Local convergence of some Newton type methods for nonlinear systems


In order to approximate the solutions of nonlinear systems \[F(x)=0,\] with \(F:D\subseteq {\mathbb R}^n \rightarrow {\mathbb R}^n\), \(n\in {\mathbb N}\), we consider the method

\begin{align*} x_{k+1} & =x_{k}-A_{k}F(x_{k})\\A_{k+1} & =A_{k}\big(2I-F^{\prime}(x_{k+1}\big)A_{k}),\;k=0,1,…, \,A_{0}\in M_{n}({\Bbb R}), x_0 \in D,\end{align*} and we study its local convergence.


Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)


nonlinear systems of equations in R^n; Schulz method.


PDF-LaTeX file (on the journal website).

Cite this paper as:

I. Păvăloiu, Local convergence of some Newton type methods for nonlinear systems, Rev. Anal. Numér. Théor. Approx., 33 (2004) 2, pp. 209-213.

About this paper

Print ISSN


Online ISSN



Cătinaş, E. and Păvăloiu, I., On approximating the eigenvalues and eigenvectors of linear continuous operators, Rev. Anal. Numér. Théor. Approx., 26, nos. 1-2, pp. 19-27, 1997, https://ictp.acad.ro/jnaat/journal/article/view/1997-vol26-nos1-2-art3

Diaconu, A., On the convergence of an iterative proceeding of Chebyshev type, Rev. Anal. Numér. Théor. Approx., 24, nos. 1-2, pp. 19-27, 1995, https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art9

Diaconu, A. and Păvăloiu, I., Sur quelques méthodes itératives pour la résolution des equations opérationelles, Rev. Anal. Numér. Théor. Approx., 1, no. 1, pp. 45-61, 1972, https://ictp.acad.ro/jnaat/journal/article/view/1972-vol1-art3

Lazăr, I., On a Newton type method, Rev. Anal. Numér. Théor. Approx., 23, no. 2, pp. 167-174, 1994, https://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no2-art5

Păvăloiu, I., Introduction in the Theory of Approximating the Solutions of Equations, Ed. Dacia, Cluj-Napoca, 1976 (in Romanian).

Ulm, S., On the iterative method with simultaneous approximation of the inverse of the operator, Izv. Nauk. Estonskoi S.S.R., 16, no. 4, pp. 403-411, 1967.

Zehnder, J. E., A remark about Newton’s method, Comm. Pure Appl. Math., 37, pp. 361-366, 1974 https://doi.org/10.1002/cpa.3160270305.


Related Posts