A generalization of the Newton method

Let \(X\) be a Banach space, \(Y\) a normed space, \(G:X\rightarrow Y\) a nonlinear operator, and \(G\left( x\right) =0\) a nonlinear equation. We denote by \(F:X^{2}\rightarrow Y\) a nonlinear operator for which the restriction to the diagonal of \(X^{2}\) coincide with \(G\). We first prove a Taylor type formula for operators with two variables. Next we consider the following two-step Newton type method: \[F\left( x_{n},x_{n-1}\right) +F_{x}^{\prime}\left( x_{n},x_{n-1}\right) \left( x_{n+1}-x_{n}\right) +F_{y}^{\prime}\left( x_{n},x_{n-1}\right) \left( x_{n}-x_{n-1}\right)=0.\] We study the convergence to the solution of the above sequence.

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Original title (in French)

Une généralisation de methode de Newton

English translation of the title

A generalization of the Newton method

Taylor polynomial with two variables; two-step Newton type method

Scanned paper.

I. Păvăloiu, Une généralisation de methode de Newton, Mathematica, 20(43) (1978) no. 1, pp. 45-52 (in French).

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