Abstract

In 1986, I. Păvăloiu [6] has considered a Banach space and the fixed point problem \[x=\lambda D\left( x\right) +y, \qquad D:X\rightarrow X \ \textrm{nonlinear},\ \lambda\in {\mathbb R},\ y\in X \ \textrm{given}\]written in the equivalent form \(F(x):=x -\lambda D\left( x\right) -y=0\) and solved by the general quasi-Newton method\[x_{n+1}=x_n-A(x_n) \left[ x_n-\lambda D(x_n) -y\right] ,\qquad n=0,1,\ldots\]Semilocal convergence results were obtained, ensuring linear convergence of these iterates. Further results were obtained for the iterates \[x_{n+1}=x_n-[I+\lambda D^\prime(x_n)] \left[x_n+\lambda D(x_n) -y\right] ,\qquad n=0,1,\ldots\] In this note, we analyze the local convergence of these iterates, and, using the Ostrowski local attraction theorem, we give some sufficient conditions such that the iterates converge locally either linearly or with higher convergence orders. The local convergence results require fewer differentiability assumptions for \(D\).

Authors

Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Keywords

nonlinear equations in Banach spaces; inexact Newton method; quasi-Newton method; Ostrowski local attraction theorem; local convergence; convergence order.

References

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About this paper

Cite this paper as:

E. Cătinaş, On the convergence of some quasi-Newton iterates studied by I. Păvăloiu, J. Numer. Anal. Approx. Theory, 44 (2015) no. 1, pp. 38-41.

Print ISSN

2457-6794

Online ISSN

2501-059X

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http://ictp.acad.ro/jnaat/journal/article/view/1972-vol1-art3

[5]  J.M. Ortega,  W.C. Rheinboldt, Iterative  solution  of  nonlinear  equations  in  several Variables , Academic Press, New York, 1970.

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