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Multihomogeneous Newton methods


Authors: Jean-Pierre Dedieu and Mike Shub
Journal: Math. Comp. 69 (2000), 1071-1098
MSC (1991): Primary 65H10, 15A99
DOI: https://doi.org/10.1090/S0025-5718-99-01114-X
Published electronically: March 10, 1999
MathSciNet review: 1752092
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Abstract | References | Similar Articles | Additional Information

Abstract: We study multihomogeneous analytic functions and a multihomogeneous Newton's method for finding their zeros. We give a convergence result for this iteration and we study two examples: the evaluation map and the generalized eigenvalue problem.


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  • [1] E. Allgower, K. Georg, Numerical Continuation Methods, Springer Verlag (1990). MR 92a:65165
  • [2] L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation, Springer Verlag (1997). MR 99a:68070
  • [3] J. P. Dedieu, Condition Operators, Condition Numbers, and Condition Number Theorem for the Generalized Eigenvalue Problem, Linear Algebra and its Applications 263, 1-24 (1997). MR 98e:15008
  • [4] J. P. Dedieu, Condition Number Analysis for Sparse Polynomial Systems, in : Foundations of Computational Mathematics, F. Cucker, M. Shub Eds. Springer (1997).
  • [5] T. Y. Li, Numerical Solution of Multivariate Polynomial Systems by Homotopy Continuation Methods, Acta Numerica 6, 399-436 (1997). CMP 98:06
  • [6] G. Malajovich, On Generalized Newton Algorithms, Theoretical Computer Science 133, 65-84 (1994). MR 95g:65073
  • [7] H. Royden. Newton's Method. Preprint.
  • [8] M. Shub, Some Remarks on Bézout's Theorem and Complexity, in : Proceedings of the Smalefest, M. V. Hirsch, J. E. Marsden, M. Shub Eds., Springer, 443-455 (1993). MR 95a:14002
  • [9] M. Shub, S. Smale, Complexity of Bézout's Theorem I : Geometric Aspects, J. Am. Math. Soc. 6, 459-501 (1993). MR 93k:65045
  • [10] 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, 267-285 (1993). MR 94m:68086
  • [11] M. Shub, S. Smale, Complexity of Bézout's Theorem III : Condition Number and Packing, J. of Complexity, 9, 4-14 (1993). MR 94g:65152
  • [12] 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
  • [13] M. Shub, S. Smale, Complexity of Bézout's Theorem V : Polynomial Time, Theoretical Computer Science, 133, 141-164 (1994). MR 96d:65091
  • [14] 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, 185-196 (1986). MR 88e:65076
  • [15] G. Stewart, J. Sun, Matrix Perturbation Theory, Academic Press, 1990. MR 92a:65017
  • [16] X. Wang, Some results relevant to Smale's reports, in : Proceedings of the Smalefest, M. V. Hirsch, J. E. Marsden, M. Shub Eds., Springer, 456-465 (1993). CMP 94:03

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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, Yorktown Heights, NY 10598-0218, USA
Email: mshub@us.ibm.com

DOI: https://doi.org/10.1090/S0025-5718-99-01114-X
Received by editor(s): October 16, 1997
Received by editor(s) in revised form: July 22, 1998
Published electronically: March 10, 1999
Additional Notes: This paper was completed when the first author was visiting at the IBM T.J. Watson Research Center in August 1997.
The second author was partially supported by an NSF grant
Article copyright: © Copyright 2000 American Mathematical Society

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