Two modifications of efficient newton-type iterative method and two variants of Super-Halley’s method for solving nonlinear equations

Abstract

The root-fnding problem of a univariate nonlinear equation is a fundamental and long-studied problem, and has wide applications in mathematics and engineering computation. In this paper, two modified Newton-type methods and two variants of Super-Halley’s method to solve nonlinear equations are brought forward. Analytical discussions are reported and the theoretical efficiency of the methods is studied. Both of Algorithms 1 and 2 require three evaluations of the functions and two evaluations of derivatives at each iteration. Therefore the efficiency indices of Algorithms 1 and 2 are 1.431 and 1.476, respectively. The presented variants of Super-Halley’s methods require two evaluations of the functions and two evaluations of derivatives at each iteration. Therefore the efficiency indices of Algorithms 3 and 4 are 1.565. Hence, all efficiency indices of the proposed algorithms are better than that of classical Newton’s method 1.414. All of the proposed algorithms in this paper are free from second derivatives. Some numerical results are finally provided to support the theoretical discussions of the proposed methods.

Keywords

Iterative method nonlinear equations order of convergence efficiency index Newton’s method

1. Introduction

One of the most important questions in mathematics is how to find a solution of the nonlinear equation

$\displaystyle f(x)=0,$ (1)

where $f:D\subseteq R\to R$ for an open interval $D$ has sufficient number of continuous derivatives in a neighborhood of $\alpha$ . There are very few functions for which the roots can be expressed explicitly in closed form. Thus, the solutions must be obtained approximately, relying on numerical techniques based on iterative processes. Solving nonlinear equations is a classical problem which has interesting applications in various branches of science, and engineering computation. The nonlinear equations play an important role in many fields of science, and many numerical methods are developed. In this paper, we apply iterative methods to find a simple root $\alpha$ of the above problem.

It is well known that Newton’s method (NM for simplicity) is one of the most famous and important methods for computing approximations $\alpha$ by the following iterative scheme [5]

$\displaystyle y_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}.$ (2)

The Newton’s method converges quadratically in some neighborhood of $\alpha$ for some appropriate start value $x_{0}$ . The main advantage of this method is that the computation of the second derivative not required.

An acceleration of Newton’s method, called Super-Halley method (SHM for simplicity) [39], which is defined by

$\displaystyle x_{n+1}=x_{n}-\left(1+\frac{1}{2}\frac{L_{f}(x_{n})}{1-L_{f}(x_{% n})}\right)\frac{f(x_{n})}{f^{\prime}(x_{n})},$ (3)

where

$\displaystyle L_{f}(x_{n})=\frac{f^{\prime\prime}(x_{n})f(x_{n})}{f^{\prime}(x% _{n})^{2}}.$

This method is, in general, an iterative process with order of convergence three although the method converges with fourth order when it is applied to quadratic equations. From a practical point of view, it is interesting and expected to research higher-order variants of Super-Halley method for general non-linear equations.

In the past decades, much attention has been paid to develop iterative methods for solving nonlinear equations, and a large number of researchers try to improve Newton’s method in order to get a method with a higher order of convergence and more accuracy in open literatures, see for example [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] and the references therein for more details.

Motivated and inspired by the ongoing activities in the direction, in this paper, to improve the local order of convergence properties, we present two modified Newton-type iterative methods and two variants of Super-Halley method for solving nonlinear equations. All of the proposed algorithms in this paper are free from second derivatives. At each iteration, Algorithms 1 and 2 require three evaluations of the functions and two evaluations of derivatives, and Algorithms 3 and 4 require two evaluations of the functions and two evaluations of derivatives. Some numerical results are given to illustrate the advantage and effectiveness of the methods.

The rest of the paper is organized as follows: In Section 2 we describe two new modified Newton-type iterative methods and we analyze their convergence. In Section 3, we give two variant of Super-Halley method and analyze their convergence. Different numerical tests confirm the theoretical results and allow us to compare our new methods with some other known methods in Section 4. Finally, the conclusions are given in Section 5.

2. Two new modified Newton-type iterative methods and analysis of their convergence

Firstly, we consider the following iterative scheme.

