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<title>Ball convergence for an Aitken-Newton method: Ball convergence for an Aitken-Newton method</title>
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<h1>Ball convergence for an Aitken-Newton method</h1>
<p class="authors">
<span class="author">Ioannis K. Argyros\(^\ast \), Munish Kansal\(^\bullet \) Vinay Kanwar\(^+\)</span>
</p>
<p class="date">March 18, 2016. Accepted: February 18, 2017. Published online: February 17, 2019.</p>
</div>
<p>\(^\ast \)Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA, e-mail: <span class="tt">iargyros@cameron.edu</span>. </p>
<p>\(^\bullet \) School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, Punjab-147004, India, e-mail: <span class="tt">mkmaths@gmail.com</span>. </p>
<p>\(^+\) University Institute of Engineering and Technology, Panjab University, Chandigarh-160 014, India, e-mail: <span class="tt">vmithil@yahoo.co.in</span>. </p>

<div class="abstract"><p> We present a local convergence analysis of an eighth-order Aitken-Newton method for approximating a locally unique solution of a nonlinear equation. Earlier studies have shown convergence of these methods under hypotheses up to the eighth derivative of the function although only the first derivative appears in the method. In this study, we expand the applicability of these methods using only hypotheses up to the first derivative of the function. This way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study. </p>
<p><b class="bf">MSC.</b> 65D10, 65D99 </p>
<p><b class="bf">Keywords.</b> Nonlinear equations, Aitken-Newton method, local convergence, eighth order of convergence. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(X\), \(Y\) be Banach spaces and \(D\) be a convex subset of \(X.\) Let also \(L(X,Y)\) denote the set of bounded linear operators from \(X\) into \(Y\). Many problems can be written in the form </p>
<div class="equation" id="I1">
<p>
  <div class="equation_content">
    \begin{equation} \label{I1} F(x)=0 \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> using Mathematical Modelling <span class="cite">
	[
	<a href="#Adomian" >1</a>
	]
</span>, <span class="cite">
	[
	<a href="#Arg2" >6</a>
	]
</span>, <span class="cite">
	[
	<a href="#Arg3" >7</a>
	]
</span>, <span class="cite">
	[
	<a href="#Arg5" >9</a>
	]
</span>, <span class="cite">
	[
	<a href="#Traub" >23</a>
	]
</span> where \(F:D\subseteq X\to Y\) is a Fréchet-differentiable operator. Most solution methods for finding a solution \(x^{*}\) of equation <a href="#I1" class="eqref">1</a> are iterative, since closed form solutions can be found only in special cases <span class="cite">
	[
	<a href="#Adomian" >1</a>
	]
</span>-<span class="cite">
	[
	<a href="#Traub" >23</a>
	]
</span>. In this paper, we study the local convergence of Aitken-Newton method <span class="cite">
	[
	<a href="#Pavalo" >21</a>
	]
</span> defined for each \(n=0,1,2,\ldots \) by </p>
<div class="equation" id="M1">
<p>
  <div class="equation_content">
    \begin{equation} \label{M1} \left\{ \,  \begin{aligned}  y_n=& x_n-F’(x_n)^{-1}F(x_n),\\ z_n=& y_n-F’(y_n)^{-1}F(y_n),\\ x_{n+1}=& z_n-[z_n,y_n;F]^{-1}F(z_n) \\ &  -[y_n,z_n;F]^{-1}[z_n,y_n,y_n;F][y_n,z_n;F]^{-1}F(z_n)F’(y_n)^{-1}F(y_n), \end{aligned} \right. \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> where \([\cdot ,\cdot ~ ;F],~ [\cdot ,\cdot ,\cdot ~ ;F]\) are divided differences of order one and two, respectively and </p>
<div class="equation" id="I2">
<p>
  <div class="equation_content">
    \begin{equation} \label{I2} [x,y;F]=\int _0^{1}F'(y+\theta (x-y))d\theta . \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p>The above method <a href="#M1" class="eqref">2</a> attains eighth-order of convergence using five functional evaluations, viz. \(F(x_n), ~ F'(x_n), ~ F(y_n)\), \(F'(y_n)\) and \(F(z_n)\), per iteration. Therefore, the efficiency index <span class="cite">
	[
	<a href="#Traub" >23</a>
	]
</span> of the proposed method is \(E=\sqrt[5]{8}\approx 1.51\), when \(X=Y=\mathbb {R}\). The convergence of method <a href="#M1" class="eqref">2</a> was shown in <span class="cite">
	[
	<a href="#Pavalo" >21</a>
	]
</span> using Taylor expansions and hypotheses reaching up to the eighth derivative of the function \(F\) although only first derivative appears in the method. We will show that method <a href="#M1" class="eqref">2</a> is well-defined and convergent using hypotheses only on the first derivative in the more general setting of a Banach space. Notice that the method <a href="#M1" class="eqref">2</a> was not shown to be well defined in <span class="cite">
	[
	<a href="#Pavalo" >21</a>
	]
