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<title>On the development and extensions of some classes of optimal three–point iterations for solving nonlinear equations: On the development and extensions of some classes of optimal three–point iterations for solving nonlinear equations</title>
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<h1>On the development and extensions of some classes of optimal three–point iterations for solving nonlinear equations</h1>
<p class="authors">
<span class="author">T. Zhanlav\(^\ast \) Kh. Otgondorj\(^{\ast \ast }\)</span>
</p>
<p class="date">August 20, 2021; accepted: January 4, 2021; published online: February 17, 2022.</p>
</div>
<div class="abstract"><p> 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 <span class="cite">
	[
	<a href="#de14" >20</a>
	]
</span>. Numerical examples are considered to check the convergence order of new families and extensions of some well-known methods. </p>
<p><b class="bf">MSC.</b> 65H05. </p>
<p><b class="bf">Keywords.</b> Multipoint methods; order of convergence; nonlinear equations </p>
</div>
<p>\(^\ast \)Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Mongolia e-mail: <span class="tt">tzhanlav@yahoo.com</span>. </p>
<p>\(^{\ast \ast }\)School of Applied Sciences, Mongolian University of Science and Technology, Mongolia e-mail: <span class="tt">otgondorj@gmail.com</span>. </p>
<h1 id="sec1">1 Introduction</h1>

<p>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 <span class="cite">
	[
	<a href="#de1" >11</a>
	, 
	<a href="#de2" >19</a>
	, 
	<a href="#de3" >17</a>
	, 
	<a href="#de4" >1</a>
	, 
	<a href="#de5" >10</a>
	, 
	<a href="#de6" >3</a>
	, 
	<a href="#de7" >12</a>
	, 
	<a href="#de8" >4</a>
	, 
	<a href="#de9" >2</a>
	, 
	<a href="#de10" >13</a>
	, 
	<a href="#de11" >8</a>
	, 
	<a href="#de12" >14</a>
	, 
	<a href="#de13" >5</a>
	, 
	<a href="#de14" >20</a>
	, 
	<a href="#de15" >21</a>
	, 
	<a href="#de16" >22</a>
	, 
	<a href="#de17" >23</a>
	, 
	<a href="#den16" >6</a>
	]
</span> 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 <span class="cite">
	[
	<a href="#de14" >20</a>
	]
</span>. </p>
<p>We consider the following standard three–point iterative methods: </p>
<div class="displaymath" id="it1.1">
  \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}
</div>
<p> In <span class="cite">
	[
	<a href="#de14" >20</a>
	]
</span> was proven that the order of convergence iterations <a href="#it1.1" class="eqref">1</a> is eight if and only if the parameters \(\bar{\tau }_n\) and \(\alpha _n\) satisfy the conditions </p>
<div class="displaymath" id="it1.2">
  \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}
</div>
<p> and </p>
<div class="equation" id="it1.3">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> where \(~ \upsilon _n=\frac{f(z_n)}{f(y_n)}.\) The optimal methods <a href="#it1.1" class="eqref">1</a> 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 <span class="cite">
	[
	<a href="#de1" >11</a>
	, 
	<a href="#de2" >19</a>
	, 
	<a href="#de3" >17</a>
	, 
	<a href="#de4" >1</a>
	, 
	<a href="#de5" >10</a>
	, 
	<a href="#de6" >3</a>
	, 
	<a href="#de7" >12</a>
	, 
	<a href="#de8" >4</a>
	, 
	<a href="#de9" >2</a>
	, 
	<a href="#de10" >13</a>
	, 
	<a href="#de11" >8</a>
	, 
	<a href="#de12" >14</a>
	, 
	<a href="#de13" >5</a>
	, 
	<a href="#de18" >15</a>
	, 
	<a href="#de19" >18</a>
	, 
	<a href="#de20" >9</a>
	]
</span>, or the nonlinear residuals <span class="cite">
	[
	<a href="#de14" >20</a>
	, 
	<a href="#de15" >21</a>
	, 
	<a href="#de16" >22</a>
	, 
	<a href="#de17" >23</a>
	]
</span>. An more detailed explanation of various aspects of convergence order based on error analysis, corrections and nonlinear residuals was given in the excellent surveys <span class="cite">
	[
	<a href="#den16" >6</a>
	, 
	<a href="#den16666" >7</a>
	]
</span>. 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. </p>
<p>In <a href="#sec2">section 2</a>, we propose new families of optimal three-point methods. In <a href="#sec3">section 3</a>, we suggested extension of classes of optimal eighth-order methods. The numerical experiments and dynamic behavior of methods are discussed in <a href="#sec4">section 4</a>. Finally, short conclusions are included in <a href="#sec5">section 5</a>. </p>
<h1 id="sec2">2 Development of the new families of optimal three–point methods</h1>
<p> First, we consider iterations <a href="#it1.1" class="eqref">1</a> with parameter \(\alpha _n\) given by </p>
<div class="displaymath" id="it1.4">
  \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}
</div>
<p> where </p>
<div class="displaymath" id="it1.5">
  \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}
</div>
<p> To show the convergence analysis of methods <a href="#it1.1" class="eqref">1</a>, <a href="#it1.4" class="eqref">6</a>, the following results is proven. </p>
<p><div class="theorem_thmwrapper theorem-style-plain" id="Dth1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> 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 <a href="#it1.1" class="eqref">1</a> satisfies the condition <a href="#it1.2" class="eqref">2</a>. Then the order of convergence of the methods <a href="#it1.1" class="eqref">1</a>, <a href="#it1.4" class="eqref">6</a> is eight. </p>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000002">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>Using the relations </p>
<div class="displaymath" id="it1.6">
  \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}
</div>
<p> in <a href="#it1.4" class="eqref">6</a>, we obtain </p>
<div class="displaymath" id="it1.7">
  \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}
</div>
<p> Using <a href="#it1.2" class="eqref">2</a> and well-known expansion </p>
<div class="displaymath" id="it1.8">
  \begin{eqnarray} \label{it1.8} \frac{1}{1-x}=1+x+x^2+x^3+\dots ,~  |x|{\lt}1, \end{eqnarray}
</div>
<p> in <a href="#it1.7" class="eqref">9</a> we obtain </p>
<div class="displaymath" id="it1.9">
  \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}
</div>
<p> From <a href="#it1.2" class="eqref">2</a> and <a href="#it1.9" class="eqref">11</a> it follows that \(\alpha _n\) defined by <a href="#it1.9" class="eqref">11</a> satisfies the condition <a href="#it1.3" class="eqref">3</a> that completes the proof of theorem. </p>

