On the development and extensions of some classes of optimal three–point iterations for solving nonlinear equations

T. Zhanlav$^\ast $ Kh. Otgondorj$^{\ast \ast }$

August 20, 2021; accepted: January 4, 2021; published online: February 17, 2022.

We develop new families of optimal eight–order methods for solving nonlinear equations. We also extend some classes of optimal methods for any suitable choice of iteration parameter. Such development and extension was made using sufficient convergence conditions given in [ 20 ] . Numerical examples are considered to check the convergence order of new families and extensions of some well-known methods.

MSC. 65H05.

Keywords. Multipoint methods; order of convergence; nonlinear equations

$^\ast $Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Mongolia e-mail: tzhanlav@yahoo.com.

$^{\ast \ast }$School of Applied Sciences, Mongolian University of Science and Technology, Mongolia e-mail: otgondorj@gmail.com.

1 Introduction

Finding solution of nonlinear equations $f(x)=0$ is an important problem in science and engineering. In last years, many optimal eight–order iterative methods were developed, see [ 11 , 19 , 17 , 1 , 10 , 3 , 12 , 4 , 2 , 13 , 8 , 14 , 5 , 20 , 21 , 22 , 23 , 6 ] and references therein. But many of them work only for special choices of iteration parameter and absolutely not clear how changed the structure of iterations for another choice of parameter. Therefore, it is very desirable to construct the optimal iterations that work well for any suitable choice of parameter. Our aim is to develop and to extend some classes of optimal three–point iterations using sufficient convergence conditions given in [ 20 ] .

We consider the following standard three–point iterative methods:

\begin{eqnarray} \label{it1.1} & & y_n=x_n-\frac{f(x_n)}{f'(x_n)},\nonumber \\ & & z_n=y_n-\bar{\tau }_n\frac{f(y_n)}{f'(x_n)},\\ & & x_{n+1}=z_n-\alpha _n\frac{f(z_n)}{f'(x_n)},~ n=0,1\dots \nonumber \end{eqnarray}

In [ 20 ] was proven that the order of convergence iterations 1 is eight if and only if the parameters $\bar{\tau }_n$ and $\alpha _n$ satisfy the conditions

\begin{eqnarray} \label{it1.2} \bar{\tau }_n=1+2\theta _n+\widetilde{\beta }\theta _n^2+\widetilde{\gamma }\theta _n^3+\dots ,~ \theta _n=\frac{f(y_n)}{f(x_n)} \end{eqnarray}

and

\begin{equation} \label{it1.3} \begin{split} \alpha _n=& 1+2\theta _n+(\widetilde{\beta }+1)\theta _n^2+(2\widetilde{\beta }+\widetilde{\gamma }-4)\theta _n^3\\ & +(1+4\theta _n)\upsilon _n+O(f(x_n)^4), \end{split} \end{equation}

where $~ \upsilon _n=\frac{f(z_n)}{f(y_n)}.$ The optimal methods 1 distinguish each other only by choices of parameters $\bar{\tau }_n$ and $\alpha _n$. It should be pointed out that to establish the convergence order of iterative methods often used either the error equation, see for example [ 11 , 19 , 17 , 1 , 10 , 3 , 12 , 4 , 2 , 13 , 8 , 14 , 5 , 15 , 18 , 9 ] , or the nonlinear residuals [ 20 , 21 , 22 , 23 ] . An more detailed explanation of various aspects of convergence order based on error analysis, corrections and nonlinear residuals was given in the excellent surveys [ 6 , 7 ] . In this paper, we propose a new family of optimal three–point methods and extensions of some classes of optimal methods. The rest of this paper is organized as follows.

In section 2, we propose new families of optimal three-point methods. In section 3, we suggested extension of classes of optimal eighth-order methods. The numerical experiments and dynamic behavior of methods are discussed in section 4. Finally, short conclusions are included in section 5.

2 Development of the new families of optimal three–point methods

First, we consider iterations 1 with parameter $\alpha _n$ given by

\begin{eqnarray} \label{it1.4} \alpha _n= \frac{f'(x_n)}{f[y_n,z_n]+2(f[x_n,z_n]-f[x_n,y_n])+(y_n-z_n)f[y_n,x_n,x_n]}, \end{eqnarray}

where

\begin{eqnarray} \label{it1.5} f[y_n,x_n,x_n]=\frac{f[y_n,x_n]-f'(x_n)}{y_n-x_n}. \end{eqnarray}

To show the convergence analysis of methods 1, 6, the following results is proven.

Theorem 1

Let the function $f(x)$ be sufficiently smooth and have a simple root $x^*$ on the open interval $I\subset R$. Furthermore, let the initial approximation $x_0$ be sufficiently close to $x^*$ and the parameter $\bar{\tau }_n$ in 1 satisfies the condition 2. Then the order of convergence of the methods 1, 6 is eight.

