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<title>A two-point eighth-order method based on the weight function for solving nonlinear equations: A two-point eighth-order method based on the weight function for solving nonlinear equations</title>
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<h1>A two-point eighth-order method based on the weight function for solving nonlinear equations</h1>
<p class="authors">
<span class="author">Vali Torkashvand\(^{\ast \,  1}\)</span>
</p>
<p class="date">February 7, 2020; accepted: August 31, 2021; published online: November 8, 2021.</p>
</div>
<div class="abstract"><p> In this work, we have designed a family of with-memory methods with eighth-order convergence. We have used the weight function technique. The proposed methods have three parameters. Three self-accelerating parameters are calculated in each iterative step employing only information from the current and all previous iteration. Numerical experiments are carried out to demonstrate the convergence and the efficiency of our iterative method. </p>
<p><b class="bf">MSC.</b> 65H05, 65B99. </p>
<p><b class="bf">Keywords.</b> Method with memory; Accelerator parameter; Weight function; Newton’s interpolatory polynomial; Order of convergence. </p>
</div>
<p>\(^\ast \)Young Researchers and Elite Club, Shahr-e-Qods Branch, Islamic Azad University, e-mail: <span class="tt">torkashvand1978@gmail.com</span>. </p>
<p>\(^1\)Department of Mathematics, Farhangian University, Tehran, Iran. </p>
<h1 id="intro">1 Introduction</h1>