  </div>
</div>
<p>Of course, there are many possibility for choice \(\bar{\tau }_n\) in <a href="#it1.1" class="eqref">1</a> satisfying the condition <a href="#it1.2" class="eqref">2</a>. In particular, we give \(\bar{\tau }_n\) as </p>
<div class="displaymath" id="it1.10">
  \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}
</div>
<p> that includes four free parameters. In <a href="#Dtable1">table 1</a>, we list some well-known choices. <div class="table"  id="Dtable1">
    <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">1</span> 
  <span class="caption_text">The choices of parameter \(\bar{\tau }_n\)</span> 
</figcaption> <table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p> Cases </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> Methods </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> Special case of <a href="#it1.10" class="eqref">12</a> </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\bar{\tau }_n\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\widetilde{\beta }\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\widetilde{\gamma }\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>i</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>Potra-Ptack’s</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(c=\omega =1,~ d=b=0\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(1+2\theta _n+\theta _n^2\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>1</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>ii</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>Maheshwari’s</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(c=1,~  b=0,~  d=\omega =-1\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\frac{1+\theta _n-\theta _n^2}{1-\theta _n}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>1</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>1</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>iii</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>Kung-Traub’s</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(c=b=1,~  d=-2,~  \omega =0\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\frac{1}{(1-\theta _n)^2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>4 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-left:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>iv</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>King’s type</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(c=1,~  \omega =b=0, ~ d=\beta -2\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\frac{1+\beta \theta _n}{1+(\beta -2)\theta _n}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(2(2-\beta )\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(2(2-\beta )^2\)</p>