Proof â–¼

Using the relations

\begin{eqnarray} \label{it1.6} & & f[x_n,y_n]=f’(x_n)(1-\theta _n),\nonumber \\ & & f[y_n,z_n]=f’(x_n)\frac{1-\upsilon _n}{\bar{\tau }_n},\\ & & f[x_n,z_n]=f’(x_n)\frac{1-\theta _n\upsilon _n }{1+\bar{\tau }_n\theta _n},\nonumber \end{eqnarray}

in 6, we obtain

\begin{eqnarray} \label{it1.7} \alpha _n= \frac{\bar{\tau }_n}{1-\upsilon _n +2\frac{\bar{\tau }_n\theta _n}{1+\bar{\tau }_n\theta _n}(1-\upsilon _n-\bar{\tau }_n+\bar{\tau }_n\theta _n)+\bar{\tau }_n^2\theta _n^2}. \end{eqnarray}

Using 2 and well-known expansion

\begin{eqnarray} \label{it1.8} \frac{1}{1-x}=1+x+x^2+x^3+\dots ,~ |x|{\lt}1, \end{eqnarray}

in 9 we obtain

\begin{eqnarray} \label{it1.9} \alpha _n= \bar{\tau }_n(1+\theta _n^2-(6-2\widetilde{\beta })\theta _n^3+(1+2\theta _n)\upsilon _n)+ {\mathcal O}(f(x_n)^4). \end{eqnarray}

From 2 and 11 it follows that $\alpha _n$ defined by 11 satisfies the condition 3 that completes the proof of theorem.

Of course, there are many possibility for choice $\bar{\tau }_n$ in 1 satisfying the condition 2. In particular, we give $\bar{\tau }_n$ as

\begin{eqnarray} \label{it1.10} \bar{\tau }_n=\frac{c+(2c+d)\theta _n+\omega \theta _n^2}{c+d\theta _n+b\theta _n^2},~ c+d+b\ne 0, \end{eqnarray}

that includes four free parameters. In table 1, we list some well-known choices.

Table 1 The choices of parameter $\bar{\tau }_n$

Cases	Methods	Special case of 12	$\bar{\tau }_n$	$\widetilde{\beta }$	$\widetilde{\gamma }$
i	Potra-Ptack’s	$c=\omega =1,~ d=b=0$	$1+2\theta _n+\theta _n^2$	1	0
ii	Maheshwari’s	$c=1,~ b=0,~ d=\omega =-1$	$\frac{1+\theta _n-\theta _n^2}{1-\theta _n}$	1	1
iii	Kung-Traub’s	$c=b=1,~ d=-2,~ \omega =0$	$\frac{1}{(1-\theta _n)^2}$	3	4
iv	King’s type	$c=1,~ \omega =b=0, ~ d=\beta -2$	$\frac{1+\beta \theta _n}{1+(\beta -2)\theta _n}$	$2(2-\beta )$	$2(2-\beta )^2$

Note that similar theorem for iteration 1, 6 for Kung-Traub’s type iteration were proved by Petković et al. [ 11 ] and by Zhanlav et al. [ 22 ] for Kings type iteration and by Wang et al. [ 19 ] for Ostrowski’s type method. Thus, theorem 1 extend essentially the class of families of optimal eight-order iterations 1, 6. Now we consider the iterations 1 with $\alpha _n$ given by

\begin{eqnarray} \label{it1.30} \alpha _n=\frac{f'(x_n)(1+A\theta _n+B\theta _n^2+C\theta _n^3+(\delta +\Delta \theta _n)\upsilon _n)}{\omega _1 f[x_n,z_n]+\omega _2 f[z_n,y_n]+\omega _3 f[x_n,y_n]}, \end{eqnarray}

where $ \omega _1 +\omega _2 +\omega _3 =1$ and $A,B,C,\delta ,\Delta ,\omega _1,\omega _2$ and $\omega _3$ are free parameters to be determined such that the iterations 1 with $\alpha _n$ given by 13 has optimal eight-order of convergence. Namely we can prove

Theorem 2

Let all assumptions of theorem 1 be fulfilled. Then the order of convergence of the iterations 1, 13 is eight when

\begin{eqnarray} \label{it1.31} & & A=\delta =1-\omega _2,~ B=(\widetilde{\beta }-2)(1-\omega _2)+1-\omega _1,~ \Delta =3-\omega _1-\omega _2,\nonumber \\ & & C=\widetilde{\gamma }(1-\omega _2)+\widetilde{\beta }(1+\omega _2-\omega _1)+\omega _1-\omega _2-5. \end{eqnarray}

