Research Article
On Generalization Based on Bi et al. Iterative Methods with
Eighth-Order Convergence for Solving Nonlinear Equations
Taher Lotfi,
1
Alicia Cordero,
2
Juan R. Torregrosa,
2
Morteza Amir Abadi,
1
and Maryam Mohammadi Zadeh
1
1
Department of Applied Mathematics, Hamedan Branch, Islamic Azad University, Hamedan 65188, Iran
2
Instituto de Matem´ atica Multidisciplinar, Universitat Polit` ecnica de Val` encia, Camino de Vera, s/n, 46022 Valencia, Spain
Correspondence should be addressed to Taher Loti; lotitaher@yahoo.com
Received 15 August 2013; Accepted 29 October 2013; Published 19 January 2014
Academic Editors: A. Agouzal and J. Sun
Copyright © 2014 Taher Loti et al. his is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
he primary goal of this work is to provide a general optimal three-step class of iterative methods based on the schemes designed by
Bi et al. (2009). Accordingly, it requires four functional evaluations per iteration with eighth-order convergence. Consequently, it
satisies Kung and Traub’s conjecture relevant to construction optimal methods without memory. Moreover, some concrete methods
of this class are shown and implemented numerically, showing their applicability and eiciency.
1. Introduction
Multipoint methods for solving nonlinear equations () =
0, where :⊂→, possess an important advantage
since they overcome theoretical limits of one-point methods
concerning the convergence order and computational ei-
ciency [1–5].
During the last years, there have been many attempts
to construct optimal three-step iterative methods without
memory for solving nonlinear equations. Indeed, Bi et al.
[6, 7] are pioneers in this case, ater Kung and Traub [8].
Some other optimal methods are due to Cordero et al. [9–
11], Dzunic et al. [12, 13], Heydari et al. [14], Geum and Kim
[15–17], Kou et al. [18], Liu and Wang [19–21], Sharma and
Sharma [22], Soleimani et al. [4], Soleymani [23], Soleymani
et al. [24–27], hukral [28–30], and hukral and Petkovi´ c[31].
Recently, iterative methods for root inding have been used
for inding matrix inversion arising from linear systems; for
more details consult Wang [32], Babajee et al. [33], Montazeri
et al. [34], Soleymani [35, 36], hukral [37], and the references
therein.
In this paper we present a new optimal class of three-step
methods without memory, which employs the idea of weight
functions in the second and third steps. he order of this
class is eight requiring four functional evaluations per step
and therefore it supports Kung and Traub’s conjecture [8]. he
proposed class includes the Bi et al. methods [6, 7].
In order to design the new methods, we will use the
divided diferences. Let () be a function deined on
an interval , where is the smallest interval containing
+1 distinct nodes
1
,
2
,...,
. he divided diference
[
0
,
1
,...,
] with th-order is deined as follows: [
0
]=
(
0
),
[
0
]=
[
1
]−[
0
]
1
−
0
,...,[
0
,
1
,...,
]
=
[
1
,
2
,...,
]−[
0
,
1
,...
−1
]
−
0
.
(1)
It is clear that the divided diference [
0
,
1
,...
] is a
symmetric function of its arguments
0
,
1
,...,
. Moreover,
if we assume that ∈
(+1)
(
), where
is the smallest
interval containing the nodes
0
,
1
,...
, and , then
[
0
,
1
,...,
,]=
(+1)
()/( + 1)! for a suitable ∈
.
Specially, if
0
=
1
=⋅⋅⋅=
=, then
[,,...,,]=
(+1)
()
(+1)!
.
(2)
Hindawi Publishing Corporation
e Scientific World Journal
Volume 2014, Article ID 272949, 8 pages
http://dx.doi.org/10.1155/2014/272949