Local convergence of difference Newton-like methods

Author:
T. J. Ypma

Journal:
Math. Comp. **41** (1983), 527-536

MSC:
Primary 65H10

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717700-4

MathSciNet review:
717700

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using affine invariant terms, we give a local convergence analysis of difference Newton-like methods for solving the nonlinear equation . The convergence conditions are weaker than those standardly required for methods of this class. The technique and results are valid for all currently known difference Newton-like methods which require evaluation of all components of *F* at the same point. Radius of convergence and rate of convergence results for particular difference Newton-like methods may easily be derived from the results reported here.

**[1]**J. C. P. Bus, "Newton-like methods for solving nonlinear simultaneous equations,"*Proc. Third Symp. Operations Research*, Mannheim, 1978, pp. 143-152. MR**541195 (80f:65056)****[2]**J. C. P. Bus,*Numerical Solution of Systems of Nonlinear Equations*, Mathematical Centre Tract 122, Mathematisch Centrum, Amsterdam, 1980. MR**589340 (81m:65072)****[3]**P. Deuflhard & G. Heindl, "Affine invariant convergence theorems for Newton's method and extensions to related methods,"*SIAM J. Numer. Anal.*, v. 16, 1979, pp. 1-10. MR**518680 (80i:65068)****[4]**J. Jankowska, "Theory of multivariate secant methods,"*SIAM J. Numer. Anal.*, v. 16, 1979, pp. 547-562. MR**537271 (80i:65056)****[5]**J. M. Ortega & W. C. Rheinboldt,*Iterative Solution of Nonlinear Equations in Several Variables*, Academic Press, New York, 1970. MR**0273810 (42:8686)****[6]**H. Schwetlick,*Numerische Lösung Nichtlinearer Gleichungen*, VEB-DVW, Berlin, 1979. MR**519682 (80f:65061)****[7]**T. J. Ypma, "Affine invariant convergence results for Newton's method,"*BIT*, v. 22, 1982, pp. 108-118. MR**654747 (84a:58018)****[8]**T. J. Ypma, "Following paths through turning points,"*BIT*, v. 22, 1982, pp. 368-383. MR**675671 (84e:65051)****[9]**T. J. Ypma,*Numerical Solution of Systems of Nonlinear Algebraic Equations*, D. Phil. thesis, Oxford, 1982.

Retrieve articles in *Mathematics of Computation*
with MSC:
65H10

Retrieve articles in all journals with MSC: 65H10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717700-4

Article copyright:
© Copyright 1983
American Mathematical Society