Convergence behaviour

of inexact Newton methods

Author:
Benedetta Morini

Journal:
Math. Comp. **68** (1999), 1605-1613

MSC (1991):
Primary 65H10

DOI:
https://doi.org/10.1090/S0025-5718-99-01135-7

Published electronically:
March 10, 1999

MathSciNet review:
1653970

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**J.M. Ortega, W.C. Rheinboldt,*Iterative solution of nonlinear equation in several variables*, Academic Press, New York, 1970. MR**42:8686****2.**J.E. Dennis, R.B. Schnabel,*Numerical methods for unconstrained optimization and nonlinear equations*, Prentice Hall, Englewood Cliff, NJ, 1983. MR**85j:65001****3.**R.S. Dembo, S.C. Eisenstat, T. Steihaug,*Inexact Newton methods*, SIAM J. Numer. Anal.,19, 1982, pp. 400-408. MR**83b:65056****4.**T. Steihaug,*Quasi-Newton methods for large scale nonlinear problems*, Ph.D. Thesis, School of Organization and Management, Yale University, 1981.**5.**J.M. Martinez, L.Qi,*Inexact Newton methods for solving nonsmooth equations*, J. Comput. Appl. Math., 60, 1995, pp. 127-145. MR**96h:65076****6.**P. Deuflhard,*Global inexact Newton methods for very large scale nonlinear problems*, Impact Comput. Sci. and Engrg., 3, 1991, pp. 366-393. MR**92i:65093****7.**P.N. Brown, A.C. Hindmarsh,*Reduced storage matrix methods in stiff ODE systems*, Appl. Math. Comput., 31, 1989, pp. 40-91. MR**90f:65104****8.**K.R. Jackson,*The numerical solution of large systems of stiff IVPs for ODEs*, Appl. Numer. Math., 20, 1996, pp. 5-20. MR**97a:65061****9.**T.J.Ypma,*Local convergence of inexact Newton methods*, SIAM, J. Numer. Anal., 21, 1984, pp. 583-590. MR**85k:65043****10.**T.J.Ypma,*Local convergence of difference Newton-like methods*, Math. Comp. 41, 1983, pp. 527-536. MR**85f:65053****11.**J.L.M. van Dorsselaer, M.N. Spijker,*The error committed by stopping the Newton iteration in the numerical solution of stiff initial value problems*, IMA J. of Numer. Anal., 14, 1994, pp 183-209. MR**95c:65097****12.**T.J.Ypma,*Affine invariant convergence for Newton's methods*, BIT, 22, 1982, pp. 108-118. MR**84a:58018****13.**P.Deuflhard, G. Heindl,*Affine invariant convergence theorem for Newton methods and extension to related methods*, SIAM J. Numer. Anal., 16, 1979, pp. 1-10. MR**80i:65068**

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Additional Information

**Benedetta Morini**

Affiliation:
Dipartimento di Energetica “Sergio Stecco”, via C. Lombroso 6/17, 50134 Firenze, Italia

Email:
morini@riscmat.de.unifi.it

DOI:
https://doi.org/10.1090/S0025-5718-99-01135-7

Keywords:
Systems of nonlinear equations,
inexact methods,
affine invariant conditions

Received by editor(s):
January 23, 1997

Received by editor(s) in revised form:
January 6, 1998

Published electronically:
March 10, 1999

Article copyright:
© Copyright 1999
American Mathematical Society