Algorithm 1. For given $x_{0}$ , we consider the following iteration method for solving nonlinear Eq. (1)

$\displaystyle y_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})},$ (4) $\displaystyle z_{n}=y_{n}-\frac{2f(y_{n})}{f^{\prime}(x_{n})}+\frac{f(y_{n})f^% {\prime}(y_{n})}{f^{\prime}(x_{n})^{2}},$ (5) $\displaystyle x_{n+1}=z_{n}-\frac{3f^{\prime}(x_{n})^{2}+f^{\prime}(y_{n})^{2}% }{2f^{\prime}(x_{n})f^{\prime}(y_{n})+2f^{\prime}(y_{n})^{2}}\frac{f(z_{n})}{f% ^{\prime}(x_{n})}.$ (6)

For Algorithm 1, we have the following convergence result.

Theorem 1. Assume that the function $f:D\subseteq R\to R$ has a single root $\alpha\in D$ , where $D$ is an open interval. If $f(x)$ has first, second and third derivatives in the interval $D$ , then Algorithm 1 defined by Eqs (4)–(6) is sixth-order convergent in a neighborhood of $\alpha$ and it satisfies the following error equation

$\displaystyle e_{n+1}=5c_{2}^{5}e_{n}^{6}+O(e_{n}^{7}),$ (7)

where

$\displaystyle e_{n}=x_{n}-\alpha,$ (8) $\displaystyle c_{k}=\frac{f^{(k)}(\alpha)}{k!f^{\prime}(\alpha)},\quad k=1,2,\ldots.$ (9)

Proof. Let $\alpha$ be the simple root of $f(x)$ , and

$\displaystyle e_{n}=x_{n}-\alpha,c_{k}=\frac{f^{(k)}(\alpha)}{k!f^{\prime}(% \alpha)},\quad k=1,2,\ldots.$

Consider the iteration function $F(x)$ defined by

$\displaystyle F(x)=z(x)-\frac{3f^{\prime}(x)^{2}+f^{\prime}(y(x))^{2}}{2f^{% \prime}(x)f^{\prime}(y(x))+2f^{\prime}(y(x))^{2}}\frac{f(z(x))}{f^{\prime}(x)}$ (10)

where

$\displaystyle z(x)=y(x)-\frac{3f^{\prime}(x)^{2}+f^{\prime}(y(x))^{2}}{2f^{% \prime}(x)f^{\prime}(y(x))+2f^{\prime}(y(x))^{2}}\frac{f(y(x))}{f^{\prime}(x)},$ (11) $\displaystyle y(x)=x-\frac{f(x)}{f^{\prime}(x)}.$ (12)

By some computations using Maple we can obtain

$\displaystyle F(\alpha)=\alpha,$ $\displaystyle F^{(i)}(\alpha)=0,\quad i=1,2,3,4,5,$ (13) $\displaystyle F^{(6)}(\alpha)=\frac{225f^{(2)}(\alpha)^{5}}{2f^{\prime}(\alpha% )^{5}}.$

Furthermore, from the Taylor expansion of $F(x_{n})$ at $\alpha$ , we have

$\displaystyle x_{n+1}=F(x_{n})=F(\alpha)+\sum_{k=1}^{6}\frac{f^{(k)}(\alpha)}{% k!}(x_{n}-\alpha)^{k}+O\left((x_{n}-\alpha)^{7}\right).$ (14)

Substituting Eq. (2) into Eq. (14) yields

$\displaystyle x_{n+1}=\alpha+e_{n+1}=\alpha+5c_{2}^{5}e_{n}^{6}+O(e_{n}^{7}).$

Therefore, we have the error expression of the algorithm

$\displaystyle e_{n+1}=5c_{2}^{5}e_{n}^{6}+O(e_{n}^{7}),$

which means the order of convergence of Algorithm 1 is six. The proof is completed.

Now, we propose the following seventh-order convergent iterative method and give its convergence analysis.

