Abstract and Applied Analysis

Volume 2013 (2013), Article ID 720640, 2 pages

http://dx.doi.org/10.1155/2013/720640

Letter to the Editor

Comment on “A New Second-Order Iteration Method for Solving Nonlinear Equations”

Haibin Li

School of Mechanical, College of Science, Inner Mongolia University of Technology, Hohhot 010051, China

Received 21 May 2013; Accepted 1 September 2013

Academic Editor: Allan Peterson

Copyright © 2013 Haibin Li. 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.

Kang et al. claimed that they obtained a new iteration formulation for nonlinear algebraic equations; however the “new” formulation was first derived in 2007 by the variational iteration method.

Recently Kang et al. studied the following algebraic equation:

and obtained the following iteration formulation [1, Equation ]:

Kang et al. claimed that this was a new iteration formulation [1]; however, a more general iteration formulation has appeared in 2007 [2].

Consider a nonlinear algebraic equation

By the variational iteration method [2], the following iteration formulation was obtained [2, Equation ]:

where is an auxiliary function.

Choosing and in (4), we have

This is exactly (2).

Using the basic idea of the variational iteration method as illustrated in [2] (see in [2]), we can construct an iteration formulation for (1) in the form

where is a Lagrange multiplier. To identify the multiplier, we set (see [2, Equation ])

from which the multiplier can be identified, which is

This results in

Remark 1. Equation (9) is exactly (2) or (4) when and .

Remark 2. Equation (7) is exactly equivalent to in [1].

Remark 3. in [1] is exactly equivalent to the Lagrange multiplier in (7).

Remark 4. The derivation process is the same as that given in [2].

It can be concluded that the so-called new iteration method is a special case of He 2007 formulation; various modifications of Newton iteration formulations are available in [2–7].

References

S. M. Kang, A. Rafiq, and Y. C. Kwun, “A new second-order iteration method for solving nonlinear equations,” Abstract and Applied Analysis, vol. 2013, Article ID 487062, 4 pages, 2013.

J. H. He, “Variational iteration method-Some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007.

J.-H. He, Y.-Q. Wan, and Q. Guo, “An iteration formulation for normalized diode characteristics,” International Journal of Circuit Theory and Applications, vol. 32, no. 6, pp. 629–632, 2004.

J.-H. He, “A new iteration method for solving algebraic equations,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 81–84, 2003.

X.-G. Luo, “A note on the new iteration method for solving algebraic equations,” Applied Mathematics and Computation, vol. 171, no. 2, pp. 1177–1183, 2005.

J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006.

J. H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012.