Convergence behaviour of inexact Newton methods
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Abstract:
In this paper we investigate local convergence properties of inexact Newton and Newton-like methods for systems of nonlinear equations. Processes with modified relative residual control are considered, and new sufficient conditions for linear convergence in an arbitrary vector norm are provided. For a special case the results are affine invariant.References
- J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
- John E. Dennis Jr. and Robert B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, Prentice Hall Series in Computational Mathematics, Prentice Hall, Inc., Englewood Cliffs, NJ, 1983. MR 702023
- Ron S. Dembo, Stanley C. Eisenstat, and Trond Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19 (1982), no. 2, 400–408. MR 650059, DOI 10.1137/0719025
- T. Steihaug, Quasi-Newton methods for large scale nonlinear problems, Ph.D. Thesis, School of Organization and Management, Yale University, 1981.
- José Mario Martínez and Li Qun Qi, Inexact Newton methods for solving nonsmooth equations, J. Comput. Appl. Math. 60 (1995), no. 1-2, 127–145. Linear/nonlinear iterative methods and verification of solution (Matsuyama, 1993). MR 1354652, DOI 10.1016/0377-0427(94)00088-I
- Peter Deuflhard, Global inexact Newton methods for very large scale nonlinear problems, Impact Comput. Sci. Engrg. 3 (1991), no. 4, 366–393. MR 1141306, DOI 10.1016/0899-8248(91)90004-E
- Peter N. Brown and Alan C. Hindmarsh, Reduced storage matrix methods in stiff ODE systems, Appl. Math. Comput. 31 (1989), 40–91. Numerical ordinary differential equations (Albuquerque, NM, 1986). MR 996040, DOI 10.1016/0096-3003(89)90110-0
- Kenneth R. Jackson, The numerical solution of large systems of stiff IVPs for ODEs, Appl. Numer. Math. 20 (1996), no. 1-2, 5–20. Workshop on the method of lines for time-dependent problems (Lexington, KY, 1995). MR 1385232, DOI 10.1016/0168-9274(95)00114-X
- T. J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (1984), no. 3, 583–590. MR 744174, DOI 10.1137/0721040
- T. J. Ypma, Local convergence of difference Newton-like methods, Math. Comp. 41 (1983), no. 164, 527–536. MR 717700, DOI 10.1090/S0025-5718-1983-0717700-4
- J. L. M. van Dorsselaer and M. N. Spijker, The error committed by stopping the Newton iteration in the numerical solution of stiff initial value problems, IMA J. Numer. Anal. 14 (1994), no. 2, 183–209. MR 1268991, DOI 10.1093/imanum/14.2.183
- T. J. Ypma, Affine invariant convergence results for Newton’s method, BIT 22 (1982), no. 1, 108–118. MR 654747, DOI 10.1007/BF01934400
- P. Deuflhard and G. Heindl, Affine invariant convergence theorems for Newton’s method and extensions to related methods, SIAM J. Numer. Anal. 16 (1979), no. 1, 1–10. MR 518680, DOI 10.1137/0716001
Additional Information
- Benedetta Morini
- Affiliation: Dipartimento di Energetica “Sergio Stecco”, via C. Lombroso 6/17, 50134 Firenze, Italia
- MR Author ID: 608586
- Email: morini@riscmat.de.unifi.it
- Received by editor(s): January 23, 1997
- Received by editor(s) in revised form: January 6, 1998
- Published electronically: March 10, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 68 (1999), 1605-1613
- MSC (1991): Primary 65H10
- DOI: https://doi.org/10.1090/S0025-5718-99-01135-7
- MathSciNet review: 1653970