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<title>Extending the radius of convergence for a class of Euler-Halley type methods: Extending the radius of convergence for a class of Euler-Halley type methods</title>
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<h1>Extending the radius of convergence for a class of Euler-Halley type methods</h1>
<p class="authors">
<span class="author">Ioannis K. Argyros\(^\ast \) Santhosh George\(^\bullet \)</span>
</p>
<p class="date">March 17, 2017; accepted: January 10, 2019; published online: January 21, 2020.</p>
</div>
<div class="abstract"><p> The aim of this paper is to extend the radius of convergence and improve the ratio of convergence for a certain class of Euler-Halley type methods with one parameter in a Banach space. These improvements over earlier works are obtained using the same functions as before but more precise information on the location of the iterates. Special cases and examples are also presented in this study. </p>
<p><b class="bf">MSC.</b> 65G99, 65H10, 65J15, 47H17, 49M15. </p>
<p><b class="bf">Keywords.</b> Euler-Halley methods, Banach space, local convergence. </p>
</div>
<p>\(^\ast \)1Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA, e-mail: <span class="tt">iargyros@cameron.edu</span>. </p>
<p>\(^\bullet \)Department of Mathematical and Computational Sciences, NIT Karnataka, India-575 025, e-mail: <span class="tt">sgeorge@nitk.ac.in</span>. </p>
<h1 id="a0000000002">1 Introduction</h1>
<p> Let \(\mathcal{B}_1, \mathcal{B}_2\) be Banach spaces and \(\Omega \) be an open and convex subset of \(\mathcal{B}_1.\) The problem of finding a solution of equation </p>
<div class="equation" id="1.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.1} F(x)=0.\end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p>where \(F:\Omega \longrightarrow \mathcal{B}_2\) is differentiable in the sense of Fréchet is important problem in applied mathematics due its wide applications. </p>
<p>In this paper we study the local convergence of the Euler-Halley-type method (EHTM) defined for each \(n=0,1,2...\) by <span class="cite">
	[
	<a href="#9" >9</a>
	]
</span>–<span class="cite">
	[
	<a href="#14" >14</a>
	]
</span> </p>
<div class="displaymath" id="1.2">
  \begin{eqnarray} \label{1.2} x_{n+1,\alpha }& =& T_{F,\alpha }(x_{n,\alpha }) \end{eqnarray}
</div>
<p> where </p>
<div class="displaymath" id="a0000000003">
  \begin{eqnarray} \nonumber T_{F,\alpha }(x)& =& x-[I+\frac{1}{2}(I-\alpha L_F(x))^{-1}L_F(x)]F’(x)^{-1}F(x)\\ \nonumber K_F(x)& =& F’(x)^{-1}F”(x)F’(x)^{-1}F(x) \end{eqnarray}
</div>
<p> with \(x_0\) being an initial guess and \(\alpha \in (-\infty , +\infty ).\) </p>
<p>Notice that, method (<a href="#1.2">2</a>) becomes Halley method when \(\alpha =\frac{1}{2}\), becomes Chebyshev-Euler method when \(\alpha =0\) and super-Halley method when \(\alpha =1.\) </p>
<p>The local convergence of the EHTM (<a href="#1.2">2</a>) was studied in <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span>, using the second-order generalized Lipschitz assumption with \(L-\)average (see Section2 in <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span>). The radius of the optimal convergence ball and the error estimation of method (<a href="#1.2">2</a>) corresponding to the parameter \(\alpha \) are also estimated for each \(\alpha \in (-\infty , +\infty )\) in <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span>. Huang and Guocham in <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span> also shown that the method (<a href="#1.2">2</a>) with \(\alpha \) is better than the one corresponding to \(-\alpha \) for each \(\alpha {\gt}0\) and the Chebyshev-Euler method is best among all methods in the family with \(\alpha \in (-\infty , 0]\) as far as the choice of initial point and error estimates are concerned. </p>
<p>In this study we use second-order generalized Lipschitz condition with \(K_0-\) average (to be precised in Definition <a href="#D2.1">1</a>) to study the local convergence of method (<a href="#1.2">2</a>). Using second-order generalized Lipschitz condition with \(K_0-\) average we improved the results in <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span>. Moreover, our radius of convergence is better than the one in <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span> and the information on the location of the iterates in our study is more precise than that of <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span>. </p>
<p>The paper is structured as follows. In Section 2 we present the local convergence analysis. We also provide a radius of convergence, computable error bounds and uniqueness result not given in the earlier studies <span class="cite">
	[
	<a href="#1" >1</a>
	]
</span>–<span class="cite">
	[
	<a href="#17" >17</a>
	]
</span>. Special cases and numerical examples are presented in the concluding Section 3. </p>

