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

  • Emil Cătinaş Tiberiu Popoviciu Institute of Numerical Analysis
Keywords: quasi-Newton iterates, local convergence

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.

References

E. Catinas , On the superlinear convergence of the successive approximations method,J. Optim. Theory Appl., 113 (2002) no. 3, pp. 473-485.

E. Catinas, The inexact, inexact perturbed and quasi-Newton methods are equivalent models, Math. Comp., 74 (2005) no. 249, pp. 291-301.

E. Catinas, On the convergence orders, manuscript. https://ictp.acad.ro/a-survey-on-the-high-convergence-orders-and-computational-convergence-orders-of-sequences/

Diaconu, A., Pavaloiu, I., Sur quelque methodes iteratives pour la resolution des equations operationnelles, Rev. Anal. Num ́er. Theor. Approx., vol. 1, 45-61 (1972).

J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in severalvariables, Academic Press, New York, 1970.

I. Pavaloiu, La convergence de certaines methodes iteratives pour resoudre certaines equations operationnelles, Seminar on functional analysis and numerical methods, Preprint no. 1 (1986), pp. 127-132 (in French).

I. Pavaloiu, A unified treatment of the modified Newton and chord methods, Carpathian J. Math. 25 (2009) no. 2, pp. 192-196.

Published
2015-12-18
How to Cite
Cătinaş, E. (2015). On the convergence of some quasi-Newton iterates studied by I. Păvăloiu. J. Numer. Anal. Approx. Theory, 44(1), 38-41. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2015-vol44-no1-art3
Section
Articles