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

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.

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Recently Kang et al. studied the following algebraic equation: 𝑥=𝑔(𝑥) (1) and obtained the following iteration formulation [1, Equation (13)]: 𝑥_𝑛+1=−𝑥_𝑛𝑔^′𝑥_𝑛+𝑔𝑥_𝑛1−𝑔^′𝑥_𝑛,𝑔^′𝑥_𝑛≠1. (2)

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 𝑓(𝑥)=0. (3)

By the variational iteration method [2], the following iteration formulation was obtained [2, Equation (7)]: 𝑥_𝑛+1=𝑥_𝑛−ℎ𝑥_𝑛𝑓𝑥_𝑛ℎ𝑥_𝑛𝑓^′𝑥_𝑛+ℎ^′𝑥_𝑛𝑓𝑥_𝑛,ℎ𝑥_𝑛𝑓^′𝑥_𝑛+ℎ^′𝑥_𝑛𝑓𝑥_𝑛≠0, (4) where ℎ(𝑥) is an auxiliary function.

Choosing ℎ(𝑥)=1 and 𝑓(𝑥)=𝑥−𝑔(𝑥) in (4), we have 𝑥_𝑛+1=𝑥_𝑛−𝑥_𝑛−𝑔𝑥_𝑛1−𝑔^′𝑥_𝑛=−𝑥_𝑛𝑔^′𝑥_𝑛+𝑔𝑥_𝑛1−𝑔^′𝑥_𝑛. (5) This is exactly (2).

Using the basic idea of the variational iteration method as illustrated in [2] (see (1)--(8) in [2]), we can construct an iteration formulation for (1) in the form 𝑥_𝑛+1=𝑔𝑥_𝑛+𝜆𝑔𝑥_𝑛−𝑥_𝑛, (6) where 𝜆 is a Lagrange multiplier. To identify the multiplier, we set (see [2, Equation (4)]) 𝑑𝑥_𝑛+1𝑑𝑥_𝑛=𝑔^′𝑥_𝑛+𝜆𝑔^′𝑥_𝑛−1=0, (7) from which the multiplier can be identified, which is 𝑔𝜆=−^′𝑥_𝑛𝑔^′𝑥_𝑛.−1 (8) This results in 𝑥_𝑛+1=𝑔𝑥_𝑛−𝑔^′𝑥_𝑛𝑔^′𝑥_𝑛𝑔𝑥−1^𝑛−𝑥_𝑛=−𝑔𝑥_𝑛+𝑥_𝑛𝑔^′𝑥_𝑛𝑔^′𝑥_𝑛.−1 (9)

Remark 1. Equation (9) is exactly (2) or (4) when ℎ(𝑥)=1 and 𝑓(𝑥)=𝑥−𝑔(𝑥).

Remark 2. Equation (7) is exactly equivalent to 𝑔^′_𝜃(𝑥)=0 in [1].

Remark 3. 𝜃 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].