Proof â–¼

The proof is the same as that of theorem 1. For convenience, here we only give the main step of proof. As before, using 8 and 10 after some manipulations we obtain

\begin{equation} \label{it1.32} \begin{split} \alpha _n=& \frac{f'(x_n)}{\omega _1 f[x_n,z_n]+\omega _2 f[z_n,y_n]+\omega _3 f[x_n,y_n]}\\ =& 1+(1+\omega _2)\theta _n+(\omega _1+\widetilde{\beta }\omega _2+(1-\omega _2)^2)\theta _n^2\\ & +\big(\widetilde{\gamma }\omega _2+\widetilde{\beta }(2(1-\omega _2)^2-\omega _1-2\omega _3)+\omega _1\\ & +2(1-\omega _2)(2-\omega _1)-2(1-\omega _2)^2-(1-\omega _2)^3\big)\theta _n^3\\ & +(\omega _2+(\omega _1+2\omega _2^2)\theta _n)\upsilon _n+O(f(x_n)^4). \end{split} \end{equation}

Substituting 15 into 13 and comparing 13 with 3 we arrive at 14.

The expression in the numerator of 13 can be expressed through first order divided differences $f[x_n,y_n]$, $f[x_n,z_n]$ and $f[z_n,y_n]$ within accuracy ${\mathcal O}(f(x_n)^4)$. Indeed using the iterations

\begin{eqnarray} \label{it1.n41} f[x_n,y_n]-f[x_n,z_n]=f’(x_n)(\theta _n^2+(\widetilde{\beta }-3)\theta _n^3+\theta _n\upsilon _n), \end{eqnarray}

and

\begin{equation} \label{it1.n42} \begin{split} f[z_n,x_n]-f[y_n,z_n]=& f’(x_n)(\theta _n+(\widetilde{\beta }-5)\theta _n^2\\ & +(\widetilde{\gamma }-5\widetilde{\beta }+11)\theta _n^3+(1-3\theta _n)\upsilon _n), \end{split} \end{equation}

in 13 we obtain

\begin{eqnarray} \label{it1.n43} \alpha _n=\frac{(3\omega _2+\omega _1-5)(f[x_n,z_n]-f[x_n,y_n])+F_n+Q_n}{\omega _1 f[x_n,z_n]+\omega _2 f[z_n,y_n]+\omega _3 f[x_n,y_n]}, \end{eqnarray}

where

\begin{eqnarray*} Q_n=f’(x_n)(1+(\omega _2-2)\theta _n^2+2(1-\omega _1-\omega _2)\theta _n^3)\end{eqnarray*}

and

\begin{eqnarray*} F_n=(1-\omega _2)(f[x_n,y_n]-f[y_n,z_n]). \end{eqnarray*}

From 13, 14 we see that $\alpha _n$ includes two free parameters $\omega _1$ and $\omega _2$. Thus, we develop the class of optimal eight-order iterations 1, 13, 14. We consider some choices of parameters $\omega _1$ and $\omega _2$.

Let $\omega _1= \omega _3=0$, $\omega _2=1$. Then 25 converted to

\begin{eqnarray} \label{it1.33} \alpha _n=\frac{(3+\theta _n) f[x_n,y_n]-2 f[z_n,x_n]}{ f[y_n,z_n]}. \end{eqnarray}
Let $\omega _1=1$, $\omega _2= \omega _3=0$. Then 25 converted to

\begin{eqnarray} \label{it1.34} \alpha _n=\frac{5f[x_n,y_n]-4f[z_n,x_n]- f[y_n,z_n]+f'(x_n)(1-2\theta _n^2)}{ f[x_n,z_n]}, \end{eqnarray}
Let $\omega _1= \omega _2=0$, $\omega _3=1$. Then 25 converted to

\begin{eqnarray} \label{it1.35} \alpha _n=\frac{6f[x_n,y_n]-5f[z_n,x_n]- f[y_n,z_n]+f'(x_n)(1-2\theta _n^2+2\theta _n^3)}{ f[y_n,x_n]}, \end{eqnarray}
Let $\omega _1=-1$, $ \omega _2=2$, $\omega _3=0$. Then 25 converted to

\begin{eqnarray} \label{it1.36} \alpha _n=\frac{f[z_n,y_n]-f[x_n,y_n]+f'(x_n)}{ 2f[z_n,y_n]-f[z_n,x_n]}. \end{eqnarray}

The iteration 1, 29 can be considered as another variant of iterations given by Sharma et al. in [ 12 , 13 , 14 ] and given by Zhanlav et al. in [ 23 ] .
Let $\omega _1=\omega _2=1$, $\omega _3=-1$. Then 25 converted to

\begin{eqnarray} \label{it1.37} \alpha _n=\frac{f[y_n,x_n]-f[z_n,x_n]+f'(x_n)(1-\theta _n^2-2\theta _n^3)}{f[z_n,x_n]+ f[z_n,y_n]-f[y_n,x_n]}. \end{eqnarray}

It can be rewritten as:

\begin{eqnarray*} \label{it1.377} \alpha _n\approx \frac{1}{\left(1-\frac{f(z_n)}{f(x_n)}\right)\left(1+(5-\widetilde{\beta })\left(\frac{f(y_n)}{f(x_n)}\right)^3\right)}\frac{f'(x_n)}{f[z_n,x_n]+ f[z_n,y_n]-f[y_n,x_n]}. \end{eqnarray*}