Algorithm 2. For given $x_{0}$ , we consider the following iteration scheme

$\displaystyle y_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})},$ (15) $\displaystyle z_{n}=y_{n}-\frac{2f(y_{n})}{f^{\prime}(x_{n})}+\frac{f(y_{n})f^% {\prime}(y_{n})}{f^{\prime}(x_{n})^{2}},$ (16) $\displaystyle x_{n+1}=z_{n}-\frac{2f^{\prime}(y_{n})^{2}}{f^{\prime}(x_{n})^{2% }-4f^{\prime}(x_{n})f^{\prime}(y_{n})+5f^{\prime}(y_{n})^{2}}\frac{f(z_{n})}{f% ^{\prime}(x_{n})}.$ (17)

For Algorithm 2, we have the following convergence result.

Theorem 2. Assume that the function $f:D\subseteq R\to R$ has a single root $\alpha\in D$ , where $D$ is an open interval. If $f(x)$ has first, second and third derivatives in the interval $D$ , then Algorithm 2 defined by Eqs (15)–(17) is seventh-order convergent in a neighborhood of $\alpha$ and its error equation is

$\displaystyle e_{n+1}=\frac{5}{8}\left(2c_{2}^{2}-c_{3}\right)e_{n}^{7}+O(e_{n% }^{8})$ (18)

where $e_{n}$ and are $c_{k}(k=1,2,\ldots)$ defined by Eq. (6).

Proof. Let $\alpha$ be the simple root of $f(x)$ , and $c_{k}=\frac{f^{(k)}(\alpha)}{k!f^{\prime}(\alpha)},k=1,2,\ldots$ , $e_{n}=x_{n}-\alpha$ .

We consider the iteration function $F(x)$ which is defined by

$\displaystyle F(x)=z(x)-\frac{2f^{\prime}(y(x))^{2}}{f^{\prime}(x)^{2}-4f^{% \prime}(x)f^{\prime}(y(x))+5f^{\prime}(y(x))^{2}}\frac{f(z(x))}{f^{\prime}(x)}$ (19)

where

$\displaystyle z(x)=y(x)-\frac{2f(y(x))}{f^{\prime}(x)}+\frac{f(y(x))f^{\prime}% (y(x))}{f^{\prime}(x)^{2}},$ $\displaystyle y(x)=x-\frac{f(x)}{f^{\prime}(x)}.$

By computations using Maple we can obtain

$\displaystyle F(\alpha)=\alpha,$ $\displaystyle F^{(i)}(\alpha)=0,i=1,2,3,4,5,6,$ (20) $\displaystyle F^{(7)}(\alpha)=-\frac{525f^{(2)}(\alpha)^{4}[-3f^{(2)}(\alpha)^% {2}+f^{\prime}(\alpha)f^{(3)}(\alpha)]}{f^{\prime}(\alpha)^{6}}.$

Furthermore, from the Taylor expansion of $F(x_{n})$ around $\alpha$ , we get

$\displaystyle x_{n+1}=F(x_{n})=F(\alpha)+\sum_{k=1}^{7}\frac{f^{(k)}(\alpha)}{% k!}{(x_{n}-\alpha)}^{k}+O((x_{n}-\alpha)^{8}).$ (21)

Substituting Eq. (2) into Eq. (21) yields

$\displaystyle x_{n+1}=\alpha+e_{n+1}=\alpha+\frac{5}{8}(2c_{2}^{2}-c_{3})e_{n}% ^{7}+O(e_{n}^{8}).$

Therefore, we get

$\displaystyle e_{n+1}=\frac{5}{8}(2c_{2}^{2}-c_{3})e_{n}^{7}+O(e_{n}^{8}),$

which shows that the order of convergence of Algorithm 2 is seven.

3. Two variants of Super-Halley’s methods and their convergence analysis

In this section, we consider the following two sixth-order variants of Super-Halley’s method. Firstly, we give the following iterative scheme.

Algorithm 3. For given $x_{0}$ , we consider the following iteration method for solving nonlinear Eq. (1)

$\displaystyle y_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})},$ (22) $\displaystyle z_{n}=x_{n}-\frac{1}{2}\cdot\frac{3f^{\prime}(x_{n})^{2}+f^{% \prime}(y_{n})^{2}}{f^{\prime}(x_{n})^{2}+f^{\prime}(y_{n})^{2}}\cdot\frac{f(x% _{n})}{f^{\prime}(x_{n})},$ (23) $\displaystyle x_{n+1}=z_{n}-\frac{f^{\prime}(x_{n})+f^{\prime}(y_{n})}{-f^{% \prime}(x_{n})+3f^{\prime}(y_{n})}\frac{f(z_{n})}{f^{\prime}(x_{n})}.$ (24)

For Algorithm 3, we have the following convergence result.

