A Modified Ren’s Method with Memory Using a Simple Self-Accelerating Parameter

Abstract

In this paper, a self-accelerating type method is proposed for solving nonlinear equations, which is a modified Ren’s method. A simple way is applied to construct a variable self-accelerating parameter of the new method, which does not increase any computational costs. The highest convergence order of new method is $2+\sqrt{6}\approx 4.4495$. Numerical experiments are made to show the performance of the new method, which supports the theoretical results.

1. Introduction

Self-accelerating type method is a kind of efficient iterative method with memory for solving nonlinear equations, which chooses some varying parameters as self-accelerating parameters in the iteration processes. The self-accelerating parameter is calculated by using information from previous and current iterations, which does not increase the computational cost of iterative method. Thus, self-accelerating type methods possess a very high computational efficiency. Self-accelerating parameter is very important to self-accelerating type method, which can make a big difference to the efficiency of iterative method. There are two ways to construct the self-accelerating parameter, which are secant method and interpolation method. Traub’s method [1] is one of the most representative methods for self-accelerating type method, which can be written as

$$\left\{\begin{array}{cc}\hfill w_n=& x_n+T_{n}f\left(x_n\right),T_n=-f{[x_n,x_{n-1}]}^{-1},\hfill \\ \hfill x_{n+1}=& x_n-\frac{f\left(x_n\right)}{f[x_n,w_n]},\hfill \end{array}\right.$$

(1)

where $f[x_n,w_n]=\{f\left(x_n\right)-f\left(w_n\right)\}/(x_n-w_n)$ is a divided difference and the self-accelerating parameter $T_n$ is constructed by using secant method. The convergence order of method (1) is $1+\sqrt{2}$. Similar, Petković et al. [2] obtained the following self-accelerating type method with order $2+\sqrt{5}$

$$\left\{\begin{array}{cc}\hfill w_n=& x_n-T_{n}f\left(x_n\right),T_n=-f{[x_n,x_{n-1}]}^{-1},\hfill \\ \hfill y_n=& x_n-\frac{f\left(x_n\right)}{f[x_n,w_n]},\hfill \\ \hfill x_{n+1}=& y_n-\frac{f\left(y_n\right)}{f[x_n,w_n]}(1+\frac{f\left(y_n\right)}{f\left(x_n\right)}+\frac{f\left(y_n\right)}{f\left(w_n\right)}).\hfill \end{array}\right.$$

(2)

Zheng et al. [3] also proposed a self-accelerating type method with order $(3+\sqrt{3})/2\approx 3.3028$, which can be given by

$$\left\{\begin{array}{cc}\hfill w_n=& x_n+T_{n}f\left(x_n\right),T_n=f{[x_n,x_{n-1}]}^{-1},\hfill \\ \hfill y_n=& x_n-\frac{f\left(x_n\right)}{f[x_n,w_n]},\hfill \\ \hfill x_{n+1}=& x_n-\frac{f{\left(x_n\right)}^2}{f[x_n,w_n](f\left(x_n\right)-f\left(y_n\right))}.\hfill \end{array}\right.$$

(3)

Using interpolation method to construct self-accelerating parameter, we [4,5,6] obtained some self-accelerating type methods, one of them is given by

$$\left\{\begin{array}{c}{y}_n=x_n-\frac{f\left(x_n\right)}{T_{n}f\left(x_n\right)+f^{\prime }\left(x_n\right)},T_n=-\frac{{H^{″}}_2\left(x_n\right)}{f^{\prime }\left(x_n\right)},\hfill \\ x_{n+1}=y_n-\frac{f\left(y_n\right)}{2T_{n}f\left(x_n\right)+f^{\prime }\left(x_n\right)}\left(1+2\frac{f\left(y_n\right)}{f\left(x_n\right)}+{\left(\frac{f\left(y_n\right)}{f\left(x_n\right)}\right)}^2\right),\hfill \end{array}\right.$$

(4)

where $H_2\left(x\right)=f\left(x_n\right)+f^{\prime }\left(x_n\right)(x-x_n)+f[x_n,x_n,y_{n-1}]{(x-x_n)}^2$, $H_2^{″}\left(x_n\right)=2f[x_n,x_n,y_{n-1}]$ and the self-accelerating parameter $T_n$ is constructed by Hermite interpolation polynomial. Method (4) has convergence order $(5+\sqrt{17})/2.$ Using Newton interpolation to construct self-accelerating parameter, Džunić et al. [7,8] gave some efficient self-accelerating type methods, one of them can be given by

$$\left\{\begin{array}{c}{w}_n=x_n+T_{n}f\left(x_n\right),T_n=-\frac{1}{{N^{\prime }}_2\left(x_n\right)},\hfill \\ x_{n+1}=x_n-\frac{f\left(x_n\right)}{f[x_k,w_k]},\hfill \end{array}\right.$$

(5)

where $N_2\left(t\right)=N_2(t;x_n,x_{n-1},w_{n-1})$ is Newton’s interpolation polynomial of second degree and $N_2^{\prime }\left(x_n\right)=f[x_n,x_{n-1}]+f[x_n,x_{n-1},w_{n-1}](x_n-x_{n-1}).$ The convergence order of method (5) is 3. By increasing the number of self-accelerating parameters or using interpolation polynomial of high degree to construct self-accelerating parameter, the convergence order and the computational efficiency of self-accelerating type methods can be improved greatly. Recently, more and more higher order self-accelerating type methods have been presented for solving nonlinear equations. Zaka et al. [9] gave an efficient tri-parametric iterative method by using Newton interpolation polynomial to construct self-accelerating parameter. Using four self-accelerating parameters, Lotfi and Assari [10] obtained a derivative-free iterative method with efficiency index near 2. We also proposed an efficient iterative method with n self-accelerating parameters, some known methods [1,2,3,7,8,9,10,11,12,13,14] can be seen as the special cases of our method [15]. Cordero et al. [16] and Campos et al. [17] designed some new self-accelerating type methods and studied the stability of their methods. A more extensive list of references as well as a survey on progress made on the self-accelerating type methods may be found in the recent books by Petković et al. [18] and Iliev et al. [19]. It is obvious that the secant method and interpolation method are very effective ways to construct self-accelerating parameter. However, all of these researches focused their works on the design of the methods trying to improve their convergence order and computational efficiency. Only some recent works [20,21] try to use new techniques to construct the self-accelerating parameter.

