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<h1>Expanding the applicability of the Gauss-Newton method for a certain class of systems of equations</h1>
<p class="authors">
<span class="author">Ioannis K. Argyros\(^1\) Santhosh George\(^2\)</span>
</p>
<p class="date">November 28, 2013.</p>
</div>
<p>\(^1\)Department of Mathematicsal Sciences, Cameron University, Lawton, OK 73505, USA, Email: <span class="ttfamily">ioannisa@cameron.edu</span>. </p>
<p>\(^2\)Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, India-575 025, Email: <span class="ttfamily">sgeorge@nitk.ac.in</span>. </p>

<div class="abstract"><p> We present a new semi-local convergence analysis of the Gauss-Newton method in order to solve a certain class of systems of equations under a majorant condition. Using a center majorant function as well as a majorant function and under the same computational cost as in earlier studies such as <span class="cite">
	[
	<a href="#11" >11</a>
	]
</span>-<span class="cite">
	[
	<a href="#13" >13</a>
	]
</span>, we present a semilocal convergence analysis with the following advantages: weaker sufficient convergence conditions; tighter error estimates on the distances involved and an at least as precise information on the location of the solution. Special cases and applications complete this study. </p>
<p><b class="bf">MSC.</b> 65H10, 65G99, 65K10, 47H17, 49M15. </p>
<p><b class="bf">Keywords.</b> Gauss-Newton method, Newton’s method, semilocal convergence, least squares problem. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p> Let \(D\subseteq \mathbb {R}^n\) be open. Let \(F:D\rightarrow \mathbb {R}^m\) be continuously Fréchet- differentiable. We are concerned with the problem of finding least squares solutions \(x^\ast \) of the nonlinear least squares problem </p>
<div class="equation" id="1.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.1} \min _{x\in D}\| F(x)\| ^2. \end{equation}
  </div>
  <span class="equation_label">1.1</span>
</p>
</div>
<p>The least squares solutions of (<a href="#1.1">1.1</a>) are stationary points of \(G(x)=\| F(x)\| ^2.\) Diversified problems arising in applied sciences and in engineering can be expressed in a form like (<a href="#1.1">1.1</a>). For example in data fitting \(n\) is the number of parameters and \(m\) is the number of observations. Other examples can be found in <span class="cite">
	[
	<a href="#5" >5</a>
	, 
	<a href="#15" >15</a>
	, 
	<a href="#18" >18</a>
	]
</span> and the references therein. The famous Gauss-Newton method defined by </p>
<div class="equation" id="1.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.2} x_{k+1}=x_k-F'(x_k)^\dagger F(x_k),\, \, \, \, \, \,  k =0,1,\ldots , \end{equation}
  </div>
  <span class="equation_label">1.2</span>
</p>
</div>
<p>where \(x_0\) is an initial point and \(F'(x_k)^\dagger \) the Moore-Penrose inverse of the linear operator \(F'(x_k)\) has been used extensively to generate a sequence \(\{ x_k\} \) converging to \(x^\ast \) <span class="cite">
	[
	<a href="#1" >1</a>
	]
</span>–<span class="cite">
	[
	<a href="#5" >5</a>
	]
</span>, <span class="cite">
	[
	<a href="#7" >7</a>
	, 
	<a href="#9" >9</a>
	, 
	<a href="#19" >19</a>
	, 
	<a href="#13" >13</a>
	, 
	<a href="#14" >14</a>
	, 
	<a href="#16" >16</a>
	]
</span>. </p>
<p>In the present paper, we are motivated by the work of Goncalves and Oliveira in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span> (see also <span class="cite">
	[
	<a href="#11" >11</a>
	]
</span>, <span class="cite">
	[
	<a href="#12" >12</a>
	]
</span>) and optimization considerations. They provided a semilocal convergence analysis for the Gauss-Newton method (<a href="#1.2">1.2</a>) for systems of nonlinear equations where the function \(F\) satisfies </p>
<div class="displaymath" id="a0000000003">
  \[ \| F'(y)^\dagger (I_{\mathbb {R}^m}-F'(x)F'(x)^\dagger )F(x)\| \leq k\| x-y\| ,\quad \forall \,  x ,y\in D, \]
</div>
<p> where \(k\in [0, 1)\) and \(I_{\mathbb {R}^m}\) denotes the identity operator on \({\mathbb {R}^m}.\) Their semilocal convergence analysis is based on the construction of a majorant function (see \((h_3)\)). Their results unify the classical results for functions involving Lipschitz derivative <span class="cite">
	[
	<a href="#5" >5</a>
	, 
	<a href="#6" >6</a>
	, 
	<a href="#15" >15</a>
	, 
	<a href="#17" >17</a>
	]
</span> with results for analytical functions (\(\alpha \)-theory or \(\gamma \)-theory) <span class="cite">
	[
	<a href="#8" >8</a>
	, 
	<a href="#10" >10</a>
	, 
	<a href="#14" >14</a>
	, 
	<a href="#16" >16</a>
	, 
	<a href="#18" >18</a>
	, 
	<a href="#19" >19</a>
	]
</span>. </p>
<p>We introduce a center majorant function (see \((h_3)\)) which is a special case of the majorant function that can provide more precise estimates on the distances \(\| F'(x)^\dagger \| .\) This modification leads to: weaker sufficient convergence conditions; more precise error estimates on the distances \(\| x_{k+1}-x_k\| , \| x_k-x^\ast \| \) and an at least as precise information on the location of the solution. </p>
<p>The paper is organized as follows: Section 2 contains standard information on Moore-Penrose inverses added so that the paper should be as self contained as possible. The semilocal convergence analysis of the Gauss-Newton method is presented in Section 3. Special cases and applications are given in the concluding Section 4. </p>

