In this paper, we establish a two step third-order iteration method for solving nonlinear equations. The efficiency index of the method is 1.442 which is greater than Newton-Raphson method. It is important to note that our method is performing very well in comparison to fixed point method and the method discussed by Kang et al. (Abstract and applied analysis; volume 2013, Article ID 487060).
Definition 1.1. Suppose, \(\{x_n\} \to \alpha\), with the property that, \begin{equation*} \lim\limits_{n \to \infty} \left\vert \frac{x_{n+1}-\alpha }{(x_{n}-\alpha )^{q}}\right\vert = D \end{equation*} where, \(D \in \mathbb{R}^+\) and \(q \in \mathbb{Z}\), then \(D\) is called the constant of convergence and \(q\) is called the order of convergence.
Definition 1.2. [2] Let \(g \in C^{p}[a,b]\). If \(g^{(k)}(x)=0\) for \(k = 1, 2, \ldots, p-1\) and \(g^{(p)}(x) \neq 0\), then the sequence \(\{x_{n}\}\) is of order \(p\).
Algorithm 1.3. [10] For a given \(x_{0}\), Kang et. al. gave the approximate solution \(x_{n+1}\) by an iteration scheme as follows. \begin{equation*} x_{n+1} = \frac{g\left( x_{n}\right) – x_{n} g^{\prime}(x_{n})}{1 – g^{\prime}(x_{n})} \end{equation*} where, \(g^{\prime}(x_{n}) \neq 1\). This scheme has convergence of order 2.
Algorithm 2.1
Theorem 3.1. Let \(f : D \subset \mathbb{R} \rightarrow \mathbb{R}\) be a nonlinear function on an open interval \(D\), such that \(f\left(x\right) = 0\) (or equivalently \(x = g\left(x\right) )\), has a simple root \(\alpha \in D\). Here, \(g : D \subset \mathbb{R} \rightarrow \mathbb{R}\), is sufficiently smooth in the neighborhood of the root \(\alpha\). Then, the order of convergence of algorithm 2.1 is at least \(3\), where \(c_{k} = \frac{g^{(k)}(\alpha)}{k!(1 – g^{\prime}(\alpha))}\), \(k = 2, 3, \ldots\).
Proof. As \(g\left(\alpha\right) = \alpha\), let \(x_{n} = \alpha + e_{n}\) and \(x_{n + 1} = \alpha + e_{n+1}\). By Taylor’s expansion, we have, \begin{equation*} g(x_{n}) = g\left(\alpha\right) + e_{n}g^{\prime}(\alpha) + \frac{e_{n}^{2}}{2!}g^{\prime\prime}(\alpha) + \frac{e_{n}^{3}}{3!} g^{\prime \prime \prime}(\alpha ) + O(e_{n}^{4}). \end{equation*} This implies that,
Example 4.1 \(f(x)=x^{3}-23x-135\), \(g(x)=23+\frac{135}{x}\)
Methods | \(N\) | \(N_{f}\) | \(x_{0}\) | \(x_{n+1}\) | \(f(x_{n+1})\) |
---|---|---|---|---|---|
FPM | \(24\) | \(24\) | \(2\) | \(2.420536e-15\) | \(27.84778272427181476 \) |
NIM | \(7\) | \(14\) | \(2\) | \(2.781849e-20\) | \(27.84778272427181484 \) |
KIM | \(4\) | \(12\) | \(2\) | \(2.647181e-16\) | \(27.84778272427181483 \) |
Example 4.2 \(f(x)=x-\cos x,\) \(g(x)=\cos x\)
Methods | \(N\) | \(N_{f}\) | \(x_{0}\) | \(x_{n+1}\) | \(f(x_{n+1})\) |
---|---|---|---|---|---|
FPM | \(91\) | \(91\) | \(6\) | \(1.376330e-16\) | \(0.73908513321516064\) |
NIM | \(10\) | \(20\) | \(6\) | \(1.083243e-29\) | \(0.73908513321516064\) |
KIM | \(5\) | \(15\) | \(6\) | \(1.809632e-36\) | \(0.73908513321516064 \) |
Example 4.3 \(f(x)=x^{3}+4x^{2}+8x+8,g(x)=-1-1/2x^{2}-1/8×3\)
Methods | \(N\) | \(N_{f}\) | \(x_{0}\) | \(x_{n+1}\) | \(f(x_{n+1})\) |
---|---|---|---|---|---|
FPM | \(50\) | \(50\) | \(-1.7\) | \(1.416263e-15\) | \(-1.99999999999999965 \) |
NIM | \(5\) | \(10\) | \(-1.7\) | \(7.836283e-27\) | \(-2.00000000000000000 \) |
KIM | \(3\) | \(9\) | \(-1.7\) | \(4.983507e-25\) | \(-2.00000000000000000 \) |
Example 4.4 \(f(x)=\ln (x-2)+x,g(x)=2+e^{-x}\)
Methods | \(N\) | \(N_{f}\) | \(x_{0}\) | \(x_{n+1}\) | \(f(x_{n+1})\) |
---|---|---|---|---|---|
FPM | \(18\) | \(18\) | \(0.1\) | \(1.163802e-15\) | \(2.12002823898764110 \) |
NIM | \(5\) | \(10\) | \(0.1\) | \(5.972968e-22\) | \(2.12002823898764123\) |
KIM | \(3\) | \(9\) | \(0.1\) | \(8.594812e-22\) | \(2.12002823898764123 \) |
Example 4.5 \(f(x)=x^{2}\sin x-\cos x,g(x)=\sqrt{\frac{1}{\tan x}}\)
Methods | \(N\) | \(N_{f}\) | \(x_{0}\) | \(x_{n+1}\) | \(f(x_{n+1})\) |
---|---|---|---|---|---|
FPM | \(417\) | \(247\) | \(2\) | \(2.668900e-16\) | \(0.89520604538423175 \) |
NIM | \(7\) | \(14\) | \(2\) | \(3.195785e-31\) | \(0.89520604538423175 \) |
KIM | \(4\) | \(12\) | \(2\) | \(1.746045e-23\) | \(0.89520604538423175 \) |
Example 4.6 \(f(x)=x^{2}-5x-16,g(x)=5+\frac{16}{x}\)
Methods | \(N\) | \(N_{f}\) | \(x_{0}\) | \(x_{n+1}\) | \(f(x_{n+1})\) |
---|---|---|---|---|---|
FPM | \(34\) | \(34\) | \(1\) | \(6.417745e-16\) | \(7.21699056602830184 \) |
NIM | \(7\) | \(14\) | \(1\) | \(2.984439e-27\) | \(7.21699056602830191 \) |
KIM | \(4\) | \(12\) | \(1\) | \(2.797394e-26\) | \(7.21699056602830191 \) |