Finding a Cluster of Zeros of Univariate Polynomials

Journal of Complexity

Volumn 16, Issue 3, 2000, Pages 603-638

  Yakoubsohn, Jean Claude ^a  

^a UNIVERSITÉ PAUL SABATIER   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0009913240     PISSN: 0885064X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcom.2000.0555     Document Type: Article

References (28)

1. Blum L., Cucker F., Shub M., Smale S. Complexity and Real Computation. 1998;Springer-Verlag, Berlin/New York.
2. Decker D. W., Kelley C. T. Newton's method at singular points 1. SIAM J. Numer. Anal. 17:1980;66-70.
3. Decker D. W., Kelley C. T. Newton's method at singular points 2. SIAM J. Numer. Anal. 17:1980;465-471.
4. Dedieu J. P. Condition number analysis for sparse polynomial systems. Cucker F., Shub M. Foundations of Computational Mathematics. 1997;Springer-Verlag, Berlin/New York.
5. Dedieu J. P., Gourdon X., Yakoubsohn J. C. Computing the Distance from a Point to an Algebraic Hypersurface. Lectures in Applied Mathematics. 32:1996;Amer. Math. Soc. Providence.
6. J. P. Dedieu, and, M. Shub, On simple double zeros and badly conditioned zeros of analytic functions of n variables, Math. Comput, in press.
7. Dedieu J. P., Yakoubsohn J. C. Localization of an algebraic hypersurface by the exclusion algorithm. Appl. Algebra Eng. Commun. Comput. 2:1992;239-256.
8. Griewank A. On solving nonlinear equations with simple singularities or nearly singular solutions. SIAM Rev. 27:1985;537-563.
9. Griewank A., Osborne M. R. Analysis of Newton's method at irregular singularities. SIAM J. Numer. Anal. 20:1983;747-773.
10. Guillemin V., Pollack A. Differential Topology. 1981;Prentice-Hall, Englewood Cliffs.
11. Hirsch M., Smale S. On algorithms for solving f(x)=0. Comm. Pure Appl. Math. 32:1979;281-312.
12. J. Hubbard, D. Schleicher, and, S. Sutherland, How to really find roots of polynomials by Newton's method, preprint, 1998.
13. Katz I. N., Ying X. A reliable principle algorithm to find the number of zeros of an analytic function in a bounded domain. Numer. Math. 53:1988;143-163.
14. Kim M.-H., Sutherland S. Polynomial root-finding algorithms and branched covers. SIAM J. Comput. 23:1994;415-436.
15. Marden M. Geometry of Polynomials. 1966;Amer. Math. Soc. Providence.
16. Morgan A. P., Sommese A. J., Wampler C. W. Computing singular solutions to nonlinear systems. Numer. Math. 58:1991;669-684.
17. Ostrowski A. M. Solutions of Equations in Euclidean and Banach Spaces. 1973;Academic Press, San Diego.
18. Petković M. Iterative Methods for Simultaneous Inclusion of Polynomial Zeros. Lecture Notes in Mathematics. 1387:1989;Springer-Verlag, Berlin/New York.
19. Pan V. solving a polynomial equation: Some history and recent progress. SIAM Rev. 39:1997;187-220.
20. Rall L. B. Convergence of the Newton process to multiple solutions. Numerische Mathematik. 9:1966;23-37.
21. Reddien C. W. On Newton method for singular problem. SIAM J. Numer. Anal. 15:1980;993-986.
22. Renegar J. On the worst case arithmetic complexity of approximating zeros of polynomials. J. Complexity. 3:1987;90-113.
23. Schröder E. Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2:1870;317-365.
24. Smale S. Newton's estimate from data at one point. Ewing R., Goss K., Martin G. the Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics. 1986;185-196 Springer-Verlag, New York.
25. Smale S. A convergent process of price adjustment and global Newton methods. J. Math. Econ. 3:1976;107-120.
26. Wilkinson J. H. The perfidious polynomial in Studies in numerical analysis. MAA Studies Mathematics. 24:1984;1-28.
27. Yakoubsohn J. C. Contraction, robustness and numerical path-following using secant maps. J. Complexity. 16:2000;206-310.
28. Yakoubsohn J. C. Tech. Report Laboratoire MIP. 1999.