<h1 id="a0000000004">2 Background</h1>
<p> Let \(U(x, r)\) denote the open ball with center \(x\in D\) and radius \(r {\gt} 0.\) Let also \(\overline{U(x, r)}\) denote its closure. The Moore-Penrose inverse of a linear operator \(M: \mathbb {R}^n\longrightarrow \mathbb {R}^m\) is the linear operator \(M^\dagger :\mathbb {R}^m\longrightarrow \mathbb {R}^n\) which satisfies </p>
<div class="equation" id="2.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.1} MM^\dagger M = M,\,  M^\dagger M M^\dagger = M, \,  (MM^\dagger )^\ast =MM^\dagger ,\,  (M^\dagger M)^\ast = M^\dagger M, \end{equation}
  </div>
  <span class="equation_label">2.1</span>
</p>
</div>
<p>where \(M^\ast \) denotes the adjoint of \(M.\) We denote by \(\operatorname {Ker}(M)\) and \(\operatorname {Im}(M)\) the kernel and image of \(M.\) It follows from (<a href="#2.1">2.1</a>) that </p>
<div class="displaymath" id="a0000000005">
  \[ MM^\dagger = \prod _{\operatorname {Ker}(M)^\perp },\qquad M^\dagger M= \prod _{\operatorname {Im}(M),} \]
</div>
<p> where \(\prod _S\) denote the projection of \(\mathbb {R}^n\) onto the subspace \(S.\) Moreover, if \(M\) is surjective, we have </p>
<div class="displaymath" id="a0000000006">
  \[ M^\dagger =M^\ast (MM^\ast )^{-1},\quad MM^\dagger = I_{\mathbb {R}^m},\quad (MM^\dagger )^\dagger = M M^\dagger . \]
</div>
<p> We refer the reader to <span class="cite">
	[
	<a href="#7" >7</a>
	]
</span> for more properties and results on Moore-Penrose inverses. </p>
<p>Next, we list a number of usefull auxiliary results, starting with Banach’s lemma on invertible operators <span class="cite">
	[
	<a href="#5" >5</a>
	, 
	<a href="#7" >7</a>
	, 
	<a href="#15" >15</a>
	]
</span>. <div class="lemma_thmwrapper " id="l2.1">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.1</span>
  </div>
  <div class="lemma_thmcontent">
  <p><span class="cite">
	[
	<a href="#7" >7</a>
	, 
	<a href="#13" >13</a>
	]
</span> Let \(M: \mathbb {R}^n\longrightarrow \mathbb {R}^n\) be linear and continuous. Suppose that \(\| M-I\|  {\lt} 1\) then \(M\) is invertible and \(\| M^{-1}\| \leq \frac{1}{1-\| M-I_{\mathbb {R}^n}\| }.\) </p>

  </div>
</div><div class="lemma_thmwrapper " id="l2.2">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.2</span>
  </div>
  <div class="lemma_thmcontent">
  <p><span class="cite">
	[
	<a href="#7" >7</a>
	, 
	<a href="#13" >13</a>
	]
</span> Let \(M_1, M_2 : \mathbb {R}^n\longrightarrow \mathbb {R}^m\) be linear and continuous. Suppose that \(1\leq \operatorname {rank}(M_2)\leq \operatorname {rank}(M_1)\) and \(\| M_1^\dagger \| \| M_1-M_2\|  {\lt} 1.\) Then the following hold </p>
<div class="displaymath" id="a0000000007">
  \[ \operatorname {rank}(M_1)=\operatorname {rank}(M_2)\, \, \textnormal{and}\, \,  \| M_2^\dagger \| \leq \frac{\| M_1^\dagger \| }{1-\| M_1^\dagger \| \| M_1-M_2\| }. \]
</div>

  </div>
</div><div class="lemma_thmwrapper " id="l2.3">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.3</span>
  </div>
  <div class="lemma_thmcontent">
  <p><span class="cite">
	[
	<a href="#10" >10</a>
	, 
	<a href="#13" >13</a>
	]
</span> Let \(R {\gt} 0.\) Suppose \(\varphi : [0, R)\longrightarrow \mathbb {R}\) is convex. Then, the following holds </p>
<div class="displaymath" id="a0000000008">
  \[ D^{+}\varphi (0)=\lim _{u\longrightarrow 0^+}\frac{\varphi (u)-\varphi (0)}{u} =\inf _{u {\gt} 0}\frac{\varphi (u)-\varphi (0)}{u}. \]
</div>

  </div>
</div><div class="lemma_thmwrapper " id="l2.4">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.4</span>
  </div>
  <div class="lemma_thmcontent">
  <p><span class="cite">
	[
	<a href="#10" >10</a>
	, 
	<a href="#13" >13</a>
	]
</span> Let \(R {\gt} 0\) and \(\theta \in [0,1].\) Suppose \(\varphi : [0, R)\longrightarrow \mathbb {R}\) is convex. Then, \(\varphi _0:[0, R)\longrightarrow \mathbb {R}\) defined by \(\varphi _0(t)=\frac{\varphi (t)-\varphi (\theta t)}{t}\) is increasing. </p>

