Some new numerical schemes for finding the solutions to nonlinear equations

1.   Introduction

One of the oldest and most basic problems in mathematics is that of solving nonlinear equations. To solve these equations, we can use iterative methods such as Newton's method and its variants. Newton's method is one of the most powerful and well-known iterative methods known to converge quadratically.

In the Adomian decomposition method, the solution is considered in terms of an infinite series, which converges to an exact solution. Chun ^[5] and Abbasbandy ^[4] constructed and investigated different higher-order iterative methods by applying the decomposition technique of Adomian ^[3]. Darvishi and Barati ^[6] also applied the Adomian decomposition technique to develop Newton-type methods that are cubically convergent for the solution of the system of non-linear equations. Implementation of this Adomian decomposition technique required higher-order derivatives evaluation, which is the major pitfall of this method.

To overcome this drawback, several new techniques have been suggested and analyzed by many researchers. Daftardar-Gejji and Jafari ^[7] have used different modifications of the Adomian decomposition method ^[3] and proposed a simple technique that does not need the derivative evaluation of the Adomian polynomial, which is the major advantage of using this technique over Adomian decomposition method. Saqib and Iqbal ^[13] and Ali et al. ^[1,2] have used this decomposition technique and developed a family of iterative methods with better efficiency and convergence order for solving the nonlinear equations. Heydari et al. ^[15,16] proposed several iterative methods including derivative free methods and discussed their convergence. For finding multiple roots of nonlinear equations and iterative schemes using homotopy perturbation techniques, see ^[17,18].

Weerakon and Fernando ^[14] improved the convergence of the Newton method using the quadrature rule. Later on, Ozban ^[11] investigated some new variant forms of the Newton method by using the concept of harmonic mean and mid-point rule. Noor ^[10] developed the fifth-order convergent iterative method using the Gaussian quadrature formula and investigated its efficacy compared to the existing methods in the literature.

In this paper, we consider the well know fixed-point iterative method in which we rewrite the nonlinear equation $\Lambda\left(\wp\right) = 0$ as $\wp = \Upsilon\left(\wp\right)$. We present and introduce some new iterative methods. We also determine the convergence analysis of proposed methods. Some numerical examples are presented to make a comparative study of newly constructed methods with known third and fourth-order convergent iterative algorithms.

2.   Construction of iterative methods

This section comprises some new multi-step third and fourth-order convergent iterative methods in view of Simpson's one-third rule and decomposition technique ^[7].

2.1.   Simpson's one third rule

Consider the nonlinear equation

which is equivalent to

Assume that $\alpha$ is the simple root of nonlinear Eq (2.1) and $\gamma$ is the initial guess sufficiently close to the root. Using fundamental theorem of calculus and Simpson's one-third quadrature formula, we have

Now, from (2.2) and (2.3), we have

Now, using the technique of He ^[8], the nonlinear Eq (2.1) can be written as an equivalent coupled system of equations

and

from which, it follows that

where

and

It is clear that $M(\wp)$ is nonlinear operator. Now, we establish the sequence of higher-order iterative methods implementing the decomposition technique presented by Daftardar-Gejji and Jafari ^[7]. In this technique, the solution of (2.1) can be represented in terms of infinite series.

Here, the operator $M(\wp)$ can be decomposed as:

Thus, from (2.6), (2.9) and (2.10), we have

which generates the following iterative schemes:

Consequently, it follows that

and

It is noted that $\wp$ is approximated by

and

For $n = 0,$

From (2.5), it can easily be computed as:

Using (2.8), we get

For $n = 1,$

Using (2.14), we have

This fixed point formulation is used to suggest the following algorithm {for solving the nonlinear equation $\Lambda\left(\wp\right) = 0$}.

Algorithm 2.1. For a given $\wp_{0}$ (initial guess), approximate solution $\wp_{n+1}$ is computed by the following iterative scheme

Kang et al. ^[9] developed this algorithm and proved that Algorithm 2.1 has quadratic convergence.

Form (2.5) and (2.8), we have

Thus

For $n = 2,$

Take

This relation yields the following two-step method for solving the nonlinear equation {$\Lambda\left(\wp\right) = 0$}.

Algorithm 2.2. For a given initial guess $\wp_{0},$ the approximated solution $\wp_{_{n+1}}$ can be computed by the following iterative schemes:

where $n = 0, 1, 2, \ldots$.

It is noted that

Using (2.5) and (2.8), we can write

and

For $n = 3,$

Using (2.19), we have

Using this relation, we suggest the following three-step method for solving nonlinear Eq (2.1).

Algorithm 2.3. For a given initial guess $\wp_{0}$, compute the approximate solution $\wp_{n+1}$ by the following iterative scheme.

and

where $n = 0, 1, 2, \ldots.$

3.   Convergence analysis of proposed iterative methods

This section comprises convergence analysis of proposed Algorithms 2.2 and 2.3. It is shown that these methods are third and fourth-order convergent, respectively.

Theorem 3.1. Let $I\subset\mathbb{R}$ be an open interval and $\Lambda: I\to\mathbb{R}$ is differential function. If $\beta \in I$ be the {simple root} of $\Lambda\left(\wp\right) = 0$ and $\wp_{0}$ is sufficiently close to $\beta$ then multi-step method defined by Algorithm 2.2 has third order ofconvergence.