<p>Nonlinearity is of interest to physicists and mathematicians, since most physical systems are inherently nonlinear in nature. One of the most important problem in computational mathematics is solving nonlinear equations. For example, nonlinear optimization aims to find a minimum or maximum of a given nonlinear function. Nonlinear equations are difficult to solve in general. The best way to solve these equations is using iterative methods. One of the classical method to solve nonlinear equation is Newton’s method which has convergence order equal to 2. It can be said that the Secant method is the oldest with memory methods that have been studied so far. The Secant method obtain by approximating the derivative in Newton’s method <i class="it">via</i> a finite divided difference \(f'(x_{k})=\tfrac {f (x_{k-1})- f (x_{k})}{x_{k-1}-x_{k}} \). </p>
<p>The method is given as </p>
<div class="equation" id="f1">
<p>
  <div class="equation_content">
    \begin{equation} \label{f1} x_{k+1}=x_{k}-\frac{x_{k-1}-x_{k}}{f (x_{k-1})- f (x_{k})}f(x_{k}). \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> In the continuation of this work, we will first define the efficiency index (EI) of an iterative method by Ostrowski <span class="cite">
	[
	<a href="#Os" >22</a>
	]
</span>: </p>
<div class="equation" id="a0000000002">
<p>
  <div class="equation_content">
    \begin{equation}  EI=r^{\frac{1}{\theta _{f}}} \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> The Q-order of convergence \(r\) and the number of function evaluations \(\theta _{f}\) per iteration. The efficiency index of the Secant method is 1.6803. Traub in <span class="cite">
	[
	<a href="#Traub" >31</a>
	]
</span> proposed the following with-memory method (TM) </p>
<div class="equation" id="f2">
<p>
  <div class="equation_content">
    \begin{equation} \label{f2} \left\{  \begin{array}{ll} \gamma _{k}=-\frac{1}{f[x_{k},x_{k-1}]}, \quad k=1,2,3,\ldots ,\\[3mm] w_{k}=x_{k}+\gamma f(x_{k}),\quad x_{k+1}=x_{k}-\frac{f(x_{k})}{f[x_{k},w_{k}]}, \quad k=0,1,2,\ldots , \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> with the order of convergence \(2.41421\). In the following, Neta proposed (NM) three step with memory method which has the order of convergence 10.81525 <span class="cite">
	[
	<a href="#Neta" >20</a>
	]
</span>: </p>
<div class="equation" id="f3">
<p>
  <div class="equation_content">
    \begin{equation} \label{f3} \left\{  \begin{array}{ll} w_{k}=x_{k}+\tfrac {f(x_{k})}{f’(x_{k})}(f(w_{k-1})\phi _{z}-f(z_{k-1})\phi _{w})(\tfrac {f(x_{k})^{2}}{f(w_{k-1}-f(z_{k-1})}), \quad k=1,2,\ldots ,\\[3mm] z_{k}=x_{k}+\tfrac {f(x_{k})}{f’(x_{k})}(f(w_{k})\phi _{z}-f(z_{k-1})\psi _{w})(\tfrac {f(x_{k})^{2}}{f(w_{k}-f(z_{k-1})}), \quad k=1,2,3,\ldots ,\\[3mm] x_{k+1}=x_{k}+\tfrac {f(x_{k})}{f’(x_{k})}(f(w_{k})\psi _{z}-f(z_{k})\psi _{w})(\tfrac {f(x_{k})^{2}}{f(w_{k}-f(z_{k})}), \quad k=0,1,2,\ldots \\ \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> He used inverse interpolation. Neta increased the convergence order from 8 to 10.81. Therefore, he has improved the degree of convergence by \(35\% \). Also, Traub improved by \(20.71\% \) by increasing the degree of convergence from 2 to 2.41. Bassiri <i class="it">et al.</i> <span class="cite">
	[
	<a href="#Bassiri" >2</a>
	]
</span> also increased the degree of convergence of a two-step method from 4 to 7.22. Therefore, the convergence order improvement of their proposed method is \(80\% \). </p>
<p>The remaining materials of this paper are uncovered as follows. Section 2 is devoted to modifications of the two-steps method proposed by Bassiri <br /><i class="it">et al.</i> <span class="cite">
	[
	<a href="#Bassiri" >2</a>
	]
</span>. Further accelerations of convergence speed are attained in Section&#160;3. This self-accelerating parameter is calculated by the Newton interpolating polynomial. The corresponding Q-order of convergence <span class="cite">
	[
	<a href="#Cat" >8</a>
	]
</span> is increased from 4 to \(7.53113, 7.94449, 7.99315\) and \(7.99915\approx 8\). </p>
<p>Numerical examples are given in Section 4 to confirm theoretical results. </p>
<p>Finally, Section 5 is devoted to the main conclusions of this work. </p>
<h1 id="a0000000003">2 Without memory methods</h1>
<p> Bassiri <i class="it">et al.</i> proposed the following optimal iterative without memory method <span class="cite">
	[
	<a href="#Bassiri" >2</a>
	]
</span>: </p>
<div class="equation" id="f4">
<p>
  <div class="equation_content">
    \begin{equation} \label{f4} \left\{  \begin{array}{ll} \! \! \! w_{k}=x_{k}+\gamma f(x_{k}),\quad y_{k}=x_{k}-\tfrac {f(x_{k})}{f[x_{k},w_{k}]+\beta f(w_{k})}, \quad k=0,1,\ldots , \\[3mm] \! \! \! s_{k}=\tfrac {f(y_{k})}{f(x_{k})}, \quad x_{k+1}=y_{k}-H(s_{k})\tfrac {f(y_{k})}{f[w_{k},y_{k}]+\beta f(w_{k})+\lambda (y_{k}-x_{k})(y_{k}-w_{k})}, \, k=0,1,\ldots \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> This method achieves order convergence 4 when the weight functions satisfy the conditions </p>
<div class="equation" id="eq1">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq1} H(0)=H'(0)=H''(0)=1. \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> And its error expression is </p>
<div class="displaymath" id="eq2">
  \begin{align} \label{eq2} e_{k+1}\! =& ((1\! +\! \gamma f’(\alpha ))^{2}(\beta \! +\! c_{2})(2\lambda \! +\! f’(\alpha )\beta ^{2}(1\! +\! \gamma f’(\alpha ))\! +\! f’(\alpha )c_{2}(2\beta (3\! +\! \gamma f’(\alpha )) \nonumber \\*[1mm]&  +(5+\gamma f’(\alpha ))c_{2})-2f’(\alpha )c_{3}))(-2f’(\alpha ))^{-1}e^{4}_{k}+{\mathcal O} (e^{5}_{k}). \end{align}
</div>
<p> where \(c_{k}=\tfrac {f^{(k)}(\alpha )}{k!f’(\alpha )}\) for \(k=2,3,\ldots \) If the weight function is not used, the order convergence of method (<a href="#f4">5</a>) will be as follows: </p>
<div class="equation" id="eq3">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq3} e_{k+1}=((1+\gamma f'(\alpha ))^{2}(\beta +c_{2})^{2}e_{k}^{3}+{\mathcal O} (e_{k}^{4}). \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> In this case, the optimality of method (<a href="#f4">5</a>) disappears. For maintaining optimality, it must be four until it reaches an optimal without-memory method by according to Kung-Traub’s conjecture <span class="cite">
	[
	<a href="#Kung" >12</a>
	]
</span>. One way to increase the degree of convergence is by using the weight function. Refer to <span class="cite">
	[
	<a href="#Behl" >3</a>
	, 
	<a href="#Chun1" >5</a>
	, 
	<a href="#Cordero1" >6</a>
	, 
	<a href="#Geum" >10</a>
	, 
	<a href="#Lotfi1" >13</a>
	, 
	<a href="#Lotfi2" >14</a>
	, 
	<a href="#Lotfi3" >15</a>
	, 
	<a href="#Sharifi" >25</a>
	, 
	<a href="#Soleymani1" >26</a>
	]
</span> for further study. Some concrete weight functions that satisfy the conditions are </p>
<div class="equation" id="eq4">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq4} H_{1}(s)=1+s+\tfrac {s^{2}}{2}, \quad H_{2}(s)=e^{s},\quad H_{3}(s)=\tfrac {2+s}{2-s}, \quad H_{4}(s)=\tfrac {1}{1-s-\frac{s^{2}}{2}}. \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p> Here it should be noted that under the functions of weight \(H_{1}(s), H_{2}(s)\) and \(H_{3}(s)\) the error equation is (<a href="#eq2">7</a>). But the error equation of this method for the weight function \(H_{4}(s)\) is as follows: </p>
<div class="displaymath" id="eq5">
  \begin{eqnarray} \label{eq5} e_{k+1}& =& ((1+\gamma f’(\alpha ))^{2}(\beta +c_{2})(f’(\alpha )\beta ^{2}(1+\gamma f’(\alpha ))-2\lambda +f’(\alpha )c_{2}(2\beta (-1+\nonumber \\*[1mm]& & \gamma f’(\alpha )) +(-3+\gamma f’(\alpha ))c_{2})+2f’(\alpha )c_{3}))(-2f’(\alpha ))^{-1}e^{4}_{k}+{\mathcal O}(e^{5}_{k}). \end{eqnarray}
</div>
<p>It is also necessary to note that any weight function that applies only in conditions \(H(0)=1\) and \(H'(0)=1\) will converge to convergence order 4. In addition, weight function \(H_{4}(s)\) does not apply in terms of (<a href="#eq1">6</a>), and \(H''_{4}(0)=3\). </p>
<p>Error relations (<a href="#eq5">9</a>) plays the key role in our study of the convergence acceleration. For method (<a href="#f4">5</a>), we have the following result. </p>
<p><div class="theorem_thmwrapper " id="thm1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For a sufficiently good initial approximation \(x_{0}\) of a simple zero \(\alpha \) of the function \(f\), the family of two-point methods <a href="#f4" class="eqref">5</a> obtains the order at least four if the weight function \(H\) satisfies conditions <a href="#eq1" class="eqref">6</a> Then the error relation for the family <a href="#f4" class="eqref">5</a> is given by <a href="#eq5" class="eqref">9</a>. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000004">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> By using Taylor’s expansion of \(f(x)\) about \(\alpha \) and taking into account that \(f(\alpha )=0\), we obtain </p>
<div class="equation" id="eq6">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq6} f(x_{k})=f'(\alpha )(e_{k}+c_{2}e_{k}^{2}+c_{3}e_{k}^{3}+c_{4}e_{k}^{4}+{\mathcal O} (e_{k}^{5})). \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> Then, computing \(e_{k,w}=w_{k}-\alpha \), we attain \(w_{k}=x_{k}+\gamma f(x_{k})\) </p>
<div class="equation" id="eq7">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq7} {e}_{k,w}=f'(\alpha )(1+\gamma f'(\alpha ))e_{k}+\gamma f'(\alpha )c_{2}e_{k}^{2}+\gamma f'(\alpha )c_{3}e_{k}^{3}+\gamma f'(\alpha )c_{4}e_{k}^{4}+{\mathcal O} (e_{k}^{5}), \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p> and </p>
<div class="displaymath" id="eq8">
  \begin{eqnarray} \label{eq8} f(w_{k})& =& f’(\alpha )(1+\gamma f’(\alpha ))e_{k}+f’(\alpha )(1+\gamma f’(\alpha )(3+\lambda f’(\alpha )))c_{2}e_{k}^{2}\nonumber \\ & & +f’(\alpha )(2\gamma f’(\alpha )(1+\gamma f’(\alpha ))c_{2}^{2}+\gamma f’(\alpha )c_{3}+(1+\gamma f’(\alpha ))^{3}\nonumber \\ & & c_{3}) e_{k}^{3}+f’(\alpha )(c_{4}+\gamma f’(\alpha )(\gamma f’(\alpha ) c_{2}^{3}+(1+\gamma f’(\alpha ))(5+3\gamma f’(\alpha ))\nonumber \\ & &  c_{2}c_{3}+(5+\gamma f’(\alpha )(6+\gamma f’(\alpha ) (4+\gamma f’(\alpha )))) c_{4}))e_{k}^{4}+{\mathcal O} (e_{k}^{5}). \end{eqnarray}
</div>
<p>Now by the Eqs. (<a href="#eq6">9</a>) and (<a href="#eq8">11</a>), we get that </p>
<div class="displaymath" id="eq9">
  \begin{align} \label{eq9} & f[x_{k},w_{k}]=\\ \nonumber & =f’(\alpha )+f’(\alpha )(2+\gamma f’(\alpha ))c_{2}e_{k}+f’(\alpha )(\gamma f’(\alpha )c_{2}^{2}+(3+\gamma f’(\alpha ) \nonumber \\*[1mm]& (3\! +\! \gamma f’(\alpha )))c_{3})e_{k}^{2}\! +\!  f’(\alpha )(2\! +\! \gamma f’(\alpha )(2\! +\! \gamma f’(\alpha ))c_{2}c_{3}+(2+\gamma f’(\alpha )(2 \nonumber \\*[1mm]& \quad +\gamma f’(\alpha )))c_{4})e_{k}^{3}+ f’(\alpha )(5c_{5}+\gamma f’(\alpha )(\lambda f’(\alpha )c_{2}^{2}c_{3}+(3+2\gamma f’(\alpha )) \nonumber \\*[1mm]& c_{3}^{2}+(7+\gamma f’(\alpha )(8+3\gamma f’(\alpha )))c_{2}c_{4}+(10+\gamma f’(\alpha )(10+\gamma f’(\alpha )\nonumber \\*[1mm]& (5+\gamma f’(\alpha ))))c_{5}))e_{k}^{4}+{\mathcal O} (e_{k}^{5}).\nonumber \end{align}
</div>
<p>Furthermore, we have </p>
<div class="displaymath" id="eq10">
