Enumeration of all solutions of a combinatorial linear inequality system arising from the polyhedral homotopy continuation method

Journal of the Operations Research Society of Japan

Volumn 45, Issue 1, 2002, Pages 64-82

  Takeda, Akiko ^a   Kojima, Masakazu ^b   Fujisawa, Katsuki ^c  

^a TOSHIBA CORPORATION   (Japan)

^b TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

^c KYOTO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0242530938     PISSN: 04534514     EISSN: None     Source Type: Journal    
DOI: 10.15807/jorsj.45.64     Document Type: Article

References (20)

1. E. Allgower and K. Georg: Numerical Continuation Methods (Springer-Verlag, 1990).
2. D.N. Bernstein: The number of roots of a system of equations. Functional Analysis and Appl., 9 (1975) 183-185.
3. G. Björck and R. Fröberg: A faster way to count the solutions of inhomogeneous systems of algebraic equations with applications to cyclic n-roots. Journal Symbolic Computation, 12 (1991) 329-336.
4. S. N. Chow, J. Mallet-Paret and J. A. York: A homotopy method for locating all zeros of a system of polynomials. In H. O. Peitgen and H. O. Walther (eds.): Functional Differential Equations and Approximation of Fixed Points (Lecture Notes in Mathematics, 730, Springer, 1979), 77-88.
5. V. Chvatal: Linear Programming (W.H. Freeman and Co., 1983).
6. I.Z. Emiris and J.F. Canny: Efficient incremental algorithms for the sparse resultant and the mixed volume. Journal of Symbolic Computation, 20 (1995) 117-149.
7. T. Gao and T.Y. Li: Mixed volume computation via linear programming. Taiwanese Journal of Mathematics, 4 (2000) 599-619.
8. C. B. Garcia and W. I. Zangwill: Determining all solutions to certain systems of nonlinear equations. Mathematics of Operations Research, 4 (1979) 1-14.
9. C. B. Garcia and W. I. Zangwill: Global calculation methods for finding all solutions to polynomial systems of equations in N variables. In A. V. Fiacco, and K. O. Kortanek (eds.): Extremal Methods and System Analysis (Lecture Notes in Economics and Mathematical Systems, 174, Springer, 1980), 481-497.
10. C. B. Garcia and W. I. Zangwill: Pathways to Solutions, Fixed Points, and Equilibria (Prentice-Hall, 1981).
11. B. Huber and B. Sturmfels: A polyhedral method for solving sparse polynomial systems. Mathematics of Computation, 64 (1995) 1541-1555.
12. T.Y. Li: Solving polynomial systems by polyhedral homotopies. Taiwanese Journal of Mathematics, 3 (1999) 251-279.
13. T.Y. Li and X. Li: Finding mixed cells in the mixed volume computation. To appear in Foundations of Computational Mathematics.
14. A. Morgan: Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems (Prentice-Hall, 1987).
15. M. Sato, H. Nakada, S. Sekiguchi, S. Matsuoka, U. Nagashima and H. Takagi: Ninf: a network based information library for a global world-wide computing infrastructure. In B. Hertzberger and P. Sloot (eds.): High-Performance Computing and Networking (Lecture Notes in Computer Science, 1225, Springer, 1997), 491-502.
16. S. Sekiguchi, M. Sato, H. Nakada, S. Matsuoka and U. Nagashima: - Ninf -: network based information library for globally high performance computing. In Proc. of Parallel Object-Oriented Methods and Applications (POOMA) (February, 1996), 39-48.
17. R. J. Vanderbei: Linear Programming (Kluwer Academic, 1997).
18. J. Verschelde: The database of polynomial systems is in his web site: http://www.math.uic.edu/~jan/.
19. J. Verschelde: Homotopy continuation methods for solving polynomial systems. Ph.D. Thesis (Department of Computer Science, Katholieke Universiteit Leuven, 1996).
20. J. Verschelde: PHCPACK: A general-purpose solver for polynomial systems by homotopy continuation. ACM Transactions on Mathematical Software, 25 (1999) 251-276.