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

<h1 id="a0000000009">3 Semi-local convergence analysis</h1>
<p> In this section we present the semi-local convergence analysis of the Gauss-Newton method. We shall use the hypotheses given by:<br />\((H)\) Let \(D\subseteq \mathbb {R}^n\) be open and \(F:D\rightarrow \mathbb {R}^m\) be continuously Fréchet-differentiable.<br />\((h_1)\) Suppose that </p>
<div class="displaymath" id="a0000000010">
  \[ \| F'(y)^\dagger (I_{\mathbb {R}^m}-F'(x)F'(x)^\dagger )F(x)\| \leq \kappa \| x-y\| ,\quad \forall x,y\in D, \]
</div>
<p> where \(\kappa \in [0, 1).\) Let \(x_0\in D\) and set \(\beta = \| F'(x_0)^\dagger F(x_0)\|  {\gt} 0,\, \, \,  F'(x_0) \neq 0.\)<br />\((h_2)\) Suppose that </p>
<div class="displaymath" id="a0000000011">
  \[ \operatorname {rank}(F'(x))\leq \operatorname {rank}(F'(x_0))\neq 0,\quad \forall x\in D. \]
</div>
<p> \((h_3)\) Suppose that there exist \(R {\gt} 0\) and \(f_0, f : [0, R)\rightarrow \mathbb {R}\) such that \(U(x_0, R)\subseteq D,\) </p>
<div class="displaymath" id="a0000000012">
  \[ \| F'(x_0)^\dagger \| \| F'(x)-F'(x_0)\| \leq f_0'(\| x-x_0\| )-f_0'(0),\quad \forall x\in D \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000013">
  \begin{align*}  \| F’(x_0)^\dagger \| \| F’(y)-F’(x)\| \leq &  f’(\| y-x\| +\| x-x_0\| )-f’(\| x-x_0\| )\\ & \forall x, y\in U(x_0, R) \end{align*}
</div>
<p> with \(\| y-x\| +\| x-x_0\|  {\lt} R.\)<br />\((h_4)\) </p>
<div class="displaymath" id="a0000000014">
  \[  f_0(0)=f(0)=0,\, \,  f'(0)=f_0'(0) = -1 \]
</div>
<div class="displaymath" id="a0000000015">
  \[ f_0(t)\leq f(t)\, \, \, \textnormal{and}\, \, \,  f_0'(t)\leq f'(t),\quad \forall t\in [0, R). \]
</div>
<p> \((h_5)\) \(f_0', f'\) are convex and strictly increasing. </p>
<p>Let \(\lambda \geq 0\) be such that \(\lambda \geq -\kappa f'(\beta )\) and define \(h_{\beta ,\lambda }:[0, R)\rightarrow \mathbb {R}\) by </p>
<div class="displaymath" id="a0000000016">
  \[ h_{\beta ,\lambda }=\beta +\lambda t+f(t). \]
</div>
<p> \((h_6)\) \(h_{\beta ,\lambda }=0\) for some \(t\in [0, R).\)<br />\((h_7)\) For each \(s, t, u \in [0,R)\) with \(s \leq t \leq u\) </p>
<div class="displaymath" id="a0000000017">
  \[ t+\frac{h_{\beta ,\lambda }(u)}{f_0'(u)}\leq u+ \frac{h_{\beta ,\lambda }(t)-h_{\beta ,\lambda }(s)-h_{\beta ,\lambda }' (s)(t-s)}{f_0'(t)} \]
</div>
<p> <div class="remark_thmwrapper " id="r3.1">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.1</span>
  </div>
  <div class="remark_thmcontent">
  <p> If \(f_0=f,\) then hypotheses \((H)\) and our results reduce to the ones given in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span>. Notice that the second hypotheses in \((h_3)\) implies the first one but not necessarily vice versa. That is the first hypotheses is a special case of the second. From the computational point of view, the computation of function \(f\) involves the computation of function \(f_0.\) Hence, the results in this study are given under the same computational cost as in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span> (since the rest of the \((H)\) hypotheses are the same as in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span>). From now on we assume the \((H)\) conditions hold. </p>
<p>The majorizing iteration \(\{ s_k\} \) for \(\{ x_k\} \) is given by </p>
<div class="equation" id="3.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.1} s_0=0,\quad s_{k+1}=s_k-\frac{h_{\beta ,\lambda }(s_k)}{f_0'(s_k)}. \end{equation}
  </div>
  <span class="equation_label">3.1</span>
</p>
</div>
<p>The corresponding iteration \(\{ t_n\} \) used in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span> (for \(f_0 = f\)) is given by </p>
<div class="equation" id="3.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.2} t_0=0,\, \quad t_{k+1}=t_k-\frac{h_{\beta ,\lambda }(t_k)}{f'(t_k)}. \end{equation}
  </div>
  <span class="equation_label">3.2</span>
</p>
</div>
<p>The proofs in this study also for each \(k=0,1,2,\ldots \) (see Lemma 11 in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span> and our Lemma 3.8) shall show that the following iterations \(\{ r_k\} \) and \(\{ q_k\} \) are also majorizing for \(\{ x_k\} :\) </p>
<div class="displaymath" id="a0000000018">
  \begin{align} \nonumber r_0& =0,\,  r_1=\beta , \\ \label{3.3} r_{k+2}& =r_{k+1}-\frac{h_{\beta ,\lambda }(r_{k+1})-h_{\beta ,\lambda }(r_{k})- h_{\beta ,\lambda }'(r_{k})(r_{k+1}-r_k)}{f_0'(r_{k+1})},\quad k=0,1, 2,\ldots , \end{align}
</div>
<p>and for \(g_{\beta ,\lambda }=\beta +\lambda t+f_0(t),\) </p>
<div class="displaymath" id="a0000000019">
  \begin{align} \nonumber q_0& =0,\,  q_1=\beta ,\,  q_2=q_1-\frac{g_{\beta ,0}(q_{1})-g_{\beta ,0}(q_{0})- g_{\beta ,0}'(q_{0})(q_{1}-q_0)}{f_0'(q_{1})} \\ \label{3.4} q_{k+2}& =q_{k+1}-\frac{h_{\beta ,\lambda }(q_{k+1})-h_{\beta ,\lambda }(q_{k})- h_{\beta ,\lambda }'(q_{k})(q_{k+1}-q_k)}{f_0'(q_{k+1})}, \quad k=1, 2,\ldots . \end{align}
</div>
<p>New majorizing sequence \(\{ s_k\} , \{ r_k\} , \{ q_k\} \) can not only be tighter than \(\{ t_k\} \) but their sufficient convergence conditions can be weaker than the corresponding ones for \(\{ t_k\} .\) <span class="qed">â–¡</span></p>

  </div>
</div></p>
<p>Next, we first study the convergence of iteration \(\{ s_k\} \) and the properties of majorizing functions. The proofs are analogous to the corresponding ones in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span>. We simply replace \(f'(t)\) by \(f_0'(t)\) in the proofs and use \(f_0'(t) \leq f'(t).\) However, there are some differences in the proofs which are not immediate to the reader. Therefore, in what follows we will point out those differences. As already noted above for the rest of the differences, we refer the reader to <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span>. <div class="proposition_thmwrapper " id="p3.2">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">3.2</span>
  </div>
  <div class="proposition_thmcontent">
  <p><span class="cite">
	[
	<a href="#13" >13</a>
	]
</span> The following hold </p>
<ol class="enumerate">
  <li><p>\(h_{\beta ,\lambda }(0)=\beta ,\, \,  h_{\beta ,\lambda }'(0)= \lambda ^{-1};\) </p>
</li>
  <li><p>\(h_{\beta ,\lambda }'\) is convex and strictly increasing. </p>
</li>
</ol>

  </div>
</div><div class="proposition_thmwrapper " id="p3.3">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">3.3</span>
  </div>
  <div class="proposition_thmcontent">
  <p> The function \(h_{\beta ,\lambda }\) has a smallest zero \(s^\ast \in [0, R),\) is strictly convex, and the following hold: </p>
<div class="equation" id="3.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.5} h_{\beta ,\lambda }(t) > 0, \,  f_0'(t) < 0,\,  t < t-\frac{h_{\beta ,\lambda }(t)}{f_0'(t)} < t^\ast ,\quad \forall t\in [0, t^\ast ). \end{equation}
  </div>
  <span class="equation_label">3.5</span>
</p>
</div>
<p>Moreover, \(f_0'(t^\ast ) \leq h_{\beta ,\lambda }(t^\ast ) = f'(t^\ast )\leq 0.\) </p>

