Convergence of Newton’s method and inverse function theorem in Banach space

Xinghua, Wang

doi:10.1090/S0025-5718-99-00999-0

Convergence of Newton’s method and inverse function theorem in Banach space
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Math. Comp. 68 (1999), 169-186 Request permission

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

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, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
W. B. Gragg and R. A. Tapia, Optimal error bounds for the Newton-Kantorovich theorem, SIAM J. Numer. Anal. 11 (1974), 10–13. MR 343594, DOI 10.1137/0711002
A. M. Ostrowski, Solution of equations in Euclidean and Banach spaces, Pure and Applied Mathematics, Vol. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Third edition of Solution of equations and systems of equations. 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.
Florian Alexandru Potra, On the a posteriori error estimates for Newton’s method, Beiträge Numer. Math. 12 (1984), 125–138. MR 732159
Florian-A. Potra and Vlastimil Pták, Sharp error bounds for Newton’s process, Numer. Math. 34 (1980), no. 1, 63–72. MR 560794, DOI 10.1007/BF01463998
Steve Smale, Newton’s method estimates from data at one point, The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985) Springer, New York, 1986, pp. 185–196. MR 870648
Xing Hua Wang and Dan Fu Han, On dominating sequence method in the point estimate and Smale theorem, Sci. China Ser. A 33 (1990), no. 2, 135–144. MR 1055318
M. W. Hirsch, J. E. Marsden, and M. Shub (eds.), From Topology to Computation: Proceedings of the Smalefest, Springer-Verlag, New York, 1993. Held at the University of California, Berkeley, California, August 5–9, 1990. MR 1246102, DOI 10.1007/978-1-4612-2740-3
Zhong Ci Shi and Chung-Chun Yang (eds.), Computational mathematics in China, Contemporary Mathematics, vol. 163, American Mathematical Society, Providence, RI, 1994. MR 1276071, DOI 10.1090/conm/163
Xing Hua Wang and Dan Fu Han, Domain estimates and point estimates for Newton iteration, Math. Numer. Sinica 12 (1990), no. 1, 47–53 (Chinese, with English summary); English transl., Chinese J. Numer. Math. Appl. 12 (1990), no. 3, 1–8. MR 1056644
Xing Hua Wang, Shi Ming Zheng, and Dan Fu Han, Convergence of Euler’s series, Euler’s iterative family and Halley’s iterative family under a point estimate criterion, Acta Math. Sinica 33 (1990), no. 6, 721–738 (Chinese). 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.
Pengyuan Chen, Approximate zeros of quadratically convergent algorithms, Math. Comp. 63 (1994), no. 207, 247–270. MR 1240655, DOI 10.1090/S0025-5718-1994-1240655-0