Theorem 3. Assume that the function $f:D\subseteq R\to R$ has a single root $\alpha\in D$ , where $D$ is an open interval. If $f(x)$ has first, second and third derivatives in the interval $D$ , then Algorithm 3 defined by Eqs (22)–(24) is sixth-order convergent in a neighborhood of $\alpha$ and it satisfies the following error equation

$\displaystyle e_{n+1}=c_{2}\left(c_{2}^{4}-2c_{2}^{2}c_{3}-\frac{5}{4}c_{3}^{2% }\right)e_{n}^{6}+O(e_{n}^{7}),$ (25)

where

$\displaystyle e_{n}=x_{n}-\alpha,$ $\displaystyle c_{k}=\frac{f^{(k)}(\alpha)}{k!f^{\prime}(\alpha)},\quad k=1,2,\ldots.$

Proof. Let $\alpha$ be the simple root of $f(x)$ , and

$\displaystyle c_{k}=\frac{f^{(k)}(\alpha)}{k!f^{\prime}(\alpha)},\quad k=1,2,% \ldots,\quad e_{n}=x_{n}-\alpha.$

Consider the iteration function $F(x)$ defined by

$\displaystyle F(x)=z(x)-\frac{f^{\prime}(x)+f^{\prime}(y(x))}{-f^{\prime}(x)+3% f^{\prime}(y(x))}\cdot\frac{f(z(x))}{f^{\prime}(x)}$ (26)

where

$\displaystyle z(x)=y(x)-\frac{1}{2}\cdot\frac{3f^{\prime}(x)^{2}+f^{\prime}(y(% x))^{2}}{f^{\prime}(x)^{2}+f^{\prime}(y(x))^{2}}\cdot\frac{f(y(x))}{f^{\prime}% (x)},$ $\displaystyle y(x)=x-\frac{f(x)}{f^{\prime}(x)}.$

By some computations using Maple we can obtain

$\displaystyle F(\alpha)=\alpha,$ $\displaystyle F^{(i)}(\alpha)=0,i=1,2,3,4,5,$ (27) $\displaystyle F^{(6)}(\alpha)=-\frac{5f^{(2)}(\alpha)\left[-9f^{(2)}(\alpha)^{% 4}+12f^{(3)}(\alpha)f^{(2)}(\alpha)^{2}f^{\prime}(\alpha)+5f^{(3)}(\alpha)^{2}% f^{\prime}(\alpha)^{2}\right]}{2f^{\prime}(\alpha)^{5}}.$

Furthermore, from the Taylor expansion $F(x_{n})$ of at $\alpha$ , we have

$\displaystyle x_{n+1}=F(x_{n})=F(\alpha)+\sum_{k=1}^{6}\frac{f^{(k)}(\alpha)}{% k!}(x_{n}-\alpha)^{k}+O((x_{n}-\alpha)^{7}).$ (28)

Substituting Eq. (3) into Eq. (28) yields

$\displaystyle x_{n+1}=\alpha+e_{n+1}=\alpha+c_{2}\left(c_{2}^{4}-2c_{2}^{2}c_{% 3}-\frac{5}{4}c_{3}^{2}\right)e_{n}^{6}+O(e_{n}^{7}).$

Therefore, we have the error expression of the algorithm

$\displaystyle e_{n+1}=c_{2}\left(c_{2}^{4}-2c_{2}^{2}c_{3}-\frac{5}{4}c_{3}^{2% }\right)e_{n}^{6}+O(e_{n}^{7}),$

which means the order of convergence of the Algorithm 3 is six. The proof is completed.

Next, we propose the following variant of Super-Halley’s method.

