|
On simple double zeros and badly conditioned zeros of analytic functions of  variables
Author(s):
Jean-Pierre
Dedieu;
Mike
Shub.
Journal:
Math. Comp.
70
(2001),
319-327.
MSC (2000):
Primary 65H10
Posted:
March 1, 2000
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We give a numerical criterion for a badly conditioned zero of a system of analytic equations to be part of a cluster of two zeros.
References:
- [1]
- V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps. Volume I. Birkhauser, 1985. MR 86f:58018
- [2]
- C. Berenstein, R. Gay, A. Vidras, A. Yger, Residue Currents and Bézout Identities. Birkhauser, 1993. MR 94m:32006
- [3]
- L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation, Springer Verlag (1997). MR 99a:68070
- [4]
- J. P. Dedieu, Condition Number Analysis for Sparse Polynomial Systems, in : Fondations of Computationnal Mathematics, F. Cucker, M. Shub Eds. Springer (1997), pp. 75-101. MR 99j:62085
- [5]
- J. P. Dedieu, M. Shub, Multihomogeneous Newton's Method. To appear in: Math. of Computation.
- [6]
- J. P. Dedieu, M. Shub, Newton's Method for Overdetermined Systems of Equations. To appear in: Math. of Computation.
- [7]
- M. Golubitsky, V. Guillemin, Stable Mappings and their Singularities. Springer Verlag, 1973. MR 49:6269
- [8]
- H. Levine, The Singularities
 , Illinois Journal of Math. 8, 152-168 (1964). MR 28:2560 - [9]
- V. Pan, Solving a Polynomial Equation: Some History and Recent Progress, SIAM Review, 39, 187-220 (1997). MR 99b:65066
- [10]
- J. Renegar, On the Worst-Case Arithmetic Complexity of Approximating Zeros of Polynomials. Journal of Complexity 3, 90-113 (1987). MR 89a:68107
- [11]
- F. Roger, Sur les variétés critiques, C. R. Acad. Sci. Paris, 208, 29-31, 1939.
- [12]
- M. Shub, S. Smale, Complexity of Bézout's Theorem I : Geometric Aspects, J. Am. Math. Soc. 6, 459-501 (1993). MR 93k:65045
- [13]
- M. Shub, S. Smale, Complexity of Bézout's Theorem II : Volumes and Probabilities, in : F. Eyssette, A. Galligo Eds. Computational Algebraic Geometry, Progress in Mathematics. Vol. 109, Birkhäuser, (1993), pp. 267-285. MR 94m:68086
- [14]
- M. Shub, S. Smale, Complexity of Bézout's Theorem III: Condition Number and Packing, J. of Complexity, 9, 4-14 (1993). MR 94g:65152
- [15]
- M. Shub, S. Smale, Complexity of Bézout's Theorem IV : Probability of Success, Extensions, SIAM J. Numer. Anal., 33, 128-148 (1996). MR 97k:65310
- [16]
- M. Shub, S. Smale, Complexity of Bézout's Theorem V : Polynomial Time, Theoretical Computer Science, 133, 141-164 (1994). MR 96d:65091
- [17]
- S. Smale, Newton's Method Estimates from Data at One Point, in : The Merging of Disciplines : New Directions in Pure, Applied and Computational Mathematics, R. Ewing, K. Gross, C. Martin Eds., Springer (1986), pp. 185-196. MR 88e:65076
Similar Articles:
Retrieve articles in Mathematics of Computation
with MSC
(2000):
65H10
Retrieve articles in all Journals with MSC
(2000):
65H10
Additional Information:
Jean-Pierre
Dedieu
Affiliation:
Laboratoire Approximation et Optimisation, Université Paul Sabatier, 31062 Toulouse Cedex 04, France
Email:
dedieu@cict.fr
Mike
Shub
Affiliation:
IBM T.J. Watson Research Center, Yorktowns Heights, New York 10598-0218
Email:
mshub@us.ibm.com
DOI:
10.1090/S0025-5718-00-01194-7
PII:
S 0025-5718(00)01194-7
Keywords:
Systems of equations,
multiple zeros,
condition numbers
Received by editor(s):
January 12, 1999
Posted:
March 1, 2000
Additional Notes:
This work was done while both authors were at MSRI, Berkeley, in fall 1998, for the Foundations of Computational Mathematics program.
Partially supported by the National Science Foundation
Copyright of article:
Copyright
2000,
American Mathematical Society
|
Comments: webmaster@ams.org