  </div>
</div><div class="proof_wrapper" id="a0000000020">
  <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 until the first estimate in (<a href="#3.5">3.5</a>) have been given in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span>. It was also shown that \(f'(t) = h_{\beta ,0}'(t) {\lt} 0.\) But by hypotheses \((h_4)\) \(f_0'(t) \leq f'(t).\) Hence, we get \(f_0'(t) {\lt}0.\) By convexity of \(h_{\beta ,\lambda },\) we have that </p>
<div class="displaymath" id="a0000000021">
  \[ 0=h_{\beta ,\lambda }(t^\ast ) {\gt} h_{\beta ,\lambda }(t) + h_{\beta ,\lambda }'(t)(t^\ast -t), \quad \forall t\in [0, R),\,  t\neq t^\ast . \]
</div>
<p> Then, the preceeding inequality can be written as </p>
<div class="displaymath" id="a0000000022">
  \[ t-\frac{h_{\beta ,\lambda }(t)}{h_{\beta ,\lambda }'(t)} {\lt} t^\ast , \quad \forall t\in [0, t^\ast ), \]
</div>
<p> (since \(-h_{\beta ,\lambda }'(t) {\gt} 0\)). We also have that </p>
<div class="displaymath" id="a0000000023">
  \[  0 {\lt} -h_{\beta ,\lambda }'(t) \leq -h_{\beta ,0}'(t)=-f'(t)\leq -f_0'(t),  \]
</div>
<p> which together with the preceeding estimates shows the third estimate in (<a href="#3.5">3.5</a>). Moreover, we have that \(h_{\beta ,\lambda }(t) {\gt} 0\) for each \(t\in [0, t^\ast )\) and \(h_{\beta ,\lambda }(t^\ast )=0.\) Then, we must have \(h_{\beta ,\lambda }'(t)\leq 0\) for each \(t\in [0, t^\ast ).\) Hence, the last estimate follows from </p>
<div class="displaymath" id="a0000000024">
  \[ 0 \geq h_{\beta ,\lambda }'(t^\ast )= \lambda +h_{\beta ,0}'(t^\ast )=\lambda +f'(t^\ast ) \geq f_0'(t^\ast ). \]
</div>

<p>It follows from the second estimate in (<a href="#3.5">3.5</a>) that the map \(\psi _{h_{\beta ,\lambda }}:\) </p>
<div class="displaymath" id="a0000000025">
  \begin{eqnarray*}  \psi _{h_{\beta ,\lambda }}:[0, t^\ast )\rightarrow \mathbb {R}, \quad t\rightarrow t-\frac{\psi _{h_{\beta ,\lambda }}}{f_0'(t)} \end{eqnarray*}
</div>
<p> is well defined. Notice that if \(f_0 = f\) this map reduces to the one used in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span>. Moreover, if \(\lambda = 0\) this map reduces to the one given in <span class="cite">
	[
	<a href="#11" >11</a>
	]
</span>. </p>
<p>We can define sequence \(\{ s_k\} \) given in (<a href="#3.1">3.1</a>) by </p>
<div class="equation" id="3.6">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.6} s_0=0,\,  s_{k+1}=\psi _{h_{\beta ,\lambda }}(s_k), \quad k = 0,1,2, \ldots . \end{equation}
  </div>
  <span class="equation_label">3.6</span>
</p>
</div>
<p><div class="proposition_thmwrapper " id="p3.4">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">3.4</span>
  </div>
  <div class="proposition_thmcontent">
  <p> The following hold </p>
<ol class="enumerate">
  <li><p>\(\beta \leq \psi _{h_{\beta ,\lambda }}(t) {\lt} t^\ast , \forall t\in [0, t^\ast ).\) </p>
</li>
  <li><p>The map \( \psi _{h_{\beta ,\lambda }}\) maps \([0, t^\ast )\) in \([0, t^\ast )\) and \(t {\lt} \psi _{h_{\beta ,\lambda }}(t),\, \, \, \,  \textnormal{for each}\, \,  \, t\in [0, t^\ast ).\) If \(\lambda =0\) or \(\lambda =0\) and \(f_0'(t^\ast ) {\lt} 0,\) then, the following hold, respectively </p>
<div class="displaymath" id="a0000000026">
  \begin{eqnarray*} \nonumber t^\ast -\psi _{h_{\beta ,\lambda }}(t)& \leq & \tfrac {1}{2}(t^\ast -t),\\ t^\ast -\psi _{h_{\beta ,\lambda }}(t)& \leq & \frac{D^-f_0'(t^\ast )}{-2f_0'(t^\ast )}(t^\ast -t)^2, \, \, \, \textnormal{for each}\, \,  t\in [0, t^\ast ) \end{eqnarray*}
</div>
<p>where \(D^-\) stands for the left directional derivative <span class="cite">
	[
	<a href="#5" >5</a>
	, 
	<a href="#13" >13</a>
	]
</span>. </p>
</li>
  <li><p>The sequence \(\{ s_k\} \) is: well defined; strictly increasing; contained in \([0, t^\ast )\) and converges to \(t^\ast .\) </p>
<p>If \(\lambda =0\) or \(\lambda =0\) and \(f_0'(t^\ast ) {\lt} 0,\) the sequence \(\{ s_k\} \) converges \(Q\)-linearly or \(Q\)-quadratically to \(t^\ast ,\) respectively and </p>
<div class="displaymath" id="a0000000027">
  \begin{eqnarray*} \nonumber t^\ast -s_{k+1}& \leq & \tfrac {1}{2}(t^\ast -s_k),\\ s^\ast -s_{k+1}& \leq & \frac{D^-f_0'(t^\ast )}{-2f_0'(t^\ast )}(t^\ast -s_k)^2, \, \, \, \textnormal{for each}\, \,  k=0,1,2,\ldots . \end{eqnarray*}
</div>
</li>
</ol>

  </div>
</div></p>
<p>In what follows we prove the convergence of Gauss-Newton method (<a href="#1.2">1.2</a>). We need the following Banach type Lemma: <div class="proposition_thmwrapper " id="p3.5">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">3.5</span>
  </div>
  <div class="proposition_thmcontent">
  <p> Suppose that \(x\in U(x_0, t)\) for \(t\in [0, t^\ast ).\) Then, the following hold \(\operatorname {rank}(F'(x)) = \operatorname {rank}(F'(x_0)) \geq 1\) and </p>
<div class="equation" id="3.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.7} \| F'(x)^\dagger \| \leq -\frac{\| F'(x_0)^\dagger \| }{f_0'(t)}. \end{equation}
  </div>
  <span class="equation_label">3.5</span>
</p>
</div>