  \begin{align} \label{eq10} & \tfrac {f(x_{k})}{f[x_{k},w_{k}]+\beta f(w_{k})}= & \nonumber \\*[1mm]& =e_{k}-(1+\gamma f’(\alpha ))(\beta +c_{2})e_{k}^{2}+((\beta +\beta \gamma f’(\alpha ))^{2}+(2+\gamma f’(\alpha ) \nonumber \\*[1mm]&  (2+\gamma f’(\alpha )))c_{2}(\beta +c_{2})-2c_{3}-\gamma f’(\alpha )(3+\gamma f’(\alpha ))c_{3})e_{k}^{3}+ \nonumber \\*[1mm]& \quad +(-(\beta + \beta \gamma f’(\alpha ))^{3}-\beta (5+\gamma f’(\alpha )(7+\gamma f’(\alpha )(4+\gamma f’(\alpha ))))c_{2}^{2} \nonumber \\*[1mm]& \quad -(4+\gamma f’(\alpha )(5+\gamma f’(\alpha )(3+\gamma f’(\alpha ))))c_{2}^{3}+\beta (4+\gamma f’(\alpha )(7+ \nonumber \\*[1mm]& \quad +\gamma f’(\alpha )(5+\gamma f’(\alpha ))))c_{3}+c_{2}(-\beta ^{2}(1+\gamma f’(\alpha ))(3+\gamma f’(\alpha )(2 \nonumber \\*[1mm]& \quad +\gamma f’(\alpha )))+(7+\gamma f’(\alpha )(10+\gamma f’(\alpha )(7+2\gamma f’(\alpha ))))c^{3} \nonumber \\*[1mm]& \quad -(1+\gamma f’(\alpha ))(3+\gamma f’(\alpha )(3+\gamma f’(\alpha )))c_{4})e_{k}^{4}+{\mathcal O} (e_{k}^{5}). \end{align}
</div>
<p> Using the second step of (<a href="#f4">5</a>) and \(e_{k,y}=y_{k}-\alpha \), we get: </p>
<div class="displaymath" id="eq11">
  \begin{align}  \label{eq11} & y_{k}=\\ \nonumber & =\alpha \! +\!  (1\! +\! \beta f’(\alpha ))(\beta +c_{2})e_{k}^{2}+(-(\beta \! +\! \beta \gamma f’(\alpha ))^{2}\! -\! (2\! +\! \lambda f’(\alpha )(2\! +\! \gamma f’(\alpha ))) \nonumber \\*[1mm]& c_{2}(\beta +c_{2}) +2c_{3}+\gamma f’(\alpha )(3+\gamma f’(\alpha ))c_{3}e_{k}^{3}+((\beta +\beta \gamma f’(\alpha ))^{3}+\beta (5+\gamma \nonumber \\*[1mm]&  f’(\alpha )(7+\lambda f’(\alpha )(4+\lambda f’(\alpha ))))c_{2}^{2}+ (4+\gamma f’(\alpha )(5+\gamma f’(\alpha )(3+\gamma f’(\alpha )))) \nonumber \\*[1mm]& c_{2}^{3}-\beta (4+\gamma f’(\alpha )(7+\gamma f’(\alpha )(5+\gamma f’(\alpha ))))c_{3}+c_{2}(\beta ^{2} (1+\gamma f’(\alpha )) (3+\gamma \nonumber \\*[1mm]&  f’(\alpha ))(2+\gamma f’(\alpha )))-(7+\gamma f’(\alpha )(10+\gamma f’(\alpha )(7+2\gamma f’(\alpha ))))c_{3})+ \nonumber \\*[1mm]& \quad +(1+\gamma f’(\alpha ))(3+\gamma f’(\alpha )(3+\gamma f’(\alpha )))c_{4})e_{k}^{4}+ {\mathcal O} (e_{k}^{5}).\nonumber \end{align}
</div>
<p> For \(f (y_{k})\), we also have </p>
<div class="displaymath" id="eq12">
  \begin{eqnarray}  \label{eq12} f(y_{k})& =& f’(\alpha )(1+\gamma f’(\alpha ))(\beta + c_{2})e_{k}^{2}-f’(\alpha )((\beta +\beta \gamma f’(\alpha ))^{2}(2+\gamma f’(\alpha )\nonumber \end{eqnarray}
</div>
<div class="displaymath" id="a0000000005">
  \begin{eqnarray} & & (2+\gamma f’(\alpha )))c_{2}(\beta + c_{2})-2c_{3}+\gamma f’(\alpha )(3+\gamma f’(\alpha ))c_{3})e_{k}^{3}+f’(\alpha )\nonumber \\*[1mm]& & ((\beta +\beta \gamma f’(\alpha ))^{3}+\beta (7+\gamma f’(\alpha )(11+\gamma f’(\alpha )(6+\gamma f’(\alpha ))))c_{2}^{2}\nonumber \\*[1mm]& & +(5+\gamma f’(\alpha )(7+\gamma f’(\alpha )(4+\gamma f’(\alpha ))))c_{2}^{3}-\beta (4+\gamma f’(\alpha )(7+\nonumber \\*[1mm]& & \gamma f’(\alpha )(5\! +\! \gamma f’(\alpha ))))c_{3}\! +\! c_{2}(\beta ^{2}(1\! +\! \gamma f’(\alpha ))(4\! +\! \gamma f’(\alpha )(3\! +\! \gamma f’(\alpha )))\nonumber \\*[1mm]& & -(7\! +\! \gamma f’(\alpha )(10\! +\! \gamma f’(\alpha )(7\! +\! 2\gamma f’(\alpha ))))c_{3})(1\! +\! \gamma f’(\alpha ))(3\! +\! \gamma f’(\alpha )\nonumber \\*[1mm]& & (3+\gamma f’(\alpha )))c_{4})e_{k}^{4}+{\mathcal O}(e_{k}^{5}). \end{eqnarray}
</div>
<p>Additionally, by using relations (<a href="#eq7">10</a>), (<a href="#eq8">11</a>), (<a href="#eq11">2</a>) and (<a href="#eq12">-3</a>), we gain </p>
<div class="displaymath" id="eq13">
  \begin{eqnarray} \label{eq13} & & \tfrac {f(y_{k})}{f[y_{k},w_{k}]+\beta f(w_{k})+\lambda (y_{k}-w_{k})(y_{k}-x_{k})}\nonumber \\*[1mm]& & =(1+ \gamma f’(\alpha ))(\beta +c_{2})e_{k}^{2}+(-2(\beta + \beta \gamma f’(\alpha ))^{2}-\beta (4+3 \gamma f’(\alpha )(2+\nonumber \\*[1mm]& & \gamma f’(\alpha )))c_{2}-(3+2 \gamma f’(\alpha )(2+ \gamma f’(\alpha )))c_{2}^{2}+(1+\gamma f’(\alpha ))(2+\gamma f’(\alpha ))\nonumber \\*[1mm]& & c_{3})e_{k}^{3}+(\tfrac {1}{ f’(\alpha )})(\beta f’(\alpha )(11+\gamma f’(\alpha )(19+\lambda f’(\alpha )(14+5\gamma f’(\alpha ))))c_{2}^{2}\nonumber \\*[1mm]& & + f’(\alpha )(7+\gamma f’(\alpha )(11+\gamma f’(\alpha )(8+3\gamma f’(\alpha ))))c_{2}^{3}-\beta f’(\alpha )(7+3\gamma f’(\alpha )\nonumber \\*[1mm]& & (5+\gamma f’(\alpha )(4+\gamma f’(\alpha ))))c_{3}+c_{2}((1+\gamma f’(\alpha ))(\beta ^{2}f’(\alpha )(8+\gamma f’(\alpha )(9+\nonumber \\*[1mm]& & 5\gamma f’(\alpha )))\! -\! (1\! +\! \gamma f’(\alpha ))\lambda )\! -\! 2f’(\alpha )(5\! +\! \gamma f’(\alpha )(9\! +\! \gamma f’(\alpha )(7\! +\! 2\gamma f’(\alpha ))))\nonumber \\*[1mm]& & c_{3})+(1+\gamma f’(\alpha ))(\beta (1+\gamma f’(\alpha ))(3\beta ^{2}f’(\alpha )(1+\gamma f’(\alpha ))-\lambda )\nonumber \\*[1mm]& & +f’(\alpha )(3+\gamma f’(\alpha )(3+\gamma f’(\alpha )))c_{4}))e_{k}^{4}+{\mathcal O} (e_{k}^{5}). \end{eqnarray}
</div>
<p>Dividing these two relations (<a href="#eq12">-3</a>) and (<a href="#eq6">9</a>) on each other gives us </p>
<div class="displaymath" id="eq14">
  \begin{align} \label{eq14} & s_{k}=\tfrac {f(y_{k})}{f(x_{k})}=\\ \nonumber & =(1+ \gamma f’(\alpha ))(\beta + c_{2})e_{k}+(-(\beta +\beta \gamma f’(\alpha ))^{2}-(3+\gamma f’(\alpha ) \nonumber \\*[1mm]& (3+\gamma f’(\alpha )))c_{2}(\beta +c_{2})+2c_{3}+\gamma f’(\alpha )(3+\gamma f’(\alpha ))c_{3})e_{k}^{2}+\nonumber \\*[1mm]& ((\beta +\gamma f’(\alpha ))^{3}+\beta (10+\gamma f’(\alpha )(14+\gamma f’(\alpha )(7+\gamma f’(\alpha )))) \nonumber \\*[1mm]& c_{2}^{2}+(2+\gamma f’(\alpha ))(4+\gamma f’(\alpha )(3+\gamma f’(\alpha )))c_{2}^{3}-\beta (5+\gamma f’(\alpha ) \nonumber \\*[1mm]& (8+\gamma f’(\alpha )(5+\gamma f’(\alpha ))))c_{3}+c_{2}(\beta ^{2}(1+\gamma f’(\alpha ))(5+\gamma f’(\alpha ) \nonumber \\*[1mm]& (4+\gamma f’(\alpha )-2(5+\gamma f’(\alpha )(7+\gamma f’(\alpha )(4+\gamma f’(\alpha ))))c_{3}))\gamma f’(\alpha ))^{4} \nonumber \\*[1mm]& (1+\gamma f’(\alpha ))(3+\gamma f’(\alpha )(3+\gamma f’(\alpha )))c_{4})e_{k}^{3}(-(\beta +\beta \nonumber \\*[1mm]& -\beta (30+\gamma f’(\alpha )(50+\gamma f’(\alpha )(34\gamma f’(\alpha )(11+\gamma f’(\alpha ))))) \nonumber \\*[1mm]& c_{2}^{3}-(2+\gamma f’(\alpha )(10+\gamma f’(\alpha )(10+\gamma f’(\alpha )(5+\gamma f’(\alpha ))))c_{2}^{4}+\beta ^{2}\nonumber \end{align}
</div>
<div class="displaymath" id="a0000000006">
  \begin{align} & (1+\gamma f’(\alpha ))(7+\gamma f’(\alpha )(10+\gamma f’(\alpha )(6+\gamma f’(\alpha ))))c_{3}-(8+\gamma f’(\alpha )\nonumber \\*[1mm]& (15+\gamma f’(\alpha )(13+\gamma f’(\alpha )(6+\gamma f’(\alpha )))))c_{3}^{2}+c_{2}^{2}(-\beta ^{2}(20+\gamma f’(\alpha )\nonumber \\*[1mm]& (41+\gamma f’(\alpha ) (32+\gamma f’(\alpha )(11+\gamma f’(\alpha )))))+(37+\gamma f’(\alpha )(3+\nonumber \\*[1mm]& \gamma f’(\alpha ))(20+\gamma f’(\alpha )(8+3\gamma f’(\alpha ))))c_{3})-\beta (7+\gamma f’(\alpha )(3+\gamma \nonumber \\*[1mm]& f’(\alpha ))(5+\gamma f’(\alpha )(3+\gamma f’(\alpha ))))c_{4}+c_{2}(-\beta ^{3}(1+\gamma f’(\alpha ))^{2}(7+\gamma f’(\alpha )\nonumber \\*[1mm]& (5+\gamma f’(\alpha )))+\beta (2+\gamma f’(\alpha ))(3+\gamma f’(\alpha ))(5+\gamma f’(\alpha )(5+2\gamma \nonumber \\*[1mm]& f’(\alpha )))c_{3}-(14+\gamma f’(\alpha )(5+2\gamma f’(\alpha ))(5+\gamma f’(\alpha )(2+\gamma f’(\alpha )))c_{3})\nonumber \\*[1mm]& -(14+\gamma f’(\alpha )(5+2\gamma f’(\alpha ))(5+\gamma f’(\alpha )(2+\gamma f’(\alpha ))))c_{4})+\nonumber \\*[1mm]& (1+\gamma f’(\alpha ))(2+\gamma f’(\alpha ))(2+\gamma f’(\alpha )(2+\gamma f’(\alpha )))c_{5})e_{k}^{4}+{\mathcal O} (e_{k}^{5}).\nonumber \end{align}
</div>
<p> Expanding \(H\) at \(0\) yields </p>
<div class="displaymath" id="eq16">
  \begin{eqnarray} \label{eq15} H(s_{k})=H(0)+H’(0))s_{k}+H”(0)\tfrac {s_{k}^{2}}{2}; \end{eqnarray}
</div>
<p> and  </p>
<div class="displaymath" id="a0000000007">
  \begin{align*}  H(s_{k})& =H(0)+H’(0)(1+\gamma f’(\alpha ))(\beta +c_{2})e_{k}+(\tfrac {1}{2}(H”(0)(1+\gamma f’(\alpha ))^{2}\nonumber \\ & (\beta +c_{2})^{2}-H’(0)((\beta +\beta \gamma f’(\alpha ))^{2}+(3+\gamma f’(\alpha )(3+\gamma f’(\alpha )))\nonumber \\ & c_{2}(\beta )+c_{2})-2c_{3}-\gamma f’(\alpha )(3+\gamma f’(\alpha ))c_{3}))e_{k}^{2}+(-H”(0)(1+\gamma \nonumber \\ & f’(\alpha ))(\beta +c_{2})((\beta +\beta \gamma f’(\alpha ))^{2}+(3+\gamma f’(\alpha )(3+\gamma f’(\alpha )))c_{2}\nonumber \\ & (\beta +c_{2})-2c_{3}-\gamma f’(\alpha )(3+\gamma f’(\alpha ))c_{3})+H’(0)((\beta + \beta \lambda f’(\alpha ))^{3} \nonumber \\ & +\beta (10+\gamma f’(\alpha )(14+\gamma f’(\alpha )(7+\gamma f’(\alpha ))))c_{2}^{2}+(2+\gamma f’(\alpha ))(4+\nonumber \\ & \gamma f’(\alpha )(3+\lambda f’(\alpha )))c_{2}^{3}-\beta (5+\gamma f’(\alpha )(8+\gamma f’(\alpha )(5+\gamma f’(\alpha ))))c_{3}\nonumber \\ & +c_{2}(\beta ^{2}(1+\gamma f’(\alpha ))(5+\gamma f’(\alpha )(4+\gamma f’(\alpha )))-2(5+\gamma f’(\alpha )(7+\nonumber  \\ & \gamma f’(\alpha )(4+\gamma f’(\alpha ))))c_{3})(1+\gamma f’(\alpha ))(3+\gamma f’(\alpha )(3+\gamma f’(\alpha )))c_{4}))\nonumber \\ & e_{k}^{3}+\tfrac {1}{2}(-(2H’(0)-3H”(0))(\beta +\beta \gamma f’(\alpha ))^{4}+2\beta (H”(0)(3+\gamma \nonumber \\ & f’(\alpha )(3\! +\! \gamma f’(\alpha )))(9\! +\! \gamma f’(\alpha )(11\! +\! 3\gamma f’(\alpha )))-H’(0)(30+\gamma f’(\alpha )\nonumber \\ & (50\! +\! \gamma f’(\alpha )(34\! +\! \gamma f’(\alpha )(11\! +\! \gamma f’(\alpha ))))))c_{2}^{3}+(H”(0)(25\! +\! 3\gamma f’(\alpha )\nonumber \\ & (3+\gamma f’(\alpha ))(6+\gamma f’(\alpha )(3+\gamma f’(\alpha ))))-2H’(0)(2+\gamma f’(\alpha ))(10+\nonumber \\ & \gamma f’(\alpha )(10+\gamma f’(\alpha )(5+\gamma f’(\alpha )))))c_{2}^{4}+2\beta ^{2}(1+\gamma f’(\alpha ))(H’(0)(7+\nonumber \\ & \gamma f’(\alpha )(10\! +\! \gamma f’(\alpha )(6\! +\! \gamma f’(\alpha ))))-H”(0)(7\! +\! \gamma f’(\alpha )(13\! +\! \gamma f’(\alpha )\nonumber \\ & (9\! +\! 2\gamma f’(\alpha )))))c_{3}\! +\! (H”(0)(1\! +\! \gamma f’(\alpha ))^{2}(2\! +\! \gamma f’(\alpha ))^{2}-2H’(0)(8\nonumber \\ & +\gamma f’(\alpha )(15+\gamma f’(\alpha )(13+\gamma f’(\alpha )(6+\gamma f’(\alpha ))))))c_{3}^{2}+c_{2}^{2}(\beta ^{2}\nonumber \\ & (-2H’(0)(20+\gamma f’(\alpha )(41+\gamma f’(\alpha )(32+\gamma f’(\alpha )(11+\gamma f’(\alpha )))))\nonumber \\ & +H”(0)(45+\gamma f’(\alpha )(112+\gamma f’(\alpha )(105\gamma f’(\alpha )(44+7\gamma f’(\alpha ))))))\nonumber \\ & +2(H’(0)(37+\gamma f’(\alpha )(3+\gamma f’(\alpha ))(20+\gamma f’(\alpha )(8+3\gamma f’(\alpha ))))-\nonumber \\ & H”(0)(1\! +\! \gamma f’(\alpha ))(16\! +\! \gamma f’(\alpha )(23\! +\! \gamma f’(\alpha )(13\! +\! 3\gamma f’(\alpha ) f’(\alpha )))))c_{3})\nonumber \\ & \! \! +\! \! 2\beta (H”(0)(1\! \! +\! \! \gamma f’(\alpha ))^{2}(3\! \! +\! \! \gamma f’(\alpha )(3\! \! +\! \! \gamma f’(\alpha )))-H’(0)(7\! \! +\! \! \gamma f’(\alpha )\nonumber \\ & (3+\gamma f’(\alpha ))(5+\gamma f’(\alpha )(3+\gamma f’(\alpha ))))c_{4}-2c_{2}(\beta ^{3}(1+\gamma f’(\alpha ))^{2}\nonumber \\ & (\! -\! 3H”(0)(3\! +\! \gamma f’(\alpha )(3\! +\! \gamma f’(\alpha )))\! +\! H’(0)(7\! +\! \gamma f’(\alpha )(5\! +\! \gamma f’(\alpha ))))\! +\! \nonumber \\ & \beta (H’(0)(2\! +\! \gamma f’(\alpha ))(3\! +\! \gamma f’(\alpha ))(5\! +\! \gamma f’(\alpha )(5\! \! +\! \! 2\gamma f’(\alpha )))\! \! -\! \! H”(0)\nonumber \\ & (1\! \! +\! \! \gamma f’(\alpha ))(21\! \! +\! \! \gamma f’(\alpha )(31\! \! +\! \! 2\gamma f’(\alpha )(9\! \! +\! \! 2\gamma f’(\alpha )))))c_{3}|1\! +\! \! H”(0)(1\! \! +\! \! \gamma \nonumber \\ &  f’(\alpha ))^{2}(3\! +\! \gamma f’(\alpha )(3\! +\! \gamma f’(\alpha )))\! -\! H’(0)(14\! +\! \gamma f’(\alpha )(5\! +\! 2\gamma f’(\alpha ))\nonumber \\ & H’(0)(5+\gamma f’(\alpha ))(2+\gamma f’(\alpha )))))c_{4})+2H’(0)(1+\gamma f’(\alpha ))\nonumber \\ & (2+\gamma f’(\alpha ))(2+\gamma f’(\alpha )(2+\gamma f’(\alpha )))c_{5})e_{k}^{4}+{\mathcal O} (e_{k}^{5}).\nonumber \\ \end{align*}
