Convergence of Newton's method and inverse function theorem in Banach space
Author:
Wang Xinghua
Journal:
Math. Comp. 68 (1999), 169186
MSC (1991):
Primary 65H10
MathSciNet review:
1489975
Fulltext PDF Free Access
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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.
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 [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 NewtonKantorovich theorem, SIAM J. Numer. Anal., 11(1974), 1013. 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), 558559; J. of Hangzhou University, 1977, 2: 1642; 1978, 3: 2326.
 [6]
 F. A. Potra, On the a posteriori error estimates for Newton's method, Beitraege Numer. Math., 12(1984), 125138. MR 85h:65128
 [7]
 F. A. Potra and V. Ptak, Sharp error bounds for Newton's process, Numer. Math., 34(1980), 6372. 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, SpringerVarlag, 1986, 185196. MR 88e:65076
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 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), 135144. MR 91h:65081
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 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, SpringerVerlag, 1993, 456465. MR 94f:00026
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 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), 721738. 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),247270. MR 94j:65067
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Additional Information
Wang Xinghua
Affiliation:
Department of Mathematics, Hangzhou University, Hangzhou 310028 China
DOI:
http://dx.doi.org/10.1090/S0025571899009990
PII:
S 00255718(99)009990
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.
Article copyright:
© Copyright 1999
American Mathematical Society
