Abstract
Let \(X\) be a Banach space, \(Y\) a normed space and \(P:X\rightarrow Y\) a nonlinear operator. We study the convergence of the following method for solving the equation \(P\left( x\right) =0\) \[ x_{n+1}=Q\left( x_{n}\right) -\left[ P^{\prime}\left( x_{n}\right) \right] ^{-1}P\left( Q(x_n)\right),\ n=0,1,…, \ x_{0}\in X \] where \(Q\) is a nonlinear operator associated to the nonlinear equation \(P\left( x\right) =0\). We show that if the successive approximations of \(Q\) converge with order \(k\geq2\), there the above sequence converge to the solution with order \(k+1\).
Authors
Ion Păvăloiu
Title
Original title (in French)
Sur la convergence d’une classe de méthodes itératives de J.F. Traub
English translation of the title
On the convergence of a class of a iteration methods of J.F. Traub
Keywords
Traub method; iterative method; nonlinear operator equation; convergence order; semilocal convergence
References
[1] Pavaloiu, I., Interpolation dans des espaces lineaires normes et applications, Mathematica (Cluj), 12 (35), 1, 149–158 (1970).
[2] Pavaloiu, I., Sur les procedes iteratifs a un ordre eleve de convergence. Mathematica Cluj, 12 (35), 12 , 309-324 (1970).
[3] Pavaloiu, I., Asupra operatorilor iterativi, Studii si cercetari matematice, 10, 23, 1537–1544 (1971).
[4] Traub, J. F., Iterative Methods for the Solution of Equations, Prentice-Hall. Inc. Englewood Cliffs N. J., 1964.
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About this paper
Cite this paper as:
I. Păvăloiu, Sur le convergence d’une classe de méthodes itératives de J. Traub, Rev. Anal. Numér. Théor. Approx., 2 (1973), pp. 99-104 (in French).
Journal
Revue d’analyse numerique et de la Theorie de l’Approximation
Publisher Name
Academia R.S. Romane
Journal website
Print ISSN
0301-9241
Online ISSN
2457-810X