</div>
<p> Finally, by placing \(H(0)=H'(0)=H''(0)=1\) as well as using equations (<a href="#eq11">2</a>), (<a href="#eq13">-7</a>) and (<a href="#eq16">-30</a>), the error equation of the memoryless method (<a href="#f4">5</a>) will be as follows: </p>
<div class="displaymath" id="eq17">
  \begin{eqnarray} \label{eq17} e_{k+1}& =& (2f’(\alpha ))^{-1}((1\! +\! \gamma f’(\alpha ))^{2}(\beta \! +\! c_{2})( f’(\alpha )\beta ^{2}(1\! +\! \gamma f’(\alpha ))\! +\! 2\lambda \! +\! f’(\alpha )c_{2}\nonumber \\ & & (2\beta (3+\gamma f’(\alpha ))+(5+\gamma f’(\alpha ))+c_{2}-2f’(\alpha )c_{3}))e_{k}^{4}+{\mathcal O} (e_{k}^{5}) \end{eqnarray}
</div>
<p> which finishes the proof of the theorem. <div class="proof_wrapper" id="a0000000008">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="with memory">3 Acceleration of the two-point method</h1>
<p> It is easy to recognize from (<a href="#eq17">-58</a>) that the order of convergence of (<a href="#f4">5</a>) is four when \(\gamma \neq \tfrac {-1}{f’(\alpha )}, \beta \neq -\tfrac {f”(\alpha )}{2f’(\alpha )}\) and \(\lambda \neq \tfrac {f”’(\alpha )}{6}\). By taking the value of \(\gamma _{k}=\frac{-1}{f'(\alpha )}, \beta _{k}= -\tfrac {f”(\alpha )}{2f’(\alpha )}\) and \(\lambda _{k}=\tfrac {f”’(\alpha )}{6}\), it can be established that the order of the method (<a href="#f4">5</a>) would be 6, 7, 7.22 and 7.53. For this type of acceleration of convergence and in actual fact the exact values of \(f'(\alpha ), f''(\alpha )\) and \(f'''(\alpha )\) are not obtainable. We could replace the parameters \(\gamma , \beta \) and \(\lambda \) by \(\gamma _{k}, \beta _{k}\) and \(\lambda _{k}\). In the remainder of this chapter, we consider the following two-parametric methods: </p>
<ul class="itemize">
  <li><p>If we only interpolate parameter \(\gamma _{k}\) using the Newton method, a procedure by six order with memory is obtained. </p>
<div class="equation" id="f7">
<p>
  <div class="equation_content">
    \begin{equation} \label{f7} \left\{  \begin{array}{ll} \gamma _{k}=-\tfrac {1}{N’_{3}(x_{k})}, k=1,2,3,\ldots ,\\ H(0)=1, H’(0)=1, |H”(0)|<\infty , s_{k}=\tfrac {f(y_{k})}{f(x_{k})}, k=0,1,2,\ldots ,\\ y_{k}=x_{k}-\tfrac { f(x_{k})}{f[x_{k},w_{k}]+\beta f(w_{k})}, w_{k}=x_{k}+\gamma _{k} f(x_{k}),\\ x_{k+1}=y_{k}-H(s_{k})\tfrac { f(y_{k})}{f[y_{k},w_{k}]+\beta f(w_{k})+\lambda (y_{k}-x_{k})(y_{k}-w_{k})}. \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-57</span>
</p>
</div>
</li>
  <li><p>We attempt to prove that the method with memory (<a href="#f4">5</a>) has convergence order seven provided that we use accelerators \(\gamma _{k}, \lambda _{k}\). </p>
<div class="equation" id="f8">
<p>
  <div class="equation_content">
    \begin{equation} \label{f8} \left\{  \begin{array}{ll} \gamma _{k}=-\tfrac {1}{N’_{3}(x_{k})}, \beta _{k}=-\frac{N'_{4}(w_{k})}{2N''_{4}(w_{k})}, k=1,2,3,\ldots ,\\ H(0)=1, H’(0)=1, |H”(0)|<\infty , s_{k}=\tfrac {f(y_{k})}{f(x_{k})}, k=0,1,2,\ldots ,\\ y_{k}=x_{k}-\tfrac { f(x_{k})}{f[x_{k},w_{k}]+\beta _{k} f(w_{k})}, w_{k}=x_{k}+\gamma _{k} f(x_{k}),\\ x_{k+1}=y_{k}-H(s_{k})\tfrac { f(y_{k})}{f[y_{k},w_{k}]+\beta _{k} f(w_{k})+\lambda (y_{k}-x_{k})(y_{k}-w_{k})}. \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-56</span>
</p>
</div>
</li>
  <li><p>Bassiri <i class="it">et al.</i> approximated self-accelerator parameters as </p>
<div class="equation" id="f9">
<p>
  <div class="equation_content">
    \begin{equation} \label{f9} \left\{  \begin{array}{ll} \gamma _{k}=-\tfrac {1}{N’_{3}(x_{k})}\simeq \frac{-1}{f'(\alpha )},\\ \beta _{k}=-\tfrac {N”_{4}(w_{k})}{2N’_{4}(w_{k})}\simeq -\tfrac {f”(\alpha )}{2f’(\alpha )},\\ \lambda _{k}=\tfrac {N”’_{5}(w_{k})}{6}\simeq f’(\alpha )c_{3}=\tfrac {f”’(\alpha )}{6}, \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-55</span>
</p>
</div>
<p> and thus three parameters family with memory is given by (BBAM) </p>
<div class="equation" id="f10">
<p>
  <div class="equation_content">
    \begin{equation} \label{f10} \left\{  \begin{array}{ll} \gamma _{k}=-\tfrac {1}{N’_{3}(x_{k})}, \beta _{k}=-\tfrac {N’_{4}(w_{k})}{2N”_{4}(w_{k})}, \lambda _{k}=\tfrac {N”’_{5}(w_{k})}{6}, k=1,2,3,\ldots ,\\ H(0)=1, H’(0)=1, |H”(0)|<\infty , s_{k}=\tfrac {f(y_{k})}{f(x_{k})}, k=0,1,2,\ldots ,\\ y_{k}=x_{k}-\frac{ f(x_{k})}{f[x_{k},w_{k}]+\beta _{k} f(w_{k})}, w_{k}=x_{k}+\gamma _{k} f(x_{k}),\\ x_{k+1}=y_{k}-H(s_{k})\tfrac { f(y_{k})}{f[y_{k},w_{k}]+\beta _{k} f(w_{k})+\lambda _{k}(y_{k}-x_{k})(y_{k}-w_{k})}. \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-54</span>
</p>
</div>
</li>
  <li><p>The self-accelerating parameters \( \gamma _{k}, \beta _{k}\) and \(\lambda _{k}\) are calculated by using of the formula: </p>
<div class="equation" id="f11">
<p>
  <div class="equation_content">
    \begin{equation} \label{f11} \left\{  \begin{array}{ll} \gamma _{k}=-\tfrac {1}{N’_{3}(x_{k})}\simeq \frac{-1}{f'(\alpha )},\\ \beta _{k}=-\tfrac {N”_{4}(w_{k})}{2N’_{4}(w_{k})}\simeq -\tfrac {f”(\alpha )}{2f’(\alpha )},\\ \lambda _{k}=\tfrac {N”’_{5}(y_{k})}{6}\simeq f’(\alpha )c_{3}=\tfrac {f”’(\alpha )}{6}, \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-53</span>
</p>
</div>
<p> where \(N_{3}(x_{k});N_{4}(w_{k})\) and \(N_{5}(y_{k})\) defined by: </p>
<div class="equation" id="a0000000009">
<p>
  <div class="equation_content">
    \begin{equation}  \left\{  \begin{array}{ll} N_{3}(x_{k})=N_{3}(t;x_{k},x_{k-1},w_{k-1},y_{k-1}),\\ N_{4}(w_{k})=N_{4}(t;w_{k},x_{k},x_{k-1},w_{k-1},y_{k-1}),\\ N_{5}(y_{k})=N_{5}(t;y_{k},w_{k},x_{k},x_{k-1},w_{k-1},y_{k-1}). \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-52</span>
</p>
</div>
<p> Now, we obtain the following three-parameter iterative with memory(TM1)method : </p>
<div class="equation" id="f12">
<p>
  <div class="equation_content">
    \begin{equation} \label{f12} \left\{  \begin{array}{ll} \gamma _{k}=-\tfrac {1}{N’_{3}(x_{k})}, \beta _{k}=-\tfrac {N’_{4}(w_{k})}{2N”_{4}(w_{k})}, \lambda _{k}=\tfrac {N”’_{5}(y_{k})}{6}, k=1,2,3,\ldots ,\\ H(0)=1, H’(0)=1, |H”(0)|<\infty , s_{k}=\tfrac {f(y_{k})}{f(x_{k})}, k=0,1,2,\ldots ,\\ y_{k}=x_{k}-\tfrac { f(x_{k})}{f[x_{k},w_{k}]+\beta _{k} f(w_{k})}, w_{k}=x_{k}+\gamma _{k} f(x_{k}),\\ x_{k+1}=y_{k}-H(s_{k})\frac{ f(y_{k})}{f[y_{k},w_{k}]+\beta _{k} f(w_{k})+\lambda _{k}(y_{k}-x_{k})(y_{k}-w_{k})}. \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-51</span>
</p>
</div>
</li>
</ul>
<p>It should be noted that the convergence order varies as the iteration go ahead. First, we need the following lemma: </p>
<p><div class="lemma_thmwrapper " id="lmm1">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2</span>
  </div>
  <div class="lemma_thmcontent">
  <p> If \(\gamma _{k}=-\tfrac {1}{N’_{3}(x_{k})}, \beta _{k}=-\tfrac {N”_{4}(w_{k})}{2N’_{4}(w_{k})},\) and \(\lambda _{k}=\tfrac {N”’_{5}(y_{k})}{6} ,\) then </p>
<div class="equation" id="f13">
<p>
  <div class="equation_content">
    \begin{equation} \label{f13} (1+\gamma _{k}f'(\alpha ))\sim e_{k-1}e_{k-1,w}e_{k-1,y}, \end{equation}
  </div>
  <span class="equation_label">-50</span>
</p>
</div>
<div class="equation" id="f14">
<p>
  <div class="equation_content">
    \begin{equation} \label{f14} (c_{2}+\beta _{k})\sim e_{k-1}e_{k-1,w}e_{k-1,y}, \end{equation}
  </div>
  <span class="equation_label">-49</span>
</p>
</div>
<div class="displaymath" id="f15">
  \begin{align} \label{f15} & ( f’(\alpha )\beta ^{2}(1+\gamma f’(\alpha ))+2\lambda +f’(\alpha )c_{2}(2\beta (3+\gamma f’(\alpha ))+\\ \nonumber & +(5+\gamma f’(\alpha ))+c_{2}-2f’(\alpha )c_{3}))\sim e_{k-1}e_{k-1,w}e_{k-1,y}\nonumber \end{align}
</div>
<p> where \(e_{k}=x_{k}-\alpha ,e_{k,w}=w_{k}-\alpha ,e_{k,y}=y_{k}-\alpha .\) </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000010">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The proof is similar to Lemma \(3.1\) in <span class="cite">
	[
	<a href="#Tor2" >30</a>
	]
</span>. <div class="proof_wrapper" id="a0000000011">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Now we state the following convergence theorem: </p>
<p><div class="theorem_thmwrapper " id="thm2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If an initial approximation \(x_{0}\) is sufficiently close to the zero \(\alpha \) of \(f(x)\) and the parameters \(\gamma _{k}, \beta _{k}\) and \(\lambda _{k}\) in the iterative schemes <a href="#f7" class="eqref">-57</a>, <a href="#f8" class="eqref">-56</a>, <a href="#f10" class="eqref">-54</a> and <a href="#f12" class="eqref">-51</a> are recursively calculated by the forms given in <a href="#f9" class="eqref">-55</a> and <a href="#f11" class="eqref">-53</a>. Then, the R-order of convergence of the three-point methods <a href="#f7" class="eqref">-57</a>, <a href="#f8" class="eqref">-56</a>, <a href="#f10" class="eqref">-54</a> and <a href="#f12" class="eqref">-51</a> with the corresponding expressions \(\gamma _{k}, \beta _{k}\) and \(\lambda _{k}\) are at least \(6\), \(7\), \(7.22\) and \(7.53\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000012">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Here, we obtain the convergence order of 6 and 7.5 for the methods (<a href="#f7">-57</a>) and (<a href="#f12">-51</a>). Bassiri and his colleagues <span class="cite">
	[
	<a href="#Bassiri" >2</a>
	]
</span> have achieved the degree of convergence of the method mentioned in Equation (<a href="#f10">-54</a>). Proof of convergence of method (<a href="#f8">-56</a>) is similar to these three cases. </p>
<p>First we assume that the \(C\)-order of convergence of sequence \(x_{k},w_{k},y_{k}\) is at least \(r,p\) and \(q\), respectively. Hence: </p>
<div class="equation" id="f16">
<p>
  <div class="equation_content">
    \begin{equation} \label{f16} e_{k+1}\sim e_{k}^{r}\sim e_{k-1}^{r^{2}}, \end{equation}
  </div>
  <span class="equation_label">-48</span>
</p>
</div>
<div class="equation" id="f17">
<p>
  <div class="equation_content">
    \begin{equation} \label{f17} e_{k,w}\sim e_{k}^{p}\sim e_{k-1}^{rp}, \end{equation}
  </div>
  <span class="equation_label">-47</span>
</p>
</div>
<p> and </p>
<div class="equation" id="f18">
<p>
  <div class="equation_content">
    \begin{equation} \label{f18} e_{k,y}\sim e_{k}^{q}\sim e_{k-1}^{rq}. \end{equation}
  </div>
  <span class="equation_label">-46</span>
</p>
</div>
<p> By (<a href="#f16">-48</a>), (<a href="#f17">-47</a>), (<a href="#f18">-46</a>), and lemma(<a href="#lmm1">2</a>), we obtain </p>
<div class="equation" id="f19">
<p>
  <div class="equation_content">
    \begin{equation} \label{f19} 1+\gamma _{k}f'(\alpha )\sim e_{k-1}^{p+q+1}. \end{equation}
  </div>
  <span class="equation_label">-45</span>
</p>
</div>
<p> On the other hand, we get </p>
<div class="equation" id="f20">
<p>
  <div class="equation_content">
    \begin{equation} \label{f20} e_{k,w}\sim (1+\gamma _{k}f'(\alpha ))e_{k}, \end{equation}
  </div>
  <span class="equation_label">-44</span>
</p>
</div>
<div class="equation" id="f21">
<p>
  <div class="equation_content">
    \begin{equation} \label{f21} e_{k,y}\sim (1+\gamma _{k}f'(\alpha )) e_{k}^{2}, \end{equation}
  </div>
  <span class="equation_label">-43</span>
</p>
</div>
<div class="equation" id="f22">
<p>
  <div class="equation_content">