The main purpose of this paper is that a new way is proposed to construct a self-accelerating parameter, which is different from the secant method and interpolation method. In Section 2, firstly, based on Ren’s method [22], a modified optimal fourth-order method is proposed for solving nonlinear equations. Then, the modified iterative method is extended into a new self-accelerating type method by using a self-accelerating parameter. Using the interpolation method to construct the self-accelerating parameter, the convergence order of the new method reaches 4.2361. In Section 3, a new way is applied to construct the self-accelerating parameter. The maximal convergence order of new method is 4.4495. Numerical examples are given in Section 4 to confirm theoretical results. Section 5 is a short conclusion.

2. A New Self-Accelerating Type Method

In order to obtain a new self-accelerating type method, we first propose a modified fourth-order method without memory. Based on the Ren’s method [22],

$$\left\{\begin{array}{c}\begin{array}{c}{w}_n=x_n+f\left(x_n\right),\hfill \\ y_n=x_n-\frac{f\left(x_n\right)}{f[x_n,w_n]},\hfill \end{array}\hfill \\ x_{n+1}=y_n-\frac{f\left(y_n\right)}{f[x_n,y_n]+f[y_n,w_n]-f[x_n,w_n]+\alpha (y_n-x_n)(y_n-w_n)},\alpha \in R,\hfill \end{array}\right.$$

(6)

we construct the following modified iterative method

$$\left\{\begin{array}{c}{w}_n=x_n+f\left(x_n\right),\hfill \\ z_n=x_n-\frac{f\left(x_n\right)}{f[x_n,w_n]},\hfill \\ y_n=z_n-T{(z_n-x_n)}^2,\hfill \\ x_{n+1}=y_n-\frac{f\left(y_n\right)}{f[x_n,y_n]+f[y_n,w_n]-f[x_n,w_n]},\hfill \end{array}\right.$$

(7)

where $T\in R$ is a parameter. For method (7), we have the following convergence analysis.

Theorem 1.

If function $f:I\subset R\to R$ is sufficiently differentiable and has a simple zero ξ on an open interval I, then iterative method (7) is of fourth-order convergence and its error equation is as follows

$$e_{n+1}=(c_2+c_2{f}^{\prime }\left(\xi \right)-T)[c_2^2(1+f^{\prime }\left(\xi \right))-c_3(1+f^{\prime }\left(\xi \right))-c_{2}T]e_n^4+O\left(e_n^5\right),$$

(8)

where $e_n=x_n-\xi$, $T\in R$ and $c_n=(1/n!)f^{\left(n\right)}\left(\xi \right)/f^{\prime }\left(\xi \right)$, $n=2,3,\cdots$.

Proof. 

Using Taylor expansion of $f\left(x\right)$, we have

$$f\left(x_n\right)=f^{\prime }\left(\xi \right)[e_n+c_2{e}_n^2+c_3{e}_n^3+c_4{e}_n^4+c_5{e}_n^5+O\left(e_n^6\right)],$$

(9)

$$e_{n,w}=w_n-\xi =e_n+f\left(x_n\right)=(1+f^{\prime }\left(\xi \right))e_n+c_2{f}^{\prime }\left(\xi \right)e_n^2+c_3{f}^{\prime }\left(\xi \right)e_n^3+O\left(e_n^4\right),$$

(10)

$$\begin{array}{c}f\left(w_n\right)=f^{\prime }\left(\xi \right)[(1+f^{\prime }\left(\xi \right))e_n+c_2(1+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)e_n^2+(2c_2^2{f}^{\prime }\left(\xi \right)(1+f^{\prime }\left(\xi \right))\\ +c_3(1+4f^{\prime }\left(\xi \right)+3f^{\prime }{\left(\xi \right)}^2+f^{\prime }{\left(\xi \right)}^3))e_n^3+O\left(e_n^4\right)],\end{array}$$

(11)

$$\begin{array}{c}f[x_n,w_n]=f^{\prime }\left(\xi \right)[1+c_2(2+f^{\prime }\left(\xi \right))e_n+(c_2^2{f}^{\prime }\left(\xi \right)+c_3(3+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))e_n^2\end{array}+(2+f^{\prime }\left(\xi \right))(2c_2{c}_3{f}^{\prime }\left(\xi \right)+c_4(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))e_n^3+O\left(e_n^4\right)].$$

(12)

According to (7), (9) and (12), we get

$$e_{n,z}=z_n-\xi =c_2(1+f^{\prime }\left(\xi \right))e_n^2+(-c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))e_n^3+O\left(e_n^4\right).$$

(13)

From (7) and (13), we obtain

$$\begin{array}{c}{e}_{n,y}=y_n-\xi =(c_2+c_2{f}^{\prime }\left(\xi \right)-T)e_n^2+(-c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)\\ +c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+2c_2(1+f^{\prime }\left(\xi \right))T)e_n^3+O\left(e_n^4\right).\end{array}$$

(14)

By a similar argument to that of (9), we get

$$\begin{array}{c}f\left(y_n\right)=f^{\prime }\left(\xi \right)(c_2+c_2{f}^{\prime }\left(\xi \right)-T)e_n^2+f^{\prime }\left(\xi \right)(-c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)\\ +c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+2c_2(1+f^{\prime }\left(\xi \right))T)e_n^3+O\left(e_n^4\right).\end{array}$$

(15)

Using (9), (11), (14) and (15), we have

$$f[x_n,y_n]=f^{\prime }\left(a\right)+c_2{f}^{\prime }\left(\xi \right)e_n+f^{\prime }\left(\xi \right)(c_3+c_2^2(1+f^{\prime }\left(\xi \right))-c_{2}T)e_n^2+O\left(e_n^3\right),$$

(16)

$$f[y_n,w_n]=f^{\prime }\left(\xi \right)+c_2{f}^{\prime }\left(\xi \right)(1+f^{\prime }\left(\xi \right))e_n+f^{\prime }\left(\xi \right)(c_3{(1+f^{\prime }\left(\xi \right))}^2+c_2^2(1+2f^{\prime }\left(\xi \right))-c_{2}T)e_n^2+O\left(e_n^3\right).$$

(17)

Together with (7) and (14)–(17), we obtain the error equation

$$e_{n+1}=x_{n+1}-\xi =(c_2+c_2{f}^{\prime }\left(\xi \right)-T)[c_2^2(1+f^{\prime }\left(\xi \right))-c_3(1+f^{\prime }\left(\xi \right))-c_{2}T]e_n^4+O\left(e_n^5\right).$$

(18)

The proof is completed. □

Remark 1.