    </td>
  </tr>
</table>
</div> </p>
<p>Note that similar theorem for iteration <a href="#it1.1" class="eqref">1</a>, <a href="#it1.4" class="eqref">6</a> for Kung-Traub’s type iteration were proved by Petković <i class="it">et al.</i> <span class="cite">
	[
	<a href="#de1" >11</a>
	]
</span> and by Zhanlav <i class="it">et al.</i> <span class="cite">
	[
	<a href="#de16" >22</a>
	]
</span> for Kings type iteration and by Wang <i class="it">et al.</i> <span class="cite">
	[
	<a href="#de2" >19</a>
	]
</span> for Ostrowski’s type method. Thus, theorem <a href="#Dth1">1</a> extend essentially the class of families of optimal eight-order iterations <a href="#it1.1" class="eqref">1</a>, <a href="#it1.4" class="eqref">6</a>. Now we consider the iterations <a href="#it1.1" class="eqref">1</a> with \(\alpha _n\) given by </p>
<div class="displaymath" id="it1.30">
  \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}
</div>
<p> 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 <a href="#it1.1" class="eqref">1</a> with \(\alpha _n\) given by <a href="#it1.30" class="eqref">13</a> has optimal eight-order of convergence. Namely we can prove </p>
<p><div class="theorem_thmwrapper theorem-style-plain" id="Dth2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let all assumptions of <a href="#Dth1">theorem 1</a> be fulfilled. Then the order of convergence of the iterations <a href="#it1.1" class="eqref">1</a>, <a href="#it1.30" class="eqref">13</a> is eight when </p>
<div class="displaymath" id="it1.31">
  \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}
</div>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000003">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>The proof is the same as that of <a href="#Dth1">theorem 1</a>. For convenience, here we only give the main step of proof. As before, using <a href="#it1.6" class="eqref">8</a> and <a href="#it1.8" class="eqref">10</a> after some manipulations we obtain </p>
<div class="equation" id="it1.32">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> Substituting <a href="#it1.32" class="eqref">15</a> into <a href="#it1.30" class="eqref">13</a> and comparing <a href="#it1.30" class="eqref">13</a> with <a href="#it1.3" class="eqref">3</a> we arrive at <a href="#it1.31" class="eqref">14</a>. </p>

  </div>
</div>
<p> The expression in the numerator of <a href="#it1.30" class="eqref">13</a> 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 </p>
<div class="displaymath" id="it1.n41">
  \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}
</div>
<p> and </p>
<div class="equation" id="it1.n42">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p> in <a href="#it1.30" class="eqref">13</a> we obtain </p>
<div class="displaymath" id="it1.n43">
  \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}
</div>
<p> where </p>
<div class="displaymath" id="a0000000004">
  \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*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000005">
  \begin{eqnarray*} F_n=(1-\omega _2)(f[x_n,y_n]-f[y_n,z_n]). \end{eqnarray*}
</div>
<p> From <a href="#it1.30" class="eqref">13</a>, <a href="#it1.31" class="eqref">14</a> we see that \(\alpha _n\) includes two free parameters \(\omega _1\) and \(\omega _2\). Thus, we develop the class of optimal eight-order iterations <a href="#it1.1" class="eqref">1</a>, <a href="#it1.30" class="eqref">13</a>, <a href="#it1.31" class="eqref">14</a>. We consider some choices of parameters \(\omega _1\) and \(\omega _2\). </p>
<ol class="enumerate">
  <li><p>Let \(\omega _1= \omega _3=0\), \(\omega _2=1\). Then <a href="#it1.n43" class="eqref">25</a> converted to </p>
<div class="displaymath" id="it1.33">
  \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}
</div>
</li>
  <li><p>Let \(\omega _1=1\), \(\omega _2= \omega _3=0\). Then <a href="#it1.n43" class="eqref">25</a> converted to </p>
<div class="displaymath" id="it1.34">
  \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}
</div>
</li>
  <li><p>Let \(\omega _1= \omega _2=0\), \(\omega _3=1\). Then <a href="#it1.n43" class="eqref">25</a> converted to </p>
<div class="displaymath" id="it1.35">
  \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}
</div>
</li>
  <li><p>Let \(\omega _1=-1\), \( \omega _2=2\), \(\omega _3=0\). Then <a href="#it1.n43" class="eqref">25</a> converted to </p>
<div class="displaymath" id="it1.36">
  \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}
</div>
<p> The iteration <a href="#it1.1" class="eqref">1</a>, <a href="#it1.36" class="eqref">29</a> can be considered as another variant of iterations given by Sharma <i class="it">et al.</i> in <span class="cite">
	[
	<a href="#de7" >12</a>
	, 
	<a href="#de10" >13</a>
	, 
	<a href="#de12" >14</a>
	]
</span> and given by Zhanlav <i class="it">et al.</i> in <span class="cite">
	[
	<a href="#de17" >23</a>
	]
</span>. </p>
</li>
  <li><p>Let \(\omega _1=\omega _2=1\), \(\omega _3=-1\). Then <a href="#it1.n43" class="eqref">25</a> converted to </p>
<div class="displaymath" id="it1.377">
  \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}
</div>
<p>It can be rewritten as: </p>
<div class="displaymath" id="a0000000006">
  \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*}
</div>
<p>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 <span class="cite">
	[
	<a href="#de19" >18</a>
	]
</span> and by Khattri <i class="it">et al.</i> in <span class="cite">
	[
	<a href="#de20" >9</a>
	]
</span>. We also note that the iteration <a href="#it1.1" class="eqref">1</a>, <a href="#it1.377" class="eqref">30</a> for \(\widetilde{\beta }=4\) was considered by Sharma <i class="it">et al.</i> in <span class="cite">
	[
	<a href="#de18" >15</a>
	]
</span>. </p>
</li>
  <li><p>Let \(\omega _1=-1\), \(\omega _2=\omega _3=1\). Then <a href="#it1.n43" class="eqref">25</a> converted to </p>
<div class="displaymath" id="it1.38">
  \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}
</div>
</li>
</ol>
<p>In each iteration step the methods <a href="#it1.1" class="eqref">1</a>, <a href="#it1.4" class="eqref">6</a> and <a href="#it1.1" class="eqref">1</a>, <a href="#it1.30" class="eqref">13</a> 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 <a href="#it1.1" class="eqref">1</a>, <a href="#it1.4" class="eqref">6</a> and <a href="#it1.1" class="eqref">1</a>, <a href="#it1.30" class="eqref">13</a> is that they work well for any choice of parameter \(\bar{\tau }_n\) satisfying the condition <a href="#it1.2" class="eqref">2</a>. </p>
<h1 id="sec3">3 Extensions of some classes of optimal eight-order methods</h1>

