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<title>Optimal inequality factor <br />for Durand–Kerner’s and Tanabe’s methods: Optimal inequality factor <br />for Durand–Kerner’s and Tanabe’s methods</title>
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<h1>Optimal inequality factor <br />for Durand–Kerner’s and Tanabe’s methods</h1>
<p class="authors">
<span class="author">Octavian Cira\(^\ast \) Cristian Mihai Cira\(^\S \)</span>
</p>
<p class="date">April 11, 2011</p>
</div>
<p>\(^\ast \)Department of Mathematics and Informatics, “Aurel Vlaicu" University, Bulevardul Revoluţiei, no. 75, Arad, România, e-mail: <span class="ttfamily">octavian.cira@uav.ro</span> </p>
<p>\(^\S \)Auburn University, Samuel Ginn College of Engineering, Auburn, AL 36849, e-mail: <span class="ttfamily">cmc0031@auburn.edu</span> </p>

<div class="abstract"><p> The convergence condition for the simultaneous inclusion methods is \(w^{(0)}{\lt}c(a,b,n)d^{(0)}\), where \(w^{(0)}\) is the maximum Weierstrass factor \(W^{(0)}_k\), \(k\in I_n\), and \(d^{(0)}\) is the minimum distance between \(z^{(0)}_1\), \(z^{(0)}_2\), …\(z^{(0)}_n\), the distinct approximations of the simple roots of the polynomial \(\zeta _1\), \(\zeta _2\), …\(\zeta _n\). The <em>i-factor</em> (inequality-factor) is the positive real function \(c(a,b,n)=\tfrac {1}{an+b}\). The article presents the optimum <em>i-factor</em> for the simultaneous inclusion methods Durand–Kerner and Tanabe. </p>
<p><b class="bf">MSC.</b> 65H10 </p>
<p><b class="bf">Keywords.</b> Root-finding methods; Polynomial zeros; Simultaneous inclusion methods; Durand–Kerner’s method, Tanabe’s method, Convergence; Computational efficiency </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p> Let </p>
<div class="equation" id="PComplex">
<p>
  <div class="equation_content">
    \begin{equation} \label{PComplex} P(z)=z^n+a_{n-1}z^{n-1}+\cdots +a_1z+a_0,\  a_k\in \mathbb {C}~ ,\  k\in I^*_{n-1}~ , \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> be a polynomial, where \(I^*_{n-1}=\left\{ 0,1,\ldots ,n-1\right\} \) and let </p>
<div class="equation" id="EcuatiaAlgebrica">
<p>
  <div class="equation_content">
    \begin{equation} \label{EcuatiaAlgebrica} P(z)=0~ , \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> be its attached algebraic equation. </p>
<p>Abel’s impossibility theorem states that: “<em>In general, polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions</em>”, <span class="cite">
	[
	<a href="#Abel1826" >1</a>
	]
</span>. This was also shown by Ruffini in 1813, <span class="cite">
	[
	<a href="#Wells1986" >11</a>
	, 
	pp. 59
	]
</span>. </p>
<p>Galois theorem states the same thing, namely that: “<em>An algebraic equation is algebraically solvable necessary and sufficient its group is solvable. In order that an irreducible equation of prime degree be solvable by radicals, it is necessary and sufficient that all its roots be rational functions of two roots</em>", <span class="cite">
	[
	<a href="#Birkhoff+MacLane1996" >18</a>
	, 
	<a href="#Dumit+Foote1998" >27</a>
	]
</span>. </p>
<p>Modern softwares such as <em>Maple</em>, <em>Mathematica</em>, <em>Mathcad</em> or <em>Matlab</em> solve the algebraic equation of degree 2, 3 and 4 using symbolic computation and classical formulas for the second degree equation or Cardano’s formulas, <span class="cite">
	[
	<a href="#Boyer+Merzbach1991" >14</a>
	, 
	pp. 282-284
	]
</span> and Ferrari–Cardano, <span class="cite">
	[
	<a href="#Boyer+Merzbach1991" >14</a>
	, 
	pp. 284-287
	]
</span> for the \(3^{rd}\) and \(4^{th}\) degree equations. This methods always give the exact solutions but have the disadvantage that, sometimes, the solutions have a complicated representation. </p>
<p>Arbitrary-precision calculus in <em>Mathcad</em> prints the solution with up to 250 decimals, while <em>Mathematica</em>, <span class="cite">
	[
	<a href="#Wolfram" >33</a>
	]
</span> permits the display of any number of decimals for a result. Computing a result with \(n\) exact decimals is equivalent to finding the solution of the equation (<a href="#EcuatiaAlgebrica">2</a>) with a numerical method with the precision \(10^{-n}\). </p>
<p>Therefore obtaining the solutions for an algebraic equation of degree 2, 3 and 4 is a solved problem. According to Abel’s impossibility theorem, in general for equation of degree greater than 4, we need to apply a numerical method to approximate the solutions. </p>