It is worth to note that similar results for derivative-free case and for some choices of $\bar{\tau }_n$ were obtained by Thukral in [ 18 ] and by Khattri et al. in [ 9 ] . We also note that the iteration 1, 30 for $\widetilde{\beta }=4$ was considered by Sharma et al. in [ 15 ] .
Let $\omega _1=-1$, $\omega _2=\omega _3=1$. Then 25 converted to

\begin{eqnarray} \label{it1.38} \alpha _n=\frac{3f[x_n,y_n]-3f[z_n,x_n]+f'(x_n)(1-\theta _n^2+2\theta _n^3)}{f[y_n,x_n]+ f[z_n,y_n]-f[z_n,x_n]}. \end{eqnarray}

In each iteration step the methods 1, 6 and 1, 13 require three function evaluations and one evaluation of first derivative. Based on the conjecture of Kung and Traub, the methods reached the optimality with higher efficiency index $E=8^{1/4}=1.68179$. One of main advantageous of the proposed iterative methods 1, 6 and 1, 13 is that they work well for any choice of parameter $\bar{\tau }_n$ satisfying the condition 2.

3 Extensions of some classes of optimal eight-order methods

Now we consider the iterations 1 with parameter $\alpha _n$ given by

\begin{eqnarray} \label{it1.11} \alpha _n= (p(t_n)+\hat{\gamma }\theta _n^3)\frac{f'(x_n)}{f[z_n,y_n]+(z_n-y_n)f[z_n,x_n,x_n]}, \end{eqnarray}

where $t_n=\frac{f(z_n)}{f(x_n)}=\theta _n\upsilon _n$ and $\hat{\gamma }$ constant and $p(t_n)$ some function of $t$.

Namely, we have

Theorem 3

Let all assumptions of theorem 1 be fulfilled. Then the order of convergence of the iterations 1 and 32 is eight when

\begin{eqnarray} \label{it1.12} p(0)=1,~ p’(0)=2, ~ \hat{\gamma }=2(\widetilde{\beta }-5). \end{eqnarray}

Proof â–¼

We denote the second factor in 32 by $\hat{ \alpha }_n$. That is

\begin{eqnarray} \label{it1.13} \hat{ \alpha }_n= \frac{f'(x_n)}{f[z_n,y_n]+(z_n-y_n)f[z_n,x_n,x_n]}. \end{eqnarray}

Then using the relations 8 we obtain

\begin{eqnarray} \label{it1.14} \hat{ \alpha }_n= \frac{\bar{\tau }_n}{1-\upsilon _n -\bar{\tau }_n\left(\frac{\bar{\tau }_n\theta _n}{1+\bar{\tau }_n\theta _n}\right)^2}. \end{eqnarray}

Using the expansion 10 in 35 and taking into account $\upsilon _n={\mathcal O}(f(x_n)^2)$, $\theta _n={\mathcal O}(f(x_n))$, we obtain

\begin{eqnarray} \label{it1.15} \hat{ \alpha }_n= \bar{\tau }_n\left(1+\upsilon _n +\bar{\tau }_n\left(\frac{\bar{\tau }_n\theta _n}{1+\bar{\tau }_n\theta _n}\right)^2\right)+{\mathcal O}(f(x_n)^4). \end{eqnarray}

By using 2 and 10 it is easy to show that

\begin{eqnarray} \label{it1.16} \left(\frac{\bar{\tau }_n\theta _n}{1+\bar{\tau }_n\theta _n}\right)^2=\theta _n^2+2\theta _n^3+{\mathcal O}(f(x_n)^4). \end{eqnarray}

Substituting 2 and 37 into 36 we get

\begin{eqnarray} \label{it1.17} \hat{ \alpha }_n= 1+2\theta _n+(\widetilde{\beta }+1)\theta _n^2+(\widetilde{\gamma }+6)\theta _n^3+(1+2\theta _n)\upsilon _n+{\mathcal O}(f(x_n)^4). \end{eqnarray}

Then 32 is written as

\begin{equation} \label{it1.18} \begin{split} \alpha _n=& (p(t_n)+\hat{\gamma }\theta _n^3)(1+2\theta _n+(\widetilde{\beta }+1)\theta _n^2\\ & +(\widetilde{\gamma }+6)\theta _n^3+(1+2\theta _n)\upsilon _n)+{\mathcal O}(f(x_n)^4), \end{split} \end{equation}

which satisfies the condition 3 provided that 33.

