On the convergence of some quasi-Newton iterates studied by I. Păvăloiu

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 \(D\).

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

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

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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.

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