<h1 id="a0000000004">2 Local Convergence </h1>
<p> Denote by \(S(\lambda ,\xi ), \bar{S}(\lambda ,\xi ),\) respectively the open and closed balls in \(\mathcal{B}_1\) with center \(\lambda \in \mathcal{B}_1\) and of radius \(\xi {\gt} 0.\) </p>
<p>Let \(R {\gt} 0.\) Set \(R_0=\sup \{ t\in [0,R):S(t, R) \subset \Omega \} .\) Let also \(K_0, \bar{K}, K\) be real valued \(C^1\) functions defined on the interval \([0, R_0],\) increasing on \([0, R_0]\) with \(K_0'(t) \geq 0, \bar{K}'(t) \geq 0, K'(t) \geq 0, K_0(0) {\gt} 0, \bar{K}(0) {\gt} 0\) and \(K(0) {\gt} 0.\) Denote by \(\rho \) the smallest positive solution of equation </p>
<div class="equation" id="2.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.1} \int _0^1K_0(t)dt=1. \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p>Define function \(h_0\) by </p>
<div class="displaymath" id="a0000000005">
  \[ h_0(t)=-t+\int _0^t(t-u)K_0(u)du. \]
</div>
<p> Notice that \(h_0'(\rho ) =0\) and \(h_0'(t) {\lt} 0\) for all \(t\in [0,\rho ).\) We need the notion of the second-order generalized center-Lipschitz condition with \(K_0-\)average in \(S(p,\rho ).\) <div class="definition_thmwrapper " id="D2.1">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1</span>
  </div>
  <div class="definition_thmcontent">
  <p> We say that \(F\) satisfies the second-order generalized Lipschitz condition with \(K_0-\) average in \(S(p,\rho ),\) if there exists \(p\in \Omega \) such that \(F(p)=0\) and \(F'(p)^{-1}\in L(\mathcal{B}_2,\mathcal{B}_1);\) </p>
<div class="displaymath" id="a0000000006">
  \[ \| F'(p)^{-1}F''(p)\leq K_0(0) \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000007">
  \[ \| F'(p)^{-1}(F''(x)-F''(p))\| \leq \int _0^{\| x-p\| }K_0'(u)du \]
</div>
<p> for all \(x\in S(p,\rho ).\) </p>

  </div>
</div><div class="definition_thmwrapper " id="D2.2">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2</span>
  </div>
  <div class="definition_thmcontent">
  <p> We say that \(F\) satisfies the second-order generalized Lipschitz condition with \(K-\)average in \(S(p,R_0),\) if there exists \(p\in \Omega \) such that \(F(p)=0\) and \(F'(p)^{-1}\in L(\mathcal{B}_2, \mathcal{B}_1);\) </p>
<div class="displaymath" id="a0000000008">
  \[ \| F'(p)^{-1}F''(p)\| \leq K(0) \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000009">
  \[ \| F'(p)^{-1}(F''(x)-F''(p+\theta (x-p)))\| \leq \int _{\theta \| x-p\| }^{\| x-p\| }K'(u)du \]
</div>
<p> for all \(x\in S(p, R_0)\) and \(\theta \in [0,1].\) </p>

