On the convergence of a class of a iteration methods of J. F. Traub

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 \(k+1\).

Ion Păvăloiu

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

Traub method; iterative method; nonlinear operator equation; convergence order; semilocal convergence

[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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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).

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