On the convergence of a class of 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
(Tiberiu Popoviciu Institute of Numerical Analysis)

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 iteration methods of J.F. Traub

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

Scanned paper (on the journal website).

PDF-Latex version of the paper.

I. Păvăloiu, Sur le convergence d’une classe de méthodes itératives de J. Traub, Rev. Anal. Numér. Théor. Approx., 2 (1973), pp. 99-104, https://doi.org/10.33993/jnaat21-15 (in French).

[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.