  </div>
</div>Next, we introduce the notion of second-order generalized \(K_0-\)restricted Lipschitz condition with \(\bar{K}-\)average in \(S(p,\rho ).\) <div class="definition_thmwrapper " id="D2.3">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">3</span>
  </div>
  <div class="definition_thmcontent">
  <p> We say that \(F\) satisfies the second-order generalized \(K_0-\)restricted Lipschitz condition with \(\bar{K}-\)average in \(S(p,\rho ),\) if there exists \(p\in \Omega \) such that \(F(p)=0\) and \(F'(p)^{-1}\in L(\mathcal{B}_2, \mathcal{B}_1);\) </p>
<div class="displaymath" id="a0000000010">
  \[ \| F'(p)^{-1}F''(p)\| \leq \bar{K}(0) \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000011">
  \[ \| F'(p)^{-1}(F''(x)-F''(p+\theta (x-p)))\| \leq \int _{\theta \| x-p\| }^{\| x-p\| }\bar{K}'(u)du \]
</div>
<p> for all \(x\in S(p, \rho )\) and \(\theta \in [0,1].\) </p>

  </div>
</div><div class="remark_thmwrapper " id="R2.4">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">4</span>
  </div>
  <div class="remark_thmcontent">
  <p> The introduction of function was not possible before, since \(K=\bar{K}(K_0).\) Clearly, we have </p>
<div class="equation" id="2.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.2} K_0(t)\leq K(t) \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<div class="equation" id="2.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.3} \bar{K}(t)\leq K(t) \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p>for all \(t\in I\subseteq [0, R_0].\) We have noticed that iterates \(\{ x_n\} \) lie in \(S(p,\rho )\) which is a more accurate location than \(S(p,R_0),\) since \(\rho \leq R_0\) and the estimate </p>
<div class="equation" id="2.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.4} \| F'(x)^{-1}F'(p)\| \leq -\tfrac {1}{h_0’(\| x-p\| )} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p>(obtained using Definition <a href="#D2.1">1</a>) is more precise than </p>
<div class="equation" id="2.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.5} \| F'(x)^{-1}F'(p)\| \leq -\tfrac {1}{h’(\| x-p\| )} \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p>(using Definition <a href="#D2.3">3</a> (see <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span>)), where </p>
<div class="displaymath" id="a0000000012">
  \[ h(t)=-t+\int _0^t(t-u)K(u)du. \]
</div>

