Numerical parameterization of affine varieties using

ABSTRACT

In the present work, we extend the standard idea of numerical parameterization (i.e., parameterization by the numerical solution of initial-value problems (IVPs) for ordinary differential equations (ODEs) to affine varieties in ℂ;ⁿ for n≥2. We use these results with an efficient implementation in Maple to explore the use of numerical parameterization for the visualization of Riemann surfaces.

REFERENCES

• 1: R. M. Corless. A new view of the computational complexity of initial value problems for ordinary differential equations. Numerical Algorithms, 31:115--124, 2002.
• 2: Robert M. Corless , Mark Giesbrecht , Ilias S. Kotsireas , Stephen M. Watt, Numerical Implicitization of Parametric Hypersurfaces with Linear Algebra, Revised Papers from the International Conference on Artificial Intelligence and Symbolic Computation, p.174-183, July 17-19, 2000
• 3: Robert M. Corless , Mark W. Giesbrecht , Mark van Hoeij , Ilias S. Kotsireas , Stephen M. Watt, Towards factoring bivariate approximate polynomials, Proceedings of the 2001 international symposium on Symbolic and algebraic computation, p.85-92, July 2001, London, Ontario, Canada [doi>10.1145/384101.384114]
• 4: Robert M. Corless , David J. Jeffrey, Graphing elementary Riemann surfaces, ACM SIGSAM Bulletin, v.32 n.1, p.11-17, March 1998 [doi>10.1145/294833.294839]
• 5: R. M. Corless, H. Kai, F. Lemaire, and G. Reid. The Wilkinson polynomial and monodromy. Technical report, University of Western Ontario, April 2003. Maple worksheet.
• 6: D. Cox, J. Little, and D. O'Shea. Ideals, Varieties, and Algorithms. Springer-Verlag, New York, 1996.
• 7: R. L. Faber. Differential Geometry and Relativity Theory. Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, 1983.
• 8: Erich Hartmann, Numerical parameterization of curves and surfaces, Computer Aided Geometric Design, v.17 n.3, p.251-266, March 2000 [doi>10.1016/S0167-8396(99)00050-3]
• 9: Nicholas J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2002
• 10: J. Rafael Sendra , Franz Winkler, Symbolic parametrization of curves, Journal of Symbolic Computation, v.12 n.6, p.607-631, Dec. 1991 [doi>10.1016/S0747-7171(08)80144-7]
• 11: Josef Schicho, Rational parametrization of surfaces, Journal of Symbolic Computation, v.26 n.1, p.1-29, July 1998 [doi>10.1006/jsco.1997.0199]
• 12: Tom Sederberg , Ron Goldman , Hang Du, Implicitizing rational curves by the method of moving algebraic curves, Journal of Symbolic Computation, v.23 n.2-3, p.153-175, Feb./March 1997 [doi>10.1006/jsco.1996.0081]
• 13: M. Spivak. Calculus on Manifolds: a Modern Approach to Classical Theorems of Advanced Calculus. Addison-Wesley, 1965.
• 14: H. J. Stetter. Numerical Polynomial Algebra: Concepts and Algorithms. In Proceed. 5th Asian Technology Conf. in Math., pages 22--36. ATCM, 2000.
• 15: M. Trott. Visualization of Riemann surfaces of algebraic functions. Mathematica in Education and Research, 6(4):15--36, 1997.
• 16: J. H. Wilkinson. The Perfidious Polynomial, pages 1--28. Mathematical Association of America, 1984.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation
          Subjects: Approximation of surfaces and contours

General Terms:
Algorithms, Design, Experimentation

Keywords:
numerical polynomial algebra, symbolic-numeric algorithms