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Numerische Mathematik

, Volume 129, Issue 1, pp 181–209 | Cite as

Computing the common zeros of two bivariate functions via Bézout resultants

  • Yuji Nakatsukasa
  • Vanni NoferiniEmail author
  • Alex Townsend
Article

Abstract

The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bézout matrices with polynomial entries. Using techniques including domain subdivision, Bézoutian regularization, and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (\(\ge \!1{,}000\)). We analyze the resultant method and its conditioning by noting that the Bézout matrices are matrix polynomials. Two implementations are available: one on the Matlab Central File Exchange and another in the roots command in Chebfun2 that is adapted to suit Chebfun’s methodology.

Mathematics Subject Classification (2000)

65D15 65F15 65F22 

Notes

Acknowledgments

We are grateful to John Boyd for making us aware of contouring algorithms and to Rob Corless for a fruitful discussion. We would like to thank Nick Higham, Françoise Tisseur, and Nick Trefethen for their support throughout our collaboration. We thank the anonymous referees for their useful comments that have led to improvements in the paper.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Yuji Nakatsukasa
    • 1
  • Vanni Noferini
    • 2
    Email author
  • Alex Townsend
    • 3
  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan
  2. 2.School of MathematicsUniversity of ManchesterManchesterUK
  3. 3.Mathematical InstituteUniversity of OxfordOxfordUK

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