  </div>
</div>Define also function \(\bar{h}\) by </p>
<div class="displaymath" id="a0000000013">
  \[ \bar{h}(t)=-t+\int _0^1(t-u)\bar{K}(u)du. \]
</div>
<p> Then, we have that </p>
<div class="equation" id="2.6">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.6} h_0(t)\leq h(t) \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p>and </p>
<div class="equation" id="2.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.7} \bar{h}(t)\leq h(t)\, \, \textnormal{for all}\, \, \, t\in I. \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p>Suppose from now on that </p>
<div class="equation" id="2.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.8} h_0(t)\leq \bar{h}(t) \, \, \textnormal{for all}\, \, \, t\in I. \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>Then, the results in <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span> can be written with \(\bar{h}\) replacing \(h\) and estimate (<a href="#2.4">4</a>) replacing (<a href="#2.5">5</a>). If </p>
<div class="equation" id="2.9">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.9} \bar{h}(t)\leq h_0(t) \, \, \textnormal{for all}\, \, \, t\in I. \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p>Then, the results in <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span> can be written with \(h_0\) replacing \(h.\) Hence, we arrived at: <div class="theorem_thmwrapper " id="T2.5">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Suppose: \(F\) satisfies the second-order generalized \(K_0-\)restricted Lipschitz condition with \(\bar{K}-\)average in \(S(p,\rho ).\) </p>
<ol class="enumerate">
  <li><p>Let \(\alpha \leq 0.\) Then, \(\bar{\rho }_\alpha \) is the unique solution of equation </p>
<div class="equation" id="2.10">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.10} 1+(\tfrac {1}{2}-\alpha )K_{\bar{h}}(t)=0 \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p>in \((0, \rho ).\) Moreover, \(\bar{\rho }_\alpha \) is the closest repelling extraneous fixed point of \(T_{\bar{h},\alpha }(t)\) to zero for \(t\) being a real number. Furthermore, if \(\bar{K}(t)\) exists and \(\bar{h}(t)\) satisfies hypotheses of Definition <a href="#D2.3">3</a> in \(S(p,\rho )\subseteq \mathbb {C},\) then \(\bar{\rho }_\alpha \) is the closest repelling extraneous fixed point of \(\{ T_{\bar{h},\alpha }(t)\} \) to \(0\) for \(t\in S(0, \rho )\subseteq \mathbb {C}.\) </p>
</li>
  <li><p>\(\bar{\rho }_\alpha \) increases, if \(\alpha \) increases in \((-\infty , 0].\) </p>
</li>
  <li><p>\(\bar{\rho }_{-\alpha } \leq \bar{\rho }_\alpha \) for all \(\alpha {\gt}0.\) </p>
</li>
  <li><p>Sequence \(\{ T^n_{F,\alpha }(x_{0,\alpha })\} \) defined by \(x_{0,\alpha }=x_0\in S(p,\bar{\rho }_{-\alpha } )-\{ p\} \) converges to \(p\) such that for all \(n=0,1,2,\ldots , \alpha \in (-\infty , +\infty )\) </p>
<div class="displaymath" id="a0000000014">
  \[ \| x_{n+1,\alpha }-p\| \leq y_{n+1,-|\alpha |}\leq \bar{q}_\alpha ^{3^{n+1}-1} y_{0, \alpha } \]
</div>
<p> where \(y_{n+1,-|\alpha |}= T_{\bar{h},-|\alpha |}(y_{n,-|\alpha |}), y_{0,\alpha }=y_0=\| x_0-p\| \in S(0, \bar{\rho }_{-|\alpha |})\) and </p>
<div class="equation" id="2.11">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.11} \bar{q}_\alpha =\sqrt{\tfrac {T_{\bar{h},-|\alpha |}(y_0)}{y_0}}\in (0,1). \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
</li>
  <li><p>Sequence \(\{ y_{n,\alpha }\} \) converges optimaly to zero for all \(\alpha {\lt} 0.\) Moreover, if \(\alpha _2 {\lt} \alpha _1 {\lt} 0,\) then </p>
<div class="displaymath" id="a0000000015">
  \[ 0 {\lt} y_{n,\alpha _1} {\lt} y_{n,\alpha _2} \]
</div>
<p> holds for all \(y_{0,\alpha _1}=y_{0,\alpha _2}=y_0\in (0, \bar{\rho }_{\alpha _2}).\) </p>
</li>
</ol>