</span>. However, the eighth order of convergence was shown assuming that method <a href="#M1" class="eqref">2</a> is well defined which may not be the case. These hypotheses limit the applicability of method <a href="#M1" class="eqref">2</a>. </p>
<p>As a motivational example, define function \(F\) on \(D=[-\frac{1}{2},\frac{5}{2}]\) by </p>
<div class="equation" id="a0000000003">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber F(x)=\begin{cases}  x^3\ln {x^2}+x^5-x^4,\qquad x\neq 0,\\ 0,~ ~ ~ \qquad \qquad \qquad \qquad x=0. \end{cases} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> We have \(x^{*}=1\), </p>
<div class="equation" id="I2">
<p>
  <div class="equation_content">
    \begin{equation} \label{I2} \begin{aligned} & F’(x)=3x^2\ln {x^2}+5x^4-4x^3+2x^2,\\ & F”(x)=6x\ln {x^2}+20x^3-12x^2+10x,\\ & F”’(x)=6\ln {x^2}+60x^2-24x+22.\\ \end{aligned} \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> Clearly, function \(F'''(x)\) is unbounded on \(D\). Hence, the results in <span class="cite">
	[
	<a href="#Pavalo" >21</a>
	]
</span> cannot be applied to solve equation \(F(x)=0,\) where \(F\) is given by <a href="#I2" class="eqref">6</a>. Moreover, the results in <span class="cite">
	[
	<a href="#Pavalo" >21</a>
	]
</span> do not provide computable convergence radii, error bounds on the distances \(|x_n-x^{*}|\) and uniqueness of the solution results. We address all these problems using only hypotheses on the first derivative. We use <i class="itshape">the computational order of convergence</i> (COC) to approximate the convergence order (which does not depend upon the solution \(x^{*}\)). Moreover, we present the results in a more general setting of a Banach space. </p>
<p>The rest of the paper is organized as follows: In Section <a href="#sec:2">2</a>, we present the local convergence of method <a href="#M1" class="eqref">2</a>. The numerical examples are presented in the concluding Section <a href="#sec:3">3</a>. </p>
<h1 id="sec:2">2 Local convergence analysis</h1>
<p> We present the local convergence analysis of method <a href="#M1" class="eqref">2</a> in this section using some scalar functions and parameters.</p>
<p>Let \(L_0{\gt}0,~ L{\gt}0,~ K{\gt}0,\) and \(M\geq 1 \) be given parameters. Define function \(g_1\) on the interval \([0,\frac{1}{L_0})\) by </p>
<div class="equation" id="a0000000004">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber \begin{aligned}  g_1(t)& =\tfrac {Lt}{2(1-L_0t)},\\ \end{aligned} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> and parameter \(r_1\) by </p>
<div class="displaymath" id="a0000000005">
  \[ r_1=\tfrac {2}{2L_0+L}. \]
</div>
<p> We have that \(g_1(r_1)=1\) and \(0\leq g_1(t){\lt}1\) for each \(t\in [0,r_1)\). Define functions \(p_1\) and \(h_{p_1}\) on the interval \([0,\frac{1}{L_0})\) by </p>
<div class="displaymath" id="a0000000006">
  \[ p_1(t)=L_0g_1(t)t \]
</div>
<p> and \(h_{p_1}(t)={p_1}(t)-1.\)</p>
<p>We get that \(h_{p_1}(0)=-1{\lt}0\) and \(h_{p_1}(t)\to +\infty \) as \(t\to \frac{1^{-}}{L_0}\). It follows from the intermediate value theorem that function \(h_{p_1}\) has zeros in the interval \((0,\frac{1}{L_0}).\) Denote by \(r_{p_1}\) the smallest such zero. Moreover, define functions \(g_2\) and \(h_{2}\) on the interval \([0,r_{p_1})\) by </p>
<div class="equation" id="a0000000007">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber \begin{aligned}  g_2(t)=\tfrac {Lg_1^2(t)t^3}{2(1-p(t))} \end{aligned} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> and \(h_{2}(t)=g_2(t)-1.\) </p>
<p>We get that \(h_{2}(0)=-1{\lt}0\) and \(h_{2}(t)\to +\infty \) as \(t\to r_{p_1}^{-}.\) Denote by \(r_{2}\) the smallest zero of function \(h_2\) in the interval \((0,r_2)\). Furthermore, define functions \(p_2\) and \(h_{p_2}\) on the interval \([0,r_{p_1})\) by </p>
<div class="displaymath" id="a0000000008">
  \[ p_2(t)=\tfrac {L_0}{2}\Big(g_1(t)+g_2(t)\Big)t \]
</div>
<p> and \(h_{p_2}(t)={p_2}(t)-1.\) </p>
<p>We have that \(h_{p_2}(0)=-1{\lt}0\) and \(h_{p_2}(t)\to +\infty \) as \(t\to r_{p_1}^{-}\). Denote by \(r_{p_2}\) the smallest such zero of function \(h_{p_2}\) in the interval \((0,r_{p_1}).\) Finally, define functions \(g_3\) and \(h_{3}\) on the interval \([0,r_{p_2})\) by </p>
<div class="equation" id="a0000000009">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber \begin{aligned} & g_3(t)=\left(1+\tfrac {M}{1-p_2(t)}+\tfrac {KM^2 g_1(t)t}{(1-p_2(t))^2(1-p_1(t))}\right)g_2(t) \end{aligned} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> and \(h_3(t)=g_3(t)-1.\) </p>