From (8), we can see that the convergence order of method (7) is at least five provided that $T=c_2(1+f^{\prime }\left(\xi \right))$. Hence, in order to obtain a new self-accelerating type method, we will use a self-accelerating parameter $T_n$ to replace the parameter T if the parameter $T_n$ satisfies the relation ${lim}_{n\to \infty }{T}_n=T=c_2(1+f^{\prime }\left(\xi \right))$. Similar to methods (4) and (5), we can use interpolation method to construct self-accelerating parameter. For example, the self-accelerating parameter $T_n$ can be given by

$$T_n=\frac{{N^{″}}_2\left(x_n\right)}{2{N^{\prime }}_2\left(x_n\right)}(1+N_2^{\prime }\left(x_n\right)),$$

(19)

where $N_2\left(t\right)=N_2(t;x_n,x_{n-1},w_{n-1})$ is Newton’s interpolatory polynomial of second degree, $N_2^{\prime }\left(x_n\right)=f[x_n,x_{n-1}]+f[x_n,x_{n-1},w_{n-1}](x_n-x_{n-1})$ and $N_2^{″}\left(x_n\right)=2f[x_n,x_{n-1},w_{n-1}]$ Now, we obtain a new self-accelerating type method as follows:

$$\left\{\begin{array}{c}{w}_n=x_n+f\left(x_n\right),\hfill \\ z_n=x_n-\frac{f\left(x_n\right)}{f[x_n,w_n]},\hfill \\ y_n=z_n-T_n{(z_n-x_n)}^2,\hfill \\ x_{n+1}=y_n-\frac{f\left(y_n\right)}{f[x_n,y_n]+f[y_n,w_n]-f[x_n,w_n]}.\hfill \end{array}\right.$$

(20)

Theorem 2.

Let the varying parameter $T_n$ be calculated by (19) in method (20). If an initial value $x_0$ is sufficiently close to a simple zero ξ of function $f\left(x\right)$, then the R-order of convergence of self-accelerating type method (20) is at least $2+\sqrt{5}\approx 4.2361$.

Proof. 

If an iterative method (IM) generates sequence $\left\{x_n\right\}$ that converges to the zero $\xi$ of $f\left(x\right)$ with the R-order $O_R(\mathrm{IM},a)\ge r,$ then we can write

$$e_{n+1}\sim D_{n,r}{e}_n^r,$$

(21)

where $e_n=x_n-\xi$ and the limit of $D_{n,r}$ is the asymptotic error constant of iterative method, as $n\to \infty$. So,

$$e_{n+1}\sim D_{n,r}{\left(D_{n-1,r}{e}_{n-1}^r\right)}^r=D_{n,r}{D}_{n-1,r}^r{e}_{n-1}^{r^2}.$$

(22)

Similar to (22), if the R-order of iterative sequence $\left\{y_n\right\}$ is p, then

$$e_{n,y}\sim D_{n,p}{e}_n^p\sim D_{n,p}{\left(D_{n-1,r}{e}_{n-1}^r\right)}^p=D_{n,p}{D}_{n-1,r}^p{e}_{n-1}^{rp}.$$

(23)

According to (14) and (18), we obtain

$$e_{n,y}=y_n-\xi \sim (c_2+c_2{f}^{\prime }\left(\xi \right)-T_n)e_n^2,$$

(24)

$$e_{n+1}=x_{n+1}-\xi \sim (c_2+c_2{f}^{\prime }\left(\xi \right)-T_n)[c_2^2(1+f^{\prime }\left(\xi \right))-c_3(1+f^{\prime }\left(\xi \right))-c_2{T}_n]e_n^4.$$

(25)

Here, we omit the higher-order terms in (24)–(25). Let $N_2\left(t\right)$ be the Newton interpolating polynomial of degree two that interpolates the function f at nodes $x_n,w_{n-1},x_{n-1}$ contained in interval. Then, the error of the Newton interpolation can be expressed as follows:

$$f\left(t\right)-N_2\left(t\right)=\frac{f^{\left(3\right)}\left(\zeta \right)}{3!}(x-x_n)(x-w_{n-1})(x-x_{n-1}),\zeta \in I.$$

(26)

Differentiating (26) at the point $t=x_n$, we get

$$N_2^{\prime }\left(x_n\right)\sim f^{\prime }\left(\xi \right)(1-c_3(1+f^{\prime }\left(\xi \right))e_{n-1}^2+O\left(e_{n-1}^3\right)),$$

(27)

$$N_2^{″}\left(x_n\right)\sim 2f^{\prime }\left(\xi \right)(c_2+c_3(2+f^{\prime }\left(\xi \right))e_{n-1}+O\left(e_{n-1}^2\right)),$$

(28)

$$\begin{array}{c}{T}_n=\frac{{N^{″}}_2\left(x_n\right)}{2{N^{\prime }}_2\left(x_n\right)}(1+N_2^{\prime }\left(x_n\right))\\ \sim c_2(1+f^{\prime }\left(\xi \right))+c_3(1+f^{\prime }\left(\xi \right))(2+f^{\prime }\left(\xi \right))e_{n-1}+O\left(e_{n-1}^2\right).\end{array}$$

(29)

Using (24), (25) and (29), we get

$$e_{n,y}\sim (c_2+c_2{f}^{\prime }\left(\xi \right)-T_n)e_n^2$$

$$\sim -c_3(1+f^{\prime }\left(\xi \right))(2+f^{\prime }\left(\xi \right))e_{n-1}{\left(D_{n-1,r}{e}_{n-1}^r\right)}^2$$

$$\sim -c_3(1+f^{\prime }\left(\xi \right))(2+f^{\prime }\left(\xi \right))D_{n-1,r}^2{e}_{n-1}^{2r+1},$$