  </div>
</div><div class="remark_thmwrapper " id="R2.6">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">6</span>
  </div>
  <div class="remark_thmcontent">
  <p> (a) Let \(\rho _\alpha , q_\alpha \) be the radius of convergence and ratio of convergence, respectively corresponding to \(\bar{\rho }_\alpha , \bar{q}_\alpha \) <i class="it">i.e.</i> \(\rho _\alpha \) satisfies </p>
<div class="equation" id="2.12">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.12} 1+(\tfrac {1}{2}-\alpha )K_h(t)=0 \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p>and </p>
<div class="equation" id="2.13">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.13} q_\alpha =\sqrt{\tfrac {T_{h,-|\alpha |}(y_0)}{y_0}}\in (0,1). \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p>Then, in view of (<a href="#2.3">3</a>) and (<a href="#2.7">7</a>), we have that </p>
<div class="equation" id="2.14">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.14} \rho _\alpha \leq \bar{\rho }_\alpha \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p>and </p>
<div class="equation" id="2.15">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.15} \bar{q}_\alpha \leq q_\alpha . \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p>Hence, (<a href="#2.14">14</a>) and (<a href="#2.15">15</a>) justify the advantages claimed in the introduction (see also the numerical examples). </p>
<p>(b) Radius \(\rho \) and function \(\bar{K}\) can be introduced in a different way as follows:<br />Suppose: There exists function \(w_0\) defined on \([0, R_0)\) with \(w_0(0)=0\) such that </p>
<div class="equation" id="2.16">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.16} \| F'(p)^{-1}(F'(x)-F'(p))\| \leq w_0(\| x-p\| ) \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p>for all \(x\in S(p,R_0).\) Let \(r\) be the smallest positive solution of equation </p>
<div class="equation" id="2.17">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.17} w_0(t)=1. \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p>If \(x\in S(p, r),\) then we have \(F'(x)^{-1}\in L(\mathcal{B}_2,\mathcal{B}_1)\) and </p>
<div class="equation" id="2.18">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.18} \| F'(x)^{-1}F'(p)\| \leq \tfrac {1}{1-w_0(\| x-p\| )}. \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p>Suppose that </p>
<div class="equation" id="2.19">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.19} w_0(t)\leq 1+h_0'(t) \, \, \textnormal{for all}\, \,  t\in I\subseteq [0,\rho ]. \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p>Then, (<a href="#2.18">18</a>) gives a better upper bound on \(\| F'(x)^{-1}F'(p)\| \) than (<a href="#2.4">4</a>). Then, since the iterates \(\{ x_n\} \) stay in \(S(p,r)\) this ball can be used in Definition <a href="#D2.3">3</a>) to introduce function \(K^1=K^1(r)\) replacing \(\bar{K}.\) Then, clearly \(r, K^1\) can replace \(\rho , \bar{K}\) in Theorem <a href="#T2.5">5</a>. Let </p>
<div class="equation" id="2.20">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.20} 1+(\tfrac {1}{2}-\alpha )K_{h^1}^1(t) =0 \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p>in \((0,r)\) and </p>
<div class="equation" id="2.21">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.21} q_\alpha ^1=\sqrt{\tfrac {T_{h^1,-|\alpha |}(y_0)}{y_0}}. \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p>where </p>
<div class="equation" id="2.22">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.22} h^1(t)=-t+\int _0^t(t-u)K^1(u)du. \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p>Suppose that \(r\leq \rho \) and </p>
<div class="equation" id="2.23">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.23} K^1(t)\leq \bar{K}(t) \, \, \textnormal{for all}\, \,  t\in I\subseteq [0,r], \end{equation}
  </div>
  <span class="equation_label">23</span>
</p>
</div>
<p>then, we have </p>
<div class="equation" id="2.24">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.24} \bar{\rho }_\alpha \leq r_\alpha \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p>and </p>
<div class="equation" id="2.25">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.25} q_\alpha ^1\leq \bar{q}_\alpha . \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p>Estimates (<a href="#2.24">24</a>) and (<a href="#2.25">25</a>) show that the radius of convergence can be enlarged even further and the error bounds can be improved even further too (see also the numerical examples). </p>

  </div>
</div></p>

<h1 id="a0000000016">3 Special cases and examples</h1>
<p> The numerical examples are presented in this section. </p>
<h2 id="a0000000017">3.1 Special case:</h2>
<p> Kantorovich-type hypothesis. Let \(K(t)=\beta t+\gamma \) for some \(\beta \geq 0\) and \(\gamma {\gt} 0.\) The other “\(K\)“ functions can be defined similarly (see also the numerical examples). </p>
<h2 id="a0000000018">3.2 Special case:</h2>
<p>(Smale-Wang-type hypothesis). Let \(K(t)=\tfrac {2\delta }{(1-\delta t)^3}\) for some \(\delta {\gt} 0.\) The other ”\(K\)“ functions can be defined similarly. </p>
<p><div class="example_thmwrapper " id="tab:1">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">7</span>
  </div>
  <div class="example_thmcontent">
  <p>Let \(X=Y=\mathbb {R}^3, D=\bar{U}(0,1), p=(0, 0, 0)^T.\) Define function \(F\) on \(D\) for \(w=(x,y, z)^T\) by </p>
<div class="displaymath" id="a0000000019">
  \[  F(w)=(e^x-1, \tfrac {e-1}{2}y^2+y, z)^T.  \]
</div>
<p> Then, the Fréchet-derivative is given by </p>
<div class="displaymath" id="a0000000020">
  \[ F'(v)=\left[ \begin{array}{ccc} e^x& 0& 0\\ 0& (e-1)y+1& 0\\ 0& 0& 1 \end{array}\right].  \]
</div>
<p> In this case \(K(t)=2et+1, K_0(t)(e-1)t+1,\,  \bar{K}(t)=2e^\rho t+1, w_0(t)=(e-1)t,\,  K_1(t)=2e^{\frac{1}{e-1}}t+1, \rho =\frac{-1+\sqrt{1+2(e-1)}}{e-1}.\) </p>
<p>Notice that \(w_0(t) {\lt} K_0(t) {\lt}K_1(t) {\lt} \bar{K}(t) {\lt} K(t).\) Then the parameters are given in Table <a href="#tab. 1">1</a>. </p>
<div class="table"  id="tab. 1">
    <div class="centered"> <small class="footnotesize"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(-\alpha \) </p>