<p>Now we consider the iterations <a href="#it1.1" class="eqref">1</a> with parameter \(\alpha _n\) given by </p>
<div class="displaymath" id="it1.11">
  \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}
</div>
<p> 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 \(t\).</p>
<p>Namely, we have <div class="theorem_thmwrapper theorem-style-plain" id="Dth3">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let all assumptions of <a href="#Dth1">theorem 1</a> be fulfilled. Then the order of convergence of the iterations <a href="#it1.1" class="eqref">1</a> and <a href="#it1.11" class="eqref">32</a> is eight when </p>
<div class="displaymath" id="it1.12">
  \begin{eqnarray} \label{it1.12} p(0)=1,~ p’(0)=2, ~ \hat{\gamma }=2(\widetilde{\beta }-5). \end{eqnarray}
</div>

  </div>
</div> </p>
<div class="proof_wrapper" id="a0000000007">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  <p>We denote the second factor in <a href="#it1.11" class="eqref">32</a> by \(\hat{ \alpha }_n\). That is </p>
<div class="displaymath" id="it1.13">
  \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}
</div>
<p> Then using the relations <a href="#it1.6" class="eqref">8</a> we obtain </p>
<div class="displaymath" id="it1.14">
  \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}
</div>
<p> Using the expansion <a href="#it1.8" class="eqref">10</a> in <a href="#it1.14" class="eqref">35</a> and taking into account \(\upsilon _n={\mathcal O}(f(x_n)^2)\), \(\theta _n={\mathcal O}(f(x_n))\), we obtain </p>
<div class="displaymath" id="it1.15">
  \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}
</div>
<p> By using <a href="#it1.2" class="eqref">2</a> and <a href="#it1.8" class="eqref">10</a> it is easy to show that </p>
<div class="displaymath" id="it1.16">
  \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}
</div>
<p> Substituting <a href="#it1.2" class="eqref">2</a> and <a href="#it1.16" class="eqref">37</a> into <a href="#it1.15" class="eqref">36</a> we get </p>
<div class="displaymath" id="it1.17">
  \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}
</div>
<p> Then <a href="#it1.11" class="eqref">32</a> is written as </p>
<div class="equation" id="it1.18">
<p>
  <div class="equation_content">
    \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}
  </div>
  <span class="equation_label">39</span>
</p>
</div>
<p> which satisfies the condition <a href="#it1.3" class="eqref">3</a> provided that <a href="#it1.12" class="eqref">33</a>. </p>