(30)

$$e_{n+1}\sim (c_2+c_2{f}^{\prime }\left(\xi \right)-T_n)[c_2^2(1+f^{\prime }\left(\xi \right))-c_3(1+f^{\prime }\left(\xi \right))-c_2{T}_n]e_n^4$$

$$\sim c_3^2{(1+f^{\prime }\left(\xi \right))}^2(2+f^{\prime }\left(\xi \right))e_{n-1}{\left(D_{n-1,r}{e}_{n-1}^r\right)}^4$$

$$\sim c_3^2{(1+f^{\prime }\left(\xi \right))}^2(2+f^{\prime }\left(\xi \right))D_{n-1,r}^4{e}_{n-1}^{4r+1}.$$

(31)

By comparing exponents of $e_{n-1}$ in relations (23), (30)) and ((22), (31), we have

$$\left\{\begin{array}{c}2r+1=rp,\\ 4r+1=r^2.\end{array}\right.$$

(32)

Solving system (32), we obtain $r=2+\sqrt{5}\approx 4.2361$ and $p=\sqrt{5}\approx 2.2361$. Therefore, the R-order of method (20) is at least 4.2361, when $T_n$ is calculated by (19). The proof is completed. □

3. A New Technique to Construct the Self-Accelerating Parameter of New Method

In this Section, we will give a new way to construct the self-accelerating parameter. It is known that Steffensen method (SM) without memory [23] converges quadratically, which can be written as

$$\left\{\begin{array}{c}{w}_n=x_n+f\left(x_n\right),\hfill \\ x_{n+1}=x_n-\frac{f\left(x_n\right)}{f[x_n,w_n]},\hfill \end{array}\right.$$

(33)

which satisfies the following expression

$$\underset{n\to \infty }{lim}\frac{x_{n+1}-\xi }{{(x_n-\xi )}^2}=\underset{n\to \infty }{lim}\frac{e_{n+1}}{e_n^2}=c_2(1+f^{\prime }\left(\xi \right)).$$

(34)

From (34), we know that the asymptotic error constant of SM is $c_2(1+f^{\prime }\left(\xi \right))$. Coincidentally, the first two steps of our method (20) is Steffensen method, so Equation (34) can be the self-accelerating parameter, which satisfies ${lim}_{n\to \infty }{T}_n=T=c_2(1+f^{\prime }\left(\xi \right))$. Since the root $\xi$ of function is unknown, we could use sequence information from the current and previous iterations to approximate the root $\xi$ and construct the following formulas for $T_n$:

Formula 1:

$$T_n=\frac{z_{n-1}-x_n}{{(x_n-x_{n-1})}^2}.$$

(35)

Formula 2:

$$T_n=\frac{(z_{n-1}-x_n)(y_{n-1}-x_{n-1})}{{(x_n-x_{n-1})}^3}.$$

(36)

Formula 3:

$$\begin{gathered} \begin{array}{c}{T}_n=\frac{z_{n-1}-x_n}{{(x_n-x_{n-1})}^2}\left\{\frac{x_n+y_{n-1}-w_{n-1}-z_{n-1}}{z_{n-1}-w_{n-1}}-\frac{x_n-y_{n-1}}{z_{n-1}-x_{n-1}}\right\}-\frac{2z_n-x_n-w_n}{(x_n-y_{n-1})(z_{n-1}-x_{n-1})}\end{array} \\ -\frac{{(x_n-z_{n-1})}^2{(w_{n-1}-x_{n-1})}^2}{{(z_{n-1}-w_{n-1})}^2{(z_{n-1}-x_{n-1})}^3}. \end{gathered}$$

(37)

Formula 4:

$$\begin{gathered} T_n=\frac{(z_{n-1}-x_n)(y_{n-1}-x_{n-1})}{{(x_n-x_{n-1})}^3}\left\{\frac{x_n+y_{n-1}-w_{n-1}-z_{n-1}}{z_{n-1}-w_{n-1}}\right\}-\frac{2z_n-x_n-w_n}{(x_n-y_{n-1})(z_{n-1}-x_{n-1})} \\ -\frac{{(x_n-z_{n-1})}^2{(w_{n-1}-x_{n-1})}^2}{{(z_{n-1}-w_{n-1})}^2{(z_{n-1}-x_{n-1})}^3}. \end{gathered}$$

(38)

The self-accelerating parameter $T_n$ is calculated by using one of the formulas (35)–(38).

Theorem 3.

Let varying parameters $T_n$ be calculated by (35) or (36) in method (20), respectively. If an initial value $x_0$ is sufficiently close to a simple zero ξ of $f\left(x\right)$, then the R-order of convergence of iterative method (20) is at least $2+\sqrt{5}\approx 4.2361$.

Proof. 

From (13), (14) and (18), we get

$$\begin{array}{c}{z}_{n-1}-x_n=c_2(1+f^{\prime }\left(\xi \right))e_{n-1}^2+(-c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))e_{n-1}^3+\left(c_2^3\right(3\\ +3f^{\prime }\left(\xi \right)+2f^{\prime }{\left(\xi \right)}^2+f^{\prime }{\left(\xi \right)}^3)+2c_2^2(1+f^{\prime }\left(\xi \right))T_{n-1}+(1+f^{\prime }\left(\xi \right))(c_4(3+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)\\ -c_3{T}_{n-1})-c_2(2c_3(3+4f^{\prime }\left(\xi \right)+3f^{\prime }{\left(\xi \right)}^2+f^{\prime }{\left(\xi \right)}^3)+T_{n-1}^2))e_{n-1}^4+O\left(e_{n-1}^5\right),\end{array}$$

(39)

$$x_n-x_{n-1}=-e_{n-1}+(c_2+c_2{f}^{\prime }\left(\xi \right)-T_{n-1})(c_2^2(1+f^{\prime }\left(\xi \right))-c_3(1+f^{\prime }\left(\xi \right))-c_2{T}_{n-1})e_{n-1}^4+O\left(e_{n-1}^5\right),$$