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

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

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

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

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

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

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.4</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2236</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.89587717</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2515</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.70428172</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2566</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:center" 
        rowspan=""
        colspan="">
      <p>0.67821226</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.5</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2157</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.70376365</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2422</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.58159283</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2469</p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.56343970</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.6</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2085</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.51959637</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2337</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.46428071</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2382</p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.45348383 </p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>1.0</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1850 </p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.48909481</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2063</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.29397157</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.2101</p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.27074561</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>2.0</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1473 </p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.82647906</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1628</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.61788060</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1655</p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.59280075</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>3.0</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1241 </p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.91927180</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1368</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.70577233</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1382</p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.68001607</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>4.0</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1079 </p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.96383321</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1175</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.74855766</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1194</p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.72259296</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.0</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0958 </p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.99009945</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1040</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.77402979</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1054</p>

    </td>
    <td  style="text-align:center" 
        rowspan=""
        colspan="">
      <p>0.74796064</p>

    </td>
  </tr>
  <tr>
    <td  style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>6.0</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>0.0863 </p>

    </td>
    <td  style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>1.60074361</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>0.0933</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>0.79090015</p>

    </td>
    <td  style="text-align:center; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>0.0945</p>

    </td>
    <td  style="text-align:center; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>0.76482714 </p>

    </td>
  </tr>
</table> </small> <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">1</span> 
  <span class="caption_text">Comparison table for the parameters.</span> 
</figcaption>  </div> 
</div>
<p>Clearly, the new results appearing in columns 4–7 are such that the radii are larger leading to a wider choice of initial points and the ratio is smaller implying fewer iterates to arrive at a desired error tolerance than in columns 2 and 3. It is worth noticing that these advantages are obtained under the same computational cost, since in practice the computation of \(K\) requires the computation of \(w_0, K_0, K_1, \bar{K}\) as special cases. Hence, the claims made previously are justified. </p>

  </div>
</div></p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
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</dd>
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</dd>
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</dd>
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</a> </p>
</dd>
  <dt><a name="13">13</a></dt>
  <dd><p><i class="sc">Z. Huang</i>, <i class="it">On a family of Chebyshev-Halley type methods in Banach space under weaker Smale condition</i>, Numer. Math. JCU <b class="bf">9</b> (2000), pp.&#160;37–44. </p>
</dd>
  <dt><a name="14">14</a></dt>
  <dd><p><i class="sc">Z. Huang, M. Guochun</i>, <a href ="https://doi.org/10.1007/s11075-009-9284-1"> <i class="it">On the local convergence of a family of Euler-Halley type iteration with a parameter</i>, Numer. Algor., <b class="bf">52</b> (2009), pp.&#160;419–433. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="15">15</a></dt>
  <dd><p><i class="sc">A.A. Magrenãn</i>, <a href ="https://doi.org/10.1016/j.amc.2014.01.037"> <i class="it">Different anomalies in a Jarratt family of iterative root-finding methods</i>, Appl.Math.Comput.<b class="bf">233</b> (2014), pp.&#160;29-38. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="16">16</a></dt>
  <dd><p><i class="sc">A.A. Magrenãn</i>, <a href ="https://doi.org/10.1016/j.amc.2014.09.061"> <i class="it">A new tool to study real dynamics: The convergence plane</i>, Appl. Math. Comput. <b class="bf">248</b> (2014), pp.&#160;215-224. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="17">17</a></dt>
  <dd><p><i class="sc">X. Wang</i>, <a href ="https://doi.org/10.1007/bf02901952"> <i class="it">Convergence of the iteration of Halley’s family and Smale operator class in Banach space</i>, Sci. China Ser. A 41, (1998), pp.&#160;700–709. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
</dl>


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