Thus, we develop the family of optimal three-point iterative methods 1 with $ \alpha _n$ given by

\begin{eqnarray} \label{it1.19} \alpha _n= (p(t_n)+2(\widetilde{\beta }-5)\theta _n^3)\frac{f'(x_n)}{f[z_n,y_n]+(z_n-y_n)f[z_n,x_n,x_n]}. \end{eqnarray}

Similar results were obtained in [ 3 , 2 ] for the iterations 1 with

\begin{eqnarray} \label{it1.20} \bar{\tau }_n=\frac{1-\theta _n/2}{1-5\theta _n/2}, \end{eqnarray}

and

\begin{eqnarray} \label{it1.21} \bar{\tau }_n=\frac{1}{1-2\theta _n-\theta _n^2-\theta _n^3/2}, \end{eqnarray}

respectively. The parameters $\bar{\tau }_n$ given by 43 and 44 satisfy the condition 2 with $\widetilde{\beta }=5$. In this case $\hat{\gamma }=0$ by 33 and the $ \alpha _n$ given by 32 leads to

\begin{eqnarray} \label{it1.22} \alpha _n= p(t_n)\frac{f'(x_n)}{f[z_n,y_n]+(z_n-y_n)f[z_n,x_n,x_n]}. \end{eqnarray}

This means that our iterations 1 and 42 include the iterations proposed by Bi et al. [ 2 ] and by Cordero et al. [ 3 ] as particular cases.

Now we consider the expression

\begin{eqnarray} \label{it1.23} \tilde{\alpha }_n=\frac{f'(x_n)}{ f[z_n,y_n]+(z_n-y_n)f[y_n,x_n,x_n]}. \end{eqnarray}

As before, using the relations 8 in 46 we obtain

\begin{eqnarray} \label{it1.24} \tilde{\alpha }_n= \frac{\bar{\tau }_n}{1-\upsilon _n -\bar{\tau }_n^2\theta _n^2}=\bar{\tau }_n(1+\upsilon _n+\bar{\tau }_n^2\theta _n^2)+{\mathcal O}(f(x_n)^4). \end{eqnarray}

By 2 one can easily to check that

\begin{eqnarray} \label{it1.25} \tilde{\alpha }_n=\hat{ \alpha }_n +{\mathcal O}(f(x_n)^4). \end{eqnarray}

It means that instead of 32 one can also use

\begin{eqnarray} \label{it1.26} \alpha _n= (p(t_n)+\hat{\gamma }\theta _n^3)\frac{f'(x_n)}{f[z_n,y_n]+(z_n-y_n)f[y_n,x_n,x_n]}, \end{eqnarray}

Therefore, theorem 3 holds true for iterations 1, 49. Similar extension can be done for all optimal eight-order iterations. As examples, we present in table 2 some of methods and their extension $\tilde{\alpha }_n=\xi _n\cdot \alpha _n$ with extension factor $\xi _n$.
===

Table 2 The extraneous fixed points

===

Thus, we obtain extensions of some well-known optimal methods that work well for any suitable parameter $\bar{\tau }_n$, satisfying the condition 2. This allows us to expand the applicability of the original methods.

4 Numerical experiments

In order to show the convergence behavior and to check the validity of theoretical results of the presented family (1) with parameters $\bar{\tau }_n$ and $\alpha _n$, we make some numerical experiments. We also compare our methods with existing methods of same order in [ 13 ] , [ 14 ] and [ 23 ] that denoted by (SAWN8) and (ZO8). Here all the computations are performed using the programming package MATHEMATICA with multiple-precision arithmetic and 1000 significant digits. As a test, we consider the following sample functions.

\begin{eqnarray*} & & f_1(x)=e^{x^3-3x}\sin x+\log (x^2+1), \quad x^*=0,\\ & & f_2(x)=x^2-\exp (x)-3x+2, \quad x^*\approx 0.25. \end{eqnarray*}

In Tables 3–5, we present the necessary iterations $(n)$, absolute error $|x_n-x^*|$ and computational order of convergence, which is calculated by the following formula [ 11 , 16 ] :

\begin{equation*} \rho \approx \frac{\ln (|x_{n-1}-x_n| / | x_{n}-x_{n-1}|)}{\ln (| x_{n}-x_{n-1}| /|x_{n-2}-x_{n-1}|)}, \nonumber \end{equation*}

where $x_{n} , x_{n-1} , x_{n-2}$ are three consecutive approximations of iterations. The convergence orders and their computational variants have been thoroughly treated in [ 6 , 7 ] . Outcomes of the numerical experiments are calculated so as to satisfy the criterion $|x_n-x^*|{\lt}10^{-30}$. For $\bar{\tau }_n$ parameter, we choose the cases i–iv listed in table 1. Table 3 gives some numerical results in order to show convergence behaviour of method 1 with $\alpha _n$ parameter given by (6), (26)–(31). We observe from table 3 that the methods 1 with parameters $\bar{\tau }_n$ given by case iv and $\alpha _n$ given by 6, 30 produce approximations of higher accuracy compared to the eight-order methods SAWN8, ZO8.

