|
Convergence of Newton's method and inverse function theorem in Banach space
Author(s):
Wang
Xinghua.
Journal:
Math. Comp.
68
(1999),
169-186.
MSC (1991):
Primary 65H10
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
Under the hypothesis that the derivative satisfies some kind of weak Lipschitz condition, a proper condition which makes Newton's method converge, and an exact estimate for the radius of the ball of the inverse function theorem are given in a Banach space. Also, the relevant results on premises of Kantorovich and Smale types are improved in this paper.
References:
- [1]
- Wang Xinghua, Convergence of Newton's method and uniqueness of the solution of equations in Banach space, Hangzhou University, preprint.
- [2]
- L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, 1982. MR 83h:46002
- [3]
- W.B. Gragg and R.A. Tapia, Optimal error bounds for the Newton-Kantorovich theorem, SIAM J. Numer. Anal., 11(1974), 10-13. MR 49:8334
- [4]
- A.M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces, Academic Press, New York, 1973. MR 50:11760
- [5]
- Wang Xinghua, Convergence of an iterative procedure, KeXue TongBao, 20(1975), 558-559; J. of Hangzhou University, 1977, 2: 16-42; 1978, 3: 23-26.
- [6]
- F. A. Potra, On the a posteriori error estimates for Newton's method, Beitraege Numer. Math., 12(1984), 125-138. MR 85h:65128
- [7]
- F. A. Potra and V. Ptak, Sharp error bounds for Newton's process, Numer. Math., 34(1980), 63-72. MR 81c:65027
- [8]
- 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 and C. Martin eds, New York, Springer-Varlag, 1986, 185-196. MR 88e:65076
- [9]
- Wang Xinghua and Han Danfu, On the dominating sequence method in the point estimates and Smale's theorem, Science in China(Ser. A.), 33(1990), 135-144. MR 91h:65081
- [10]
- Wang Xinghua, Some results relevant to Smale's reports, in From Topology to Computation: Proceedings of the Smalefest, M.W. Hirsch, J. E. Marsden and M. Shub eds., New York, Springer-Verlag, 1993, 456-465. MR 94f:00026
- [11]
- Wang Xinghua, A summary on complexity theorey, Contemporary Mathematics, 163(1994), 155-170. MR 94m:65007
- [12]
- Wang Xinghua, Han Danfu and Sun Fangyu, Point estimations on deformated Newton's iteration, Math. Num. Sin., 12(1990), 145-156; Chinese J. Num. Math. Appl., 12(1990), 1-13. MR 91d:58014
- [13]
- Wang Xinghua, Zheng Shiming and Han Danfu, Convergence on Euler's series, the iterations of Euler's and Halley's families, Acta Mathematica Sinica, 33(1990), 721-738. MR 92b:65041
- [14]
- Wang Xinghua and Han Danfu, The convergence of Euler's series and combinatorial skills, preprint, Hangzhou University, 1996.
- [15]
- L. Blum, F. Cucker, M. Shub and S. Smale, Complexity and Real Computation, Part II: Some Geometry of Numerical Algorithms, City University of Hong Kong, preprint, 1996.
- [16]
- Chen Pengyuan, Approximate zeros of quadratically convergent algorithms, Mathematics of Computation, 63(1994),247-270. MR 94j:65067
Similar Articles:
Retrieve articles in Mathematics of Computation
with MSC
(1991):
65H10
Retrieve articles in all Journals with MSC
(1991):
65H10
Additional Information:
Wang
Xinghua
Affiliation:
Department of Mathematics, Hangzhou University, Hangzhou 310028 China
DOI:
10.1090/S0025-5718-99-00999-0
PII:
S 0025-5718(99)00999-0
Received by editor(s):
March 12, 1997
Received by editor(s) in revised form:
June 6, 1997
Additional Notes:
Supported by the China State Major Key Project for Basic Research and the Zhejiang Provincial Natural Science Foundation.
Copyright of article:
Copyright
1999,
American Mathematical Society
|
Comments: webmaster@ams.org