(40)

$$\begin{array}{c}{y}_{n-1}-x_{n-1}=-e_{n-1}+(c_2+c_2{f}^{\prime }\left(\xi \right)-T_{n-1})e_{n-1}^2+(-c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)\\ +2c_2(1+f^{\prime }\left(\xi \right))T_{n-1})e_{n-1}^3+(c_2^3(4+5f^{\prime }\left(\xi \right)+3f^{\prime }{\left(\xi \right)}^2+f^{\prime }{\left(\xi \right)}^3)-c_2{c}_3(7+10f^{\prime }\left(\xi \right)\\ +7f^{\prime }{\left(\xi \right)}^2+2f^{\prime }{\left(\xi \right)}^3)-c_2^2(5+6f^{\prime }\left(\xi \right)+3f^{\prime }{\left(\xi \right)}^2)T_{n-1}+(1+f^{\prime }\left(\xi \right))\left(c_4\right(3+3f^{\prime }\left(\xi \right)\\ +f^{\prime }{\left(\xi \right)}^2)+2c_3(2+f^{\prime }\left(\xi \right))T_{n-1}\left)\right)e_{n-1}^4+O\left(e_{n-1}^5\right).\end{array}$$

(41)

From (39)–(41), we get

$$T_n=\frac{z_{n-1}-x_n}{{(x_n-x_{n-1})}^2}$$

$$=c_2(1+f^{\prime }\left(\xi \right))+(-c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))e_{n-1}+O\left(e_{n-1}^2\right),$$

(42)

$$c_2(1+f^{\prime }\left(\xi \right))-T_n=(c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)-c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))e_{n-1}+O\left(e_{n-1}^2\right),$$

(43)

$$\begin{gathered} \begin{array}{c}{T}_n=\frac{(z_{n-1}-x_n)(y_{n-1}-x_{n-1})}{{(x_n-x_{n-1})}^3}\\ =c_2(1+f^{\prime }\left(\xi \right))+(c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)-c_2^2(3+4f^{\prime }\left(\xi \right)+2f^{\prime }{\left(\xi \right)}^2)\end{array} \\ +c_2(1+f^{\prime }\left(\xi \right))T_{n-1})e_{n-1}+O\left(e_{n-1}^2\right), \end{gathered}$$

(44)

$$\begin{gathered} \begin{array}{c}{c}_2(1+f^{\prime }\left(\xi \right))-T_n=-(c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)-c_2^2(3+4f^{\prime }\left(\xi \right)+2f^{\prime }{\left(\xi \right)}^2)\end{array} \\ +c_2(1+f^{\prime }\left(\xi \right))T_{n-1})e_{n-1}+O\left(e_{n-1}^2\right). \end{gathered}$$

(45)

Using (24), (25) and (43), we get

$$e_{n,y}\sim (c_2+c_2{f}^{\prime }\left(\xi \right)-T_n)e_n^2$$

$$\sim -(-c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))e_{n-1}{\left(D_{n-1,r}{e}_{n-1}^r\right)}^2$$

$$\sim -(-c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))D_{n-1,r}^2{e}_{n-1}^{2r+1},$$

(46)

$$e_{n+1}\sim (c_2+c_2{f}^{\prime }\left(\xi \right)-T_n)[c_2^2(1+f^{\prime }\left(\xi \right))-c_3(1+f^{\prime }\left(\xi \right))-c_2{T}_n]e_n^4$$

$$\sim -c_3(1+f^{\prime }\left(\xi \right))(c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)-c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))e_{n-1}{\left(D_{n-1,r}{e}_{n-1}^r\right)}^4$$

$$\sim -c_3(1+f^{\prime }\left(\xi \right))(c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)-c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))D_{n-1,r}^4{e}_{n-1}^{4r+1}.$$

(47)

Comparing exponents of $e_{n-1}$ in relations ((23), (46)) and ((22), (47)), we get

$$\left\{\begin{array}{c}2r+1=rp,\\ 4r+1=r^2.\end{array}\right.$$

(48)

Solving system (48), we obtain $r=2+\sqrt{5}\approx 4.2361$ and $p=\sqrt{5}\approx 2.2361$. Therefore, the R-order of method (20), when $T_n$ is calculated by (35), is at least 4.2361. From (43) and (45), we know that self-accelerating parameters (35) and (36) have the same error level. So, the R-order of method (20), when $T_n$ is calculated by (36), is at least 4.2361.

The proof is completed. □

Theorem 4.

Let varying parameters $T_n$ be calculated by (37) or (38) in method (20), respectively. If an initial value $x_0$ is sufficiently close to a simple zero ξ of $f\left(x\right)$, then the R-order of convergence of iterative method (20) is at least $2+\sqrt{6}\approx 4.4495$.

Proof. 

According to (10), (13), (14) and (18), we have

$$\begin{gathered} \begin{array}{c}{x}_n-y_{n-1}=(-c_2(1+f^{\prime }\left(\xi \right))+T_{n-1})e_{n-1}^2+(c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)-c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)\\ -2c_2(1+f^{\prime }\left(\xi \right))T_{n-1})e_{n-1}^3+(-c_2^3(3+3f^{\prime }\left(\xi \right)+2f^{\prime }{\left(\xi \right)}^2+f^{\prime }{\left(\xi \right)}^3)+c_2^2(3+4f^{\prime }\left(\xi \right)\\ +3f^{\prime }{\left(\xi \right)}^2)T_{n-1}-(1+f^{\prime }\left(\xi \right))(c_4(3+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+c_3(3+2f^{\prime }\left(\xi \right))T_{n-1})\end{array} \\ +c_2(2c_3(3+4f^{\prime }\left(\xi \right)+3f^{\prime }{\left(\xi \right)}^2+f^{\prime }{\left(\xi \right)}^3)+T_{n-1}^2))e_{n-1}^4+O\left(e_{n-1}^5\right), \end{gathered}$$

(49)

$$\begin{gathered} \begin{array}{c}{z}_{n-1}-x_{n-1}=-e_{n-1}+c_2(1+f^{\prime }\left(\xi \right))e_{n-1}^2+(-c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+c_3(2+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2))e_{n-1}^3\\ +(c_2^3(4+5f^{\prime }\left(\xi \right)+3f^{\prime }{\left(\xi \right)}^2+f^{\prime }{\left(\xi \right)}^3)+c_4(3+6f^{\prime }\left(\xi \right)+4f^{\prime }{\left(\xi \right)}^2+f^{\prime }{\left(\xi \right)}^3)\end{array} \\ -c_2{c}_3(7+10f^{\prime }\left(\xi \right)+7f^{\prime }{\left(\xi \right)}^2+2f^{\prime }{\left(\xi \right)}^3))e_{n-1}^4+O\left(e_{n-1}^5\right), \end{gathered}$$

