Let \(X\) be a Banach space and \(Y\) a normed space, and \(P:X\rightarrow Y\) a nonlinear operator. In order to solve the equation \(P\left( x\right)=0\), we consider the iterative method \(x_{n+1}=x_{n}+\varphi \left(x_{n}\right) \), where \(\varphi:X\rightarrow X\). We give some sufficient semilocal conditions relating \(\varphi\) and \(P\) for these iterations to converge to a solution with a given convergence order. As particular instances, we obtain convergence results for the Newton, Chebyshev and Steffensen mehods.


Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)


Original title (in French)

Sur les procedées itératifs à un ordre élevé de convergence

English translation of the title

On the iterative methods with high convergence orders


iterative methods in normed spaces; convergence order; Newton type method; Chebyshev type method; Steffensen type method; semilocal convergence


Cite this paper as:

I. Păvăloiu, Sur les procedées itérative à un order élevé de convergence, Mathématica, 12(35) (1970) no. 2, pp. 309-324 (in French).

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Academia Republicii S.R.


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