The convergence ball of Newton’s method and the uniqueness ball of equations under Hölder-type continuous derivatives. (English) Zbl 1052.65054

Huang, Zhengda

Comput. Math. Appl. 47, No. 2-3, 247-251 (2004).

For Newton’s method applied to a nonlinear operator equation with a Fréchet differentiable operator in a Banach space, the optimal radius of the convergence ball is investigated. The optimality of the radius is proved for equations with a continuous derivative operator of Hölder-type. Also, another optimal estimate of the radius of the uniqueness ball of the equation, in the same type of hypothesis, is obtained.

Reviewer: Iulian Coroian (Baia Mare)

MSC:

65J15	Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
47J25	Iterative procedures involving nonlinear operators

Keywords:

Newton’s method; nonlinear operator equation; convergence ball; uniqueness ball; radius estimates; Fréchet differentiable operator; Banach space

References:

[1]	Traub, J.F.; ozniakowski, H., Convergence and complexity of Newton iteration for operator equation, J. assoc. comput. mech., 26, 250-258, (1979) · Zbl 0403.65019
[2]	Wang, X., The convergence on Newton’s method (in Chinese), Kexue tongbao, Mathematics, physics, & chemistry, 25, 36-37, (1980), A Special Issue of
[3]	Smale, S., (), 185-196
[4]	Wang, X.; Han, D., Criterion a and Newton’s method in the weak conditions, Math. numer. sinica, 19, 103-112, (1997), (in Chinese) · Zbl 0879.65031
[5]	Wang, X., Convergence of Newton’s method and uniqueness of the solution of equations in Banach space, IMA J. numer. anal., 20, 123-134, (2000) · Zbl 0942.65057
[6]	Wang, X.; Li, C., Local and global behavior for algorithm for solving equations, Chinese sci. bull., 46, 444-448, (2001)
[7]	Argyros, I.K., On Newton’s method under mild differentiability conditions and applications, Appl. math. comput., 102, 177-183, (1999) · Zbl 0930.65061
[8]	Argyros, I.K., A convergence analysis for Newton-like methods in Banach space under weak hypotheses and applications, Tamkang J. math., 30, 255-263, (1999)
[9]	Davis, H.T., Introduction to nonlinear differential and integral equations, (1962), Dover New York
[10]	Hernández, M.A., Relaxing convergence conditions for Newton’s method, J. math. anal. app., 249, 463-475, (2000) · Zbl 0965.65082
[11]	Rokne, J., Newton’s method under mild differentiability conditions with error analysis, Numer. math., 18, 401-412, (1972) · Zbl 0221.65084
[12]	Kantorovich, L.V.; Akilov, G.P., Functional analysis, (1982), Pergamon Oxford · Zbl 0484.46003