Research Article
Efficient Four-Parametric with-and-without-Memory Iterative
Methods Possessing High Efficiency Indices
Alicia Cordero ,
1
Moin-ud-Din Junjua ,
2
Juan R. Torregrosa ,
1
Nusrat Yasmin ,
2
and Fiza Zafar
1,2
1
Instituto de Matem´ atica Multidisciplinar, Universitat Polit` ecnica de Val` encia, Val` encia, Spain
2
Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan
Correspondence should be addressed to Juan R. Torregrosa; jrtorre@mat.upv.es
Received 12 December 2017; Accepted 8 March 2018; Published 10 April 2018
Academic Editor: Hang Xu
Copyright © 2018 Alicia Cordero et al. Tis 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.
We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear
function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-
memory class. Te extension of new family to the with-memory one is also presented which attains the convergence order 15.5156
and a very high efciency index 15.5156
1/4
≈ 1.9847. Some particular schemes of the with-memory family are also described.
Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.
1. Introduction
In this manuscript, the problem of fnding numerical solu-
tions of nonlinear equations ()=0 is addressed. Iterative
procedures are widely used to solve this problem, (see, e.g.,
[1–3]). Traub [3] classifed the iterative methods as one-
step and multistep schemes. One-step Stefensen’s iterative
method is a known improvement of Newton’s method as it
avoids using the derivative unlike in the case of Newton’s
method. Te concept of optimal iterative method was given
by Kung and Traub [4]; that is, a multistep iterative scheme,
without memory, based on +1 functional evaluations could
attain an optimal order of convergence 2
.
According to Ostrowski [2], if is the convergence
order of an iterative method and is the total number of
functional evaluations per iteration, then the index =
1/
is known as efciency index of an iterative method.
Since multistep iterative methods overcome theoretical limits
of one-step methods concerning the order of convergence
and the efciency index, therefore several multistep iterative
schemes have been developed for solving nonlinear equations
(see, e.g., [5–7] and the overview [8]). Some optimal eighth-
order methods without memory can be found in [9–14];
these methods, among others, have been designed by using
diferent techniques as composition of known schemes and
elimination of functional evaluations using interpolation,
rational approximations, and so on or by freezing the deriva-
tives and using weight function procedure.
Multistep iterative methods with memory, which use
information from the current and previous iterations, can
increase the convergence order and the efciency index
of the multistep iterative methods without memory with
no additional functional evaluations. Te increasing in the
order of convergence is based on one or more accelerator
parameters which appear in the error equations of methods
without memory. For this reason, several multistep memory
iterative methods have been developed in recent years. For a
background study regarding the acceleration of convergence
order with memorization, one should see, for example,
[8, 15].
Traub [3] developed the frst method with memory by a
slight change in Stefensen’s scheme:
=
+
(
),
̸ =0,
+1
=
−
(
)
[
,
]
, ≥0,
(1)
Hindawi
Mathematical Problems in Engineering
Volume 2018, Article ID 8093673, 12 pages
https://doi.org/10.1155/2018/8093673