<p>We obtain that \(h_3(0)=-1{\lt}0\) and \(h_3(t)\to +\infty \) as \(t \to r_{p_2}^{-}\). Denote by \(r_{3}\) the smallest zero of function \(h_3\) in the interval \((0,r_{p_2}).\) Define the radius of convergence \(r\) by </p>
<div class="equation" id="L1">
<p>
  <div class="equation_content">
    \begin{equation} \label{L1} r=\min \{ r_i\} ,~  i=1,2,3. \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> Then, we have that </p>
<div class="displaymath" id="L2">
  \begin{align} \label{L2} 0{\lt}r\leq r_1{\lt}\tfrac {1}{L_0} \end{align}
</div>
<p>and for each \(t\in [0,r)\) </p>
<div class="equation" id="L3">
<p>
  <div class="equation_content">
    \begin{equation} \label{L3} 0\leq g_i(t)<1,~  i=1,2,3 \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> and </p>
<div class="equation" id="L4">
<p>
  <div class="equation_content">
    \begin{equation} \label{L4} 0\leq p_j(t)<1, ~ j=1,2. \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p>Let \(U(z,\rho )~ \text{and}~ \bar{U}(z,\rho )\) stand respectively for the open and closed balls in \(X\) with center at \(z\in X\) and of radius \(\rho {\gt}0.\) Next, the local convergence analysis of method <a href="#M1" class="eqref">2</a> shall be presented using previous notations. </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> Let \(F:{D} \subseteq X \to Y\) be a Fréchet-differentiable operator. Let \([\cdot ,\cdot ~ ;F],~ [\cdot ,\cdot ,\cdot ~ ;F]\) are divided differences of order one and two on \(D\), respectively. Suppose there exist \(x^{*}\in D\) and \(L_0{\gt}0,\) such that for each \(x\in D\) </p>
<div class="equation" id="L5">
<p>
  <div class="equation_content">
    \begin{equation} \label{L5} F(x^{*})=0,~ F'(x^{*})^{-1}\in L(Y,X)\quad \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p> and </p>
<div class="equation" id="L6">
<p>
  <div class="equation_content">
    \begin{equation} \label{L6} \lVert F'(x^{*})^{-1}(F'(x)-F'(x^{*}))\rVert \leq L_0\lVert x-x^{*}\rVert . \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p> Moreover, suppose there exist \(L{\gt}0,~ K{\gt}0\) and \(M\geq 1\) such that for each \(x,y\in D_0:=D\cap U\left(x^{*},\frac{1}{L_0}\right)\), we have </p>
<div class="equation" id="L7">
<p>
  <div class="equation_content">
    \begin{equation} \label{L7} \lVert F'(x^{*})^{-1}(F'(x)-F'(y))\rVert \leq L\lVert x-y \rVert ,\quad , \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<div class="equation" id="L8">
<p>
  <div class="equation_content">
    \begin{equation} \label{L8} \lVert F'(x^{*})^{-1}F'(x)\rVert \leq M, \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<div class="equation" id="L9">
<p>
  <div class="equation_content">
    \begin{equation} \label{L9} \lVert F'(x^{*})^{-1}[x,y,y;F] \rVert \leq K, \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> and </p>
<div class="equation" id="L10">
<p>
  <div class="equation_content">
    \begin{equation} \label{L10} \bar{U}(x^{*},r)\subseteq D, \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p> where radius of convergence \(r\) is defined by <a href="#L1" class="eqref">7</a>. Then, the sequence \(\{ x_n\} \) generated for \(x_0\in U(x^{*},r)\)-\(\{ x^{*}\} \) by method <a href="#M1" class="eqref">2</a> is well defined, remains in \(U(x^{*},r)\) for each \(n=0,1,2,\ldots \) and converges to the solution \(x^{*}\). Moreover, the following error estimates hold </p>
<div class="displaymath" id="L11">
  \begin{align} \label{L11} \lVert y_n-x^{*} \rVert \leq g_1(\lVert x_n-x^{*}\rVert )\lVert x_n-x^{*}\rVert \leq \lVert x_n-x^{*}\rVert {\lt}r, \end{align}
</div>
<div class="equation" id="L12">
<p>
  <div class="equation_content">
    \begin{equation} \label{L12} \lVert z_n-x^{*} \rVert \leq g_2(\lVert x_n-x^{*}\rVert )\lVert x_n-x^{*}\rVert \leq \lVert x_n-x^{*}\rVert \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p> and </p>
<div class="equation" id="L13">
<p>
  <div class="equation_content">
    \begin{equation} \label{L13} \lVert x_{n+1}-x^{*} \rVert \leq g_3(\lVert x_n-x^{*}\rVert )\lVert x_n-x^{*}\rVert \leq \lVert x_n-x^{*}\rVert , \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p> where the \(``g_i,~ i=1,2,3"\) functions are defined previously. For \(T\in [r,\frac{2}{L_0}),\) the limit point \(x^{*}\) is the only solution of equation \(F(x)=0\) in \(D_1:=D\cap \bar{U}(x^{*},T).\) </p>

  </div>
</div> </p>
<p><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>We shall show using mathematical induction that sequence \(\{ x_n\} \) of iterates generated by <a href="#M1" class="eqref">2</a> is well defined, remains in \(U(x^{*},r)\) and satisfies estimations <a href="#L11" class="eqref">17</a>–<a href="#L13" class="eqref">19</a>. By hypothesis, \(x_0\in U(x^{*},r)-x^{*}\), <a href="#L2" class="eqref">8</a> and <a href="#L6" class="eqref">12</a>, we have that </p>