Algorithm 4. For given $x_{0}$ , we consider the following iteration method for solving nonlinear Eq. (1)

$\displaystyle y_{n}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})},$ (29) $\displaystyle z_{n}=x_{n}-\frac{1}{2}\cdot\frac{3f^{\prime}(x_{n})^{2}+f^{% \prime}(y_{n})^{2}}{f^{\prime}(x_{n})^{2}+f^{\prime}(y_{n})^{2}}\cdot\frac{f(x% _{n})}{f^{\prime}(x_{n})},$ (30) $\displaystyle x_{n+1}=z_{n}+\frac{2f^{\prime}(x_{n})^{2}}{f^{\prime}(x_{n})^{2% }-4f^{\prime}(x_{n})f^{\prime}(y_{n})+f^{\prime}(y_{n})^{2}}\frac{f(z_{n})}{f^% {\prime}(x_{n})}.$ (31)

For Algorithm 4, we have the following convergence result.

Theorem 4. Assume that the function $f:D\subseteq R\to R$ has a single root $\alpha\in D$ , where $D$ is an open interval. If $f(x)$ has first, second and third derivatives in the interval $D$ , then Algorithm 4 defined by Eq. (29)–(31) is sixth-order convergent in a neighborhood of $\alpha$ and it satisfies the following error equation

$\displaystyle e_{n+1}=\frac{1}{4}c_{2}\left(2c_{2}^{2}+c_{3}\right)\left(6c_{2% }^{2}-5c_{3}\right)e_{n}^{6}+O(e_{n}^{7}),$ (32)

where

$\displaystyle e_{n}=x_{n}-\alpha,$ $\displaystyle c_{k}=\frac{f^{(k)}(\alpha)}{k!f^{\prime}(\alpha)},\quad k=1,2,\ldots.$

Proof. Let $\alpha$ be the simple root of $f(x)$ , and

$\displaystyle c_{k}=\frac{f^{(k)}(\alpha)}{k!f^{\prime}(\alpha)},\quad k=1,2,\ldots,$ $\displaystyle e_{n}=x_{n}-\alpha.$

Consider the iteration function $F(x)$ defined by

$\displaystyle F(x)=z(x)-\frac{f^{\prime}(x)+f^{\prime}(y(x))}{-f^{\prime}(x)+3% f^{\prime}(y(x))}\cdot\frac{f(z(x))}{f^{\prime}(x)}$ (33)

where

By some computations using Maple we can obtain

$\displaystyle F(\alpha)=\alpha,$ $\displaystyle F^{(i)}(\alpha)=0,\quad i=1,2,3,4,5,$ (34) $\displaystyle F^{(6)}(\alpha)=-\frac{5f^{(2)}(\alpha)\left[3f^{(2)}(\alpha)^{2% }+f^{\prime}(\alpha)f^{(3)}(\alpha)\right]\left[-9f^{(2)}(\alpha)^{2}+5f^{% \prime}(\alpha)f^{(3)}(\alpha)\right]}{2f^{\prime}(\alpha)^{5}}.$

Furthermore, from the Taylor expansion of $F(x_{n})$ at $\alpha$ , we have

$\displaystyle x_{n+1}=F(x_{n})=F(\alpha)+\sum_{k=1}^{6}\frac{f^{(k)}(\alpha)}{% k!}(x_{n}-\alpha)^{k}+O((x_{n}-\alpha)^{7}).$ (35)

Substituting Eq. (3) into Eq. (35) yields

$\displaystyle x_{n+1}=\alpha+e_{n+1}=\alpha+\frac{1}{4}c_{2}\left(2c_{2}^{2}+c% _{3}\right)\left(6c_{2}^{2}-5c_{3}\right)e_{n}^{6}+O(e_{n}^{7}).$

Therefore, we have the error expression of the algorithm

$\displaystyle e_{n+1}=\frac{1}{4}c_{2}\left(2c_{2}^{2}+c_{3}\right)\left(6c_{2% }^{2}-5c_{3}\right)e_{n}^{6}+O(e_{n}^{7}),$

which means the order of convergence of Algorithm 4 is six. The proof is completed.

