic time, versus the classical cubic time and the quadratic time of the previous advanced algorithms. We extend this progress to decrease the arithmetic time of the known numerical algorithms from quadratic to nearly linear for computations with matrices that have structure of Cauchy or Vandermonde type and for the evaluation and interpolation of polynomials and rational functions. We detail and analyze the new algorithms, and in [21] we extend them further.

Some preliminary results of this paper have been presented at CASC 2013. Our research has been supported by the NSF Grant CC 1116736 and the PSC CUNY Awards 64512–0042 and 65792–0043.

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Author information

Authors and Affiliations

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

    Victor Y. Pan

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  1. Victor Y. Pan
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Editors and Affiliations

  1. Steklov Institute of Mathematics at St. Petersburg, Russian Academy of Sciences, 27 Fontanka,, 191023, St. Petersburg, Russia

    Edward A. Hirsch

  2. Higher School of Economics, National Research University, 109028, Moscow, Russia

    Sergei O. Kuznetsov

  3. CNRS and University Paris Diderot, Paris, France

    Jean-Éric Pin

  4. Moscow State University, 119992, Moscow, Russia

    Nikolay K. Vereshchagin

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Pan, V.Y. (2014). Fast Approximate Computations with Cauchy Matrices, Polynomials and Rational Functions. In: Hirsch, E.A., Kuznetsov, S.O., Pin, JÉ., Vereshchagin, N.K. (eds) Computer Science - Theory and Applications. CSR 2014. Lecture Notes in Computer Science, vol 8476. Springer, Cham. https://doi.org/10.1007/978-3-319-06686-8_22

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  • DOI: https://doi.org/10.1007/978-3-319-06686-8_22

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06685-1

  • Online ISBN: 978-3-319-06686-8

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