    \begin{equation} \label{f22} e_{k+1}\sim (1+\gamma _{k}f'(\alpha ))^{2}e_{k}^{4}. \end{equation}
  </div>
  <span class="equation_label">-42</span>
</p>
</div>
<p> Combining (<a href="#f16">-48</a>)–(<a href="#f22">-42</a>), (<a href="#f17">-47</a>)–(<a href="#f20">-44</a>), and (<a href="#f18">-46</a>)–(<a href="#f21">-43</a>), we conclude </p>
<div class="equation" id="f23">
<p>
  <div class="equation_content">
    \begin{equation} \label{f23} e_{k,w}\sim e_{k-1}^{(1+p+q)+r}, \end{equation}
  </div>
  <span class="equation_label">-41</span>
</p>
</div>
<div class="equation" id="f24">
<p>
  <div class="equation_content">
    \begin{equation} \label{f24} e_{k,y}\sim e_{k-1}^{(1+p+q)+2r}, \end{equation}
  </div>
  <span class="equation_label">-40</span>
</p>
</div>
<p> and </p>
<div class="equation" id="f25">
<p>
  <div class="equation_content">
    \begin{equation} \label{f25} e_{k+1}\sim e_{k-1}^{2(1+p+q)+4r}. \end{equation}
  </div>
  <span class="equation_label">-39</span>
</p>
</div>
<p> Equating the powers of error exponents of \(e_{k-1}\) in pair relations (<a href="#f16">-48</a>)–(<a href="#f25">-39</a>), (<a href="#f17">-47</a>)–(<a href="#f23">-41</a>), and (<a href="#f18">-46</a>)–(<a href="#f24">-40</a>), we have </p>
<div class="equation" id="f26">
<p>
  <div class="equation_content">
    \begin{equation} \label{f26} \left\{  \begin{array}{ll} r p-r-(p+q+1)=0,\\ r q-2r-(p+q+1)=0,\\ r^{2}-4r-2(p+q+1)=0.\\ \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-38</span>
</p>
</div>
<p> This system has the solution \(p=2, q=4\) and \(r=6\) which specifies the \(C\)-order of convergence of the derivative-free scheme with memory (<a href="#f7">-57</a>). Varying parameters \(\gamma _{k}, \beta _{k}\) and \(\lambda _{k}\) in (<a href="#f12">-51</a>) using (<a href="#f11">-53</a>), we obtain the family of two-point with-memory methods of order 7.53, which is the improvement of the convergence rate of \(88.25\% \). Similar to the first part of the <a href="#thm2">theorem 3</a>: </p>
<div class="equation" id="al7">
<p>
  <div class="equation_content">
    \begin{equation} \label{al7} e_{k+1}\sim e_{k}^{r}\sim e_{k-1}^{r^{2}}, \end{equation}
  </div>
  <span class="equation_label">-37</span>
</p>
</div>
<div class="equation" id="al8">
<p>
  <div class="equation_content">
    \begin{equation} \label{al8} e_{k,w}\sim e_{k}^{p}\sim e_{k-1}^{rp}, \end{equation}
  </div>
  <span class="equation_label">-36</span>
</p>
</div>
<p> and </p>
<div class="equation" id="al9">
<p>
  <div class="equation_content">
    \begin{equation} \label{al9} e_{k,y}\sim e_{k}^{q}\sim e_{k-1}^{rq}. \end{equation}
  </div>
  <span class="equation_label">-35</span>
</p>
</div>
<p> By (<a href="#al7">-37</a>), (<a href="#al8">-36</a>), (<a href="#al9">-35</a>), and <a href="#lmm1">lemma 2</a>, we obtain </p>
<div class="equation" id="al10">
<p>
  <div class="equation_content">
    \begin{equation} \label{al10} 1+\gamma _{k}f'(\alpha )\sim e_{k-1}^{p+q+1}. \end{equation}
  </div>
  <span class="equation_label">-34</span>
</p>
</div>
<p> On the other hand, we get </p>
<div class="equation" id="al11">
<p>
  <div class="equation_content">
    \begin{equation} \label{al11} e_{k,w}\sim (1+\gamma _{k}f'(\alpha ))e_{k}, \end{equation}
  </div>
  <span class="equation_label">-33</span>
</p>
</div>
<div class="equation" id="al12">
<p>
  <div class="equation_content">
    \begin{equation} \label{al12} e_{k,y}\sim (1+\gamma _{k}f'(\alpha ))(\beta _{k}+c_{2}) e_{k}^{2}, \end{equation}
  </div>
  <span class="equation_label">-32</span>
</p>
</div>
<div class="displaymath" id="al13">
  \begin{eqnarray} \label{al13} e_{k+1}& \sim & ((1+\gamma f’(\alpha ))^{2}(\beta +c_{2})( f’(\alpha )\beta ^{2}(1+\gamma f’(\alpha ))+2\lambda +f’(\alpha )c_{2}\nonumber \\ & & (2\beta (3+\gamma f’(\alpha ))+(5+\gamma f’(\alpha ))+c_{2}-2f’(\alpha )c_{3}))e_{k}^{4} \end{eqnarray}
</div>
<p> Combining (<a href="#al10">-34</a>)–(<a href="#al11">-33</a>), (<a href="#al10">-34</a>)–(<a href="#al12">-32</a>), and (<a href="#al10">-34</a>)–(<a href="#al13">-31</a>), we conclude </p>
<div class="equation" id="al14">
<p>
  <div class="equation_content">
    \begin{equation} \label{al14} e_{k,w}\sim e_{k-1}^{(1+p+q)+r}, \end{equation}
  </div>
  <span class="equation_label">-30</span>
</p>
</div>
<div class="equation" id="al15">
<p>
  <div class="equation_content">
    \begin{equation} \label{al15} e_{k,y}\sim e_{k-1}^{2(1+p+q)+2r}, \end{equation}
  </div>
  <span class="equation_label">-29</span>
</p>
</div>
<p> and </p>
<div class="equation" id="al16">
<p>
  <div class="equation_content">
    \begin{equation} \label{al16} e_{k+1}\sim e_{k-1}^{4(1+p+q)+4r}. \end{equation}
  </div>
  <span class="equation_label">-28</span>
</p>
</div>
<p> Considering the error equations of \(e_{k-1},e_{w},e_{y}\) in pair relations of (<a href="#al8">-36</a>), (<a href="#al14">-30</a>), (<a href="#al9">-35</a>), (<a href="#al15">-29</a>), and (<a href="#al7">-37</a>), (<a href="#al16">-28</a>), we have </p>
<div class="equation" id="f27">
<p>
  <div class="equation_content">
    \begin{equation} \label{f27} \left\{  \begin{array}{ll} r p-r-(p+q+1)=0,\\ r q-2r-2(p+q+1)=0,\\ r^{2}-4r-4(p+q+1)=0.\\ \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-27</span>
</p>
</div>
<p> Th positive solution for the system is \(p=\frac{1}{8}(7+\sqrt{65})\approx 1.88278 , q=\frac{1}{4}(7+\sqrt{65})\approx 3.76556\) and \(r=\frac{1}{2}(7+\sqrt{65})\approx 7.53113\). It specifies that the \(C\)-order for convergence of the derivative-free scheme with memory (<a href="#f12">-51</a>) is 7.53113. <div class="proof_wrapper" id="a0000000013">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="a0000000014">4 Maximum improvement of convergence order</h1>
<p>Now, we can propose our iteration schemes (TM2) with memory, </p>
<div class="equation" id="f28">
<p>
  <div class="equation_content">
    \begin{equation} \label{f28} \left\{  \begin{array}{ll} \gamma _{k}=-\tfrac {1}{N’_{6}(x_{k})},\beta _{k}=-\tfrac {N”_{7}(w_{k})}{2N’_{7}(w_{k})},\lambda _{k}=\tfrac {N”’_{8}(y_{k})}{6}, k=2,3,4,\ldots ,\\ w_{k}=x_{k}+\gamma _{k} f(x_{k}), y_{k}=x_{k}-\tfrac {f(x_{k})}{f[x_{k},w_{k}]+\beta _{k} f(w_{k})}, s_{k}=\tfrac {f(y_{k})}{f(x_{k})}, k=0,1,2,\ldots ,\\ x_{k+1}=y_{k}-H(s_{k})\tfrac {f(y_{k})}{f[w_{k},y_{k}]+\beta _{k} f(w_{k})+\lambda _{k}(y_{k}-x_{k})(y_{k}-w_{k})}, k=0,1,2,\ldots , \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-26</span>
</p>
</div>
<p> and, similarly, the following ones with better interpolation degrees (TM3): </p>
<div class="equation" id="f29">
<p>
  <div class="equation_content">
    \begin{equation} \label{f29} \left\{  \begin{array}{ll} \gamma _{k}=-\tfrac {1}{N’_{9}(x_{k})},\beta _{k}=-\tfrac {N”_{10}(w_{k})}{2N’_{10}(w_{k})},\lambda _{k}=\tfrac {N”’_{11}(y_{k})}{6}, k=3,4,5,\ldots ,\\ w_{k}=x_{k}+\gamma _{k} f(x_{k}), y_{k}=x_{k}-\tfrac {f(x_{k})}{f[x_{k},w_{k}]+\beta _{k} f(w_{k})}, s_{k}=\frac{f(y_{k})}{f(x_{k})}, k=0,1,2,\ldots ,\\ x_{k+1}=y_{k}-H(s_{k})\tfrac {f(y_{k})}{f[w_{k},y_{k}]+\beta _{k} f(w_{k})+\lambda _{k}(y_{k}-x_{k})(y_{k}-w_{k})}, k=0,1,2,\ldots , \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-25</span>
</p>
</div>
<p> Also, we get our proposed two-step iterative method given for \(k =4;5;6; ...\), by(denoted (TM4)) </p>
<div class="equation" id="f30">
<p>
  <div class="equation_content">
    \begin{equation} \label{f30} \left\{  \begin{array}{ll} \gamma _{k}=-\tfrac {1}{N’_{12}(x_{k})},\beta _{k}=-\frac{N''_{13}(w_{k})}{2N'_{13}(w_{k})},\lambda _{k}=\tfrac {N”’_{14}(y_{k})}{6}, k=4,5,6,\ldots ,\\ y_{k}=x_{k}-\tfrac {f(x_{k})}{f[x_{k},w_{k}]+\beta _{k} f(w_{k})}, s_{k}=\tfrac {f(y_{k})}{f(x_{k})}, k=0,1,2,\ldots ,\\ x_{k+1}=y_{k}-H(s_{k})\frac{f(y_{k})}{f[w_{k},y_{k}]+\beta _{k} f(w_{k})+\lambda _{k}(y_{k}-x_{k})(y_{k}-w_{k})}, k=0,1,2,\ldots . \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-24</span>
</p>
</div>
<p> As an illustration, here we can also define: </p>
<ul class="itemize">
  <li><p>\(N_{6}(t)= N_{6}(t; x_{k}, x_{k-1}, w_{k-1}, y_{k-1}, x_{k-2}, w_{k-2}, y_{k-2})\), as an interpolation polynomial of sixth degree, passing through the best seven saved points </p>
<p>\(x_{k}, x_{k-1}, w_{k-1}, y_{k-1}, x_{k-2}, w_{k-2}, y_{k-2}\), for any \(k \in \lbrace 2,3,4,...\rbrace \). </p>
</li>
  <li><p>\(N_{7}(t)= N_{7}(t; w_{k}, x_{k}, x_{k-1}, w_{k-1}, y_{k-1}, x_{k-2}, w_{k-2}, y_{k-2})\), as an interpolation polynomial of seventh degree, passing through the best eight saved points \(w_{k}, x_{k}, x_{k-1}, w_{k-1}, y_{k-1}, x_{k-2}, w_{k-2}, y_{k-2}\), for any <br />\(k \in \lbrace 2,3,4,...\rbrace \). </p>
</li>
  <li><p>\(N_{8}(t)= N_{8}(t; y_{k}, w_{k}, x_{k}, x_{k-1}, w_{k-1}, y_{k-1}, x_{k-2}, w_{k-2}, y_{k-2})\), as an interpolation polynomial of eighth degree, passing through the best nine saved points \(y_{k}, w_{k}, x_{k}, x_{k-1}, w_{k-1}, y_{k-1}, x_{k-2}, w_{k-2}, y_{k-2}\), for any \(k \in \lbrace 2,3,4,...\rbrace \). </p>
</li>
</ul>
<p>The convergence analysis of (<a href="#f14">-49</a>), (<a href="#f15">-48</a>) and (<a href="#f16">-48</a>) can now be established as follows. <div class="theorem_thmwrapper " id="thm3">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Consider the same assumptions as in <a href="#thm2">theorem 3</a>. Then, the C-order of the improved Steffensen’s methods with memory <a href="#f28" class="eqref">-26</a>, <a href="#f29" class="eqref">-25</a> and <a href="#f30" class="eqref">-24</a> are \(7.94449\), \(7.99315\) and \(7.99915\), respectively. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000015">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The convergence of each of the methods mentioned in equations (<a href="#f28">-26</a>), (<a href="#f29">-25</a>) and (<a href="#f30">-24</a>) is given below separately. </p>
<p><b class="bfseries">Method TM2</b>: </p>
<p>Let \(\lbrace x_{k}\rbrace ,\lbrace w_{k}\rbrace \), and \(\lbrace y_{k}\rbrace \), be convergent with orders \(r, p\), and \(q\), respectively. Then: </p>
<div class="equation" id="al17">
<p>
  <div class="equation_content">
    \begin{equation} \label{al17} e_{k+1}\sim e_{k}^{r}\sim e_{k-1}^{r^{2}}\sim e_{k-2}^{r^{3}}\sim e_{k-3}^{r^{4}} \sim e_{k-4}^{r^{5}}, \end{equation}
  </div>
  <span class="equation_label">-23</span>
</p>
</div>
<div class="equation" id="al18">
<p>
  <div class="equation_content">
    \begin{equation} \label{al18} e_{k,w}\sim e_{k}^{p}\sim e_{k-1}^{p r}\sim e_{k-2}^{r^{2}p} \sim e_{k-3}^{r^{3}p} \sim e_{k-4}^{r^{4}p}, \end{equation}
  </div>
  <span class="equation_label">-22</span>
</p>
</div>
<div class="equation" id="al19">
<p>
  <div class="equation_content">
    \begin{equation} \label{al19} e_{k,y}\sim e_{k}^{ q}\sim e_{k-1}^{q r}\sim e_{k-2}^{r^{2}q} \sim e_{k-3}^{r^{3}q}\sim e_{k-4}^{r^{4}q}. \end{equation}
  </div>
  <span class="equation_label">-21</span>
</p>
</div>
<p> Using Theorem (<a href="#thm1">1</a>) and error equations (<a href="#eq7">10</a>), (<a href="#eq11">2</a>) and (<a href="#eq17">-58</a>) we obtain: </p>
<div class="equation" id="al20">
<p>
  <div class="equation_content">
    \begin{equation} \label{al20} e_{k,w}\sim (1+\gamma _{k}f'(\alpha )) e_{k}, \end{equation}
  </div>
  <span class="equation_label">-20</span>
</p>
</div>
<div class="equation" id="al21">
<p>
  <div class="equation_content">
    \begin{equation} \label{al21} e_{k,y}\sim (1+\gamma _{k} f'(\alpha ))(\beta _{k}+c_{2}) e_{k}^{2}, \end{equation}