As the order of an iterative method increases, so does the number of function evaluations per step. The efficiency index gives a measure of balance between those quantities, according to the formula $p^{1/\omega}$ , where $p$ is the order of the method and $\omega$ is the number of function evaluations per iteration required by the method. It is not hard to see that the efficiency indices of Algorithms 1 and 2 are 1.431 and 1.476, respectively, and the efficiency indices of Algorithms 3 and 4 are 1.565, which are all better than that of classical Newton’s method (NM) 1.414.

4. Numerical results

This section is devoted to checking the effectiveness and efficiency of our proposed methods Algorithms 1–4 with NM, SHM, and PPM method defined by

$\displaystyle x_{n+1}=x_{n}-\frac{f(x_{n})+f(y_{n})}{f^{\prime}(x_{n})}$ (36)

which is third-order convergent with efficiency index 1.442.

Table 1 shows the number of iterations (ITs) required to satisfy the stopping criterion. All computations were done by using MATLAB 7.0 and using 64 digit floating point arithmetics. In Table 1, we use the following functions.

$\displaystyle f_{1}(x)=\sin^{2}(x)-x^{2}+1,\alpha\approx 1.404492.$ $\displaystyle f_{2}(x)=2\sin x-x,\alpha\approx 1.895494.$ $\displaystyle f_{3}(x)=x^{3}+4x^{2}-10,\alpha\approx 1.365230.$ $\displaystyle f_{4}(x)=\cos x-x,\alpha\approx 0.739085.$ $\displaystyle f_{5}(x)=(x+2)e^{x}-1,\alpha\approx-0.442854.$ $\displaystyle f_{6}(x)=(x-1)^{3}-1,\alpha=2.$ $\displaystyle f_{7}(x)=x^{2}-e^{x}-3x+1,\alpha\approx 1.404492.$ $\displaystyle f_{8}(x)=x^{3}-10,\alpha\approx 2.154435.$ $\displaystyle f_{9}(x)=\cos x-xe^{x}+x^{2},\alpha\approx 0.639154.$ $\displaystyle f_{10}(x)=e^{x}-\arctan x-1.5,\alpha\approx 0.767653.$ $\displaystyle f_{11}(x)=8x-\cos(x)-2x^{2},\alpha\approx 0.128077.$ $\displaystyle f_{12}(x)=\arctan(x),\alpha=0.$

Table 1

Comparison of various methods

Functions	$x_{0}$	NM	PPM	SHM	Algorithm 1	Algorithm 2	Algorithm 3	Algorithm 4
$f_{1}(x)$	1.5	5	4	4	3	3	3	3
	2.0	6	5	4	4	3	4	4
$f_{2}(x)$	2.3	5	4	4	3	4	3	3
	1.6	6	5	5	4	4	4	4
$f_{3}(x)$	1.2	5	4	4	3	3	3	3
	1.4	4	3	3	3	2	3	3
$f_{4}(x)$	0.6	4	3	3	3	3	3	3
	3.1	7	3	3	3	3	3	3
$f_{5}(x)$	7.8	12	6	5	5	3	4	4
	$-$ 0.7	6	4	4	3	4	4	4
$f_{6}(x)$	2.4	6	4	3	3	3	3	3
	1.4	7	56	6	5	4	4	5
$f_{7}(x)$	0.7	6	5	4	3	3	3	3
	0.75	8	6	5	4	3	4	3
$f_{8}(x)$	2	16	11	8	6	6	5	6
	1.8	12	8	6	4	4	4	4
$f_{9}(x)$	1	11	8	5	4	3	4	4
	1.2	14	6	5	4	4	4	4
$f_{1}0(x)$	1	10	5	4	3	3	3	3
	0.95	15	8	5	4	4	4	4
$f_{1}1(x)$	1	16	10	7	6	4	5	5
	1.5	18	11	8	6	4	5	6
$f_{1}2(x)$	0.5	8	5	4	3	3	3	3
	0.8	11	7	6	5	4	5	5

The computational results in Table 1 show that Algorithms 1–4 require less ITs than NM. Therefore, the proposed four methods are of practical interest and can compete with NM and PPM.