<p>Let us denote by \(\mathbf{d}_n=\left\{ n,n+1,\ldots \right\} \), \(n\in \mathbb {N}\) the polynomial degree greater or equal \(n\). </p>
<h1 id="a0000000003">2 Simultaneous methods with corrections</h1>
<p>Since \(k\in I_n=\left\{ 1,2,\ldots ,n\right\} \) we denote </p>
<div class="displaymath" id="a0000000004">
  \[  \sum \tfrac {W_j}{\widehat{z}_k-z_j}=\sum _{j=1}^n\tfrac {W_j}{\widehat{z}_k-z_j},\   \prod _{j\ne k}(z_k-z_j)=\prod _{j=1\atop {k\ne j}}^n (z_k-z_j),  \]
</div>
<div class="displaymath" id="a0000000005">
  \[  \max \left\vert W_k\right\vert =\max _{k\in I_n}\left\vert W_k\right\vert ,\   \min _{k{\lt}j}\left\vert z_k-z_j\right\vert =\min _{k,j\in I_n\atop k{\lt}j}\left\vert z_k-z_j\right\vert ,\    \]
</div>
<p>Weierstrass correction factor is denoted by </p>
<div class="equation" id="FactorulWeierstrass">
<p>
  <div class="equation_content">
    \begin{equation} \label{FactorulWeierstrass} W_k=W(z_k)=\tfrac {P(z_k)}{\displaystyle \prod _{j\ne k}(z_k-z_j)}, \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> where \(z_k\) is the approximation for the simple zero \(\zeta _k\) of the polynomial (<a href="#PComplex">1</a>), \(w\) is the absolute maximum value of the Weierstrass correction factors and \(d\) the minimum distance between two approximations \(z_1\), \(z_2\), …, \(z_n\). </p>
<div class="displaymath" id="a0000000006">
  \[  w=\max \left\vert W_k\right\vert ~ ,\  \  d=\min _{k{\lt}j}\left\vert z_k-z_j\right\vert .  \]
</div>
<p>We will also note by \(z\), \(w\), \(d\), \(W\) the current iteration \(z^{(m)}\), \(w^{(m)}\), \(d^{(m)}\), \(W^{(m)}\) and by \(\widehat{z}\), \(\widehat{w}\), \(\widehat{d}\), \(\widehat{W}\) the next iteration \(z^{(m+1)}\), \(w^{(m+1)}\), \(d^{(m+1)}\), \(W^{(m+1)}\). </p>
<p>We will consider a class \(C\) of simultaneous method with corrections, \(C(z)=P(z)/F(z)\), with \(F(z)\ne 0\) for any zero \(\zeta _k\) with \(k\in I_n\), of polynomial \(P\), and for any approximation \(z^{(m)}_k\) with \(k\in I_n\) and \(m=0,1,\ldots \) obtained by the iterative process. We will denote by </p>
<div class="equation" id="FormaCorectiilor">
<p>
  <div class="equation_content">
    \begin{equation} \label{FormaCorectiilor} C^{(m)}_k=C_k(z^{(m)}_1,z^{(m)}_2,\ldots ,z^{(m)}_k), \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> the correction factor for the \(m^{th}\) iteration. The vast majority of the iterative methods that simultaneously approximate the simple zeros of an polynomial can be expressed as: </p>
<div class="equation" id="MetodaCorectii">
<p>
  <div class="equation_content">
    \begin{equation} \label{MetodaCorectii} z^{(m+1)}_k=z^{(m)}_k-C^{(m)}_k,\  \  k\in I_n\,  \  m=0,1,\ldots , \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> where \(z^{(m)}_1\), \(z^{(m)}_2\), \(\ldots \), \(z^{(m)}_n\) are distinct approximations of the simple zeros \(\zeta _1\), \(\zeta _2\), …, \(\zeta _n\). </p>
<p>We try to find a convergence condition of the form \(w{\lt}c(n)d\), where \(c(n)\) is a real positive function that depends on \(n\) (the polynomial degree) asymptotic to the function \(\phi (n)=1/(n+2\sqrt{n-1})\) or \(\psi (n)=1/(2n)\). We consider the initial condition satisfied </p>
<div class="equation" id="ConditiaInitiala">
<p>
  <div class="equation_content">
    \begin{equation} \label{ConditiaInitiala} w^{(0)} < c(n)d^{(0)}. \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> The function \(c(n)\) is called the <em>i-factor</em> (inequality factor) in <span class="cite">
	[
	<a href="#Petkovic1997" >20</a>
	]
</span>, <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	]
</span> and many other works, and it depends on the degree \(n\) of the polynomial. For the choice of \(c(n)\), Petković and its collaborators propose that \(c(n)\) has the form: </p>
<div class="equation" id="I-FactorPetkovic">
<p>
  <div class="equation_content">
    \begin{equation} \label{I-FactorPetkovic} c(n)=c(a,b,n)=\tfrac {1}{an+b}~ \  \textnormal{with}\  a,b\in \mathbb R_+. \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p><div class="theorem_thmwrapper " id="Teorema14Petkovic">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(\eta _k=z_k-W_k\in \mathbb {C}\backslash \left\{ z_1,z_2,\ldots ,z_n\right\} \) and set </p>
