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

Abstract

In a paper from 1986, Păvăloiu \cite{Pavaloiu 1986} has considered the fixed point problem (in Banach space setting)

\[
x=\lambda D\left( x\right) +y, \qquad \lambda\in {\mathbb R},
\]
written in the form \(x -\lambda D\left( x\right) -y\) and solved by the quasi-Newton iterates
\[
x_{n+1}=x_{n}-A\left( x_{n}\right) \left[ x_{n}-\lambda D\left(
x_{n}\right) -y\right] ,\qquad n=0,1,\ldots
\]

Semilocal convergence results were obtained, ensuring linear convergence of these iterates as well as for those resulting from a special instance of \(A(x),\) when given by a certain approximation of the inverse of the Jacobian of \(x-\lambda D\left( x\right) -y:\)
\[
x_{n+1}=x_{n}-[1+\lambda D^\prime\left( x_{n}\right)] \left[x_{n}+\lambda D\left( x_{n}\right) -y\right] ,\qquad n=0,1,\ldots
\]

In this note, we analyze the local convergence of those iterates, and, using the Ostrowski local attraction theorem, we give some sufficient conditions such that the iterates converge locally both linearly and with higher convergence orders.