<div class="displaymath" id="L14">
  \begin{align} \label{L14} \lVert F’(x^{*})^{-1}(F’(x_0)-F’(x^{*}))\rVert \leq L_0\lVert x_0-x^{*}\rVert {\lt}L_0r{\lt}1. \end{align}
</div>
<p> It follows from estimate <a href="#L14" class="eqref">20</a> and the Banach lemma on invertible operators <span class="cite">
	[
	<a href="#Arg2" >6</a>
	, 
	<a href="#Arg5" >9</a>
	, 
	<a href="#Potra" >20</a>
	, 
	<a href="#Rhein" >22</a>
	, 
	<a href="#Traub" >23</a>
	]
</span> that \(F'(x_0)^{-1}\in L(Y,X)\) and </p>
<div class="equation" id="L15">
<p>
  <div class="equation_content">
    \begin{equation} \label{L15} \lVert F'(x_0)^{-1}F'(x^{*})\rVert \leq \tfrac {1}{1-L_0\Vert x_0-x^{*}\rVert }. \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p> Hence, \(y_0\) is well defined. Using <a href="#L2" class="eqref">8</a>, <a href="#L3" class="eqref">9</a>, <a href="#L7" class="eqref">13</a>, <a href="#L15" class="eqref">21</a> and the first substep of method <a href="#M1" class="eqref">2</a> for \(n=0\), we get in turn that </p>
<div class="displaymath" id="L16">
  \begin{align} & \lVert y_0-x^{*}\rVert \leq \label{L16} \\ & \leq \lVert x_0-x^{*}-F’(x_0)^{-1}F’(x_0)\rVert \nonumber \\ & \leq \lVert F’(x_0)^{-1}F’(x^{*})\rVert \Big\Vert \int _{0}^{1}F’(x^{*})^{-1}[F’(x^{*}+\theta (x_0-x^{*}))-F’(x_0))](x_0-x^{*})d\theta \Big\Vert \nonumber \\ & \leq \dfrac {L\lVert x_0-x^{*}\rVert ^2}{2(1-L_0\lVert x_0-x^{*}\rVert )}= g_1(\lVert x_0-x^{*}\rVert )\lVert x_0-x^{*}\rVert \leq \lVert x_0-x^{*}\rVert {\lt}r, \nonumber \end{align}
</div>
<p> which shows <a href="#L11" class="eqref">17</a> for \(n=0\) and \(y_0\in U(x^{*},r)\). Then, we have as in <a href="#L14" class="eqref">20</a> that </p>
<div class="displaymath" id="L17">
  \begin{align} \label{L17} \lVert F’(x^{*})^{-1}(F’(y_0)-F’(x^{*}))\rVert \leq L_0\lVert y_0-x^{*} \rVert & \leq L_0g_1(\lVert x_0-x^{*}\rVert )\lVert x_0-x^{*}\rVert \\ & =p_1(\lVert x_0-x^{*}\rVert ){\lt}p_1(r){\lt}1, \end{align}
</div>
<p> so \(F'(y_0)^{-1}\in L(Y,X)\). Therefore, we have </p>
<div class="equation" id="L18">
<p>
  <div class="equation_content">
    \begin{equation} \label{L18} \lVert F'(y_0)^{-1}F'(x^{*})\rVert \leq \tfrac {1}{1-p_1\Vert x_0-x^{*}\rVert } \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p> and as in <a href="#L16" class="eqref">22</a> </p>
<div class="displaymath" id="L19">
  \begin{align} \label{L19} \lVert z_0-x^{*}\rVert & \leq \tfrac {L\lVert y_0-x^{*}\rVert ^2}{2(1-L_0\lVert y_0-x^{*}\rVert )}\\ & \leq g_2(\lVert x_0-\xi \rVert ) \lVert x_0-x^{*}\rVert \leq \lVert x_0-x^{*}\rVert {\lt}r, \end{align}
</div>
<p> which implies that <a href="#L12" class="eqref">18</a> holds for \(n=0\) and \(z_0\in U(x^{*},r).\) </p>
<p>Next, we show that \([y_0,z_0;F]^{-1}\) exists. In view of the definition of divided difference \([\cdot ,\cdot ;F]\), <a href="#L2" class="eqref">8</a>, <a href="#L3" class="eqref">9</a>, <a href="#L6" class="eqref">12</a>, <a href="#L16" class="eqref">22</a> and <a href="#L18" class="eqref">25</a> that </p>
<div class="displaymath" id="L20">
  \begin{align} \label{L20} & \lVert F’(x^{*})^{-1}\left([z_0,y_0;F]-F’(x^{*})\right)\rVert \leq \\ & \leq \tfrac {L_0}{2}\left(\lVert z_0-x^{*}\rVert +\lVert y_0-x^{*}\rVert \right) \nonumber \\ & \leq \tfrac {L_0}{2}\Big(g_1(\lVert x_0-x^{*}\rVert )\! +\! g_2(\lVert x_0-x^{*}\rVert )\Big)\lVert x_0-x^{*}\rVert \nonumber \\ & =p_2(\lVert x_0-x^{*}\rVert ){\lt}p_2(r){\lt}1, \nonumber \end{align}
</div>
<p> so </p>
<div class="equation" id="L21">
<p>
  <div class="equation_content">
    \begin{equation} \label{L21} \lVert [z_0,y_0;F]^{-1}-F'(x^{*})\rVert \leq \tfrac {1}{1-p_2\Vert x_0-x^{*}\rVert }. \end{equation}
  </div>
  <span class="equation_label">29</span>
</p>
</div>
<p> Hence, \(x_1\) is well defined. Notice that \(\lVert x^{*}+\theta (x_0-x^{*})-x^{*}\rVert \leq \theta \lVert x_0-x^{*}\rVert {\lt}r\) for each \(\theta \in [0,1],\) so \(x^{*}+\theta (x_0-x^{*})\in U(x^{*},r)\) for each \(\theta \in [0,1].\) Then, by <a href="#L5" class="eqref">11</a> and <a href="#L8" class="eqref">14</a>, we get that </p>
<div class="displaymath" id="L22">
  \begin{align} \label{L22} & \lVert F’(x^{*})^{-1} F(x_0) \rVert = \\ & =\lVert F’(x^{*})^{-1}( F(x_0)-F(x^{*}) \rVert \nonumber \\ & =\lVert \int _0^1 F’(x^{*})^{-1} F’(x^{*}+\theta (x_0-x^{*}))(x_0-x^{*}) \rVert \leq M \lVert x_0-x^{*} \rVert \nonumber \end{align}
</div>
<p> and similarly </p>
<div class="equation" id="L23">
<p>
  <div class="equation_content">
    \begin{equation} \label{L23} \begin{aligned}  \lVert F’(x^{*})^{-1} F(y_0) \rVert & \leq M \lVert y_0-x^{*} \rVert \leq M g_1(\lVert x_0-x^{*} \rVert )\lVert x_0-x^{*} \rVert . \end{aligned} \end{equation}
