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

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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.

**[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**

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Additional Information

**Wang Xinghua**

Affiliation:
Department of Mathematics, Hangzhou University, Hangzhou 310028 China

DOI:
https://doi.org/10.1090/S0025-5718-99-00999-0

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.

Article copyright:
© Copyright 1999
American Mathematical Society