  </div>
</div><div class="proof_wrapper" id="a0000000028">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Using \((h_2),\,  (h_4),\,  (h_5),\) the first hypothesis in \((h_3)\) and the second hypothesis in (<a href="#3.5">3.5</a>), we obtain in turn that </p>
<div class="displaymath" id="a0000000029">
  \begin{eqnarray*}  \| F’(x_0)^\dagger \| \| F’(x)-F’(x_0)\| & \leq & f_0’(\| x-x_0\| )-f_0’(0)\\ & \leq & f_0’(t)+1=h’_{\beta , 0}(t)+1 {\lt} 1. \end{eqnarray*}
</div>
<p> In view of the preceeding estimate and Lemma <a href="#l2.2">2.2</a> we deduce that \(\operatorname {rank}(F'(x)) = \operatorname {rank}(F'(x_0)) \geq 1\) and </p>
<div class="displaymath" id="a0000000030">
  \[ \| F'(x)^\dagger \| \leq -\frac{\| F'(x_0)^\dagger \| }{1-(f_0'(t)+1)}= -\frac{\| F'(x_0)^\dagger \| }{f_0'(t)}. \]
</div>
<p> <div class="remark_thmwrapper " id="r3.6">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.6</span>
  </div>
  <div class="remark_thmcontent">
  <p> It is worth noticing that the second inequality in \((h_3)\) is used in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span> to obtain instead of (<a href="#3.7">3.5</a>) that </p>
<div class="equation" id="3.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.8} \| F'(x)^\dagger \| \leq -\frac{\| F'(x_0)^\dagger \| }{f'(t)} \end{equation}
  </div>
  <span class="equation_label">3.6</span>
</p>
</div>
<p>which is more expensive to arrive at and less precise than (<a href="#3.7">3.5</a>) if \(f_0'(t) {\lt} f'(t)\) for each \(t\in [0, t^\ast ).\) This important observation makes the main difference in our approach over the one used in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span> and the earlier studies <span class="cite">
	[
	<a href="#11" >11</a>
	, 
	<a href="#12" >12</a>
	, 
	<a href="#14" >14</a>
	, 
	<a href="#16" >16</a>
	, 
	<a href="#19" >19</a>
	]
</span>. </p>
<p>As in earlier studies of this type <span class="cite">
	[
	<a href="#4" >4</a>
	, 
	<a href="#5" >5</a>
	, 
	<a href="#6" >6</a>
	, 
	<a href="#9" >9</a>
	, 
	<a href="#10" >10</a>
	, 
	<a href="#11" >11</a>
	, 
	<a href="#13" >13</a>
	]
</span> we study the linearization error of \(F\) at each point in \(D\) by defining </p>
<div class="equation" id="3.9">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.9} E_F(x,y):=F(y)-[F(x)+F'(x)(y-x)],\quad \forall x, y \in D \end{equation}
  </div>
  <span class="equation_label">3.7</span>
</p>
</div>
<p>and the error in the linearization on majorant function \(f\) by </p>
<div class="equation" id="3.11">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.11} e_{f}(t,u):=f(u)-[f(t)+f'(t)(u-t)], \quad \forall t, u \in [0, R). \end{equation}
  </div>
  <span class="equation_label">3.8</span>
</p>
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div><div class="lemma_thmwrapper " id="l3.7">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.7</span>
  </div>
  <div class="lemma_thmcontent">
  <p><span class="cite">
	[
	<a href="#11" >11</a>
	]
</span> Let \(x, y \in U(x_0, R)\) and \(0 \leq t {\lt} r {\lt} R.\) Suppose that \(\| x-x_0\| \leq t\) and \(\| y-x\| \leq r-t.\) Then, </p>
<div class="equation" id="3.11*">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.11*} \| F'(x_0)^\dagger \| \| E_F(x,y)\| \leq e_f(t,u)\frac{\| y-x\| ^2}{(r-t)^2}. \end{equation}
  </div>
  <span class="equation_label">3.9</span>
</p>
</div>

  </div>
</div></p>
<p>It follows from Proposition <a href="#p3.5">3.5</a> that Gauss-Newton map \(G_F\) for \(F:\) </p>
<div class="displaymath" id="a0000000031">
  \begin{eqnarray} \nonumber G_F: U(x_0, t^\ast )& \longrightarrow &  \mathbb {R}^n\\ \label{3.13} x& \longrightarrow &  x-F’(x)^\dagger F(x) \end{eqnarray}
</div>
<p> is well-defined. In order to guarantee that the Gauss-Newton iteration can be repeated indefinitely we need to introduce certain sets: </p>
<div class="displaymath" id="3.14">
  \begin{eqnarray} \label{3.14} K_0(t) & :=& \Big\{ x\in D: \| x-x_0\| \leq t,\,  \| F’(x)^\dagger F(x)\| \leq -\frac{h_{\beta ,\lambda }(t)}{f_0'(t)}\Big\} \\ \label{3.15} K_0& =& \bigcup _{t\in [0, t^\ast )}K_0(t),\\ \label{3.16} K(t) & :=& \Big\{ x\in D: \| x-x_0\| \leq t,\,  \| F’(x)^\dagger F(x)\| \leq -\frac{h_{\beta ,\lambda }(t)}{f'(t)}\Big\} \\ \nonumber \textnormal{and}& & \\ \label{3.17} K& =& \bigcup _{t\in [0, t^\ast )}K(t). \end{eqnarray}
</div>
<p> The sets \(K(t)\) and \(K\) were defined in <span class="cite">
	[
	<a href="#11" >11</a>
	, 
	<a href="#13" >13</a>
	]
</span>. It follows from the estimate \(f_0'(t)\leq f'(t)\) for each \(t\in [0, t^\ast )\) and definitions (<a href="#3.14">3.11</a>)-(<a href="#3.17">3.14</a>) that </p>
<div class="equation" id="3.18">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.18} K(t)\subset K_0(t) \end{equation}
  </div>
  <span class="equation_label">3.15</span>
</p>
</div>
<p>and </p>
<div class="equation" id="3.19">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.19} K\subset K_0. \end{equation}
  </div>
  <span class="equation_label">3.16</span>
</p>
</div>
<p>In view of (<a href="#3.18">3.15</a>) and (<a href="#3.19">3.16</a>) the next two results improve the corresponding ones in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span>. The proofs are given in exactly the same way but replacing \(K(t), K, \{ t_n\} , h'_{\beta ,0}\) by \(K_0(t), K_0, \{ s_n\} , f'_{0},\) respectively. <div class="lemma_thmwrapper " id="l3.8">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.8</span>
  </div>
  <div class="lemma_thmcontent">
  <p> The following hold for each \(t\in [0, t^\ast ):\) </p>
<ol class="enumerate">
  <li><p>\(K_0(t) \subset U(x_0,t^\ast );\) </p>
</li>
  <li><p>\(\| G_F(G_F(x))-G_F(x)\| \leq -\frac{h_{\beta ,\lambda }(\psi _{h_{\beta ,\lambda }}(t))}{f_0'(\psi _{h_{\beta ,\lambda }}(t))}(\frac{\| G_F(x)-x\| }{\psi _{h_{\beta ,\lambda }}(t)-t})^2\), \(\forall x\in K_0(t)\) </p>
</li>
  <li><p>\(G_F(K_0(t))\subset K_0(\psi _{h_{\beta ,\lambda }}(t))\) </p>
</li>
  <li><p>\(K_0 \subset U(x_0, t^\ast )\) and \(G_F(K_0)\subset K_0.\) </p>
</li>
</ol>