  </div>
  <span class="equation_label">-19</span>
</p>
</div>
<div class="displaymath" id="al22">
  \begin{eqnarray} \label{al22} e_{k+1}& \sim & ((1+\gamma f’(\alpha ))^{2}(\beta +c_{2})( f’(\alpha )\beta ^{2}(1+\gamma f’(\alpha ))+2\lambda +f’(\alpha )c_{2}\nonumber \\ & & (2\beta (3+\gamma f’(\alpha ))+(5+\gamma f’(\alpha ))+c_{2}-2f’(\alpha )c_{3}))e_{k}^{4}. \end{eqnarray}
</div>
<p> And </p>
<div class="equation" id="al23">
<p>
  <div class="equation_content">
    \begin{equation} \label{al23} (1+\gamma _{k} f'(\alpha ))\sim e_{k-2}e_{k-1}e_{k-2,w}e_{k-1,w}e_{k-2,y}e_{k-1,y}, \end{equation}
  </div>
  <span class="equation_label">-17</span>
</p>
</div>
<div class="equation" id="al24">
<p>
  <div class="equation_content">
    \begin{equation} \label{al24} (\beta _{k}+c_{2})\sim e_{k-2}e_{k-1}e_{k-2,w}e_{k-1,w}e_{k-2,y}e_{k-1,y}, \end{equation}
  </div>
  <span class="equation_label">-16</span>
</p>
</div>
<div class="displaymath" id="al25">
  \begin{align} \label{al25} ( f’(\alpha )\beta ^{2}(1+\gamma f’(\alpha ))& + 2\lambda +f’(\alpha )c_{2}(2\beta (3+\gamma f’(\alpha ))+ \\ \nonumber & +(5+\gamma f’(\alpha ))+c_{2}-2f’(\alpha )c_{3})) \\ \nonumber & \sim e_{k-2}e_{k-1}e_{k-2,w}e_{k-1,w}e_{k-2,y}e_{k-1,y}. \end{align}
</div>
<p> Combining (<a href="#al18">-22</a>), (<a href="#al20">-20</a>), (<a href="#al23">-17</a>) and (<a href="#al19">-21</a>), (<a href="#al21">-19</a>), (<a href="#al23">-17</a>), (<a href="#al24">-16</a>) also (<a href="#al17">-23</a>), (<a href="#al22">-18</a>), (<a href="#al23">-17</a>), (<a href="#al24">-16</a>), (<a href="#al25">-15</a>) we get </p>
<div class="equation" id="al26">
<p>
  <div class="equation_content">
    \begin{equation} \label{al26} e_{k,w}\sim e_{k-2}^{1+r+p+rp+q+qr}, \end{equation}
  </div>
  <span class="equation_label">-14</span>
</p>
</div>
<p> and </p>
<div class="equation" id="al27">
<p>
  <div class="equation_content">
    \begin{equation} \label{al27} e_{k,y}\sim e_{k-2}^{2(1+r+p+rp+q+qr)}, \end{equation}
  </div>
  <span class="equation_label">-13</span>
</p>
</div>
<p> also </p>
<div class="equation" id="al28">
<p>
  <div class="equation_content">
    \begin{equation} \label{al28} e_{k+1}\sim e_{k-2}^{4(1+r+p+rp+q+qr)}, \end{equation}
  </div>
  <span class="equation_label">-12</span>
</p>
</div>
<p>Comparing the right and left side of error equations (<a href="#al18">-22</a>), (<a href="#al26">-14</a>) and (<a href="#al19">-21</a>), (<a href="#al27">-13</a>), and (<a href="#al20">-20</a>), (<a href="#al28">-12</a>), we have: </p>
<div class="equation" id="f31">
<p>
  <div class="equation_content">
    \begin{equation} \label{f31} \left\{  \begin{array}{ll} r^{2} p=(1+r+r^{2}+p+pr+q+qr),\\ r^{2} q=2(1+r+r^{2}+p+pr+q+qr),\\ r^{3}=4(1+r+r^{2}+p+pr+q+qr) . \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-11</span>
</p>
</div>
<p>The positive real solution of this system is \( p_{1}\simeq {1.98612} ,p_{2}\simeq {3.97225} \) and \( r\simeq {7.94449}\). We conclude that the C-order of the methods with memory (<a href="#f28">-26</a>) is at least 7.94449. </p>
<p><b class="bfseries">Method TM3</b>: </p>
<p>Similar to the previous method: </p>
<div class="equation" id="al29">
<p>
  <div class="equation_content">
    \begin{equation} \label{al29} (1+\gamma _{k} f'(\alpha ))\sim e_{k-3}e_{k-2}e_{k-1}e_{k-3,w}e_{k-2,w}e_{k-1,w}e_{k-3,y}e_{k-2,y}e_{k-1,y}, \end{equation}
  </div>
  <span class="equation_label">-10</span>
</p>
</div>
<div class="equation" id="al30">
<p>
  <div class="equation_content">
    \begin{equation} \label{al30} (\beta _{k}+c_{2})\sim e_{k-3}e_{k-2}e_{k-1}e_{k-3,w}e_{k-2,w}e_{k-1,w}e_{k-3,y}e_{k-2,y}e_{k-1,y}, \end{equation}
  </div>
  <span class="equation_label">-9</span>
</p>
</div>
<div class="displaymath" id="al31">
  \begin{align} \label{al31} ( f’(\alpha )\beta ^{2}(1+\gamma f’(\alpha ))& +\\ \nonumber & +2\lambda +f’(\alpha )c_{2}(2\beta (3+\gamma f’(\alpha ))+\\ \nonumber & +(5+\gamma f’(\alpha ))+c_{2}-2f’(\alpha )c_{3}))\nonumber \\ & \sim e_{k-3}e_{k-2}e_{k-1}e_{k-3,w}e_{k-2,w}e_{k-1,w}e_{k-3,y}e_{k-2,y}e_{k-1,y}.\nonumber \end{align}
</div>
<p> Combining (<a href="#al18">-22</a>), (<a href="#al20">-20</a>), (<a href="#al29">-10</a>) and (<a href="#al19">-21</a>), (<a href="#al21">-19</a>), (<a href="#al29">-10</a>), (<a href="#al30">-9</a>) also (<a href="#al17">-23</a>), (<a href="#al22">-18</a>), (<a href="#al23">-17</a>), (<a href="#al24">-16</a>), (<a href="#al25">-15</a>), (<a href="#al29">-10</a>), (<a href="#al30">-9</a>), (<a href="#al31">-8</a>) we get </p>
<div class="equation" id="al32">
<p>
  <div class="equation_content">
    \begin{equation} \label{al32} e_{k,w}\sim e_{k-3}^{1+r+r^{2}+p+rp+r^{2}p+q+qr+r^{2}q}, \end{equation}
  </div>
  <span class="equation_label">-8</span>
</p>
</div>
<p> and </p>
<div class="equation" id="al33">
<p>
  <div class="equation_content">
    \begin{equation} \label{al33} e_{k,y}\sim e_{k-3}^{1+r+r^{2}+p+rp+r^{2}p+q+qr+r^{2}q} \end{equation}
  </div>
  <span class="equation_label">-7</span>
</p>
</div>
<p> also </p>
<div class="equation" id="al34">
<p>
  <div class="equation_content">
    \begin{equation} \label{al34} e_{k+1}\sim e_{k-3}^{1+r+r^{2}+p+rp+r^{2}p+q+qr+r^{2}q} \end{equation}
  </div>
  <span class="equation_label">-6</span>
</p>
</div>
<p>Comparing the right and left side of error equations (<a href="#al20">-20</a>), (<a href="#al32">-8</a>) and (<a href="#al21">-19</a>), (<a href="#al33">-7</a>), also (<a href="#al22">-18</a>), (<a href="#al34">-6</a>), we obtained the following system of equations: </p>
<div class="equation" id="f32">
<p>
  <div class="equation_content">
    \begin{equation} \label{f32} \left\{  \begin{array}{ll} r^{3} p=(1+r+r^{2}+p+pr+r^{2}p+q+qr+r^{2}q),\\ r^{3} q=2(1+r+r^{2}+p+pr+r^{2}p+q+qr+r^{2}q),\\ r^{4}=4(1+r+r^{2}+p+pr+r^{2}p+q+qr+r^{2}q). \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-5</span>
</p>
</div>
<p> Positive solution of this system is \( p\simeq {1.99829} ,q\simeq {3.99657} \) and \( r\simeq {7.99315}\). Therefore, the C-order of the methods with memory (<a href="#f28">-26</a>) is at least 7.99315. </p>
<p><br /><b class="bfseries">Method TM4</b> </p>
<p>Using the result of the two methods TM2, TM3 and lemma (<a href="#lmm1">2</a>) we have </p>
<div class="displaymath" id="al35">
  \begin{align} \label{al35} & (1+\gamma _{k} f’(\alpha ))\sim \prod _{s=0}^{k-4}e_{s}e_{s,w}e_{s,y},\nonumber \\ & (\beta _{k} +c_{2})\sim \prod _{s=0}^{k-4}e_{s}e_{s,w}e_{s,y},\nonumber \\ & ( f’(\alpha )\beta ^{2}(1+\gamma f’(\alpha ))+2\lambda +f’(\alpha )c_{2}(2\beta (3+\gamma f’(\alpha ))+\nonumber \\*[1mm]& (5+\gamma f’(\alpha ))+c_{2}-2f’(\alpha )c_{3}))\nonumber \\*[1mm]&  \sim \prod _{s=0}^{k-4}e_{s}e_{s,w}e_{s,y}. \end{align}
</div>
<p>Compare the right and left side of error equations (<a href="#al20">-20</a>), (<a href="#al35">-4</a>) and (<a href="#al21">-19</a>), (<a href="#al35">-4</a>), and (<a href="#al22">-18</a>), (<a href="#al35">-4</a>), we have: </p>
<div class="equation" id="al36">
<p>
  <div class="equation_content">
    \begin{equation} \label{al36} e_{k,w}\sim e_{k-4}^{1+r+r^{2}+r^{3}+p+pr+r^{2}p+r^{3}p+q+qr+r^{2}q+r^{3}q}, \end{equation}
  </div>
  <span class="equation_label">-5</span>
</p>
</div>
<p> and </p>
<div class="equation" id="al37">
<p>
  <div class="equation_content">
    \begin{equation} \label{al37} e_{k,y}\sim e_{k-4}^{1+r+r^{2}+r^{3}+p+pr+r^{2}p+r^{3}p+q+qr+r^{2}q+r^{3}q} \end{equation}
  </div>
  <span class="equation_label">-4</span>
</p>
</div>
<p> also </p>
<div class="equation" id="al38">
<p>
  <div class="equation_content">
    \begin{equation} \label{al38} e_{k+1}\sim e_{k-4}^{1+r+r^{2}+r^{3}+p+pr+r^{2}p+r^{3}p+q+qr+r^{2}q+r^{3}q} \end{equation}
  </div>
  <span class="equation_label">-3</span>
</p>
</div>
<p> Comparing the right and left side of error equations (<a href="#al20">-20</a>), (<a href="#al36">-5</a>) and (<a href="#al21">-19</a>), (<a href="#al37">-4</a>), also (<a href="#al22">-18</a>), (<a href="#al38">-3</a>), we have. In the similar fashion we find the final system equation which is given by </p>
<div class="equation" id="f33">
<p>
  <div class="equation_content">
    \begin{equation} \label{f33} \left\{  \begin{array}{ll} r^{4} p=(1+r+r^{2}+r^{3}+p+pr+r^{2}p+r^{3}p+q+qr+r^{2}q+r^{3}q),\\ r^{4} q=2(1+r+r^{2}+r^{3}+p+pr+r^{2}p+r^{3}p+q+qr+r^{2}q+r^{3}q),\\ r^{5}=4(1+r+r^{2}+r^{3}+p+pr+r^{2}p+r^{3}p+q+qr+r^{2}q+r^{3}q) . \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">-2</span>
</p>
</div>
<p> Since positive solution of this system is \(p=1.99979, q=3.99957\) and, \(r=7.99915 \approx 8\), and therefore, we conclude that the C-order of the methods with memory (<a href="#f30">-24</a>) is at least \(7.99915\approx 8\). </p>
<p>Therefore, the proof of the Theorem is completed. <div class="proof_wrapper" id="a0000000016">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="a0000000017">5 Numerical results and comparisons</h1>
<p> Let \(e_{k} = x_{k}-\alpha \) be the error in the \(k^{th}\) iteration, we call the relation </p>
<div class="equation" id="eq30">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq30} e_{k+1}=C e_{k}^{p}+{\mathcal O} (e_{k}^{p+1}) \end{equation}
  </div>
  <span class="equation_label">-1</span>
</p>
</div>
<p> as the error equation. If we can obtain error equation for any iterative method, then the value of \(p\) is the order of convergence. </p>
<p>Suppose that \(x_{k},x_{k-1}\) and \(x_{k+1}\) are three successive iterations closer to the root \(\alpha \). Then, the computational order of convergence \(\rho \) (see <span class="cite">
	[
	<a href="#Cat" >8</a>
	]
</span>) is approximated by using (<a href="#eq3">7</a>) as </p>
<div class="equation" id="eq40">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq40} \rho =\frac{\log \frac{|x_{k+1}-\alpha |}{ |x_{k}-\alpha |}}{\log \frac{|x_{k}-\alpha |}{ |x_{k-1}-\alpha |}}. \end{equation}
  </div>
  <span class="equation_label">0</span>
</p>
</div>
<p>We now present some examples to comparison the performance of present methods \(TM1, TM2, TM3,TM4, TM, NM, BBAM\). And two-step with memory methods of Soleymani <i class="it">et al.</i> (SLTKM) with convergence order of 7.22 <span class="cite">
	[
	<a href="#Soleymani2" >27</a>
	]
</span> are defined as: </p>
<div class="equation" id="eq50">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq50} \left\{  \begin{array}{ll} \gamma _{k}=-\tfrac {1}{N’_{3}(x_{k})},\beta _{k}=-\tfrac {N”_{4}(w_{k})}{2N’_{4}(w_{k})},\lambda _{k}=\tfrac {N”’_{5}(w_{k})}{6}, k=1,2,3,\ldots ,\\ y_{k}=x_{k}-\tfrac {f(x_{k})}{f[x_{k},w_{k}]+\beta _{k} f(w_{k})}, k=0,1,2,\ldots ,\\ x_{k+1}=y_{k}-\tfrac {f(y_{k})}{f[w_{k},y_{k}]+\beta _{k} f(w_{k})+\lambda _{k}(y_{k}-x_{k})(y_{k}-w_{k})}(1+\tfrac {f(y_{k})}{f(x_{k})}).\\ \end{array} \right. \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> Computations are performed using MATHEMATICA software. A comparison between without memory and with memory methods in terms of the maximum convergence order alongside the number of steps per cycle are given in <a href="#fig.1">figure 1</a> </p>
<div class="table"  id="a0000000018">
   <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">1</span> 
  <span class="caption_text">Test functions</span> 
</figcaption> <small class="footnotesize"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p> Nonlinear function </p>