(50)

$$\begin{gathered} \begin{array}{c}{y}_{n-1}-w_{n-1}=(-1-f^{\prime }\left(\xi \right))e_{n-1}+(c_2-T_{n-1})e_{n-1}^2+(-c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+c_3(2+2f^{\prime }\left(\xi \right)\end{array} \\ +f^{\prime }{\left(\xi \right)}^2)+2c_2(1+f^{\prime }\left(\xi \right))T_{n-1})e_{n-1}^3+O\left(e_{n-1}^4\right), \end{gathered}$$

(51)

$$\begin{array}{c}{z}_{n-1}-w_{n-1}=(-1-f^{\prime }\left(\xi \right))e_{n-1}+c_2{e}_{n-1}^2-(c_2^2-c_3)(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)e_{n-1}^3+\left(c_2^3\right(4\\ +5f^{\prime }\left(\xi \right)+3f^{\prime }{\left(\xi \right)}^2+f^{\prime }{\left(\xi \right)}^3)+c_3(3+5f^{\prime }\left(\xi \right)+4f^{\prime }{\left(\xi \right)}^2+f^{\prime }{\left(\xi \right)}^3)\\ -c_2{c}_3(7+10f^{\prime }\left(\xi \right)+7f^{\prime }{\left(\xi \right)}^2+2f^{\prime }{\left(\xi \right)}^3))e_{n-1}^4+O\left(e_{n-1}^5\right),\end{array}$$

(52)

$$w_{n-1}-x_{n-1}=f^{\prime }\left(\xi \right)e_{n-1}+c_2{f}^{\prime }\left(\xi \right)e_{n-1}^2+c_3{f}^{\prime }\left(\xi \right)e_{n-1}^3+c_4{f}^{\prime }\left(\xi \right)e_{n-1}^4+O\left(e_{n-1}^5\right),$$

(53)

$$z_n-x_n=-(c_2+c_2{f}^{\prime }\left(\xi \right)-T_{n-1})(c_2^2(1+f^{\prime }\left(\xi \right))-c_3(1+f^{\prime }\left(\xi \right))-c_2{T}_{n-1})e_{n-1}^4+O\left(e_{n-1}^5\right),$$

(54)

$$z_n-w_n=-(1+f^{\prime }\left(\xi \right))(c_2+c_2{f}^{\prime }\left(\xi \right)-T_{n-1})(c_2^2(1+f^{\prime }\left(\xi \right))-c_3(1+f^{\prime }\left(\xi \right))-c_2{T}_{n-1})e_{n-1}^4+O\left(e_{n-1}^5\right).$$

(55)

According to (37) and (49)–(55), we get

$$\begin{gathered} \begin{array}{c}{T}_n=\frac{z_{n-1}-x_n}{{(x_n-x_{n-1})}^2}\left\{\frac{x_n+y_{n-1}-w_{n-1}-z_{n-1}}{z_{n-1}-w_{n-1}}-\frac{x_n-y_{n-1}}{z_{n-1}-x_{n-1}}\right\}\\ -\frac{2z_n-x_n-w_n}{(x_n-y_{n-1})(z_{n-1}-x_{n-1})}-\frac{{(x_n-z_{n-1})}^2{(w_{n-1}-x_{n-1})}^2}{{(z_{n-1}-w_{n-1})}^2{(z_{n-1}-x_{n-1})}^3}\end{array} \\ =c_2(1+f^{\prime }\left(\xi \right))-A_1{e}_{n-1}^2+O\left(e_{n-1}^3\right), \end{gathered}$$

(56)

where

$$\begin{gathered} \begin{array}{c}{A}_1=(-c_2^3(1+3f^{\prime }\left(\xi \right)+6f^{\prime }{\left(\xi \right)}^2+3f^{\prime }{\left(\xi \right)}^3)+c_2^2(2+2f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)T_{n-1}+(1+f^{\prime }\left(\xi \right))(c_4\end{array} \\ +c_4{f}^{\prime }\left(\xi \right)+c_3{T}_{n-1})+c_2(c_3{f}^{\prime }\left(\xi \right)(3+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)+T_{n-1}^2)), \end{gathered}$$

(57)

$$c_2(1+f^{\prime }\left(\xi \right))-T_n=A_1{e}_{n-1}^2+O\left(e_{n-1}^3\right),$$

(58)

Using (24), (25) and (58), we obtain

$$e_{n,y}\sim (c_2+c_2{f}^{\prime }\left(\xi \right)-T_n)e_n^2\sim A_1{e}_{n-1}^2{\left(D_{n-1,r}{e}_{n-1}^r\right)}^2$$

$$\sim A_1{D}_{n-1,r}^2{e}_{n-1}^{2r+2},$$

(59)

$$e_{n+1}\sim (c_2+c_2{f}^{\prime }\left(\xi \right)-T_n)[c_2^2(1+f^{\prime }\left(\xi \right))-c_3(1+f^{\prime }\left(\xi \right))-c_2{T}_n]e_n^4$$

$$\sim -A_1{c}_3(1+f^{\prime }\left(\xi \right))e_{n-1}^2{\left(D_{n-1,r}{e}_{n-1}^r\right)}^4$$

$$\sim -A_1{c}_3(1+f^{\prime }\left(\xi \right))D_{n-1,r}^4{e}_{n-1}^{4r+2}.$$

(60)

By comparing exponents of $e_{n-1}$ in relations ((23), (59)) and ((22), (60)), we get

$$\left\{\begin{array}{c}2r+2=rp,\\ 4r+2=r^2.\end{array}\right.$$

(61)

Solving system (61), we get $r=2+\sqrt{6}\approx 4.4495$ and $p=\sqrt{6}\approx 2.4495$. Therefore, the R-order of method (20) is at least 4.4495, when ${\lambda }_n$ is calculated by (37). According to (38) and (49)–(55), we get

$$\begin{gathered} \begin{array}{c}{T}_n=\frac{(z_{n-1}-x_n)(y_{n-1}-x_{n-1})}{{(x_n-x_{n-1})}^3}\left\{\frac{x_n+y_{n-1}-w_{n-1}-z_{n-1}}{z_{n-1}-w_{n-1}}\right\}\\ -\frac{2z_n-x_n-w_n}{(x_n-y_{n-1})(z_{n-1}-x_{n-1})}-\frac{{(x_n-z_{n-1})}^2{(w_{n-1}-x_{n-1})}^2}{{(z_{n-1}-w_{n-1})}^2{(z_{n-1}-x_{n-1})}^3}\end{array} \\ =c_2(1+f^{\prime }\left(\xi \right))-A_2{e}_{n-1}^2+O\left(e_{n-1}^3\right), \end{gathered}$$

