Communications in Numerical Analysis 2013 (2013) 1-3
Available online at www.ispacs.com/cna Volume 2013, Year 2013 Article ID cna-00185, 3 Pages
doi:10.5899/2013/cna-00185 Research Article
A note on the paper “A family of optimal iterative methods
with fifth and tenth order convergence for solving nonlinear
convergence”
Taher Lotfi1∗, Mehdi Salimi2
(1)Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran
(2)Department of Mathematics, Toyserkan Branch, Islamic Azad University, Toyserkan, Iran
Copyright 2013 c⃝Taher Lotfi and Mehdi Salimi. This 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.
Abstract
In this paper we point to serious errors that appeared in the paper [ISPACS, Journal of Interpolation and Approximation in Scientific Computing, Volume 2012, Year 2012, Article ID jiasc-00012, 11 pages, doi:10.5899/2012/jiasc-00012].
Keywords:Nonlinear equations, iterative methods, multipoint methods, convergence.
AMS Mathematical Subject Classification (2010):65H05.
1 Main result
The aim of this note is to pay attention to the assertions given in the paper [1] of Matinfar and Aminzadeh that rank the proposed root-solvers as the best methods in the considered class. However, such a high status is unjustified since the mentioned assertions about the order of convergence of these methods are wrong.
First, we give a short introduction to the results recently published in the paper [1]. Letx∗be a simple zero of a scalar
function f :D⊂R→Rdefined on an open intervalD.Matinfar and Aminzadeh [1] have developed the following
two-point method for solving nonlinear equationf(x) =0 :
yn=xn−
f(xn)
f′(x
n)
,
zn=yn−G(µn)
f(xn)
f′(xn)
, µn=
f(yn)
xn
(n=0,1, . . .), (1.1)
denoted as the equation (2.7) in [1]. It is assumed thatGis a real-valued weight function. The authors have “proved”
the following convergence theorem for the method (1.1):
Theorem(Theorem 2.1 in [1]). Assume that f∈C5(D).Suppose f(x∗) =0and f′(x∗)̸=0.If the initial point x0is sufficiently close to x∗,then the sequence{xn} generated by the iteration scheme (1.1) converges to x∗.If G is any
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function with G(0) =0,G′(0) =1,G′′(0) =2,G(k)=0(k≥3),then the convergence order of the family (1.1) arrives
to five.
There is a lot of serious flaws and incorrect results in the paper [1], including the assertions of the above theorem (that is, Theorem 2.1 in [1]) and Theorem 2.1 (not presented here). They are listed below through several comments.
Comment 1:The scheme (1.1) does not have a form of an iterative cycle. It is necessary either to replaceznin (1.1)
byxk+1or to setzn:=xnafter the second step to create an iterative process. The notationzn(instead of correctxk+1) is
also used in the proof of Theorem 2.1. This wrong notation could be regarded as a minor error and is not of essential importance.
Comment 2: Dealing with a parameter defined as µ= f(y)/xis extremely unusual, without any motivation and clear idea; moreover, it has never been applied in the existing methods. Very likely such an approach led to a great confusion and incorrect results.
Comment 3:Properties of the weight functionGare not discussed in the paper [1], although it could be assumed from
the authors proof of their original Theorem 2.1 thatGis represented by its Taylor seriesG(µ) =G(0) +G′(0)µ+
1 2G′′(0)µ
2+· · · to search for the undermined coefficientsG(k)(0) (k≥0)in order to provide as high as possible order
of convergence.
Comment 4:The conclusion that the proposed method (1.1) is of the fifth order arose from the error-relation presented in [1] in the form
zn−x∗= (2c3c22−3c42)en5+O(e6n), en=xn−x∗,ck=
1
k!
f(k)(x∗)
f′(x∗) (k≥2), (1.2)
which is wrong! Note again that one should writexn+1instead ofzn. Starting from the iterative method (1.1) and
keep-ingG(µ) =G(0) +G′(0)µ+12G′′(0)µ2+· · · temporarily undetermined, it is easy to derive by symbolic computation (in Mathematica or Maple, for example) the correct error-relation
en+1 = xn+1−x∗=−G(0)en+c2(1+G(0))e2n+
(
(2c3−2c22)(1+G(0))
−f
′(x∗)G′(0)c
2 xn
)
e3n+Ane4n. (1.3)
Let us emphasize that the expressionAnin (1.3) depends only on the normalized Taylor coefficients ck,G(0)and
G′(0),but not onG(k)(0)fork≥2.
Substituting the authors conditionsG(0) =0 andG′(0) =1 in (1.3) one obtains
en+1=xn+1−x∗=c2e2n+
(
2c3−2c22−
f′(x∗)c2 xn
)
e3n+O(e4n). (1.4)
Therefore, the proposed method (1.1) has onlyquadratic convergence!A short analysis of the expressions (1.3) and
(1.4) shows that the presence of the term withxnin the denominators is the main obstacle in obtaining higher order,
which points to a senseless definition of the parameterµnwithxnin the denominator.
Comment 5:It is obvious that the serious drawback in the construction of the two-point method (1.1) makes that the three-point method (2.14) in [1] is also pointless. If all extensive calculations concerned with this method (2.14) are performed correctly, its expected order is four.
Comment 6:Without any checking of the assertions given in [1] that the proposed two-point methods (2.1) and (2.14) have the order five and ten, respectively, it can be concluded that these assertions (given in Theorem 2.1 and 2.2 in
[1]) are wrong. Namely, according to the results of Wo´zniakowski [2], it follows that ann-point method, which is
based on Hermitian type of information and requiresn+1 function evaluations, has the order at most 2n. This is,
actually, the proof of the Kung-Traub hypothesis from 1974 for a wide class of iterative methods. Since both iterative
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schemes (2.1) and (2.14) use Hermitian type of information, their order (assuming that they are correctly designed) cannot exceed four and eight, respectively.
Comment 7:Having in mind all aforementioned comments, a mystery concerning numerical results given in Section 3 arises. How was it possible to produce approximations of very high accuracy by implementing the modest methods (2.1) and (2.14)? This is a question addressed to the authors rather than readers.
Acknowledgements
We would like to express our gratitude to the anonymous referees for their insightful valuable comments and suggestions.
References
[1] M. Marintar, M. Aminzadeh, A family of optimal iterative methods with fifth and tenth order convergence for solving nonlinear convergence for solving nonlinear equations, Journal of Interpolation and Approximation in Scientific Comnputing, 2012 (2012) 1-11.
http://dx.doi.org/10.5899/2012/jiasc-00012
[2] H. Wo´zniakowski, Maximal order of multipoint iterations usingnevaluations, In: Analytic Computational
Com-plexity (Ed. J. F. Traub), Academic Press, New York, (1976) 75-107.
