ACCELERATING THE CONVERGENCE OF THE ITERATIVE METHODS OF INTERPOLATORY TYPE

. In this paper we deal with iterative methods of interpolatory type, for solving nonlinear equations in Banach spaces. We show that the convergence order of the iterations may considerably grow if the nodes are properly controlled.


INTRODUCTION
Let X be a Banach space, D ⊆ X a subset, and f : D → X a nonlinear mapping.Consider the equation ( 1) where θ ∈ X is the zero vector of X.
Regarding f we make the following assumptions: a) f : D → f (D) is a one to one mapping; b) equation ( 1) has a solution x * ∈ D; c) the operator f is Fréchet differentiable on D and f (x) = θ 1 , where θ 1 is the null linear operator.
In order to accelerate the convergence of the iterative methods of interpolatory type, we also consider an equation, equivalent to (1), of the form where ϕ : D → D.
We make the following assumptions regarding ϕ: the Newton identity holds (see, e.g., [3], [4]): By hypotheses a) and b) we get the relation: The relation above and (3) for y = θ attract that (5) whence we deduce an approximation u for x * , of the form: The error for this approximation is bounded by where the norm of [θ, y 0 , . . .y n ; f −1 ] is considered in the space of n + 1-linear operators.
In [2] it is shown that the convergence order of the iterative methods given by ( 6) cannot be greater than 2, even if the number of the interpolation nodes is arbitrarily increased.However, the convergence order can be increased if we use the auxilliary function ϕ considered above.
Let x 0 ∈ D be an initial approximation to x * .Denote (6) the interpolation nodes by y 0 i , i = 0, n, we obtain for x * a first approximation, denoted by x 1 : (8) then we obtain the next approximation in the following way.Denote Analogously to (8) we get (9) with the error (10) In the following we shall analyze some particular instance of (9).
If we take n = 1 in (9), we get (11) , we notice that we are lead to the Steffensen method: (12) If we take n = 2 and recall that . ., and we get In [5] we have studied the local convergence of the Steffensen method (12).In the following we shall study the local convergence of the general method (9).We shall show that the convergence order considerably grows even for p = 1, under condition a ).

LOCAL CONVERGENCE
From (10), using the finite growth formula (see, e.g., [4]), we get (15) where we have assumed that there exist M, K > 0 such that From the hypotheses a ) and b ), using the Taylor formula we get (18) where l = L p! and δ s = s−1 j=0 p j .
In case of the particular methods (12) and (14) we get the following results.
Corollary 2.1.In the case of the Steffensen method, we obtain the well known result (see, e.g., [5]) for α = 1 and β = 1 + p if p > 1 and β = 2 if p = 1.We conclude that the iterative methods of interpolatory type may attain a substantially higher convergence order if the nodes are properly controlled.