Mathematics in Computer Science

, Volume 5, Issue 2, pp 223–235 | Cite as

A Quadratic Clipping Step with Superquadratic Convergence for Bivariate Polynomial Systems

  • Bert Jüttler
  • Brian Moore


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.


Root finding Polynomial Bézier clipping 

Mathematics Subject Classification (2000)

Primary 30C15 Secondary 65H10 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute of Applied GeometryJohannes Kepler UniversityLinzAustria
  2. 2.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of ScienceLinzAustria

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