Numerical and Algebraic Properties of Bernstein Basis Resultant Matrices
Algebraic properties of the power and Bernstein forms of the companion, Sylvester and Bézout resultant matrices are compared and it is shown that some properties of the power basis form of these matrices are not shared by their Bernstein basis equivalents because of the combinatorial factors in the Bernstein basis functions. Several condition numbers of a resultant matrix are considered and it is shown that the most refined measure is NP-hard, and that a simpler, sub-optimal measure is easily computed. The transformation of the companion and Bézout resultant matrices between the power and Bernstein bases is considered numerically and algebraically. In particular, it is shown that these transformations are ill—conditioned, even for polynomials of low degree, and that the matrices that occur in these basis transformation equations share some properties.
KeywordsCondition Number Companion Matrix Bernstein Polynomial Power Basis Resultant Matrice
Unable to display preview. Download preview PDF.
- 1.Barnett, S.: Polynomials and Linear Control Systems. Marcel Dekker. New York, USA, 1983.Google Scholar
- 2.Bini, D., Gemignani, L.: Bernstein-Bezoutian matrices. Theoretical Computer Science (to appear).Google Scholar
- 3.Canny, J. F.: The Complexity of Robot Motion Planning. The MIT Press. Cambridge, USA, 1988.Google Scholar
- 4.Goldman, R. N.: Private communication.Google Scholar
- 6.Golub, G. H., Van Loan, C. F.: Matrix Computations. John Hopkins University Press. Baltimore, USA, 1989.Google Scholar
- 7.Higham, N. J.: Accuracy and Stability of Numerical Algorithms. SIAM. Philadelphia, USA, 2002.Google Scholar
- 8.Horn, R. A., Johnson, C. R.: Topics in Matrix Analysis. Cambridge University Press. Cambridge, England, 1991.Google Scholar
- 9.Kajiya, J. T.: Ray tracing parametric patches. Computer Graphics 16 (1982) 245–254.Google Scholar
- 13.Winkler, J. R.: Computational experiments with resultants for scaled Bernstein polynomials, in Mathematical Methods for Curves and Surfaces: Oslo 2000, Tom Lyche and Larry L. Schumaker (eds.). Vanderbilt University Press. Nashville, Tennessee, USA (2001) 535–544.Google Scholar
- 14.Winkler, J. R.: Properties of the companion matrix resultant for Bernstein polynomials, in Uncertainty in Geometric Computations, J. R. Winkler and M. Niranjan (eds.). Kluwer Academic Publishers. Massachusetts, USA (2002) 185–198.Google Scholar
- 17.Winkler, J. R., Goldman, R. N.: The Sylvester resultant matrix for Bernstein polynomials, in Curve and Surface Design: Saint-Malo 2002, Tom Lyche, Marie-Laurence Mazure and Larry L. Schumaker (eds.). Nashboro Press. Brentwood, Tennessee, USA (2003) 407–416.Google Scholar
- 18.Winkler, J. R., Ragozin, D. L.: A class of Bernstein polynomials that satisfy Descartes’ rule of signs exactly, in The Mathematics of Surfaces IX, Roberto Cipolla and Ralph Martin (eds.). Springer-Verlag. London, England (2000) 424–437.Google Scholar