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

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

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{1}(x)= t\log (1+x\sin (x))+e^{-1+x^{2}+x \cos (x)}\sin (\pi x) \)</p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(x_{0}=0.6\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{2}(x)= 1+\frac{1}{x^{4}}-\frac{1}{x}-x^{2}\) </p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(x_{0}=1.4\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{3}(x)= e^{x^3-x}-\cos (x^2-1)+x^3+1\)</p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(x_{0}=-1.5\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{4}(x)= \frac{-5x^2}{2}+x^4+x^5+\frac{1}{1+x^2}\) </p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(x_{0}=1.4\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{5}(x)=\log (1+x^2)+e^{-3x+x^2}\sin (x) \)</p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(x_{0}=0.4\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; 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>\(f_{6}(x)= \frac{8}{17}-\sqrt{6}+\frac{x^{3}}{1+x^{4}}+\sqrt{8+x^{4}}\sin (\frac{\pi }{2+x^2}) \)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\alpha =\sqrt{\pi }\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(x_{0}=1.7\)</p>

    </td>
  </tr>
</table> </small> 
</div>
<figure id="fig.1">
  <div class="centered"><img src="img-0001.png" alt="\includegraphics[width=\textwidth ]{mo.png}" style="None" />
 <figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">1</span> 
  <span class="caption_text">Comparison of methods without memory, with memory and adaptive (\(\% 25, \% 50, \% 75\), and \(\% 100\) of improvements) in terms of the highest possible convergence order.</span> 
</figcaption>  </div>