(62)

where

$$\begin{array}{c}{A}_2=(c_2{c}_3{f}^{\prime }\left(\xi \right)(3+3f^{\prime }\left(\xi \right)+f^{\prime }{\left(\xi \right)}^2)-c_2^3(1+4f^{\prime }\left(\xi \right)+8f^{\prime }{\left(\xi \right)}^2+4f^{\prime }{\left(\xi \right)}^3)\end{array}+c_2^2(3+4f^{\prime }\left(\xi \right)+2f^{\prime }{\left(\xi \right)}^2){\lambda }_{n-1}+(1+f^{\prime }\left(\xi \right))(c_4+c_4{f}^{\prime }\left(\xi \right)+c_3{T}_{n-1})),$$

(63)

$$c_2(1+f^{\prime }\left(\xi \right))-T_n=A_2{e}_{n-1}^2+O\left(e_{n-1}^3\right).$$

(64)

Equations (58) and (64) have the same error level. Therefore, the R-order of method (20), when $T_n$ is calculated by (38), is at least 4.4495. The proof is completed. □

Remark 2.

Theorems 3 and 4 prove that the self-accelerating parameters (35)–(38) of method (20) are efficient. In fact, the self-accelerating parameter $T_n$ can be constructed by many schemes. Here, we get some other schemes as follows:

$$T_n=\frac{z_{n-1}-x_n}{{(z_{n-1}-x_{n-1})}^2}.$$

(65)

$$T_n=\frac{z_{n-1}-x_n}{(z_{n-1}-x_{n-1})(y_{n-1}-x_{n-1})}.$$

(66)

$$T_n=\frac{z_{n-1}-x_n}{{(y_{n-1}-x_{n-1})}^2}.$$

(67)

$$T_n=\frac{z_{n-1}-x_n}{{(z_{n-1}-x_{n-1})}^2}.$$

(68)

$$T_n=\frac{z_{n-1}-x_n}{(z_{n-1}-x_{n-1})(y_{n-1}-x_{n-1})}.$$

(69)

$$T_n=\frac{z_{n-1}-z_n}{{(y_{n-1}-x_{n-1})}^2}.$$

(70)

$$T_n=\frac{z_{n-1}-z_n}{{(x_n-x_{n-1})}^2}.$$

(71)

$$T_n=\frac{z_{n-1}-z_n}{{(z_{n-1}-x_{n-1})}^2}.$$

(72)

$$T_n=\frac{z_{n-1}-x_n}{(z_{n-1}-x_{n-1})(x_n-x_{n-1})}.$$

(73)

Using the results of Theorem 3, we can prove that the R-order of method (20), is at least 4.2361, when $T_n$ is calculated by any schemes of (65)–(73). Here, we omit the proving process.

4. Numerical Results

The new methods (7) and (20) are employed to solve nonlinear equations $f_i\left(x\right)(i=1,2,3)$ and compared with Ren’s method ($\alpha =0$) (RM, (6)), Petković’s method (PM, (2)), Zheng’s method (ZM, (3)) and Wang’s method (WM, (4)).

Table 1. Numerical results for $f_1\left(x\right)$.

Methods	$|{\mathit{x}}_1-\mathit{\xi }|$	$|{\mathit{x}}_2-\mathit{\xi }|$	$|{\mathit{x}}_3-\mathit{\xi }|$	$|{\mathit{x}}_4-\mathit{\xi }|$	$\mathit{\rho }$
RM	$0.30201\times {10}^{-4}$	$0.96552\times {10}^{-20}$	$0.10086\times {10}^{-81}$	$0.12011\times {10}^{-329}$	4.0000000
(7)	$0.67349\times {10}^{-4}$	$0.51236\times {10}^{-18}$	$0.17160\times {10}^{-74}$	$0.21590\times {10}^{-300}$	4.0000000
PM	$0.63702\times {10}^{-3}$	$0.78493\times {10}^{-16}$	$0.57495\times {10}^{-70}$	$0.20389\times {10}^{-299}$	4.2384668
ZM	$0.79077\times {10}^{-3}$	$0.14418\times {10}^{-11}$	$0.25525\times {10}^{-40}$	$0.25811\times {10}^{-135}$	3.3039563
WM	$0.75634\times {10}^{-3}$	$0.25811\times {10}^{-15}$	$0.15116\times {10}^{-67}$	$0.60634\times {10}^{-289}$	4.2386876
(20) with (19)	$0.67349\times {10}^{-4}$	$0.32043\times {10}^{-20}$	$0.47317\times {10}^{-89}$	$0.10705\times {10}^{-380}$	4.2371414
(20) with (35)	$0.67349\times {10}^{-4}$	$0.13942\times {10}^{-19}$	$0.64548\times {10}^{-86}$	$0.61388\times {10}^{-367}$	4.2364379
(20) with (36)	$0.67349\times {10}^{-4}$	$0.20839\times {10}^{-19}$	$0.33951\times {10}^{-85}$	$0.70224\times {10}^{-364}$	4.2360962
(20) with (37)	$0.67349\times {10}^{-4}$	$0.21778\times {10}^{-20}$	$0.56497\times {10}^{-94}$	$0.27015\times {10}^{-421}$	4.4481352
(20) with (38)	0.67349$\times {10}^{-4}$	$0.25160\times {10}^{-20}$	$0.98512\times {10}^{-94}$	$0.33331\times {10}^{-420}$	4.4473908

,

Table 2. Numerical results for $f_2\left(x\right)$.

