On the semilocal convergence of efficient Chebyshev–Secant-type methods

Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces

“…Various modification of Newton's method are proposed to increase the order of convergence and efficiency. In literature [11,18,13,2,15,8,9,16], authors have established the semilocal convergence of higher order iterative methods under various continuity conditions. Recently, the semilocal convergence of an efficient fifth order method is established in [17] under Lipschitz condition on F .…”

A study of optimization for Steffensen-type methods with frozen divided differences

“…Observe in (2) that the same operator Θ is used in the three steps in order to minimize the computational cost. Several papers in the recent literature taking into account iterative methods with an operator repeated, that we call frozen, can be found in [1][2][3][4]12]. We give a more general expression of the first order divided difference operator setting ε A = A − x * and ε B = B − x * , where x * ∈ R m is a simple solution of nonlinear equation (1), and A and B are functions of x with A(x * ) = x * and B(x * ) = x * .…”

Semilocal Convergence of Steffensen-Type Algorithms for Solving Nonlinear Equations

“…Among them, a classic iterative process with cubic convergence is Chebyshev's method (see [5,7,13], and [15]):…”

“…There exists an interest in constructing the families of iterative processes free of derivatives. To obtain a new family in [7], we considered an approximation of the first derivative of F from a divided difference of first order, that is, F (x n ) ≈ [x n−1 , x n , F ], where [x, y; F ] is a divided difference of order one for the operator F at the points x, y ∈ . Then we introduce the (CSTM) as follows:…”