Ball analysis for an e cient sixth convergence order-scheme under weaker conditions

In this study we consider an e cient sixth order-scheme for solving Banach space valued equations. The convergence criteria in earlier studies involve higher order derivatives limiting applicability of these methods. In this study we use the rst derivative only in our analysis to expand the usage of these schemes. The technique we use can be used on other schemes to obtain the same advantages. Numerical experiments compare favorably our results to earlier ones.


Introduction
Let F : D ⊂ B 1 → B 2 be a continuously dierentiable nonlinear operator and D stand for an open non empty convex compact set of B 1 . Here B 1 and B 2 stand for Banach spaces. Consider the problem of nding a solution x * of the nonlinear equation It is desirable to obtain a unique solution x * of (1). But this can rarely be achieved, so most researchers and practitioners develop iterative schemes which converge to x * . In this paper we extend the convergence ball of a class of an ecient sixth order-scheme studied in [18]. Precisely, we consider the sixth order method dened in [18] for n = 1, 2, . . . , by where A n = F (x n ) + F (y n ). The analysis in [18] uses assumptions on the sixth order derivatives of F and when B 1 = B 2 = R m . The assumptions on higher order derivatives reduce the applicability of method (2). For example: Let Then, we get Hence, the convergence of scheme (2) is not guaranteed by the analysis in [18]. In this study we use only assumptions on the rst derivative to prove our results. The advantages of our approach include: larger radius needed on method of convergence (i.e. more initial points), tighter upper bounds on x k − x * ( i.e. fewer iterates to achieve a desired error tolerance). It is worth noting that these advantages are obtained without any additional conditions [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. Throughout this paper U (x, r) stand for open ball with center at x and radius r > 0 andŪ (x, r) denote the closure of U (x, r).
Rest of the paper is organized as follows. The convergence analysis of method (2) is given in Section 2 and examples are given in Section 3.

Ball analysis
We consider real functions and parameters to assist us in the convergence of method (2). Assume equation h 1 (s) = g 1 (s) − 1.

Assume equation
has a least root in (0, r 0 ) denoted by r p . Dene functions q, b, g 2 and h 2 on the interval [0, r p ) as Assume Then, we get again using (4) and the denitions: has a least root in (0, r p ) denoted by r 1 . Dene functions c, g 3 and h 3 on (0, r 1 ) as Notice that then, we have for s ∈ [0, R) and Set e n = x n − x * . The conditions (A) that follow shall be used in the ball convergence of method (2): (A2) There exists a continuous and increasing function ω 0 on I 0 with values on itself with ω 0 (0) = 0 such that for all , if r 0 exists and is given in (1).
(A3) There exist continuous and increasing functions ω and ω 1 on the interval I 0 with values on interval I 0 such that for each x, y ∈ I 0 and (1)-(7) are true, where R is dened in (7).
Under these denitions and conditions we present the ball convergence of method (2).
Then, the following items hold for all lim n−→∞ y n − x * ≤ g 1 (e n )e n ≤ e n < R, x n+1 − x * ≤ g 3 (e n )e n ≤ e n , and x * is the only solution of equation F (x) = 0 in the set D 1 given below condition (A5), and the functions g i , h i are dened previously.
Then, by (A1) and (A5) we obtain so the invertability is implied leading together with the estimate 0 Remark 2.2. We can compute [24] the computational order of convergence (COC) dened by or the approximate computational order of convergence This way we obtain in practice the order of convergence without resorting to the computation of higher order derivatives appearing in the method or in the sucient convergence criteria usually appearing in the Taylor expansions for the proofs of those results.