Abstract
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.
Authors
Ion Păvăloiu
Title
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
Keywords
iterative methods in normed spaces; convergence order; Newton type method; Chebyshev type method; Steffensen type method; semilocal convergence
References
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About this paper
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).
Journal
Mathematica
Publisher Name
Academia Republicii S.R.
DOI
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