Abstract

The inexact secant method \([x_{k-1},x_{k};F]s_{k}=-F(x_k) +r_k\), \( x_{k+1}=x_k+s_k\), \( k=1,2,\ldots\), \( x_0,x_1 \in {\mathbb R}^n\) for solving the nonlinear system \( F(x)=0\) is considered, where \(F:{\mathbb R}^n \rightarrow {\mathbb R}^n\) is a nonlinear mapping.

We study the setting of the inexact Newton method, i.e., when the linear system (involving the divided differences) at each step is not solved exactly, and an error term \(r_k\) (called residual) is considered.

Under certain standard assumptions, we characterize the superlinear convergence and the r-convergence order of the secant method in terms of the residuals. We also give a sufficient result for linear convergence.

Authors

E. Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)

Keywords

nonlinear system of equations; secant method; r-convergence order; inexact method; residual; error term; linear convergence.

Cite this paper as:

E. Cătinaş, A note on inexact secant methods, Rev. Anal. Numér. Théor. Approx., 25 (1996) nos. 1-2, pp. 33-41.

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Print ISSN

2457-6794

Online ISSN

2501-059X

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References

[1] G. Goldner, M. Balazs, Observații asupra diferențelor divizate și asupra metodei coardei, Revista de Analiza Numerica si Teoria Aproximatiei, 3 (1974) no. 1, pp. 19–30 (in Romanian).
[English title: Remarks on divided differences and method of chords]
article

[2] J.E. Dennis Jr., J.J. More, Quasi-Newton methods, motivation and theory, SIAM Rev., 19 (1977), pp. 46–89.
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[3] R.S. Dembo, S.C. Eisenstat, T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982) 2, pp. 400–408.
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[4] G. Goldner, M. Balazs, Asupra metodei coardei si a unei modificari a ei pentru rezolvarea ecuațiilor operationale neliniare, Stud. Cerc. Mat., 20 (1968), pp. 981–990 (in Romanian).
[English title: On the method of chord and on its modification for solving the nonlinear operator equations]

[5] I. Muntean, Analiza functionala, ”Babes-Bolyai” University, Cluj-Napoca, 1993 (in Romanian)
[English title: Functional Analysis]

[6] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
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