  </div>
</div></p>
<p>The Gauss-Newton method (<a href="#1.2">1.2</a>) can be written as </p>
<div class="equation" id="3.20">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.20} x_{k+1}=G_F(x_k), \  k= 0, 1, 2, \ldots . \end{equation}
  </div>
  <span class="equation_label">3.17</span>
</p>
</div>
<p>Next, the main semi-local convergence result for the Gauss-Newton method is presented. <div class="theorem_thmwrapper " id="t3.9">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.9</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Suppose that the \((H)\) conditions hold. Then, the following hold:<br />\(h_{\beta ,\lambda }(t)\) has a smallest zero \(t^\ast \in (0, R),\) the sequences \(\{ s_k\} \) and \(\{ x_k\} \) for solving \(h_{\beta ,\lambda }(t)=0\) and \(F(x)=0,\) with starting point \(t_0=0\) and \(x_0,\) respectively given by <span class="rm">(<a href="#3.2">3.2</a>)</span> and <span class="rm">(<a href="#3.20">3.17</a>)</span> are well defined, \(\{ s_k\} \) is strictly increasing, remains in \([0, t^\ast ),\) and converges to \(t^\ast , \,  \{ x_k\} \) remains in \(U(x_0, t^\ast ),\) converges to a point \(x^\ast \in U(x_0, t^\ast )\) such that \(F'(x^\ast )^\dagger F(x^\ast )=0.\) Moreover, the following estimates hold: </p>
<div class="displaymath" id="a0000000032">
  \[ \| x_{k+1}-x_k\| \leq s_{k+1}-s_k, \quad k= 0, 1, 2, \ldots , \]
</div>
<div class="displaymath" id="a0000000033">
  \[ \| x^\ast -x_k\| \leq t^\ast -s_k, \quad k= 0, 1, 2, \ldots , \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000034">
  \[ \| x_{k+1}-x_k\| \leq \frac{s_{k+1}-s_k}{(s_k-s_{k-1})^2}\| x_k-x_{k-1}\| ^2 , \quad k= 0, 1, 2, \ldots . \]
</div>
<p> Furthermore, if \(\lambda =0\) (\(\lambda =0\) and \(f_0'(t^\ast ) {\lt} 0\)), the sequence \(\{ s_k\} ,\,  \{ x_k\} \) converge \(Q\)-linearly and \(R\)-linearly (\(Q\)-quadratically and \(R\)-quadratically) to \(t^\ast \) and \(x^\ast ,\) respectively. </p>

  </div>
</div><div class="remark_thmwrapper " id="r3.10">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.10</span>
  </div>
  <div class="remark_thmcontent">
  <p> (i) As noted in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span> the best choice for \(\lambda \) is given by \(\lambda =-\kappa f'(\kappa ).\) </p>
<p>(ii) Sequence \(\{ r_k\} \) appears in the proof of Lemma 11 in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span> or our Lemma <a href="#l3.8">3.8</a> (see also Theorem <a href="#t4.1">4.1</a>). </p>
<p>(iii) Sequence \(\{ q_k\} \) is derived from the proof of Lemma 11 in <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span> or our Lemma <a href="#l3.8">3.8</a> if we simply use \(x=x_0\) and notice that the first instead of the second hypothesis in \((h_3)\) can be used for the corresponding estimate. Notice that \(\{ q_k\} \) is the tightest among \(\{ s_k\} \) and \(\{ r_k\} .\) Moreover \(\{ s_k\} ,\) \(\{ t_k\} \) and \(\{ q_k\} \) converges under the \((H)\) conditions. <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000035">4 Special cases and applications</h1>
<p> In this section, we present some special cases and applications. Then, we arrive at: <div class="theorem_thmwrapper " id="t4.1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Under the \((H)\) hypotheses, the conclusions of Theorem <span class="rm"><a href="#t3.9">3.9</a></span> hold. Moreover, the following estimates hold </p>
<div class="displaymath" id="4.2">
  \begin{align} \label{4.2} \| x_{k+1}-x_k\| & \leq q_{k+1}-q_k\leq r_{k+1}-r_k\leq s_{k+1}-s_k {\lt} t_{k+1}-t_k, \quad k= 0, 1, 2, \ldots \end{align}
</div>

