Computing the topology of a real algebraic plane curve whose equation is not directly available

SNC'07 - Proceedings of the 2007 International Workshop on Symbolic-Numeric Computation

Volumn , Issue , 2007, Pages 46-54

  Aruliah, D A ^a   Corless, Robert M ^b   Shakoori, Azar ^b   Gonzalez Vega, Laureano ^c   Rua, Ignacio F ^d  

^a UNIVERSITY OF ONTARIO INSTITUTE OF TECHNOLOGY   (Canada)

^b UNIVERSITY OF WESTERN ONTARIO   (Canada)

^c UNIVERSITY OF CANTABRIA   (Spain)

^d UNIVERSITY OF OVIEDO   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; POLYNOMIALS; PROBLEM SOLVING; TOPOLOGY;

BEZOUTIAN; LAGRANGE BASIS; NULLSPACE;

CURVE FITTING;

EID: 36448990386     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1277500.1277510     Document Type: Conference Paper

References (36)

1. A. Amiraslani. Algorithms for Matrices, Polynomials, and Matrix Polynomials. PhD thesis, University of Western Ontario, London, Canada, May 2006.
2. A. Amiraslani, D. A. Aruliah and R. M. Corless. The Rayleigh quotient iteration for generalized companion matrix pencils. Submitted, 2006.
3. A. Amiraslani, R. M. Corless and P. Lancaster. Linearization of matrix polynomials expressed in polynomial bases. Submitted, 2006.
4. D.A. Aruliah, R. M. Corless, L. Gonzalez-Vega, and A. Shakoori. Geometric Applications of the Bezout Matrix in the Bivariate Tensor-Product Lagrange basis. In these proceedings, 2007.
5. D.A. Aruliah, R. M. Corless, L. Gonzalez-Vega and A. Shakoori. Companion Matrix Pencils and Bezout Matrices for Hermite Interpolants. In these proceedings, 2007.
6. A. Amiraslani. Dividing polynomials when you only know their values. In Proceedings EACA, Laureano Gonzalez-Vega and Tomas Recio editors, pages 5-10, June 2004.
7. D. Arnon and S. McCallum. A polynomial time algorithm for the topological type of a real algebraic curve. Journal of Symbolic Computation 5, 213-236, 1988.
8. C. Bajaj, C. M. Hoffmann, R. E. Lynch and J. E. H. Hopcroft. Tracing surface intersections. Computer Aided Geometric Design 5, 285-307, 1988.
9. D. Bini and V. Y. Pan. Polynomial and matrix computations. Fundamental algorithms. Progress in Theoretical Computer Science, 1, Birkhäuser, 1994.
10. L. Bus, H. Khalil and B. Mourrain. Resultant-Based Method for Plane Curves Intersection Problems. Lecture Notes in Computer Science 3718, 75-92, 2005.
11. P. Cellini, P. Gianni and C. Traverso. Algorithms for the shape of semialgebraic sets: a new approach. Lecture Notes in Computer Science 539, 1-18, Springer-Verlag, 1991.
12. R. M. Corless. Generalized companion matrices in the Lagrange basis. In Proceedings EACA, L. Gonzalez-Vega and T. Recio editors, pages 317-322, June 2004.
13. R. M. Corless, N. Rezvani and A. Amiraslani. Pseudospectra of matrix polynomials that are expressed in alternative bases. Mathematics and Computer Science, (in press) 2007.
14. F. Cucker, L. Gonzalez-Vega and F. Rosello. On algorithms for real algebraic plane curves. In Effective Methods in Algebraic Geometry, Progress in Mathematics 94, 63-88, Birkhausser, 1991.
15. J. W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, 1997.
16. I. G. Gohberg, P. Lancaster, and L. Rodman. Matrix polynomials. Academic Press, 1982.
17. R. L. Graham, D. E. Knuth and O. Patashnik. Concrete Mathematics. Addison-Wesley, 1994.
18. A. Eigenwillig, M. Kerber, N. Wolpert. Fast and Exact Geometric Analysis of Real Algebraic Plane Curves. In Proceedings ISSAC'07 (in press), July 2007.
19. H. Feng. Decomposition and computation of the topology of plane real algebraic curves. PhD thesis, The Royal Institute of Technology, Stockholm, Sweden, 1992.
20. M. Fioravanti and L. Gonzalez-Vega. On the geometric extraneous components appearing when using implicitization. In Mathematical Methods for Curves and Surfaces, pages 157-168. Nashboro Press, 2005.
21. P. Gianni and C. Traverso. Shape determination of real curves and surfaces. Ann. Univ. Ferrara Sez VII Sec. Math., vol. XXIX, 87-109, 1983.
22. L. Gonzalez-Vega and I. Necula. Efficient topology determination of implicitly defined algebraic plane curves. Computer Aided Geometric Design 19, 719-743, 2002.
23. L. Gonzalez-Vega. Implicitization of Parametric Curves and Surfaces by using Multidimensional Newton Formulae. Journal of Symbolic Computation 23, 137-152, 1997.
24. L. Gonzalez-Vega and M. El Kahoui. An improved upper complexity bound for the topology computation of a real algebrad plane curve. Journal of Complexity 12, 527-544, 1996.
25. G. Heinig and K. Rost. Algebraic methods for Toeplitz-like matrices and operators. Operator Theory: Advances and Applications, 13, Birkhäuser, 1984.
26. U. Helmke and P. A. Fuhrmann. Bezoutians. Linear Algebra and Its Applications 122/123/124, 1039-1097, 1989.
27. H. Hong. An efficient method for analyzing the topology of plane real algebraic curves. Mathematics and Computers in Simulation 42, 4-6, 571-582 (1996).
28. J. Keyser , T. Culver, D. Manocha and S. Krishnan. Efficient and Exact Manipulation of Algebraic Points and Curves. Computer Aided Design 32, 649-662, 2000.
29. D. Manocha and J. Demmel. Algorithms for Intersecting Parametric and Algebraic Curves I: Simple Intersections. ACM Transactions on Graphics 13, 73-100, 1994.
30. D. Manocha and J. W. Demmel. Algorithms for intersecting parametric and algebraic curves II: multiple intersections. Computer Vision, Graphics and Image Processing: Graphical Models and Image Processing 57, 81-100, 1995.
31. M.-F. Roy. Basic algorithms in real algebraic geometry and their complexity: from Sturm's theorem to the existential theory of reals. In Lectures in real geometry, 1-67, de Gruyter Exp. Math., 23, de Gruyter, 1996.
32. M.-F. Roy and A. Szpirglas. Complexity of the computation of cylindrical decomposition and topology of real algebraic curves using Thom's Lemma. Lecture Notes in Mathematics 1420, 223-236, Springer-Verlag, 1990.
33. T. Sakkalis. The topological configuration of a real algebraic curve. Bulletin of the Australian Mathematical Society 43, 37-50, 1991.
34. R. Seidel and N. Wolpert. On the Exact Computation of the Topology of Real Algebraic Curves (Exploiting a little more Geometry and a little less Algebra). Proceedings of the 21-st Annual ACM Symposium on Computational Geometry, 107-115, 2005.
35. A. Shakoori. The Bezout matrix in the Lagrange basis. In Proceedings EACA, Laureano Gonzalez-Vega and Tomas Recio editors, 295-299, June 2004.
36. F. Tisseur and N. J. Higham. Structured pseudospectra for polynomial eigenvalue problems, with applications. SIAM J. Matrix Anal. Appl 23, 187-208, 2001.