Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On simple double zeros and badly conditioned zeros of analytic functions of $n$ variables

Authors: Jean-Pierre Dedieu and Mike Shub
Journal: Math. Comp. 70 (2001), 319-327
MSC (2000): Primary 65H10
Published electronically: March 1, 2000
MathSciNet review: 1680867
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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 [Enhancements On Off] (What's this?)

  • [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 $S_{1}^{q}$, 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

Mike Shub
Affiliation: IBM T.J. Watson Research Center, Yorktowns Heights, New York 10598-0218

Keywords: Systems of equations, multiple zeros, condition numbers
Received by editor(s): January 12, 1999
Published electronically: 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
Article copyright: © Copyright 2000 American Mathematical Society