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Computing the common zeros of two bivariate functions via Bézout resultants

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.

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Notes

  1. 1.

    Historically, this functional viewpoint of a Bézout matrix is in fact due to Cayley, who modified the original method of Bézout, both in the monomial basis [43, Lesson IX].

  2. 2.

    This includes those at infinity. The requirement can be formalized, but the algebraic details are beyond the scope of this paper.

  3. 3.

    We use \(r_x \approx -0.004\) and \(r_y\approx -0.0005\). There is no special significance of these constants apart from that they are small and arbitrary.

  4. 4.

    Strictly speaking, \(D\) needs to be allowed to have \(2\times 2\) blocks, since an \(LDL^T\) factorization with \(D\) diagonal may not exist, as the example illustrates. It is possible to extend the argument to such cases, but most symmetric matrices do permit \(D\) to be diagonal, and our purpose is to explain the behavior observed in practice.

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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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Correspondence to Vanni Noferini.

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Yuji Nakatsukasa was partially supported by EPSRC grant EP/I005293/1.

Vanni Noferini was supported by ERC Advanced Grant MATFUN (267526).

Alex Townsend was supported by EPSRC grant EP/P505666/1 and the ERC grant FP7/2007-2013 to

L. N. Trefethen.

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Nakatsukasa, Y., Noferini, V. & Townsend, A. Computing the common zeros of two bivariate functions via Bézout resultants. Numer. Math. 129, 181–209 (2015). https://doi.org/10.1007/s00211-014-0635-z

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Mathematics Subject Classification (2000)

  • 65D15
  • 65F15
  • 65F22