|
A condition number theorem for underdetermined polynomial systems
Author(s):
Jérôme
Dégot.
Journal:
Math. Comp.
70
(2001),
329-335.
MSC (2000):
Primary 65H10
Posted:
July 10, 2000
MathSciNet review:
1458220
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
The condition number of a numerical problem measures the sensitivity of the answer to small changes in the input. In their study of the complexity of Bézout's theorem, M. Shub and S. Smale prove that the condition number of a polynomial system is equal to the inverse of the distance from this polynomial system to the nearest ill-conditioned one. Here we explain how this result can be extended to underdetermined systems of polynomials (that is with less equations than unknowns).
References:
- 1.
- B. Beauzamy, J. Dégot, Differential identities, Trans. Amer. Math. Soc. 347, (1995), no. 7, pp 2607-2619. MR 96c:05009
- 2.
- J-P. Dedieu, Approximate solutions of numerical problems, condition number analysis and condition number theorems, The mathematics of numerical analysis. Lectures in Appl. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1996. MR 98a:65062
- 3.
- J. Demmel, On condition numbers and the distance to the nearest ill-posed problem, Num. Math. 51, (1987), pp 251-289. MR 88i:15014
- 4.
- C. Eckart, G. Young, The approximation of one matrix by another of lower rank, Psychometrika 1, (1936), pp 211-218.
- 5.
- B. Reznick, An inequality for products of polynomials, Proc. Amer. Math. Soc. 117 (1993), 1063-1073. MR 93e:11058
- 6.
- M. Shub, S. Smale, Complexity of Bézout's theorem: geometric aspects, Journal of the Amer. Math. Soc. 6 (1993), pp 459-501. MR 93k:65045
Similar Articles:
Retrieve articles in Mathematics of Computation
with
MSC (2000):
65H10
Retrieve articles in all Journals with
MSC (2000):
65H10
Additional Information:
Jérôme
Dégot
Affiliation:
Lycée Fénelon, 2, rue de l'éperon, 75006 Paris, France
Email:
jerome.degot@wanadoo.fr
DOI:
10.1090/S0025-5718-00-00934-0
PII:
S 0025-5718(00)00934-0
Received by editor(s):
August 13, 1996
Posted:
July 10, 2000
Copyright of article:
Copyright
2000,
American Mathematical Society
|