</figure>
<div class="table"  id="a0000000019">
    <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">2</span> 
  <span class="caption_text">Numerical results for \(f_{1}(x), f_{2}(x)\)</span> 
</figcaption> <small class="footnotesize"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left: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:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\vert x_{1}-\alpha \vert \) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\vert x_{2}-\alpha \vert \) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\vert x_{3}-\alpha \vert \) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        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; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{1}(x),\beta _{0}=\gamma _{0}=\lambda _{0}=0.1\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.21313(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.39355(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.10883(-39)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2456\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.21414(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.38636(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.93320(-40)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2457\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.21455(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.38340(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.87536(-40)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2457\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.20150(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.45682(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.37936(-39)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2453\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.21313(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.31807(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12473(-43)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5517\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.21313(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.42385(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.16123(-42)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.21414(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.31172(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.10374(-43)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5518\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.21414(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.41544(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.13741(-42)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.21455(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.30912(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.96118(-44)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.3753\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.21455(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.41199(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12855(-42)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.20150(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.37754(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.54962(-43)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5514\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.20150(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.50018(-5)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.60479(-42)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{2}(x),\beta _{0}=\gamma _{0}=\lambda _{0}=0.1\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46835(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.60268(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.27605(-108)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2388\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46282(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.58207(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.21389(-108)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2389\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46040(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.57310(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.19088(-108)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2389\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.60717(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.11532(-14)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.31089(-106)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2389\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46835(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.82793(-16)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.13403(-119)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5317\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46835(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.60206(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.93782(-120)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46282(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.79699(-16)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.10053(-119)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5317\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46282(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.58147(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.70999(-120)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46040(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.78359(-16)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.88431(-120)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5220\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46040(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.57252(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.62711(-120)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.60711(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.17133(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.29944(-117)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5318\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; 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>TM4 (<a href="#f16">-48</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(0.60711(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(0.11520(-14)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(0.16842(-117)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
</table> </small> 
</div>
<div class="table"  id="a0000000020">
   <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">3</span> 
  <span class="caption_text">Numerical results for \(f_{3}(x), f_{4}(x)\)</span> 
</figcaption> <small class="footnotesize"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left: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:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\vert x_{1}-\alpha \vert \) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\vert x_{2}-\alpha \vert \) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\vert x_{3}-\alpha \vert \) </p>

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

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{3}(x),\beta _{0}=\gamma _{0}=\lambda _{0}=0.1\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.56205(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.57879(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.14139(-179)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2417\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.56128(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.57562(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.13570(-179)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2417\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.56090(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.57408(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.13299(-179)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2417\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.70544(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.14312(-23)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12585(-175)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2416\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.56205(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12031(-25)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46980(-200)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5335\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.56205(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.57879(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.11038(-194)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.56128(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.11966(-25)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.44966(-200)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5335\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.56128(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.57562(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.10564(-194)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.56090(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.11934(-25)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.44013(-200)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.4971\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.56090(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.57408(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.10340(-194)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.70544(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.29684(-25)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.65051(-197)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5334\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.70544(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.14312(-23)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.15426(-191)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{4}(x),\beta _{0}=\gamma _{0}=\lambda _{0}=0.1\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.19289(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.37539(-3)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.46525(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2489\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.19024(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.34800(-3)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.24511(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2498\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.18803(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.32640(-3)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.14220(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2506\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.18331(-1)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.68016(-10)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.48337(-73)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2407\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.19289(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.35581(-3)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.57300(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5060\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.19289(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.32894(-3)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.25124(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.19024(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.33030(-3)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.31618(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5065\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.19024(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.30520(-3)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.13800(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.18803(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.31015(-3)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.19123(-24)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.7585\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.18803(0)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.28646(-3)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.83136(-25)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.18331(-1)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12512(-9)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.28649(-75)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5310\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; 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>TM4 (<a href="#f16">-48</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(0.18331(-1)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(0.58632(-10)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(0.25626(-78)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
</table> </small> 
</div>
<p><br />  </p>
<div class="table"  id="a0000000021">
    <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">4</span> 
  <span class="caption_text">Numerical results for \(f_{5}(x), f_{6}(x)\)</span> 
</figcaption> <small class="footnotesize"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left: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:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\vert x_{1}-\alpha \vert \) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\vert x_{2}-\alpha \vert \) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\vert x_{3}-\alpha \vert \) </p>

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

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{5}(x),\beta _{0}=\gamma _{0}=\lambda _{0}=0.1\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.72896(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.23509(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.55235(-112)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2400\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.72908(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.23480(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.54729(-112)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2400\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.72915(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.23465(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.54465(-112)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2400\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.74155(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.20479(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.19725(-112)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2173\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.72896(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.49295(-16)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12794(-122)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5330\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.72896(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.23502(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.27176(-122)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.72908(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.49241(-16)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12677(-122)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5330\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.72908(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.23473(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.26909(-122)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.72915(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.49213(-16)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12616(-122)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5056\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.72915(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.23458(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.26870(-122)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.74155(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.43552(16)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.45392(-123)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5050\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.74155(-2)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.20474(-15)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.90133(-123)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(f_{6}(x),\beta _{0}=\gamma _{0}=\lambda _{0}=0.1\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.51844(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.24676(-30)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12709(-229)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2389\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.51846(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.24680(-30)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12724(-223)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2389\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.51847(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.24682(-30)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.12731(-223)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2389\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM (<a href="#f10">-54</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.50848(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.23052(-30)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.75805(-224)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.2389\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.51844(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.43704(-33)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.29624(-251)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5307\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{1}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.51844(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.24676(-30)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.10414(-245)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.51846(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.43707(-33)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.29651(-251)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5307\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{2}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.51846(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.24680(-530)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.10427(-245)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.51847(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.43709(-33)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.29664(-251)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5370\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM4 (<a href="#f16">-48</a>), \(H_{3}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.51847(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.24682(-30)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.10433(-245)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1 (<a href="#f12">-51</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.50848(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.42256(-33)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(0.19722(-251)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(7.5365\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; 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>TM4 (<a href="#f16">-48</a>), \(H_{4}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(0.50848(-4)\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(0.23052(-30)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(0.60406(-246)\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(8.0000\) </p>

    </td>
  </tr>
</table> </small> 
</div>
<p><br />  </p>
<div class="table"  id="a0000000022">
   <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">5</span> 
  <span class="caption_text">Comparison improvement of convergence order the proposed method with other schemes</span> 
</figcaption> <small class="footnotesize"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p> with memory methods</p>

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

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

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

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>BBAM<span class="cite">
	[
	<a href="#Bassiri" >2</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.240</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 81\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>CPJM<span class="cite">
	[
	<a href="#Choubey" >4</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>4.560</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 14\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>CPJM<span class="cite">
	[
	<a href="#Choubey" >4</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>4.790</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 19.75\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>CPJM<span class="cite">
	[
	<a href="#Choubey" >4</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 20\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>CLTAMM<span class="cite">
	[
	<a href="#Cordero2" >7</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 75\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>EM<span class="cite">
	[
	<a href="#Eftekhari" >9</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>12.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 50\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>JM<span class="cite">
	[
	<a href="#Jaiswal" >11</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 75\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>JM<span class="cite">
	[
	<a href="#Jaiswal" >11</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>14.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 75\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>LMBSM<span class="cite">
	[
	<a href="#Lotfi1" >13</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>6.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 50\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>LMBSM<span class="cite">
	[
	<a href="#Lotfi1" >13</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.200</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 30\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>LAM<span class="cite">
	[
	<a href="#Lotfi3" >15</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.240</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 31\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>LAM<span class="cite">
	[
	<a href="#Lotfi3" >15</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>6.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 50\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>LSGAM<span class="cite">
	[
	<a href="#Lotfi4" >16</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.770</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 94.25\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>LSGAM<span class="cite">
	[
	<a href="#Lotfi4" >16</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>12.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 50\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>LSNKKM<span class="cite">
	[
	<a href="#Lotfi5" >17</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>6.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 50\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>LSNKKM<span class="cite">
	[
	<a href="#Lotfi5" >17</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>12.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 50\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>MLAM<span class="cite">
	[
	<a href="#Mohamadi" >19</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.700</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 42.5\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>MLAM<span class="cite">
	[
	<a href="#Mohamadi" >19</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.950</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 48.75\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>NM<span class="cite">
	[
	<a href="#Neta" >20</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>10.815</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 35.19\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>PIDM<span class="cite">
	[
	<a href="#Pe2" >24</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>4.449</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 11.23\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>SLTKM<span class="cite">
	[
	<a href="#Soleymani2" >27</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>12.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 50\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>SLTKM<span class="cite">
	[
	<a href="#Soleymani2" >27</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.240</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 81\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>SM<span class="cite">
	[
	<a href="#Soleymani3" >28</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>10.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 81\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TKM<span class="cite">
	[
	<a href="#Tor2" >30</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 75\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TKM<span class="cite">
	[
	<a href="#Tor2" >30</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>14.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 75\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TKM<span class="cite">
	[
	<a href="#Tor2" >30</a>
	]
</span></p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 16.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>28.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 75\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM<span class="cite">
	[
	<a href="#Traub" >31</a>
	]
</span> </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 2.000 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 2.410</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 20.5\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>WM<span class="cite">
	[
	<a href="#Wang" >32</a>
	]
</span> </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.449</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 11.23\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>WM<span class="cite">
	[
	<a href="#Wang2" >33</a>
	]
</span> </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 5.702</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 42.55\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM1(<a href="#f12">-51</a>)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 7.531</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 83.78\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM2(<a href="#f14">-49</a>)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>4.000 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 7.945</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 98.63\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>TM3(<a href="#f15">-48</a>)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 4.000 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.993</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\% 99.83\)</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; 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>TM4(<a href="#f16">-48</a>)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> 4.000 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>8.000</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\% 100\)</p>

    </td>
  </tr>
</table> </small> 
</div>
<h1 id="a0000000023">6 Conclusion</h1>
<p> We have proposed the new two-step tri-parametric family of iterative methods by using four weight functions. In all these techniques, the proper initial guess is necessary for the convergent <span class="cite">
	[
	<a href="#Tor1" >29</a>
	]
</span>. These techniques have been used in Tables 2-4. There are two main advantages of the adaptive-type methods. Firstly, these new methods have the best efficiency index (equal 2), and second, the convergence order of eight for the proposed methods have been formulated with only three function evaluations compared to other with and without memory methods. From Tables 2,3,4, and 5, it can obtain that the adaptive-methods perform better than the considered methods in this study, while the degree of convergence improved up to \(100\% \) <span class="cite">
	[
	<a href="#AR" >1</a>
	, 
	<a href="#Bassiri" >2</a>
	, 
	<a href="#Choubey" >4</a>
	, 
	<a href="#Cordero2" >7</a>
	, 
	<a href="#Eftekhari" >9</a>
	, 
	<a href="#Jaiswal" >11</a>
	, 
	<a href="#Lotfi1" >13</a>
	, 
	<a href="#Lotfi2" >14</a>
	, 
	<a href="#Lotfi3" >15</a>
	, 
	<a href="#Lotfi4" >16</a>
	, 
	<a href="#Lotfi5" >17</a>
	, 
	<a href="#Mah" >18</a>
	, 
	<a href="#Mohamadi" >19</a>
	, 
	<a href="#Pe2" >24</a>
	, 
	<a href="#Soleymani2" >27</a>
	, 
	<a href="#Tor2" >30</a>
	, 
	<a href="#Traub" >31</a>
	, 
	<a href="#Wang" >32</a>
	, 
	<a href="#Wang2" >33</a>
	, 
	<a href="#ZH" >34</a>
	]
</span>. </p>
<p><br />  <div class="acknowledgement_thmwrapper " id="a0000000024">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The author would like to record his sincerest thanks to an anonymous referee for the worthwhile suggestions and comments which have helped to the readability and reliability of this paper. </p>

  </div>
</div> </p>
<p><br /></p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
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</a> </p>
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</a> </p>
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</a> </p>
</dd>
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</a> </p>
</dd>
  <dt><a name="Mah">18</a></dt>
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</dd>
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</dl>


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