5. Conclusions

This paper presented and analyzed two modified Newton-type iterative methods and two variants of Super-Halley’s method for solving nonlinear equations. The methods are free from second derivatives. Algorithms 1 and 2 require three evaluations of the functions and two evaluations of derivatives at each step. Algorithms 3 and 4 require two evaluations of the functions and two evaluations of derivatives per iteration. Several numerical tests demonstrate that the methods proposed in the paper are more efficient and perform better than Newton’s method, SHM, and PPM.

Footnotes

Acknowledgments

The authors acknowledge the Project of National Natural Science Foundation of China (11241005), Research Project Supported by Shanxi Scholarship Council of China (2015-093), Fund Program for the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province, the Fund for Shanxi Key Subjects Construction (FSKSC), Natural Science Foundation of Shandong province (Grant: ZR2016AM06), Excellent Young Scientist Foundation of Shandong Province (Grant: BS2011SF024), Yuncheng University Academic Scientific Research Project (YQ-2012020).

References

Huang

and Wan

, A new nonmonotone spectral residual method for nonsmooth nonlinear equations, Journal of Computational and Applied Mathematics 313 (2017), 82–101.

Liu

Zheng

and Huang

C.-E.

, Third- and fifth-order Newton-Gauss methods for solving nonlinear equations with n variables, Applied Mathematics and Computation 290 (2016), 250–257.

Chen

X.-D.

Zhang

Shi

and Wang

, An efficient method based on progressive interpolation for solving nonlinear equations, Applied Mathematics Letters 61 (2016), 67–72.

Amat

Bermúdez

Hernández-Verón

M.A.

and Martínez

, On an efficient image k-step iterative method for nonlinear equations, Journal of Computational and Applied Mathematics 302 (2016), 258–271.

Fang

X.-W.

and Zeng

M.-L.

, A modified quasi-Newton method for nonlinear equations, Journal of Computational and Applied Mathematics 328 (2018), 44–58.

Yang

Liu

and Zhou

, Chaos optimization algorithms based on chaotic maps with different probability distribution and search speed for global optimization, Commun Nonlinear Sci Numer Simul 19(4) (2014), 1229–1246.

Ahmad

Soleymani

Khaksar Haghani

and Serra-Capizzano

, Higher order derivative-free iterative methods with and without memory for systems of nonlinear equations, Applied Mathematics and Computation 314 (2017), 199–211.

Khare

and Saxena

, Novel PT-invariant solutions for a large number of real nonlinear equations, Physics Letters A 380 (2016), 856–862.

Kima

Y.-C.

and Lee

K.-A.

, The Evans-Krylov theorem for nonlocal parabolic fully nonlinear equations, Nonlinear Analysis 160 (2017), 79–107.

10.

Song

Wang

H.-X.

and Cai

, Locating multiple optimal solutions of nonlinear equation systems based on multiobjective optimization, IEEE Trans Evol Comput 19(3) (2015), 414–31.

11.

Xiao

and Lu

, Finite-time solution to nonlinear equation using recurrent neural dynamics with a specially-constructed activation function, Neurocomputing 151(1) (2015), 246–251.

12.

Alquran

Jaradat Muhammed

H.M.

and Syam

, A modified approach for a reliable study of new nonlinear equation: Two-mode Korteweg-de Vries-Burgers equation, Nonlinear Dynamics 91(3) (2018), 1619–1626.

13.

Al-Baali

Spedicato

and Maggioni

, Broyden’s quasi-Newton methods for a nonlinear system of equations and unconstrained optimization: A review and open problems, Optim Methods Softw 29 (2014), 937–954.

14.

Wan

Chen

Huang

and Feng

D.D.

, A modified nonmonotone BFGS algorithm for solving smooth nonlinear equations, Optim Lett 8 (2014), 1845–1860.

15.

Cheng

and Chen

, Nonmonotone spectral method for large-scale symmetric nonlinear equations, Numer Algorithms 62 (2013), 149–162.

16.

Wan

and Liu

, A modifed spectral conjugate gradient projection method for solving nonlinear monotone symmetric equations, Pac J Optim 12 (2016), 603–622.

17.

Narushima

, A smoothing conjugate gradient method for solving systems of nonsmooth equations, Appl Math Comput 219 (2013), 8646–8655.

18.