Methods	$|{\mathit{x}}_1-\mathit{\xi }|$	$|{\mathit{x}}_2-\mathit{\xi }|$	$|{\mathit{x}}_3-\mathit{\xi }|$	$|{\mathit{x}}_4-\mathit{\xi }|$	$\mathit{\rho }$
RM	$0.33251\times {10}^{-3}$	$0.30709\times {10}^{-13}$	$0.22312\times {10}^{-53}$	$0.62179\times {10}^{-214}$	4.0000000
(7)	$0.29605\times {10}^{-3}$	$0.16982\times {10}^{-13}$	$0.18366\times {10}^{-54}$	$0.25128\times {10}^{-218}$	4.0000000
PM	$0.34882\times {10}^{-2}$	$0.10531\times {10}^{-10}$	$0.36075\times {10}^{-46}$	$0.14899\times {10}^{-196}$	4.2403195
ZM	$0.15866\times {10}^{-2}$	$0.46751\times {10}^{-9}$	$0.14185\times {10}^{-30}$	$0.11659\times {10}^{-101}$	3.3035264
WM	$0.10688\times {10}^{-2}$	$0.25811\times {10}^{-15}$	$0.34580\times {10}^{-66}$	$0.35950\times {10}^{-283}$	4.2408357
(20) with (19)	$0.29605\times {10}^{-3}$	$0.14719\times {10}^{-16}$	$0.16544\times {10}^{-72}$	$0.13025\times {10}^{-309}$	4.2378388
(20) with (35)	$0.29605\times {10}^{-3}$	$0.70804\times {10}^{-15}$	$0.34181\times {10}^{-64}$	$0.44018\times {10}^{-273}$	4.2357244
(20) with (36)	$0.29605\times {10}^{-3}$	$0.18175\times {10}^{-14}$	$0.18755\times {10}^{-62}$	$0.10235\times {10}^{-265}$	4.2358506
(20) with (37)	$0.29605\times {10}^{-3}$	$0.48384\times {10}^{-15}$	$0.95540\times {10}^{-68}$	$0.38865\times {10}^{-302}$	4.4472587
(20) with (38)	$0.29605\times {10}^{-3}$	$0.73524\times {10}^{-15}$	$0.45757\times {10}^{-67}$	$0.47216\times {10}^{-299}$	4.4436750

and

Table 3. Numerical results for $f_3\left(x\right)$.

Methods	$|{\mathit{x}}_1-\mathit{\xi }|$	$|{\mathit{x}}_2-\mathit{\xi }|$	$|{\mathit{x}}_3-\mathit{\xi }|$	$|{\mathit{x}}_4-\mathit{\xi }|$	$\mathit{\rho }$
RM	$0.14664\times {10}^{-4}$	$0.12289\times {10}^{-23}$	$0.60662\times {10}^{-100}$	$0.36019\times {10}^{-405}$	4.0000000
(7)	$0.10564\times {10}^{-5}$	$0.40124\times {10}^{-26}$	$0.83509\times {10}^{-108}$	$0.15669\times {10}^{-434}$	4.0000000
PM	$0.90035\times {10}^{-2}$	$0.25682\times {10}^{-10}$	$0.87635\times {10}^{-46}$	$0.33663\times {10}^{-196}$	4.2410060
ZM	$0.51016\times {10}^{-2}$	$0.26111\times {10}^{-8}$	$0.52455\times {10}^{-29}$	$0.21691\times {10}^{-97}$	3.3040229
WM	$0.84480\times {10}^{-2}$	$0.96292\times {10}^{-11}$	$0.89256\times {10}^{-48}$	$0.74204\times {10}^{-205}$	4.2416331
(20) with (19)	$0.10564\times {10}^{-5}$	$0.21218\times {10}^{-30}$	$0.70199\times {10}^{-134}$	$0.16895\times {10}^{-572}$	4.2386648
(20) with (35)	$0.10564\times {10}^{-5}$	$0.13904\times {10}^{-26}$	$0.10705\times {10}^{-116}$	$0.49497\times {10}^{-498}$	4.2317152
(20) with (36)	$0.10564\times {10}^{-5}$	$0.13529\times {10}^{-26}$	$0.95737\times {10}^{-117}$	$0.30814\times {10}^{-498}$	4.2317416
(20) with (37)	$0.10564\times {10}^{-5}$	$0.17838\times {10}^{-29}$	$0.28166\times {10}^{-135}$	$0.50483\times {10}^{-606}$	4.4493324
(20) with (38)	$0.10564\times {10}^{-5}$	$0.20284\times {10}^{-29}$	$0.47006\times {10}^{-135}$	$0.50638\times {10}^{-605}$	4.4489767

show the absolute errors $\left|x_k-\xi \right|$ in the first four iterations, where the root $\xi$ is computed with 1200 significant digits. The first iteration uses initial parameters $T=0.1$ and $T_0=0.1$. The approximate computational order of convergence $\rho$ is defined by [24]:

$$\rho \approx \frac{ln(\left|x_{n+1}-x_n\right|/\left|x_n-x_{n-1}\right|)}{ln(\left|x_n-x_{n-1}\right|/\left|x_{n-1}-x_{n-2}\right|)}.$$

(74)

Test functions are used as follows:

$$f_1\left(x\right)=cos\left(x\right)-x,\xi \approx 0.7390851332151606,x_0=0.5.$$

$$f_2\left(x\right)=10x{e}^{-x^2}-1,\xi \approx 1.6796306104284499,x_0=1.8.$$

$$f_3\left(x\right)=sin\left(x\right)-\frac{1}{3}x,\xi \approx 2.2788626600758283,x_0=2.0.$$

Table 1, Table 2 and Table 3 show that our methods have better convergence behaves than some existing methods in this paper. Numerical results prove the validity of theory.

Remark 3.

If we use high-order Newton’s interpolation polynomial to calculate the self-accelerating parameter, our method (20) will obtain higher convergence order. High-order interpolation polynomial is complex, which is disadvantageous to reduce computation. Therefore, we give a simple way to construct self-accelerating parameter.

5. Conclusions

In this paper, a new self-accelerating type method is proposed for solving nonlinear equations, which is a modified scheme of Ren’s method. The new method reaches the highest convergence order 4.4495 by using a self-accelerating parameter. More importantly, a novel technique is applied to construct a self-accelerating parameter, which is different from the secant method and interpolation method. The new self-accelerating parameter does not increase the computational cost of the new method. The new method is compared in performance with the existing methods. Numerical examples confirm the validity of theoretical results and show that the new method has a better convergence feature than some existing methods.