 
Summary: On a generalization of Steffensen method
One of the most important techniques to study nonlinear equations is the
use of iterative processes, starting from an initial approximation x0, called pivot,
successive approaches (until some predetermined convergence criterion is satisŢed)
xi are computed , i = 1, 2, . . . , with the help of certain iteration function : X
X,
xi+1 := (xi), i = 0, 1, 2 . . . (1)
Certainly Newton method (second order) is the most useful iteration for this
purpose. In this case, we need to evaluate a derivative in each step, it is the main
difficulty. Steffensen method (second order) can be considered as a simpliŢcation
of original Newton method where F
0
(xk) is replaced by a special approximation.
If we are interesting to approximate a solution of the nonlinear equation
F (x) = x, (2)
Steffensen method can be written as
xk+1 = xk + (I  [F(xk), xk; F])1
(F(xk)  xk). (3)
where [·, ·; F] denotes a divided difference of Ţrst order for the operator F : X X
(X a Banach space).
