On a generalization of the Steffensen method

We extend the Steffensen method for solving the equation \(f\left( x\right)=0\) to the setting of the Banach spaces, \(f:X\rightarrow X,\ X\) a Banach space. Considering another equation \(x-g\left( x\right) =0\), equivalent to the above one and assuming certain conditions on the first and second order divided differences of \(f\) we obtain a semilocal convergence result for the method \[x_{n+1}=x_{n}-\left[ x_{n},g\left( x_{n}\right) ;f\right]^{-1}f\left( x_{n}\right) ,~x_{0}\in X.\]

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Steffensen method in Banach spaces; semilocal convergence

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I. Păvăloiu, Sur une généralisation de la méthode de Steffensen, Rev. Anal. Numér. Théor. Approx., 21 (1992) no. 1, pp. 59-67 (in French).

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