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A Quadratic Clipping Step with Superquadratic Convergence for Bivariate Polynomial Systems

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Abstract

A new numerical approach to compute all real roots of a system of two bivariate polynomial equations over a given box is described. Using the Bernstein–Bézier representation, we compute the best linear approximant and the best quadratic approximant of the two polynomials with respect to the L 2 norm. Using these approximations and bounds on the approximation errors, we obtain a fat line bounding the zero set first of the first polynomial and a fat conic bounding the zero set of the second polynomial. By intersecting these fat curves, which requires solely the solution of linear and quadratic equations, we derive a reduced subbox enclosing the roots. This algorithm is combined with splitting steps, in order to isolate the roots. It is applied iteratively until a certain accuracy is obtained. Using a suitable preprocessing step, we prove that the convergence rate is 3 for single roots. In addition, experimental results indicate that the convergence rate is superlinear (1.5) for double roots.

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Authors and Affiliations

  1. Institute of Applied Geometry, Johannes Kepler University, Altenberger Str. 69, 4040, Linz, Austria

    Bert Jüttler

  2. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Science, Altenberger Str. 69, 4040, Linz, Austria

    Brian Moore

Authors
  1. Bert Jüttler
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  2. Brian Moore
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Correspondence to Bert Jüttler.

Additional information

B. Jüttler was supported through the Austrian Science Fund (FWF), DK W1214. B. Moore was supported by the Austrian Science Fund (FWF) under the SFB grant F1303 and the Fonds Québécois de la Recherche sur la Nature et les Technologies (FQRNT).

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Jüttler, B., Moore, B. A Quadratic Clipping Step with Superquadratic Convergence for Bivariate Polynomial Systems. Math.Comput.Sci. 5, 223–235 (2011). https://doi.org/10.1007/s11786-011-0091-4

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  • DOI: https://doi.org/10.1007/s11786-011-0091-4

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