  </div>
  <span class="equation_label">31</span>
</p>
</div>
<p> By the last substep of method <a href="#M1" class="eqref">2</a> for \(n=0\), <a href="#L1" class="eqref">7</a>, <a href="#L2" class="eqref">8</a>, <a href="#L16" class="eqref">22</a>, <a href="#L18" class="eqref">25</a>, <a href="#L19" class="eqref">26</a>, <a href="#L21" class="eqref">29</a>–<a href="#L23" class="eqref">31</a>, we obtain in turn that </p>
<div class="displaymath" id="L24">
  \begin{align} \label{L24} & \lVert x_1-x^{*} \rVert \leq \\ & \leq \lVert z_0-x^{*}\rVert +\lVert [z_0,y_0;F]^{-1}F’(x^{*})\rVert \lVert F’(x^{*})^{-1}F’(z_0)\rVert \nonumber + \\ & \quad +\lVert [z_0,y_0;F]^{-1}F’(x^{*})\rVert \lVert [z_0,y_0;F]F’(x^{*})^{-1}\rVert \lVert [z_0,y_0;F]^{-1}F’(x^{*})\rVert \cdot \nonumber \\ &  \quad \cdot \lVert F’(x^{*})^{-1} [z_0,y_0,y_0;F]^{-1}F’(x^{*})\rVert \lVert F’(y_0)^{-1}F’(x^{*})\rVert \lVert F’(x^{*})^{-1}F(y_0)\rVert \nonumber \\ & \leq \lVert z_0-x^{*} \rVert +\tfrac {M\lVert z_0-x^{*} \rVert }{1-p_2(\lVert x_0-x^{*} \rVert )}+\tfrac {KM^2\lVert z_0-x^{*} \rVert \lVert y_0-x^{*} \rVert }{(1-p_2(\lVert x_0-x^{*} \rVert ))^2(1-p_1(\lVert x_0-x^{*} \rVert ))}\nonumber \\ & =\Big(1+\tfrac {M}{1-p_2(\lVert x_0-x^{*} \rVert )}+\tfrac {KM^2\lVert y_0-x^{*} \rVert }{(1-p_2(\lVert x_0-x^{*} \rVert ))^2(1-p_1(\lVert x_0-x^{*} \rVert ))}\Big)\lVert z_0-x^{*} \rVert \nonumber \\ & \leq g_3(\lVert x_0-x^{*}\rVert )\lVert x_0-x^{*}\rVert \leq \lVert x_0-x^{*}\rVert {\lt}r, \nonumber \end{align}
</div>
<p> which implies that <a href="#L13" class="eqref">19</a> holds for \(n=0\) and \(x_1\in U(x^{*},r).\) By simply replacing \(y_0,z_0,x_1\) by \(y_n,z_n,x_{n+1}\) in the preceding estimates, we complete the induction for estimates <a href="#L11" class="eqref">17</a>–<a href="#L13" class="eqref">19</a>. Then, in view of the estimate \(\lVert x_{n+1}-x^{*}\rVert \leq c \lVert x_{n}-x^{*}\rVert {\lt}r,~ c=g_3(\lVert x_{0}-x^{*}\rVert )\in [0,1),\) we deduce that \(\lim _{n\to \infty }x_n=x^{*}\) and \(x_{n+1}\in U(x^{*},r).\) The proof of the uniqueness follows using standard arguments <span class="cite">
	[
	<a href="#Arg8" >11</a>
	]
</span>. </p>
<p><div class="remark_thmwrapper " id="a0000000011">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2</span>
  </div>
  <div class="remark_thmcontent">
  <p>1) It follows from <a href="#L6" class="eqref">12</a> that condition <a href="#L8" class="eqref">14</a> can be dropped, if we set </p>
<div class="displaymath" id="a0000000012">
  \[ M(t)=1+L_0t \]
</div>
<p> or </p>
<div class="displaymath" id="a0000000013">
  \[ M(t)=M=2,~ \text{since}~ ~ t\in \Big[0,\tfrac {1}{L_0}\Big). \]
</div>
<p>2) The results obtained here can also be used for operators \(F\) satisfying autonomous differential equations <span class="cite">
	[
	<a href="#Arg2" >6</a>
	, 
	<a href="#Arg5" >9</a>
	]
</span> of the form: </p>
<div class="displaymath" id="a0000000014">
  \[ F'(x)=P(F(x)), \]
</div>
<p> where \(P\) is a continuous operator. Then, since \(F'(x^{*})=P(F(x^{*}))=P(0)\), we can apply the results without actually knowing \(x^{*}\). For example, let \(f(x) = e^x-1\). Then, we can choose \(P(x) =x +1.\)<br />3) The radius \(\bar{r_1}=\frac{2}{2L_0+L_1}\) was shown by Argyros <span class="cite">
	[
	<a href="#Arg2" >6</a>
	]
</span> to be the convergence radius of Newton’s method </p>
<div class="equation" id="V31">
<p>
  <div class="equation_content">
    \begin{equation} \label{V31} x_{n+1}=x_n-F'(x_n)^{-1}F(x_n),~ \text{for each}~ n=0,1,2,\ldots \end{equation}
  </div>
  <span class="equation_label">33</span>
</p>
</div>
<p> under the conditions <a href="#L5" class="eqref">11</a>–<a href="#L7" class="eqref">13</a> on \(D\), where \(L_1\) is the Lipschitz constant on \(D.\) We have that \(L\leq L_1\) and \(L_0\leq L_1,\) so \(\bar{r_1}\leq r_1.\) It follows that the convergence radius \(r\) of the method <a href="#M1" class="eqref">2</a> cannot be larger than the convergence radius \(r_1\) of the second order Newton’s method <a href="#V31" class="eqref">33</a>. As already noted in <span class="cite">
	[
	<a href="#Arg2" >6</a>
	]
</span>, \(\bar{r_1}\) is at least as large as the convergence ball given by Rheinboldt <span class="cite">
	[
	<a href="#Rhein" >22</a>
	]
</span> </p>
<div class="equation" id="a0000000015">
<p>
  <div class="equation_content">
    \begin{equation} \nonumber r_R=\tfrac {2}{3L_1}. \end{equation}
  </div>
  <span class="equation_label">34</span>
</p>
</div>
<p> In particular, for \(L_0{\lt}L_1,\) we have that </p>
<div class="displaymath" id="a0000000016">
