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Convergence of Newton's method
and inverse function theorem
in Banach space
Author: Wang Xinghua
Journal: Math. Comp. 68 (1999), 169-186
MSC (1991): Primary 65H10
DOI: https://doi.org/10.1090/S0025-5718-99-00999-0
MathSciNet review: 1489975
Full-text PDF Free Access
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 [Enhancements On Off] (What's this?)
[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, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
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[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.
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Wang Xinghua, Convergence of Newton's method and uniqueness of the solution of equations in Banach space, Hangzhou University, preprint.
L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, 1982. MR 0664597
W.B. Gragg and R.A. Tapia, Optimal error bounds for the Newton-Kantorovich theorem, SIAM J. Numer. Anal., 11(1974), 10-13. MR 0343594
A.M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces, Academic Press, New York, 1973. MR 0359306
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.
F. A. Potra, On the a posteriori error estimates for Newton's method, Beitraege Numer. Math., 12(1984), 125-138. MR 0732159
F. A. Potra and V. Ptak, Sharp error bounds for Newton's process, Numer. Math., 34(1980), 63-72. MR 0560794
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 0870648
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 1055318
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 1246102
Wang Xinghua, A summary on complexity theorey, Contemporary Mathematics, 163(1994), 155-170. MR 1276071
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 1056644
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 1090621
Wang Xinghua and Han Danfu, The convergence of Euler's series and combinatorial skills, preprint, Hangzhou University, 1996.
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.
Chen Pengyuan, Approximate zeros of quadratically convergent algorithms, Mathematics of Computation, 63(1994),247-270. MR 1240655
Retrieve articles in Mathematics of Computation with MSC (1991): 65H10
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Wang Xinghua
Affiliation: Department of Mathematics, Hangzhou University, Hangzhou 310028 China
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
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