  </div>
</div><div class="proof_wrapper" id="a0000000036">
  <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 conclusions of Theorem <a href="#t3.9">3.9</a> hold under the \((H)\) conditions. Then (<a href="#4.2">4.1</a>) follows using the definition of sequences \(\{ q_k\} ,\) \(\{ r_k\} ,\) \(\{ q_k\} ,\) \(\{ t_k\} ,\) \((h_7)\) a simple induction argument and the discussion in Remark <a href="#r3.10">3.10</a> (ii) and (iii). Indeed, we have that \(s_0=r_0\) and \(s_1=r_1=\beta .\) Then it follows from (<a href="#3.1">3.1</a>) and (<a href="#3.3">3.3</a>) for \(\kappa =0\) and \((h_7)\) that </p>
<div class="displaymath" id="a0000000037">
  \begin{align*}  r_2& =r_1-\frac{h_{\beta ,\lambda }(r_1)-h_{\beta ,\lambda }(r_0)-h_{\beta ,\lambda }'(r_0)(r_1-r_0)}{f_0'(r_1)}\\ & =s_1-\frac{h_{\beta ,\lambda }(s_1)-h_{\beta ,\lambda }(s_0)-h_{\beta ,\lambda }'(s_0)(s_1-s_0)}{f_0'(s_1)}\\ & \leq s_1-\frac{h_{\beta ,\lambda }(s_1)}{f_0'(s_1)}=s_2 \end{align*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000038">
  \[  r_2-r_1 =-\frac{h_{\beta ,\lambda }(s_1)-h_{\beta ,\lambda }(s_0)-h_{\beta ,\lambda }'(s_0)(s_1-s_0)}{f_0'(s_1)} \leq s_2-s_1.  \]
</div>
<p>Suppose that \(r_i\leq s_i\) and \(r_{i+1}-r_i\leq s_{i+1}-s_i\) for \(i=0, 1, 2, \cdots k.\) We shall show that \(r_{k+2}\leq s_{k+2}\) and \(r_{k+2}-r_{k+1} \leq s_{k+2}-s_{k+1}.\) We have from the induction hypotheses and \((h_7)\) that </p>
<div class="displaymath" id="a0000000039">
  \begin{align*}  r_{k+2}& =r_{k+1}-\frac{h_{\beta ,\lambda }(r_{k+1})-h_{\beta ,\lambda }(r_k)-h_{\beta ,\lambda }'(r_k)(r_{k+1}-r_k)}{f_0'(r_{k+1})}\\ & \leq s_{k+1}-\frac{h_{\beta ,\lambda }(s_{k+1})}{f_0'(s_{k+1})}=s_{k+2} \end{align*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000040">
  \[  r_{k+2}-r_{k+1} \leq s_{k+2}-s_{k+1}.  \]
</div>
<p> Similarly, we show that \(q_k \leq r_k\) and \(q_{k+1}-q_k\leq r_{k+1}-r_k.\) </p>
<p>So, far we have shown that under the \((H)\) and \((h_7)\) hypotheses more precise majorizing sequences can be obtained for \(\{ x_k\} .\) At this point we are wondering if a direct study of the convergence of majorizing sequences \(\{ r_k\} , \{ s_k\} , \{ q_k\} \) lead to weaker sufficient convergence conditions than \((h_6).\) Let us present this in the interesting case of Newton’s method </p>
<div class="equation" id="4.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{4.3} x_{k+1}=x_k-F'(x_k)^{-1}F(x_k), \quad k=0,1,2,\ldots \end{equation}
  </div>
  <span class="equation_label">4.2</span>
</p>
</div>
<p>for solving nonlinear equation </p>
<div class="equation" id="4.4">
<p>
  <div class="equation_content">
    \begin{equation} \label{4.4} F(x)=0. \end{equation}
  </div>
  <span class="equation_label">4.3</span>
</p>
</div>
<p>That is \(\lambda =\kappa =0.\) Let us define \(f_0\) and \(f\) on \([0, R)\) by \(f_0(t)=\frac{L_0}{2}t^2-t\) and \(f(t)=\frac{L}{2}t^2-t\) for \(L_0 {\gt} 0\) and \(L {\gt} 0.\) Then, sequences \(\{ t_k\} , \{ s_k\} , \{ r_k\} \) and \(\{ q_k\} \) reduce to </p>
<div class="displaymath" id="a0000000041">
  \begin{eqnarray} \nonumber t_0& =& 0,\,  t_{k+1}=t_k-\frac{\frac{L}{2}t_k^2-t_k+\beta }{L_0t_k-1}\\ \nonumber & =&  t_k-\frac{L(t_k-t_{k-1})^2}{2(Lt_k-1)},\\ \nonumber s_0& =& 0, s_1=\beta , s_{k+1}=s_k-\frac{\frac{L}{2}s_k^2-s_k+\beta }{L_0s_k-1}\\ \nonumber r_0& =& 0, r_1=\beta ,\\ \nonumber r_{k+2}& =& r_{k+1}-\frac{\frac{L}{2}r_{k+1}^2-r_{k+1}-(\frac{L}{2}r_{k}^2-r_{k})-(Lr_k-1)(r_{k+1}-r_k)}{L_0r_{k+1}-1}\\ \nonumber q_0& =& 0,q_1=\beta ,\,  q_2=q_1-\frac{L_0(q_1-q_0)^2}{2(L_0q_1-1)},\\ \nonumber q_{k+2}& =& q_{k+1}-\frac{L(q_{k+1}-q_k)^2}{2(L_0q_{k+1}-1)}. \end{eqnarray}
</div>
<p> Then, according to \((h_3),\) and Theorem <a href="#t4.1">4.1</a> sequences \(\{ t_k\} , \{ s_k\} , \{ r_k\} ,\{ q_k\} \) converge, if the famous for its simplicity and clarity Kantorovich hypothesis <span class="cite">
	[
	<a href="#5" >5</a>
	, 
	<a href="#15" >15</a>
	]
</span> </p>
<div class="displaymath" id="a0000000042">
  \[  h=L\beta \leq \tfrac {1}{2}  \]
</div>
<p> is satisfied. We also have that </p>
<div class="displaymath" id="a0000000043">
  \[  t^\ast =\frac{1-\sqrt{1-2h}}{L}.  \]
</div>
<p> However, a direct study of sequence \(\{ q_k\} \) (see <span class="cite">
	[
	<a href="#5" >5</a>
	, 
	<a href="#6" >6</a>
	]
</span>) shows that this sequence converges provided that </p>
<div class="displaymath" id="a0000000044">
  \[  h_0=\bar{L}\beta \leq \tfrac {1}{2},  \]
</div>
<p> where </p>
<div class="displaymath" id="a0000000045">
  \[  \bar{L}=\tfrac {1}{8}\Big(4L+\sqrt{8L^2+L_0L}+\sqrt{L_0L}\Big)  \]
</div>
<p> Notice that since \(\bar{L}\leq L,\) </p>
<div class="displaymath" id="a0000000046">
  \[  h\leq \tfrac {1}{2}\Rightarrow h_0\leq \tfrac {1}{2}  \]
</div>
<p> but not necessarily vice versa unless if \(L_0 = L.\) Moreover, we have that </p>
<div class="equation" id="4.14">
<p>
  <div class="equation_content">
    \begin{equation} \label{4.14} \tfrac {h_0}{h}\rightarrow 0,\quad \textnormal{as}\, \,  \tfrac {L}{L_0}\rightarrow 0. \end{equation}
  </div>
  <span class="equation_label">4.4</span>
</p>
</div>
<p>Implication (<a href="#4.14">4.4</a>) shows by how many times (at most) the applicability of Newton’s method (<a href="#4.3">4.2</a>) is extended under our approach. Notice also that if \(L_0 {\lt} L\) </p>
<div class="displaymath" id="a0000000047">
  \[ q_k {\lt} t_k, \quad \textnormal{for each}\, \, \,  k=2, 3, \ldots , \]
</div>
<div class="displaymath" id="a0000000048">
  \[ q_{k+1}-q_k {\lt} t_{k+1}-t_k, \quad \textnormal{for each}\, \, \,  k=2, 3, \ldots , \]
</div>
<p> and \(q^\ast = \lim _{k\rightarrow \infty }q_k \leq t^\ast .\) Examples, where \(L_0 {\lt} L\) can be found in <span class="cite">
	[
	<a href="#5" >5</a>
	, 
	<a href="#6" >6</a>
	]
</span>. </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="1">1</a></dt>