The results corresponding to the same kind of experiments for the extension of methods can be found in Table 4-5. In table 4, we present the numerical results of iteration 1 with parameter $\alpha _n$ given 32 and 49, in which we used function $p(t)=\frac{1}{(1-t)^2}$. In table 5, we present the numerical results of the extension of some methods that work well any parameters $\bar{\tau }_n$ satisfies condition 2.

From the results displayed in Table 3–5, we see that the calculated values of the computational order of convergence are in complete agreement with the theoretical orders proved in Section 2, 3.

Additionally, we analyze the basin of attraction of our methods to find out what is the best choice for the parameters. To generate basin attraction for complex polynomials using the methods, we take a grid of $400 \times 400$ points $z_0$ in the square $[-3, 3]\times [-3, 3]\subset C$. We have used the method (1) for cubic polynomial $p(z)=z^3-1$ having three simple zeros.

Table 3 The numerical result for $f_i(x)$ by the methods 1 with $\bar{\tau }_n$ and $\alpha _n$

$\alpha _n$	$\bar{\tau }_n$	$n$	$\|x^*-x_n\|$	$\rho $	$n$	$\|x^*-x_n\|$	$\rho $
			$f_1(x),\quad x_0=0.5$			$f_2(x),\quad x_0=2$
	case i	3	0.5407e-222	8.00	3	0.1220e-231	8.00
	case ii	3	0.4264e-222	8.00	3	0.4412e-189	7.99
(6)	case iii	3	0.2180e-233	8.00	3	0.2054e-245	8.00
	case iv, $\beta =0$	3	0.2111e-226	8.00	3	0.4836e-229	7.99
	case i	3	0.5182e-200	8.00	3	0.3202e-231	7.99
	case ii	3	0.7800e-203	8.00	2	0.6607e-31	8.00
(26)	case iii	3	0.4084e-132	8.00	3	0.2183e-205	7.99
	case iv, $\beta =0$	3	0.5805e-127	8.00	3	0.3391e-235	8.00
	case i	3	0.3401e-122	8.00	3	0.3173e-229	7.99
	case ii	3	0.5182e-200	8.00	3	0.3976e-247	7.99
(27)	case iii	3	0.4084e-132	7.99	3	0.9235e-207	7.99
	case iv, $\beta =0$	3	0.5805e-127	8.00	3	0.1166e-232	7.99
	case i	3	0.8671e-118	8.00	3	0.2475e-221	7.99
	case ii	3	0.1363e-116	8.00	3	0.3905e-241	7.99
(28)	case iii	3	0.1213e-121	8.00	3	0.2152e-202	7.99
	case iv, $\beta =0$	3	0.3621e-119	8.00	3	0.1296e-226	7.99
	case i	3	0.1024e-139	7.99	3	0.1307e-234	7.99
	case ii	3	0.4756e-142	7.99	2	0.7314e-33	7.93
(29)	case iii	3	0.1560e-146	7.99	3	0.3104e-204	8.00
	case iv, $\beta =0$	3	0.5207e-153	8.00	3	0.1715e-172	8.00
	case i	3	0.2195e-222	8.00	3	0.4879e-248	7.99
	case ii	3	0.4844e-243	8.00	2	0.9301e-33	7.97
(30)	case iii	3	0.2720e-219	8.00	3	0.8760e-212	7.99
	case iv, $\beta =0$	3	0.8966e-220	8.00	3	0.4439e-247	8.00
	case i	3	0.1236e-187	8.00	3	0.1047e-221	7.99
	case ii	3	0.9877e-182	8.00	3	0.1326e-245	7.99
(31)	case iii	3	0.1077e-184	8.00	3	0.9069e-201	8.00
	case iv$, \beta =0$	3	0.1073e-184	8.00	3	0.1044e-227	8.00
SAWN8 [ 14 ]		3	0.5207e-153	8.00	3	0.1715e-172	8.00
ZO8 [ 23 ]		3	0.1036e-137	7.99	3	0.3317e-178	8.00

Table 4 The numerical result for $f_i(x)$ by the methods 1 with $\bar{\tau }_n$ and $\alpha _n$

$\alpha _n$	$\bar{\tau }_n$	$n$	$\|x^*-x_n\|$	$\rho $	$n$	$\|x^*-x_n\|$	$\rho $
			$f_1(x),\quad x_0=0.5$			$f_2(x),\quad x_0=2$
	case i	3	0.6992e-190	8.00	3	0.2459e-223	7.99
(32)	case ii	3	0.1915e-190	8.00	3	0.3996e-244	7.99
	case iii	3	0.2945e-186	8.00	3	0.1124e-181	7.99
	case iv, $\beta =0$	3	0.1334e-189	8.00	3	0.8535e-221	8.00
	case i	3	0.3186e-194	8.00	3	0.1093e-200	7.99
(49)	case ii	3	0.1273e-194	8.00	3	0.1051e-217	7.99
	case iii	3	0.2945e-186	8.00	3	0.3833e-158	7.99
	case iv, $\beta =0$	3	0.6747e-194	8.00	3	0.5685e-198	8.00