  \[ r_R{\lt}\bar{r_1} \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000017">
  \[ \tfrac {r_R}{\bar{r_1}}\to \tfrac {1}{3}~ \text{as}~ \tfrac {L_0}{L_1}\to 0. \]
</div>
<p> That is our convergence ball \(\bar{r_1}\) is at most three times larger than Rheinboldt’s. The same value of \(r_R\) was given by Traub <span class="cite">
	[
	<a href="#Traub" >23</a>
	]
</span>. 4) It is worth noticing that method <a href="#M1" class="eqref">2</a> is not changing when we use the conditions of Theorem <a href="#thm1">1</a> instead of stronger conditions used in previous studies. Moreover, we can consider the semi-computational order of convergence defined by </p>
<div class="displaymath" id="a0000000018">
  \begin{align*}  Q_{L}(k)= &  \tfrac {\ln |x_{n+1}-x^{*}| } {\ln |x_n-x^{*}| } \\ Q_\Lambda (k)= &  \ln \Big(\tfrac {|x_{n+1}-x^{*}|}{| x_{n}-x^{*}|}\Big)/\ln \Big(\tfrac {| x_{n}-x^{*}|}{| x_{n-1}-x^{*}|}\Big), \end{align*}
</div>
<p> for which some interesting properties were obtained in <span class="cite">
	[
	<a href="#Potra89" >19</a>
	]
</span> and <span class="cite">
	[
	<a href="#BEQ90" >12</a>
	]
</span>. We can even compute the (full) computational order of convergence defined by </p>
<div class="displaymath" id="a0000000019">
  \begin{align*}  Q^\prime _{L}(k)= &  \tfrac {\ln |x_{n}-x_{n-1}| } {\ln |x_{n-1}-x_{n-2}| } \\ Q^\prime _\Lambda (k)=& \ln \Big(\tfrac {| x_{n}-x_{n-1}|}{| x_{n-1}-x_{n-2}|}\Big)/\ln \Big(\tfrac {| x_{n-1}-x_{n-2}|}{| x_{n-2}-x_{n-3}|}\Big), \end{align*}
</div>
<p> for which interesting properties were obtained in <span class="cite">
	[
	<a href="#BEQ90" >12</a>
	]
</span>. We also recommend to the motivated reader the excellent survey on these notions <span class="cite">
	[
	<a href="#Cat19" >13</a>
	]
</span>, containing full proofs and historical aspects. </p>
<p>This way we obtain in practice the order of convergence in a way that avoids the bounds involving estimates using estimates higher than the first Fréchet derivative of operator \(F.\) Notice also that the computation of \(\xi ^{*}\) does not require knowledge of \(x^{*}.\) 5) Also, condition <a href="#L6" class="eqref">12</a> can be replaced by </p>
<div class="equation" id="L6'">
<p>
  <div class="equation_content">
    \begin{equation} \label{L6'} \lVert F'(x^{*})^{-1} ([x,y;F]-F'(x^{*}))\rVert \leq \tfrac {\bar{L_0}}{2} (\lVert x-x^{*}\rVert +\lVert y-x^{*}\rVert ). \end{equation}
  </div>
  <span class="equation_label">34</span>
</p>
</div>
<p> In this case, the choice of the divided difference given by <a href="#I2" class="eqref">6</a> can be dropped. Moreover, \(\bar{L_0}\) and <a href="#L6'" class="eqref">34</a> can replace \(L_0\) and <a href="#L6" class="eqref">12</a>, respectively in the proof of Theorem <a href="#thm1">1</a>. <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="sec:3">3 Numerical examples</h1>
<p> We present numerical examples in this section. <div class="example_thmwrapper " id="a0000000020">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">3</span>
  </div>
  <div class="example_thmcontent">
  <p>Let \(X=Y=\mathbb {R}^3\), \(D=\bar{U}(0,1),~ x^{*}=(0,0,0)^T\). Define function \(F\) on \(D\) for \(w=(x,y,z)^T\) by </p>
<div class="displaymath" id="a0000000021">
  \[ F(w)=(e^x-1,\tfrac {e-1}{2}y^2+y,z)^T. \]
</div>
<p> Then, the Fréchet derivative is given by </p>
<div class="displaymath" id="a0000000022">
  \[ F'(w)=\begin{bmatrix} {} e^x

& 0

& 0

\\ 0

& (e-1)y+1

& 0

\\ 0

& 0

& 1

\\ \end{bmatrix}. \]
</div>
<p> We have that \(L_0=e-1\), \(L=e^{\frac{1}{L_0}}\), \(M=e^{\frac{1}{L_0}}\), \(K=\tfrac {e^{\frac{1}{L_0}}}{2}\) and \(L_1=e\). The parameters using method <a href="#M1" class="eqref">2</a> are: </p>
<div class="displaymath" id="a0000000023">
  \begin{align*} & r_1=0.382692,~ r_2=0.45738,~ r_3=0.38726, \\ & r=0.38726,~ \bar{r_1}=0.324947,~ r_R=0.245253. \end{align*}
</div>

  </div>
</div> <div class="example_thmwrapper " id="a0000000024">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">4</span>
  </div>
  <div class="example_thmcontent">
  <p>Let \(X=Y=C[0,1],\) the space of continuous functions defined on \([0,1]\) and be equipped with the max norm. Let \(D=\bar{U}(0,1)\) and \(B(x)=F''(x)\) for each \(x\in D\). Define function \(F\) on \(D\) by </p>
<div class="equation" id="a0000000025">
<p>
  <div class="equation_content">
    \begin{equation}  F(\phi )(x)=\phi (x)-5\int _0^1 x\theta \phi (\theta )^3 d\theta . \end{equation}
  </div>
  <span class="equation_label">35</span>
</p>
</div>
<p> We have that </p>
<div class="equation" id="a0000000026">
<p>
  <div class="equation_content">