  </div>
</div>
<p> Thus, we develop the family of optimal three-point iterative methods <a href="#it1.1" class="eqref">1</a> with \( \alpha _n\) given by </p>
<div class="displaymath" id="it1.19">
  \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}
</div>
<p> Similar results were obtained in <span class="cite">
	[
	<a href="#de6" >3</a>
	, 
	<a href="#de9" >2</a>
	]
</span> for the iterations <a href="#it1.1" class="eqref">1</a> with </p>
<div class="displaymath" id="it1.20">
  \begin{eqnarray} \label{it1.20} \bar{\tau }_n=\frac{1-\theta _n/2}{1-5\theta _n/2}, \end{eqnarray}
</div>
<p> and </p>
<div class="displaymath" id="it1.21">
  \begin{eqnarray} \label{it1.21} \bar{\tau }_n=\frac{1}{1-2\theta _n-\theta _n^2-\theta _n^3/2}, \end{eqnarray}
</div>
<p> respectively. The parameters \(\bar{\tau }_n\) given by <a href="#it1.20" class="eqref">43</a> and <a href="#it1.21" class="eqref">44</a> satisfy the condition <a href="#it1.2" class="eqref">2</a> with \(\widetilde{\beta }=5\). In this case \(\hat{\gamma }=0\) by <a href="#it1.12" class="eqref">33</a> and the \( \alpha _n\) given by <a href="#it1.11" class="eqref">32</a> leads to </p>
<div class="displaymath" id="it1.22">
  \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}
</div>
<p> This means that our iterations <a href="#it1.1" class="eqref">1</a> and <a href="#it1.19" class="eqref">42</a> include the iterations proposed by Bi <i class="it">et al.</i> <span class="cite">
	[
	<a href="#de9" >2</a>
	]
</span> and by Cordero <i class="it">et al.</i> <span class="cite">
	[
	<a href="#de6" >3</a>
	]
</span> as particular cases. </p>
<p>Now we consider the expression </p>
<div class="displaymath" id="it1.23">
  \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}
</div>
<p> As before, using the relations <a href="#it1.6" class="eqref">8</a> in <a href="#it1.23" class="eqref">46</a> we obtain </p>
<div class="displaymath" id="it1.24">
  \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}
</div>
<p> By <a href="#it1.2" class="eqref">2</a> one can easily to check that </p>
<div class="displaymath" id="it1.25">
  \begin{eqnarray} \label{it1.25} \tilde{\alpha }_n=\hat{ \alpha }_n +{\mathcal O}(f(x_n)^4). \end{eqnarray}
</div>
<p> It means that instead of <a href="#it1.11" class="eqref">32</a> one can also use </p>
<div class="displaymath" id="it1.26">
  \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}
</div>
<p> Therefore, <a href="#Dth3">theorem 3</a> holds true for iterations <a href="#it1.1" class="eqref">1</a>, <a href="#it1.26" class="eqref">49</a>. Similar extension can be done for all optimal eight-order iterations. As examples, we present in <a href="#Dtable2">table 2</a> some of methods and their extension \(\tilde{\alpha }_n=\xi _n\cdot \alpha _n\) with extension factor \(\xi _n\). <br />=== <div class="table"  id="Dtable2">
  <div class="centered"><div class="centered"><figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">2</span> 
  <span class="caption_text">The extraneous fixed points</span> 
</figcaption>  <small class="small">
</small></div></div>
</div> <br /><p>=== </p>
<p>Thus, we obtain extensions of some well-known optimal methods that work well for any suitable parameter \(\bar{\tau }_n\), satisfying the condition <a href="#it1.2" class="eqref">2</a>. This allows us to expand the applicability of the original methods. </p>
</p>
<h1 id="sec4">4 Numerical experiments </h1>
<p>  In order to show the convergence behavior and to check the validity of theoretical results of the presented family (<a href="#it1.1">1</a>) 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 <span class="cite">
	[
	<a href="#de10" >13</a>
	]
</span>, <span class="cite">
	[
	<a href="#de12" >14</a>
	]
</span> and <span class="cite">
	[
	<a href="#de17" >23</a>
	]
</span> 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. </p>
<div class="displaymath" id="a0000000008">
  \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*}
</div>
<p> In Tables <a href="#Dnumtab3">3</a>–<a href="#Dnumtab5">5</a>, we present the necessary iterations \((n)\), absolute error \(|x_n-x^*|\) and computational order of convergence, which is calculated by the following formula <span class="cite">
	[
	<a href="#de1" >11</a>
	, 
	<a href="#de20222" >16</a>
	]
</span>: </p>
<div class="displaymath" id="a0000000009">
  \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*}
</div>
<p>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 <span class="cite">
	[
	<a href="#den16" >6</a>
	, 
	<a href="#den16666" >7</a>
	]
</span>. 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 <a href="#Dtable1">table 1</a>. <a href="#Dnumtab3">Table 3</a> gives some numerical results in order to show convergence behaviour of method <a href="#it1.1" class="eqref">1</a> with \(\alpha _n\) parameter given by (<a href="#it1.4">6</a>),&#160;(<a href="#it1.33">26</a>)–(<a href="#it1.38">31</a>). We observe from <a href="#Dnumtab3">table 3</a> that the methods <a href="#it1.1" class="eqref">1</a> with parameters \(\bar{\tau }_n\) given by case iv and \(\alpha _n\) given by <a href="#it1.4" class="eqref">6</a>, <a href="#it1.377" class="eqref">30</a> produce approximations of higher accuracy compared to the eight-order methods SAWN8, ZO8. </p>
<p>The results corresponding to the same kind of experiments for the extension of methods can be found in Table <a href="#Dnumtab4">4</a>-<a href="#Dnumtab5">5</a>. In <a href="#Dnumtab4">table 4</a>, we present the numerical results of iteration <a href="#it1.1" class="eqref">1</a> with parameter \(\alpha _n\) given <a href="#it1.11" class="eqref">32</a> and <a href="#it1.26" class="eqref">49</a>, in which we used function \(p(t)=\frac{1}{(1-t)^2}\). In <a href="#Dnumtab5">table 5</a>, we present the numerical results of the extension of some methods that work well any parameters \(\bar{\tau }_n\) satisfies condition <a href="#it1.2" class="eqref">2</a>. </p>
<p>From the results displayed in Table <a href="#Dnumtab3">3</a>–<a href="#Dnumtab5">5</a>, we see that the calculated values of the computational order of convergence are in complete agreement with the theoretical orders proved in Section <a href="#sec2">2</a>, <a href="#sec3">3</a>. </p>