Table 5 The numerical results of extension of methods for $f_2(x)$

Methods	Extension factor $\xi _n$	$\bar{\tau }_n$	$n$	$\|x^*-x_n\|$	$\rho $
		Case i	3	0.3159e-214	8.00
Sharma [ 12 ]	$1+(\widetilde{\beta }-4)\theta _n^3$	Case ii	3	0.3453e-212	8.00
		Case iv, $\beta =0$	3	0.8671e-210	7.99
		Case i	3	0.2781e-229	7.99
DP8 [ 11 ]	$1+(\widetilde{\beta }-4)\theta _n^2+(\widetilde{\gamma }-8)\theta _n^3$	Case ii	3	0.1848e-238	7.99
		Case iv, $\beta =0$	3	0.1196e-162	7.99
		Case i	3	0.2433e-214	7.99
GK8 [ 8 ]	$1+(\widetilde{\beta }-3)\theta _n^2+(\widetilde{\gamma }-6)\theta _n^3$	Case ii	3	0.5841e-236	7.99
		Case iv, $\beta =1$	3	0.6505e-172	7.99
		Case i	3	0.2100e-114	8.00
Chun [ 4 ]	$1+(\widetilde{\beta }-3)\theta _n^2+(\widetilde{\gamma }-4)\theta _n^3$	Case ii	3	0.2547e-114	8.00
		Case iii	3	0.7827e-120	8.00
		Case i	3	0.5103e-156	8.00
Thukral [ 17 ]	$1+(\widetilde{\beta }-2(2-\beta ))\theta _n^2$	Case ii	3	0.1603e-155	8.00
	$ +(\widetilde{\gamma }-2(2-\beta )^2)\theta _n^3$	Case iv, $\beta =0$	3	0.5103e-156	8.00
		Case i	3	0.1915e-234	8.00
Lotfi [ 10 ]	$1+(\widetilde{\beta }-4)\theta _n^2+(\widetilde{\gamma }-8)\theta _n^3$	Case ii	3	0.1797e-233	8.00
		Case iv, $\beta =0$	3	0.2567e-157	8.00

In Figure 1–3, the yellow, red and blue colors are assigned for the attraction basin of the three zeros and the roots of function are marked with white points. Black color is shown lack of convergence to any of the roots. In this cases, the stopping criterion $\varepsilon =10^{-3}$ and maximum of 25 iterations are used.

Based on Figure 1–3 for $p(z)$, we can see that the method (1) with $\bar{\tau }_n$ given by case iii and $\alpha _n$ given by 6 is the best one and have fewer diverging points that other cases of parameters.

\includegraphics[width=0.27\textwidth ]{m1-2-4.png} — (1)-ii-(4)

\includegraphics[width=0.27\textwidth ]{m1-3-4.png} — (1)-ii-(4)

\includegraphics[width=0.27\textwidth ]{m1-4-17.png} — (1)-iv-(17)

\includegraphics[width=0.27\textwidth ]{m1-4-18.png} — (1)-iv-(17)

\includegraphics[width=0.27\textwidth ]{m1-4-20.png} — (1)-iv-(20)

\includegraphics[width=0.27\textwidth ]{m1-4-21.png} — (1)-iv-(20)

5 Conclusion

The main contributions of this work are:

The development of wide class of optimal eight-order iterative methods and extensions of some optimal methods that work well for any suitable choice of parameter $\bar{\tau }_n$ satisfying the condition 2.

The proposed iterative methods can be regarded as an advancement in the topic and can compete with other well-known methods.

Acknowledgements

The authors wish to thank the anonymous referees for their valuable suggestions and comments, which improved paper.

Bibliography

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Cases	Methods	Special case of 12	\(\bar{\tau }_n\)	\(\widetilde{\beta }\)	\(\widetilde{\gamma }\)
i	Potra-Ptack’s	\(c=\omega =1,~ d=b=0\)	\(1+2\theta _n+\theta _n^2\)	1	0
ii	Maheshwari’s	\(c=1,~ b=0,~ d=\omega =-1\)	\(\frac{1+\theta _n-\theta _n^2}{1-\theta _n}\)	1	1
iii	Kung-Traub’s	\(c=b=1,~ d=-2,~ \omega =0\)	\(\frac{1}{(1-\theta _n)^2}\)	3	4
iv	King’s type	\(c=1,~ \omega =b=0, ~ d=\beta -2\)	\(\frac{1+\beta \theta _n}{1+(\beta -2)\theta _n}\)	\(2(2-\beta )\)	\(2(2-\beta )^2\)