Proof. Let $\beta$ be the root of nonlinear equation $\Lambda\left(\wp\right) = 0$, or equivalently $\wp = \Upsilon\left(\wp\right)$. Let $e_{n}$ and $e_{n+1}$ be the errors at $nth$ and $\left(n+1\right)$ iterations, respectively.

Now, expanding $\Upsilon\left(\wp\right)$ and $\Upsilon^{\prime }\left(\wp\right)$ by using Taylor series about $\beta,$ we have

and

Thus, we have

and

From (2.16), we have

Using (2.17), we obtain

Expanding $\Upsilon\left(v_{n}\right)$ in terms of Taylor series about $\beta$, we get

Expanding $\Upsilon^{\prime }\left(v_{n}\right)$ in terms of Taylor series about $\beta$, we get

Expanding $\Upsilon^{\prime }\left(\frac{v_{n}+\wp_{n}}{2}\right)$ in terms of Taylor series about $\beta$, we get

From (3.2), (3.4) and (3.5), we have

Now,

Using (3.3) and (3.7), we have

Using (3.6), we have

Dividing (3.8) and (3.9), we have

Using (2.18), we have

Therefore,

This shows that Algorithm 2.2 has third-order of convergence.

Theorem 3.2. Let $I\subset\mathbb{R}$ be an open interval and $\Lambda: I\to\mathbb{R}$ is differential function. If $\beta \in I$ be the {simple root} of $\Lambda\left(\wp\right) = 0$ and $\wp_{0}$ is sufficiently close to $\beta$ then multi-step method defined by Algorithm 2.3 has fourth order ofconvergence.

Proof. From (3.10), we have

Expanding $\Upsilon\left(w_{n}\right)$, in terms of Taylor's series

Expanding $\Upsilon^{\prime }\left(w_{n}\right)$, in terms of Taylor's series

Expanding $\Upsilon^{\prime }\left(\frac{w_{n}+\wp_{n}}{2}\right)$ in terms of Taylor's series

Using (3.2), (3.12) and (3.13), we have

Now,

Using (3.11) and (3.15), we have

Using (3.14), we have

Dividing (3.17) and (3.18), we have

Using (2.21), we have

Therefore,

This shows that Algorithm 2.3 has fourth order of convergence.

4.   Numerical examples and comparison results

This section elaborates on the efficacy of algorithms introduced in this paper with the support of examples. We obtain an estimated simple root rather than the exact based on the exactness of the computer. We utilize $\varepsilon = 10^{-5},$ for computational work we use the following stopping criteria $\left\vert \wp_{n+1}-\wp_{n}\right\vert < \varepsilon.$

We make a comparative representation Newton method (NM), Halley method (HM), Algorithms 2A ^[12], 2B ^[12] and 2C ^[12] with Algorithms 2.2 and 2.3. In tables, we also displayed the number of iterations $\left(\text{IT}\right)$, the approximate root $\wp_{n}$ and the value of $\Lambda\left(\wp_{n}\right).$

We solved all test problems and calculated the CPU time consumptions in second with the aid of the computer software Maple 17.

Example 4.1. For $\Lambda\left(\wp\right) = \wp^{3}-10, \Upsilon\left(\wp\right) = \sqrt{\frac{10}{\wp}}$ and $\wp_{0} = 1.5.$

Example 4.2. For $\Lambda\left(\wp\right) = \cos \left(\wp\right)-\wp, \Upsilon\left(\wp\right) = \cos \left(\wp\right)$ and $\wp_{0} = 1.7.$

Example 4.3. For $\Lambda\left(\wp\right) = \left(\wp-1\right) ^{3}-1, \Upsilon\left(\wp\right) = 1+\sqrt{\frac{1}{\left(\wp-1\right) }}$ and $\wp_{0} = 3.5.$

Example 4.4. For $\Lambda\left(\wp\right) = e^{\wp^{2}+7\wp-30}-1, \Upsilon\left(\wp\right) = \frac{1}{7}\left(30-\wp^{2}\right)$ and $\wp_{0} = 3.5.$

Example 4.5. For $\Lambda\left(\wp\right) = \sin ^{2}\left(\wp\right) -\wp^{2}+1, \Upsilon\left(\wp\right) = \sin \left(\wp\right) +\frac{1}{\sin \left(\wp\right) +\wp}$ and $\wp_{0} = 1.$

Example 4.6. For $\Lambda\left(\wp\right) = e^{\wp}-3\wp^{2}, \Upsilon\left(\wp\right) = \sqrt{\frac{e^{\wp}}{3}}$ and $\wp_{0} = 0.8.$

5.   Conclusions

We have established two new algorithms of third and fourth-order convergence by using a modified decomposition technique for the coupled systems. We have discussed the convergence analysis of these newly established algorithms. With the help of test examples, the computational comparison has been made with well-known third and fourth-order convergent iterative methods. It has been observed from Examples 4.1 to 4.6 that our CPU time and accuracy are much better than the existing algorithms in some cases.

Acknowledgements

This research was supported by the Fundamental Fund of Khon Kaen University, Thailand.

Conflict of interest

The authors declare that they have no competing interests.