<p>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 (<a href="#it1.1">1</a>) for cubic polynomial \(p(z)=z^3-1\) having three simple zeros. </p>
<div class="table"  id="Dnumtab3">
  <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">3</span> 
  <span class="caption_text">The numerical result for \(f_i(x)\) by the methods <a href="#it1.1" class="eqref">1</a> with \(\bar{\tau }_n\) and \(\alpha _n\) </span> 
</figcaption>  <small class="small"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> \(\alpha _n\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\(\bar{\tau }_n\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\(n\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\(|x^*-x_n|\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>\(\rho \) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \(n\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \(|x^*-x_n|\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> \(\rho \) </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="2">
      <p>\(f_1(x),\quad x_0=0.5\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="2">
      <p> \(f_2(x),\quad x_0=2\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.5407e-222 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1220e-231 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.4264e-222 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.4412e-189 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>(<a href="#it1.4">6</a>) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2180e-233 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2054e-245 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2111e-226 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.4836e-229 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.5182e-200 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3202e-231 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.7800e-203 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 2 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.6607e-31 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>(<a href="#it1.33">26</a>) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.4084e-132 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.2183e-205 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.5805e-127 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.3391e-235 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3401e-122 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3173e-229 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.5182e-200 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3976e-247 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>(<a href="#it1.34">27</a>) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.4084e-132 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.9235e-207 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.5805e-127 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.1166e-232 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.8671e-118 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2475e-221 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1363e-116 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3905e-241 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>(<a href="#it1.35">28</a>) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.1213e-121 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.2152e-202 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.3621e-119 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.1296e-226 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1024e-139 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1307e-234 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.4756e-142 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 2 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.7314e-33 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.93 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>(<a href="#it1.36">29</a>) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1560e-146 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3104e-204 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.5207e-153 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1715e-172 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2195e-222 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.4879e-248 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.4844e-243 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 2 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.9301e-33 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.97 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>(<a href="#it1.377">30</a>) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2720e-219 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.8760e-212 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.8966e-220 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.4439e-247 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1236e-187 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1047e-221 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.9877e-182 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1326e-245 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>(<a href="#it1.38">31</a>) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1077e-184 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.9069e-201 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iv\(, \beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.1073e-184 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.1044e-227 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>SAWN8 <span class="cite">
	[
	<a href="#de12" >14</a>
	]
</span> </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.5207e-153 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1715e-172 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>ZO8 <span class="cite">
	[
	<a href="#de17" >23</a>
	]
</span> </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>3 </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>0.1036e-137 </p>