    \begin{equation}  F'(\phi (\xi ))(x)=\xi (x)-15\int _0^1 x\theta \phi (\theta )^2 \xi (\theta ) d\theta , ~ \text{for each}~ \xi \in D. \end{equation}
  </div>
  <span class="equation_label">36</span>
</p>
</div>
<p> Then, we get that \(x^{*}=0,~ L_0=7.5,~ L_1=15,~ L=15,~ K=15\) and \(M=2.\) The parameters using method <a href="#M1" class="eqref">2</a> are: </p>
<p>\(r_1=0.0666667,~ r_2=0.114372,~ r_3=0.0783549,~ r=0.0783549,\)<br />\(\bar{r_1}=0.0666667,~ r_R=0.0444444.\) <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="example_thmwrapper " id="a0000000027">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">5</span>
  </div>
  <div class="example_thmcontent">
  <p>Returning back to the motivational example at the introduction of this paper, we have that \(L=L_0=146.6629073\), \(M=2\), \(L_1=L\) and \(K=48.3315.\) The parameters using method <a href="#M1" class="eqref">2</a> are: </p>
<div class="displaymath" id="a0000000028">
  \begin{align*} & r_1=0.00454557,~ r_2=.006787,~ r_3=0.004991287,\\ & \bar{r_1}=0.00454557,~ r_R=0.00454557,~ \xi ^{*}=7.8403. \end{align*}
</div>

  </div>
</div> </p>
<p><div class="example_thmwrapper " id="a0000000029">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">6</span>
  </div>
  <div class="example_thmcontent">
  <p>Let \(X=Y=\mathbb {R}\) and define function \(F\) on \(D=\mathbb {R}\) by </p>
<div class="equation" id="N1">
<p>
  <div class="equation_content">
    \begin{equation} \label{N1} F(x)=\beta x-\gamma \sin {(x)}-\delta ,\end{equation}
  </div>
  <span class="equation_label">37</span>
</p>
</div>
<p> where \(\beta ,~ \gamma ,~ \delta \) are given real numbers. Suppose that there exists a solution \(x^{*}\) of \(F(x)=0\) with \(F'(x^{*})\neq 0.\) Then, we have </p>
<div class="displaymath" id="a0000000030">
  \[  L_1=L_0=L=\tfrac {|\gamma |}{|\beta -\gamma \cos {x^{*}}|},~ M=\tfrac {|\gamma |+|\beta |}{|\beta -\gamma \cos {x^{*}}|}~ \text{and}~ K=\tfrac {|\gamma |}{2|\beta -\gamma \cos {x^{*}}|}.  \]
</div>
<p> Then one can find the convergence radii for different values of \(\beta ,~ \gamma \) and \(\delta \). As a specific example, let us consider Kepler’s equation <a href="#N1" class="eqref">37</a> with \(\beta =1\), \(0\leq \gamma {\lt}1\) and \(0\leq \delta \leq \pi .\) A numerical study was presented in <span class="cite">
	[
	<a href="#Danby" >14</a>
	]
</span> for different values of \(\gamma \) and \(\delta .\) </p>
<p>Let us take \(\gamma =0.9\) and \(\delta =0.1\). Then the solution is given by \(x^{*}=0.6308435.\) Hence, for method <a href="#M1" class="eqref">2</a> the parameters are: </p>
<div class="equation" id="a0000000031">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{aligned} & r_1=0.202387,~ r_2=0.261858,~ r_3=0.196578,\\ & r=0.196578, \bar{r_1}=0.202387, r_R=0.202387,~ \xi ^{*}=8.0353. \end{aligned}\end{equation}
  </div>
  <span class="equation_label">38</span>
</p>
</div>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
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</dd>
  <dt><a name="Amat1">2</a></dt>
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</dd>
  <dt><a name="Amat2">3</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s00010-004-2733-y"> <i class="sc">S. Amat, S. Busquier, S. Plaza</i>, <i class="itshape">Dynamics of the King’s and Jarratt iterations</i>, Aeq. Math., <b class="bf">69</b> (2005), 212–213. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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  <dt><a name="Amat3">4</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.amc.2008.08.050"> <i class="sc">S. Amat, M.A. Hernández, N. Romero</i>, <i class="itshape">A modified Chebyshev’s iterative method with at least sixth order of convergence</i>, Appl. Math. Comput. <b class="bf">206 (1)</b> (2008), 164–174. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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</a> </p>
</dd>
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  <dd><p><i class="sc">I.K. Argyros, D. Chen</i>, <i class="itshape">Results on the Chebyshev method in Banach spaces</i>, Proyecciones <b class="bf">12 (2)</b> (1993), 119–128. </p>
</dd>
  <dt><a name="Arg3">7</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.jmaa.2004.04.008"> <i class="sc">I.K. Argyros</i>, <i class="itshape">A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space</i>, J. Math. Anal. Appl., <b class="bf">298</b> (2004), 374–397 <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="Arg4">8</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/s1570-579x(07)x8020-0"> <i class="sc">I.K. Argyros</i>, <i class="itshape">Computational Theory of Iterative Methods</i>, Series: Studies Comput. Math. 15, Editors: C.K. Chui and L. Wuytack, Elsevier, New York, USA, (2007) <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="Arg5">9</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.jco.2011.12.003"> <i class="sc">I.K. Argyros, S. Hilout</i>, <i class="itshape">Weaker conditions for the convergence of Newton’s method</i>, J. Complexity. <b class="bf">28</b> (2012), 364–387. <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="Arg6">10</a></dt>
