zbMATH — the first resource for mathematics

Local convergence analysis of inexact Newton-like methods under majorant condition. (English) Zbl 1279.90195
The aim of this paper is to present a new local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. In this analysis, the classical Lipschitz condition is relaxed using a majorant function. This condition is equivalent to Wang’s condition introduced in [X. Wang, IMA J. Numer. Anal. 20, No. 1, 123–134 (2000; Zbl 0942.65057)] and used by J. Chen and W. Li in [J. Comput. Appl. Math. 191, No. 1, 143–164 (2006; Zbl 1092.65043)] to study the inexact Newton-like methods. The presented convergence analysis is linear in an arbitrary norm. It provides a new estimate for the convergence radius and a clear relationship between the majorant function and the nonlinear operator under consideration. The results also allow to obtain some special cases that can be evaluated as an application.

90C53 Methods of quasi-Newton type
Full Text: DOI
[1] Alvarez, F., Bolte, J., Munier, J.: A unifying local convergence result for Newton’s method in Riemannian manifolds. Found. Comput. Math. 8(2), 197–226 (2008) · Zbl 1147.58008
[2] Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1997) · Zbl 0948.68068
[3] Chen, J.: The convergence analysis of inexact Gauss-Newton methods for nonlinear problems. Comput. Optim. Appl. 40(1), 97–118 (2008) · Zbl 1192.90200
[4] Chen, J., Li, W.: Convergence behaviour of inexact Newton methods under weak Lipschitz condition. J. Comput. Appl. Math. 191(1), 143–164 (2006) · Zbl 1092.65043
[5] Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19(2), 400–408 (1982) · Zbl 0478.65030
[6] Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs (1983) · Zbl 0579.65058
[7] Deuflhard, P., Heindl, G.: Affine invariant convergence for Newtons method and extensions to related methods. SIAM J. Numer. Anal. 16(1), 1–10 (1979) · Zbl 0395.65028
[8] Ferreira, O.P.: Local convergence of Newton’s method in Banach space from the viewpoint of the majorant principle. IMA J. Numer. Anal. (2008, to appear). doi: 10.1093/imanum/drn036
[9] Ferreira, O.P., Svaiter, B.F.: Kantorovich’s majorants principle for Newton’s method. Comput. Optim. Appl. 42, 213–229 (2009) · Zbl 1191.90095
[10] Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1993)
[11] Martinez, J.M., Qi, L.: Inexact Newton methods for solving nonsmooth equations. J. Comput. Appl. Math. 60(1–2), 127–145 (1995) · Zbl 0833.65045
[12] Moret, I.: A Kantorovich-type theorem for inexact Newton methods. Numer. Funct. Anal. Optim. 10(3–4), 351–365 (1989) · Zbl 0653.65044
[13] Morini, B.: Convergence behaviour of inexact Newton methods. Math. Comput. 68(228), 1605–1613 (1999) · Zbl 0933.65050
[14] Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics, vol. 13. SIAM, Philadelphia (1994) · Zbl 0824.90112
[15] Smale, S.: Newton method estimates from data at one point. In: Ewing, R., Gross, K., Martin, C. (eds.) The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics, pp. 185–196. Springer, New York (1986)
[16] Traub, J.F., Wozniakowski, H.: Convergence and complexity of Newton iteration for operator equation. J. Assoc. Comput. Mach. 26(2), 250–258 (1979) · Zbl 0403.65019
[17] Wang, X.: Convergence of Newton methods and uniqueness of the solution of equations in Banach space. IMA J. Numer. Anal. 20(1), 123–134 (2000) · Zbl 0942.65057
[18] Wu, M.: A convergence theorem for the Newton-like methods under some kind of weak Lipschitz conditions. J. Math. Anal. Appl. 339(2), 1425–1431 (2008) · Zbl 1136.65059
[19] Ypma, T.J.: Affine invariant convergence results for Newton’s methods. BIT 22(1), 108–118 (1982) · Zbl 0481.65027
[20] Ypma, T.J.: Local convergence of inexact Newton methods. SIAM J. Numer. Anal. 21(3), 583–590 (1984) · Zbl 0566.65037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.