Ahmad

Tohidi

and Ullah

M.Z.

Carrasco

J.A.

, Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: Application to PDEs and ODEs, Comput Math Appl 70 (2015), 624–636.

19.

Budzkoa

D.A.

Cordero

and Torregrosa

J.R.

, New family of iterative methods based on the Ermakov-Kalitkin scheme for solving nonlinear systems of equations, Comput Math Math Phy 55 (2015), 1947–1959.

20.

Cordero

Soleymani

and Torregrosa

J.R.

, Dynamical analysis of iterative methods for nonlinear systems or how to deal with the dimension? Appl Math Comput 244 (2014), 398–412.

21.

Grau-Sánchez

Noguera

and Diaz-Barrero

J.L.

, Note on the efficiency of some iterative methods for solving nonlinear equations, SeMA J 71 (2015), 15–22.

22.

Petkovic

M.S.

and Sharma

J.R.

, On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations, Numer Algorithm 71 (2015), 1017–1398.

23.

Sharma

J.R.

Arora

and Petkovic

M.S.

, An efficient derivative free family of fourth order methods for solving systems of nonlinear equations, Appl Math Comput 235 (2014), 383–393.

24.

Sharma

J.R.

and Arora

, Efficient derivative-free numerical methods for solving systems of nonlinear equations, Comput Appl Math 35 (2016), 269–284.

25.

Wang

Zhang

Qian

and Teng

, Seventh-order derivative-free iterative method for solving nonlinear systems, Numer Algorithm 70 (2015), 545–558.

26.

W.P.

Zhao

and Wang

T.Z.

, Small prime solutions of an nonlinear equation, Acta Math Sinica (Chin Ser) 58 (2015), 739–764 (in Chinese).

27.

Chang-Lara

and Kriventsov

, Further time regularity for nonlocal, fully nonlinear parabolic equations, Comm Pure Appl Math 70 (2017), 950–977.

28.

Sharma

J.R.

Guha

R.K.

and Gupta

, Improved King’s methods with optimal order of convergence based on rational approximations, Appl Math Lett 26 (2013), 473–480.

29.

Chen

Argyros

I.K.

and Qian

Q.S.

, A local convergence theorem for the Super-Halley method in a Banach space, Appl Math Lett 7(5) (1994), 49–52.

30.

Song

Wang

H.-X.

and Cai

, Locating multiple optimal solutions of nonlinear equation systems based on multiobjective optimization, IEEE Trans Evol Comput 19(3) (2015), 414–31.

31.

Yang

Liu

and Zhou

, Chaos optimization algorithms based on chaotic maps with different probability distribution and search speed for global optimization, Commun Nonlinear Sci Numer Simul 19(4) (2014), 1229–1246.

32.

Shao

Wei

and Chang

, A multi-stage MPPT algorithm for PV systems based on golden section search method, in: Proceedings of the Twentyninth Annual Ieee Applied Power Electronics Conference and Exposition (APEC), 2014, pp. 676–83.

33.

Sharma

J.R.

and Arora

, On efficient weighted-Newton methods for solving systems of nonlinear equations, Appl Math Comput 222 (2013), 497–506.

34.

Deng

and Wan

, A three-term conjugate gradient algorithm for large-scale unconstrained optimization problems, Appl Numer Math 92 (2015), 70–81.

35.

Cheng

and Chen

, Nonmonotone spectral method for large-scale symmetric nonlinear equations, Numer Algorithms 62 (2013), 149–162.

36.

Dai

Y.H.

Al-Baali

and Yang

X.Q.

, A positive Barzilai-Borwein-Like stepsize and an extension for symmetric linear systems, Numer Funct Anal Optim 134 (2015), 59–75.

37.

Grau-Sánchez

Noguera

and Amat

, On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, J Comput Appl Math 237 (2013), 363–372.

38.

Soleymani

Sharifi

and Mousavi

B.S.

, An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight, J Optim Theory Appl 153(1) (2012), 225–236.

39.

Cordero

Hueso

J.L.

Martínez

and Torregrosa

J.R.

, Increasing the convergence order of an iterative method for nonlinear systems, Appl Math Lett 25 (2012), 2369–2374.