A new convergence theorem for Steffensen’s method on Banach spaces and applications. (English) Zbl 0895.65024

Argyros, Ioannis K.

Southwest J. Pure Appl. Math. 1, 23-29 (1997).

The author proves a semilocal convergence theorem for Steffensen’s method, applied for approximating a locally unique solution of the equation \(f(x)=0\), where \(f\) is a twice Fréchet-differentiable continuous operator on a closed convex domain \(D\) of a Banach space with values in a Banach space. As shown by an example, the convergence region thus obtained is larger than the ones given in other related papers.

Reviewer: V.Berinde (Baia Mare)

MSC:

65J15	Numerical solutions to equations with nonlinear operators
47J25	Iterative procedures involving nonlinear operators

Keywords:

divided differences; locally unique solution; region of accessibility; Steffensen’s method; contraction mapping principle; Banach space; semilocal convergence