  <dd><p><i class="sc">I.K. Argyros</i>, <i class="itshape">Numerical Methods in Nonlinear Analysis</i>, World Scientific Publ. Comp. New Jersey, 2013. </p>
</dd>
  <dt><a name="Arg8">11</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s40819-015-0078-y"> <i class="sc">I.K. Argyros, S. George</i>, <i class="itshape">Local convergence for an efficient eighth order iterative method with a parameter for solving equations under weaker conditions</i>, Int. J. Appl. Comput. Math. <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="BEQ90">12</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/BF00049571"> <i class="sc">W.A. Beyer, B.R. Ebanks, C.R. Qualls, </i> <i class="it">Convergence rates and convergence-order profiles for sequences</i>, Acta Appl. Math., <b class="bf">20</b> (1990), 267–284. <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="Cat19">13</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.amc.2018.08.006"> <i class="sc">E. Catinas</i>, <i class="it">A survey on the high convergence orders and computational convergence orders of sequences</i>, Appl. Math. Comput., <b class="bf">343</b> (2019), 1–20. <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="Danby">14</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/bf01686811"> <i class="sc">J.M.A. Danby, T.M. Burkardt</i>, <i class="itshape">The solution of Kepler’s equation I.</i>, Celest. Mech. <b class="bf">31</b> (1983), 95–107. <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="Ezqu1">15</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s10543-009-0226-z"> <i class="sc">J.A. Ezquerro, M.A. Hernández</i>, <i class="itshape">New iterations of R-order four with reduced computational cost</i>, BIT Numer Math. <b class="bf">49</b> (2009), 325–342. <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="GH">16</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/s0898-1221(98)00168-0"> <i class="sc">J.M. Guttiérrez, M.A. Hernández</i>, <i class="itshape">Recurrence relations for the super-Halley method</i>, Comput. Math. Appl., <b class="bf">36</b> (1998), 1–8. <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="Kantro">17</a></dt>
  <dd><p><i class="sc">L.V. Kantorovich, G.P. Akilov</i>, <i class="itshape">Functional Analysis</i>, Pergamon Press, Oxford, 1982. </p>
</dd>
  <dt><a name="Margen">18</a></dt>
  <dd><p><a href ="http:// dialnet.unirioja.es/servlet/tesis?codigo=38821"> <i class="sc">A.A. Magreñán</i>, <i class="itshape">Estudio de la dinámica del método de Newton amortiguado</i>, (PhD Thesis), Servicio de Publicaciones Universidad de La Rioja, 2013. <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="Potra89">19</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/bf00939805"> <i class="sc">F.A. Potra</i>, <i class="itshape">On Q-order and R-order of convergence</i>, J. Optim. Theory Appl., <b class="bf">63</b> (1989) no. 3, 415–431. <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="Potra">20</a></dt>
  <dd><p><i class="sc">F.A. Potra, V. Pták</i>, <i class="itshape">Nondiscrete introduction and iterative processes</i>, Research Notes in Mathematics. <b class="bf">103</b>, Pitman, Boston, MA, 1984. </p>
</dd>
  <dt><a name="Pavalo">21</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.amc.2016.03.036"> <i class="sc">I. Pavaloiu, E. Catinas</i>, <i class="itshape">On a robust Aitken-Newton method based on the Hermite polynomial</i>, Appl. Math. Comput., <b class="bf">287-288</b> (2016), 224–231. <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="Rhein">22</a></dt>
  <dd><p><i class="sc">W.C. Rheinboldt</i>, <i class="itshape">An adaptive continuation process for solving systems of nonlinear equations. Mathematical models and numerical methods</i> (A.N.Tikhonov et al. eds.) pub.3, (19), 129–142 Banach Center, Warsaw, Poland. </p>
</dd>
  <dt><a name="Traub">23</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/b978-0-12-528550-6.50017-x"> <i class="sc">J.F. Traub</i>, <i class="itshape">Iterative Methods for the Solution of Equations</i>, Prentice-Hall, Englewood Cliffs, NJ, 1964. <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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