Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency

Abstract

The aim of this paper is to study the semilocal convergence of the eighth-order iterative method by using the recurrence relations for solving nonlinear equations in Banach spaces. The existence and uniqueness theorem has been proved along with priori error bounds. We have also presented the comparative study of the computational efficiency in case of R m with some existing methods whose semilocal convergence analysis has been already discussed. Finally, numerical application on nonlinear integral equations is given to show our approach.

Erratum to: Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency

“…Consider as in [1], the nonlinear integral equation F (x) = 0, where It can be seen that The above relations implies that Table 2; for other schemes the results are unchanged. There is no change in the Tables 3-5, since these value does not depend on the value of g(t).…”

“…Here we give only the modified scalar function appearing in [1], rest are the same. The correct value of g(t) is Other scalar functions such as h(t), p 1 , p 2 and p 3 are correct.…”

Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

“…It is noted that in all examples r i are found by solving scalar equationsp i (x) = 0, i = 1, 2, 3. Then, r is obtained using Equation (6). The parameters r i have been shown to exist above Lemma 1.…”

“…for Banach space valued mappings with F : Ω ⊂ B → B, where F is differentiable in the sense of Fréchet [1,2]. For a good survey of literature on local and semilocal convergence criteria of iterative methods see [3][4][5][6][7][8][9][10][11][12][13].…”

“…0 , ρ 1 and ρ 3 given by Equations (3)-(5), respectively exist and r is defined in Equation(6). (a 5 ) There exists r * ≥ r such that…”