# Comment on “A Note on Kang-Rafiq-Kwun Iteration Method for Solving Nonlinear Equations”

Authors: Shin Min Kang; Arif Rafiq; Young Chel Kwun
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We point out in this paper that the claims made by Li in “A Note on Kang-Rafiq-Kwun Iteration Method for Solving Nonlinear Equations” are not true.


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## Body

## 1. Introduction and Explanation

Consider the following nonlinear equation:(1)f(x)=0,
which can be written in the form of the following functional equation:
(2)x=g(x),
which is the famous fixed point method.By using the following iterative relation(3)xn+1=xn+λf(xn)h(xn),
and the variational iteration method, He [1] established the following iteration method:
(4)xn+1=xn-h(xn)f(xn)h(xn)f′(xn)+h′(xn)f(xn);h(xn)f′(xn)+h′(xn)f(xn)≠0.He [1] noted that the value of the auxiliary function h(xn)  should not be zero or small value during all iteration steps, |h(xn)|>1.In [2], Li claimed that the iteration method introduced by Kang et al. [3],
(5)xn+1=-g′(xn)xn+g(xn)1-g′(xn),is not new and the formulation was first derived by He [1].We comment as follows.(1)
Their claim is wrong, because no such derivation or explanation was presented in [1].(2)
Forg(x)=x-h(x)f(x), it can be easily seen that the iteration method (5) reduces to (4). Hence, (4) is the special case of (5).(3)
On page 2 of [2], it is interesting to note that for f(x)=g(x)-x, the claimed new formulation,
(6)xn+1=g(xn)+λ(g(xn)-xn),is not new and reduced to(7)xn+1=xn+λ′f(xn);λ′=1+λ,which is the special case of relation (5).(4)
Remark  1 (see [2]). Actually Li pointed out the special case of an already derived method due to Kang et al. [3].(5)
Remark  2 (see [2]). The claim is wrong because in [3] the equation gθ′(xn)=0=θ+g′(xn) is different from the equation (7) in [2].(6)
Remark  3 (see [2]). This argument is misleading because in [3] θ=-g′(xn) which is not equal to λ=(-g′(xn))/(g′(xn)-1) appeared in equation (8) of [2].(7)
Remark  4 (see [2]). This statement is meaningless because the approach of Kang et al. [3] is different.


## 2. Conclusions

All the claims made by Li in [2] are incorrect. However, in order to obtain the variants and generalizations of the Newton-Raphson method, the approach and the performance of the variational iteration formulation can be seen in [4].



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*Source: 232618-2013-09-23.xml*