Toric Newton method for polynomial homotopies

Journal of Symbolic Computation

Volumn 29, Issue 4-5, 2000, Pages 777-793

  Verschelde, Jan ^a  

^a MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034179140     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1006/jsco.1999.0296     Document Type: Article

References (32)

1. Bernshtein D. N. The number of roots of a system of equations. Funct. Anal. Appl. 9:1975;183-185.
2. Funktsional. Anal. i Prilozhen. (1975), 9, 1, 4.
3. Blum L., Cucker F., Shub M., Smale S. Complexity and Real Computation. 1997;Springer, New York.
4. E. Cattani, A. Dickenstein, B. Sturmfels, Algorithms in Algebraic Geometry and Applications, Gonzalez-Vega, L.Recio, T. 1996, Birkhäuser, Basel, 135, 164.
5. Cattani E., Dickenstein A. A global view of residues in the torus. J. Pure Appl. Algebra. 1997;117-118: 119-144.
6. Cox D. A. The homogeneous coordinate ring of a toric variety. J. Algebr. Geom. 4:1995;17-50.
7. Cox D., Little J., O'Shea D. Using Algebraic Geometry. 1998;Springer, New York.
8. Danilov V. The geometry of toric varieties. Russ. Math. Surv. 33:1978;97-154.
9. J. P. Dedieu, Foundations of Computational Mathematics. Selected Papers of a Conference Held at IMPA in Rio de Janeiro, January 1997, Cucker, F.Shub, M. 1997, Springer, 75, 101.
10. J. P. Dedieu, M. Shub.
11. J. P. Dedieu, M. Shub.
12. Fulton W. Introduction to Toric Varieties. 1993;Princeton University Press, Princeton.
13. T. Gao, T. Y. Li, J. Verschelde, M. Wu.
14. Gel'fand I. M., Kapranov M. M., Zelevinsky A. V. Discriminants, Resultants and Multidimensional Determinants. 1994;Birkhäuser, Basel.
15. Huber B., Sturmfels B. A polyhedral method for solving sparse polynomial systems. Math. Comput. 64:1995;1541-1555.
16. Huber B., Verschelde J. Polyhedral end games for polynomial continuation. Numer. Algorithms. 18:1998;91-108.
17. T. Krick, L. M. Pardo, Algorithms in Algebraic Geometry and Applications, Gonzalez-Vega, L.Recio, T. 1996, Birkhäuser, Basel, 193, 253.
18. Li T. Y. Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta Numerica. 6:1997;399-436.
19. Li T. Y., Sauer T., Yorke J. A. The cheater's homotopy: an efficient procedure for solving systems of polynomial equations. SIAM J. Numer. Anal. 26:1989;1241-1251.
20. Meintjes K., Morgan A. P. A methodology for solving chemical equilibrium systems. Appl. Math. Comput. 22:1987;333-361.
21. Morgan A. Solving Polynomial Systems using Continuation for Engineering and Scientific Problems. 1987;Prentice-Hall, Englewood Cliffs, NJ.
22. Morgan A. P., Sommese A. J. Coefficient-parameter polynomial continuation. Appl. Math. Comput. 29:1989;123-160.
23. Errata: Appl. Math. Comput. 51:1992;207.
24. Oda T. Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties. 1988;Springer, Berlin, Heidelberg, New York.
25. Rojas J. M. Solving degenerate sparse polynomial systems faster. J. Symb. Comput. 28:1999;155-186.
26. J. Verschelde.
27. Verschelde J. PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25:1999;251-276.
28. Verschelde J., Gatermann K., Cools R. Mixed-volume computation by dynamic lifting applied to polynomial system solving. Discrete Comput. Geom. 16:1996;69-112.
29. Verschelde J., Verlinden P., Cools R. Homotopies exploiting Newton polytopes for solving sparse polynomial systems. SIAM J. Numer. Anal. 31:1994;915-930.
30. Watson L. T., Billups S. C., Morgan A. P. Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms. ACM Trans. Math. Softw. 13:1987;281-310.
31. Watson L. T., Sosonkina M., Melville R. C., Morgan A. P., Walker H. F. HOMPACK90: a suite of Fortran 90 codes for globally convergent homotopy algorithms. ACM Trans. Math. Softw. 23:1997;514-549.
32. Ziegler G. M. Lectures on Polytopes. 1995;Springer, New York.