A two-step Steffensen's method under modified convergence conditions

Abstract

A simplification of a third order iterative method is proposed. The main advantage of this method is that it does not need to evaluate neither any Fréchet derivative nor any bilinear operator. A semilocal convergence theorem in Banach spaces, under modified Kantorovich conditions, is analyzed. A local convergence analysis is also performed. Finally, some numerical results are presented.

An improved local convergence analysis for Newton–Steffensen-type method

“…We also suppose that ρ = M(μ(a) + ω((1 + α 1 ) a, α 2 a)) < 1. For every constant 1 > c > ρ, there exist γ > 0 such that, for every starting point x 0 in B γ (x * ) (with x 0 = x * ), and a sequence (x k ) defined by (2) which satisfies…”

“…We reintroduce [11,18] the iterative method 0 ∈ F (x k ) + H (x k ) + ∇F (x k ) + [g 1 (x k ), g 2 (x k ); H ] (x k+1 − x k ) + G(x k+1 ), (2) where g i : D −→ X (i = 1, 2) is a continuous mapping and [x, y; H ] ∈ L(X) is a divided difference of order one satisfying …”

“…In this study, using weaker conditions than [18] and under less computational cost, we also show the linear local convergence of Newton-Steffensen method (2). We are motivated by optimization considerations [1][2][3][4][5][6][7][8][9][10].…”

“…We are motivated by optimization considerations [1][2][3][4][5][6][7][8][9][10]. In our approach, the ratio of convergence is improved and the radius of convergence is enlarged.…”

An improved local convergence analysis for a two-step Steffensen-type method

“…Recently, Amat and Busquier [2] analyze (3) in the case g 1 (x k ) = x k − α k f (x k ) and g 2 (x k ) = x k + α k f (x k ) where the sequence α k controls the good approximation of the derivative. They use in this analysis the ω-conditioned divided difference in a neighborhood V of x * :…”