    </td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>0.3317e-178 </p>

    </td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
</table> </small> 
</div>
<div class="table"  id="Dnumtab4">
  <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">4</span> 
  <span class="caption_text">The numerical result for \(f_i(x)\) by the methods <a href="#it1.1" class="eqref">1</a> with \(\bar{\tau }_n\) and \(\alpha _n\) </span> 
</figcaption>  <div class="centered"><small class="small"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> \(\alpha _n\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\(\bar{\tau }_n\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\(n\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\(|x^*-x_n|\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>\(\rho \)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>\(n\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>\(|x^*-x_n|\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>\(\rho \) </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="2">
      <p>\(f_1(x),\quad x_0=0.5\) </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="2">
      <p> \(f_2(x),\quad x_0=2\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.6992e-190 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.2459e-223 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>(<a href="#it1.11">32</a>) </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.1915e-190 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.3996e-244 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2945e-186 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.1124e-181 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.1334e-189 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.8535e-221 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3186e-194 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.1093e-200 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>(<a href="#it1.26">49</a>) </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>0.1273e-194 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.1051e-217 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> case iii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2945e-186 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.3833e-158 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> 0.6747e-194 </p>

    </td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> 0.5685e-198 </p>

    </td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
</table> </small> </div>
</div>
<div class="table"  id="Dnumtab5">
  <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">5</span> 
  <span class="caption_text"> The numerical results of extension of methods for \(f_2(x)\) </span> 
</figcaption>  <div class="centered"><small class="small"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>  Methods</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>Extension factor \(\xi _n\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\(\bar{\tau }_n\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p>\(n\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>\(|x^*-x_n|\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>\(\rho \) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> Case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.3159e-214 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>Sharma <span class="cite">
	[
	<a href="#de7" >12</a>
	]
</span> </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\(1+(\widetilde{\beta }-4)\theta _n^3\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.3453e-212 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.8671e-210 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> Case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.2781e-229 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>DP8 <span class="cite">
	[
	<a href="#de1" >11</a>
	]
</span> </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\(1+(\widetilde{\beta }-4)\theta _n^2+(\widetilde{\gamma }-8)\theta _n^3\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.1848e-238 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case iv, \(\beta =0\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.1196e-162 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> Case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.2433e-214 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>GK8 <span class="cite">
	[
	<a href="#de11" >8</a>
	]
</span> </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\(1+(\widetilde{\beta }-3)\theta _n^2+(\widetilde{\gamma }-6)\theta _n^3\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.5841e-236 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>Case iv, \(\beta =1\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.6505e-172 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>7.99 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> Case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.2100e-114 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>Chun <span class="cite">
	[
	<a href="#de8" >4</a>
	]
</span> </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\(1+(\widetilde{\beta }-3)\theta _n^2+(\widetilde{\gamma }-4)\theta _n^3\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.2547e-114 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case iii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.7827e-120 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> Case i </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.5103e-156 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>Thukral <span class="cite">
	[
	<a href="#de3" >17</a>
	]
</span></p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\(1+(\widetilde{\beta }-2(2-\beta ))\theta _n^2\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case ii </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.1603e-155 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\( +(\widetilde{\gamma }-2(2-\beta )^2)\theta _n^3\)</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.5103e-156 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> Case i</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.1915e-234 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>Lotfi <span class="cite">
	[
	<a href="#de5" >10</a>
	]
</span></p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>\(1+(\widetilde{\beta }-4)\theta _n^2+(\widetilde{\gamma }-8)\theta _n^3\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case ii</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p> 0.1797e-233 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> Case iv, \(\beta =0\) </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 3 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.2567e-157 </p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>8.00 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px; text-align:left" 
        rowspan=""
        colspan="">&nbsp;</td>
  </tr>
</table> </small> </div>
</div>
<p>In Figure <a href="#fig1:images1">1</a>–<a href="#fig3:images3">3</a>, 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. </p>
<p>Based on Figure <a href="#fig1:images1">1</a>–<a href="#fig3:images3">3</a> for \(p(z)\), we can see that the method (<a href="#it1.1">1</a>) with \(\bar{\tau }_n\) given by case iii and \(\alpha _n\) given by <a href="#it1.4" class="eqref">6</a> is the best one and have fewer diverging points that other cases of parameters. </p>
<figure id="fig1:images1">
  <div class="centered"><figure class="subfigure" id="f1:grf">
 <img src="img-0001.png" alt="\includegraphics[width=0.27\textwidth ]{m1-2-4.png}" style="width: 360px; height: 359px" />