\(\alpha _n\)	\(\bar{\tau }_n\)	\(n\)	\(\|x^*-x_n\|\)	\(\rho \)	\(n\)	\(\|x^*-x_n\|\)	\(\rho \)
			\(f_1(x),\quad x_0=0.5\)			\(f_2(x),\quad x_0=2\)
	case i	3	0.5407e-222	8.00	3	0.1220e-231	8.00
	case ii	3	0.4264e-222	8.00	3	0.4412e-189	7.99
(6)	case iii	3	0.2180e-233	8.00	3	0.2054e-245	8.00
	case iv, \(\beta =0\)	3	0.2111e-226	8.00	3	0.4836e-229	7.99
	case i	3	0.5182e-200	8.00	3	0.3202e-231	7.99
	case ii	3	0.7800e-203	8.00	2	0.6607e-31	8.00
(26)	case iii	3	0.4084e-132	8.00	3	0.2183e-205	7.99
	case iv, \(\beta =0\)	3	0.5805e-127	8.00	3	0.3391e-235	8.00
	case i	3	0.3401e-122	8.00	3	0.3173e-229	7.99
	case ii	3	0.5182e-200	8.00	3	0.3976e-247	7.99
(27)	case iii	3	0.4084e-132	7.99	3	0.9235e-207	7.99
	case iv, \(\beta =0\)	3	0.5805e-127	8.00	3	0.1166e-232	7.99
	case i	3	0.8671e-118	8.00	3	0.2475e-221	7.99
	case ii	3	0.1363e-116	8.00	3	0.3905e-241	7.99
(28)	case iii	3	0.1213e-121	8.00	3	0.2152e-202	7.99
	case iv, \(\beta =0\)	3	0.3621e-119	8.00	3	0.1296e-226	7.99
	case i	3	0.1024e-139	7.99	3	0.1307e-234	7.99
	case ii	3	0.4756e-142	7.99	2	0.7314e-33	7.93
(29)	case iii	3	0.1560e-146	7.99	3	0.3104e-204	8.00
	case iv, \(\beta =0\)	3	0.5207e-153	8.00	3	0.1715e-172	8.00
	case i	3	0.2195e-222	8.00	3	0.4879e-248	7.99
	case ii	3	0.4844e-243	8.00	2	0.9301e-33	7.97
(30)	case iii	3	0.2720e-219	8.00	3	0.8760e-212	7.99
	case iv, \(\beta =0\)	3	0.8966e-220	8.00	3	0.4439e-247	8.00
	case i	3	0.1236e-187	8.00	3	0.1047e-221	7.99
	case ii	3	0.9877e-182	8.00	3	0.1326e-245	7.99
(31)	case iii	3	0.1077e-184	8.00	3	0.9069e-201	8.00
	case iv\(, \beta =0\)	3	0.1073e-184	8.00	3	0.1044e-227	8.00
SAWN8 [ 14 ]		3	0.5207e-153	8.00	3	0.1715e-172	8.00
ZO8 [ 23 ]		3	0.1036e-137	7.99	3	0.3317e-178	8.00

Methods	Extension factor \(\xi _n\)	\(\bar{\tau }_n\)	\(n\)	\(\|x^*-x_n\|\)	\(\rho \)
		Case i	3	0.3159e-214	8.00
Sharma [ 12 ]	\(1+(\widetilde{\beta }-4)\theta _n^3\)	Case ii	3	0.3453e-212	8.00
		Case iv, \(\beta =0\)	3	0.8671e-210	7.99
		Case i	3	0.2781e-229	7.99
DP8 [ 11 ]	\(1+(\widetilde{\beta }-4)\theta _n^2+(\widetilde{\gamma }-8)\theta _n^3\)	Case ii	3	0.1848e-238	7.99
		Case iv, \(\beta =0\)	3	0.1196e-162	7.99
		Case i	3	0.2433e-214	7.99
GK8 [ 8 ]	\(1+(\widetilde{\beta }-3)\theta _n^2+(\widetilde{\gamma }-6)\theta _n^3\)	Case ii	3	0.5841e-236	7.99
		Case iv, \(\beta =1\)	3	0.6505e-172	7.99
		Case i	3	0.2100e-114	8.00
Chun [ 4 ]	\(1+(\widetilde{\beta }-3)\theta _n^2+(\widetilde{\gamma }-4)\theta _n^3\)	Case ii	3	0.2547e-114	8.00
		Case iii	3	0.7827e-120	8.00
		Case i	3	0.5103e-156	8.00
Thukral [ 17 ]	\(1+(\widetilde{\beta }-2(2-\beta ))\theta _n^2\)	Case ii	3	0.1603e-155	8.00
	\( +(\widetilde{\gamma }-2(2-\beta )^2)\theta _n^3\)	Case iv, \(\beta =0\)	3	0.5103e-156	8.00
		Case i	3	0.1915e-234	8.00
Lotfi [ 10 ]	\(1+(\widetilde{\beta }-4)\theta _n^2+(\widetilde{\gamma }-8)\theta _n^3\)	Case ii	3	0.1797e-233	8.00
		Case iv, \(\beta =0\)	3	0.2567e-157	8.00