  <dd><p><i class="sc">I. Argyros</i>, <i class="it">On the semilocal convergence of the Gauss-Newton method</i>, Adv. Nonlinear Var. Inequal., <b class="bf">8</b> (2005) 2, pp.&#160;93–99. </p>
</dd>
  <dt><a name="2">2</a></dt>
  <dd><p><i class="sc">I. Argyros</i> and <i class="sc">S. Hilout</i>, <i class="it">On the local convergence of the Gauss-Newton method</i>, Punjab Univ. J. Math., <b class="bf">41</b> (2009), pp.&#160;23–33. </p>
</dd>
  <dt><a name="3">3</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/s12190-010-0377-8"> <i class="sc">I. Argyros</i> and <i class="sc">S. Hilout</i>, <i class="it">On the Gauss-Newton method</i>, J. Appl. Math. Comput., <b class="bf">35</b> (2011), pp.&#160;537–550. <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="4">4</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/s11075-011-9446-9"> <i class="sc">I. Argyros</i> and <i class="sc">S. Hilout</i>, <i class="it">Extending the applicability of the Gauss-Newton method under average Lipschitz-type conditions</i>, Numer. Algor., <b class="bf">58</b> (2011) 1, pp. &#160;23–52. <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="5">5</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/j.cam.2011.10.021"> <i class="sc">I. Argyros</i> and <i class="sc">S. Hilout</i>, <i class="it">Improved local convergence of Newton’s method under weak majorant condition</i>, J. Comp. Appl. Math., <b class="bf">236</b> (2012) 7, pp. 1892–1902. <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="6">6</a></dt>
  <dd><p><i class="sc">I. Argyros</i> and <i class="sc">S. Hilout</i>, <i class="it">Numerical methods in nonlinear analysis</i>, World Scientific Publ. Comp. New Jersey, USA, 2013. </p>
</dd>
  <dt><a name="7">7</a></dt>
  <dd><p><i class="sc">A. Ben-Israel</i> and <i class="sc">T.N.E. Greville</i>, <i class="it">Generalized inverses</i>, CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC, 15. Springer-Verlag, New York, 2nd ed., Theory and Applications, 2003. </p>
</dd>
  <dt><a name="8">8</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1090/S0025-5718-04-01646-1"> <i class="sc">E. Cătinaş</i>, <i class="it">The inexact, inexact perturbed, and quasi-Newton methods are equivalent models</i>, Math. Comput., <b class="bf">74</b> (2005) 249, pp.&#160;291–301. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
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  <dd><p><a href ="http://dx.doi.org/10.1006/jcom.2001.0612"> <i class="sc">J.P. Dedieu</i> and <i class="sc">M.H. Kim</i>, <i class="it">Newton’s method for analytic systems of equations with constant rank derivatives</i>, J. Complexity, <b class="bf">18</b>, (2002) 1, pp.&#160;187–209. <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="10">10</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/j.jco.2010.09.001"> <i class="sc">O.P. Ferreira</i>, <i class="sc">M.L.N. Gonçalves</i> and <i class="sc">P.R. Oliveira</i>, <i class="it">Local convergence analysis of inexact Gauss-Newton like methods under majorant condition</i>, J. Complexity, <b class="bf">27</b> (2011) 1, pp.&#160;111–125. <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="11">11</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/s10589-007-9082-4"> <i class="sc">O.P. Ferreira</i> and <i class="sc">B.F. Svaiter</i>, <i class="it">Kantorovich’s majorants principle for Newton’s method</i>, Comput. Optim. Appl., <b class="bf">42</b>(2009) 2, pp.&#160;213–229. <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="12">12</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/BF01389446"> <i class="sc">W.M. Häussler</i>, <i class="it">A Kantorovich-type convergence analysis for the Gauss-Newton-method</i>, Numer. Math., <b class="bf">48</b> (1986) 1, pp.&#160;119–125. <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="13">13</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1080/02331934.2013.778854"> <i class="sc">M.L.N. Gonçalves</i> and <i class="sc">P.R. Oliveira</i>, <i class="it">Convergence of the Gauss-Newton method for a special class of systems of equations under a majorant condition</i>, Optimization, <b class="bf">64</b> (2015) 3, pp.&#160;577–594. <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="14">14</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/j.cam.2007.07.006"> <i class="sc">N. Hu</i>, <i class="sc">W. Shen</i> and  C. Li, <i class="it">Kantorovich’s type theorems for systems of equations with constant rank derivatives</i>, J. Comput. Appl. Math., <b class="bf">219</b> (2008) 1, pp.&#160;110–122. <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">L.V. Kantorovich</i> and <i class="sc">G.P. Akilov</i>, <i class="it">Functional Analysis</i>, Pergamon Press, Oxford, 1982. </p>
</dd>
  <dt><a name="16">16</a></dt>
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</span></a> </p>
</dd>
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  <dd><p><i class="sc">F.A. Potra</i> and <i class="sc">V. Ptak</i>, <i class="it">Nondiscrete induction and iterative processes</i>, Research notes in Mathematics, 103, Pitman (Advanced Publishing Program), Boston, MA, 1984. </p>
</dd>
  <dt><a name="18">18</a></dt>
  <dd><p><i class="sc">S. Smale</i>, <i class="it">Newton’s method estimates from data at one point. The merging of disciplines: new directions in pure, applied, and computational mathematics</i> (Laramie, Wyo., 1985), pp.&#160;185–196, Springer, New York, 1986. </p>
</dd>
  <dt><a name="19">19</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1093/imanum/20.1.123"> <i class="sc">X.H. Wang</i>, <i class="it">Convergence of Newton’s method and uniqueness of the solution of equations in Banach spaces</i>, IMA J. Numer. Anal., <b class="bf">20</b> (2000), pp.&#160;123–134. <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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