<figcaption>
<span class="subcaption">(1)-ii-(4)</span>
</figcaption>
</figure>  <figure class="subfigure" id="f1:n16mu2m1">
 <img src="img-0002.png" alt="\includegraphics[width=0.27\textwidth ]{m1-3-4.png}" style="width: 360px; height: 359px" />

<figcaption>
<span class="subcaption">(1)-iii-(4)</span>
</figcaption>
</figure>  <figure class="subfigure" id="f1:n16mu2m2">
 <img src="img-0003.png" alt="\includegraphics[width=0.27\textwidth ]{m1-4-4.png}" style="width: 360px; height: 359px" />

<figcaption>
<span class="subcaption">(1)-iv-(4)</span>
</figcaption>
</figure> <figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">1</span> 
  <span class="caption_text">Basins of attraction of methods for \(z^3-1\).</span> 
</figcaption>  </div>

</figure>
<figure id="fig2:images2">
  <div class="centered"><figure class="subfigure" id="f2:grf">
 <img src="img-0004.png" alt="\includegraphics[width=0.27\textwidth ]{m1-4-17.png}" style="width: 360px; height: 359px" />

<figcaption>
<span class="subcaption">(1)-iv-(17)</span>
</figcaption>
</figure>  <figure class="subfigure" id="f2:n16mu2m1">
 <img src="img-0005.png" alt="\includegraphics[width=0.27\textwidth ]{m1-4-18.png}" style="width: 360px; height: 359px" />

<figcaption>
<span class="subcaption">(1)-iv-(18)</span>
</figcaption>
</figure>  <figure class="subfigure" id="f2:n16mu2m2">
 <img src="img-0006.png" alt="\includegraphics[width=0.27\textwidth ]{m1-3-20.png}" style="width: 360px; height: 359px" />

<figcaption>
<span class="subcaption">(1)-iii-(20)</span>
</figcaption>
</figure> <figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">2</span> 
  <span class="caption_text">Basins of attraction of methods for \(z^3-1\).</span> 
</figcaption>  </div>

</figure>
<figure id="fig3:images3">
  <div class="centered"><figure class="subfigure" id="f3:grf">
 <img src="img-0007.png" alt="\includegraphics[width=0.27\textwidth ]{m1-4-20.png}" style="width: 360px; height: 359px" />

<figcaption>
<span class="subcaption">(1)-iv-(20)</span>
</figcaption>
</figure>  <figure class="subfigure" id="f3:n16mu2m1">
 <img src="img-0008.png" alt="\includegraphics[width=0.27\textwidth ]{m1-4-21.png}" style="width: 360px; height: 359px" />

<figcaption>
<span class="subcaption">(1)-iv-(21)</span>
</figcaption>
</figure>  <figure class="subfigure" id="f3:n16mu2m2">
 <img src="img-0009.png" alt="\includegraphics[width=0.27\textwidth ]{m1-4-31.png}" style="width: 360px; height: 359px" />

<figcaption>
<span class="subcaption">(1)-iv-(31)</span>
</figcaption>
</figure> <figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">3</span> 
  <span class="caption_text">Basins of attraction of methods for \(z^3-1\).</span> 
</figcaption>  </div>

</figure>
<h1 id="sec5">5 Conclusion</h1>
<p> The main contributions of this work are: </p>
<p>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 <a href="#it1.2" class="eqref">2</a>. </p>
<p>The proposed iterative methods can be regarded as an advancement in the topic and can compete with other well-known methods. </p>
<p><div class="acknowledgement_thmwrapper theorem-style-remark" id="a0000000010">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors wish to thank the anonymous referees for their valuable suggestions and comments, which improved paper. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
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</dd>
  <dt><a name="de17">23</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.5923/j.ajcam.20180801.02"> <i class="sc">T. Zhanlav, Kh. Otgondorj</i>, <i class="it">A new family of optimal eighth-order methods for solving nonlinear equations</i>, Amer. J. Comput. Appl. Math., <b class="bf">8</b> (2018), 15–19, <a href="http://dx.doi.org/10.5923/j.ajcam.20180801.02">http://dx.doi.org/10.5923/j.ajcam.20180801.02</a>. <img src="img-0010.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width: 120px; height: 107px" />
</a> </p>
</dd>
</dl>


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