<div class="displaymath" id="a0000000007">
  \[  \gamma _k=\left\vert W_k\right\vert \cdot \max _{j\neq k}\left\vert z_j-\eta _k\right\vert ^{-1},\   \sigma _k=\sum _{j\neq k}\tfrac {\left\vert W_j\right\vert }{\left\vert z_j-\sigma _k\right\vert },\  \  k\in I_n.  \]
</div>
<p> If \(\sqrt{1+\gamma _k}{\gt}\sqrt{\gamma _k}+\sqrt{\sigma _k}\), then there is exactly one zero of \(P\) in the disk centered in \(\eta _k\) and of radius </p>
<div class="displaymath" id="a0000000008">
  \[  \left\vert W_k\right\vert \cdot \left(1-\tfrac {2(1-2\sigma _k-\gamma _k)}{1-\sigma _k-2\gamma _k+\sqrt{(1-\sigma _k-2\gamma _k)^2+ 4\sigma _k(1-2\sigma _k-\gamma _k)^2}}\right).  \]
</div>
<p> If </p>
<div class="equation" id="InegaliatiGammaSigma">
<p>
  <div class="equation_content">
    \begin{equation} \label{InegaliatiGammaSigma} \sqrt{1+\gamma _k}>\sqrt{\gamma _k}+\sqrt{\sigma _k}\  \textnormal{and}\  \gamma _k+2\sigma _k<1, \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p> then there is exactly one zero of \(P\), in disk centered in \(\eta _k\) and of radius </p>
<div class="displaymath" id="a0000000009">
  \[  \left\vert W_k\right\vert \tfrac {\gamma _k+\sigma _k}{1-\sigma _k}.  \]
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000010">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Carstensen1991" >12</a>
	]
</span> and <span class="cite">
	[
	<a href="#Borsch-Supan1963" >5</a>
	, 
	<a href="#Braess+Hadeler1973" >8</a>
	, 
	<a href="#Carstensen1991a" >13</a>
	]
</span>. <div class="proof_wrapper" id="a0000000011">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Teorema15Petkovic">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If the <em>i-factor</em> \(c(n)\) appearing in <em>(<a href="#ConditiaInitiala">6</a>)</em> is </p>
<div class="equation" id="FunctiaPhi">
<p>
  <div class="equation_content">
    \begin{equation} \label{FunctiaPhi} c(n)<\phi (n)=\tfrac {1}{n+2\sqrt{n-1}}\  \textnormal{and}\  c(n)<\psi (n)=\tfrac {1}{2n}, \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> then both inequalities <em>(<a href="#InegaliatiGammaSigma">8</a>)</em> hold and the minimal radius of the inclusion disk given in Theorem <em><a href="#Teorema14Petkovic">1</a></em> is not greater than \(\left\vert W_k\right\vert \). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000012">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	T 1.5.
	]
</span>. <div class="proof_wrapper" id="a0000000013">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Teorema16Petkovic">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(c(a,b,n)=\tfrac {1}{an+b}\), \(a\ge 2\), \(b{\gt}(2-a)n\), and let us assume that \(w{\lt}c(a,b,n)d\), holds. Then for \(n\in \mathbf{d}_3\), the disks </p>
<div class="displaymath" id="a0000000014">
  \begin{align*}  D_1=D\left(z_1-W_1;\tfrac {n}{(a-1)n+b}\left\vert W_1\right\vert \right),\ldots , D_n=D\left(z_n-W_n;\tfrac {n}{(a-1)n+b}\left\vert W_n\right\vert \right) \end{align*}
</div>
<p> are mutually disjoint and each of then contain unique zero of \(P\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000015">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	T 1.6
	]
</span>. <div class="proof_wrapper" id="a0000000016">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="corollary_thmwrapper " id="Corolarul11Petkovic">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">4</span>
  </div>
  <div class="corollary_thmcontent">
  <p> Under the conditions of Theorem <em><a href="#Teorema16Petkovic">3</a></em>, each of disks \(D^*_k\) defined by </p>
<div class="displaymath" id="DiscuriD*">
  \begin{align} \label{DiscuriD*} D^*_k& =D\left(z_k-W_k;\tfrac {n}{(a-1)n+b}\left\vert W_k\right\vert \right)\\ & \nonumber =D\left(z_k;\tfrac {1}{1-nc(a,b,n)}\left\vert W_k\right\vert \right),\  k\in I_n \end{align}
</div>
<p> contains exactly one zero of \(P\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000017">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	Corollary 1.1
	]
</span>. <div class="proof_wrapper" id="a0000000018">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Let \(g:(0,1)\to \mathbb R\) </p>
<div class="equation" id="Functiag">
<p>
  <div class="equation_content">
    \begin{equation} \label{Functiag} g(t)= \begin{cases}  1+2t, &  0 < t\le \tfrac {1}{2}, \\ \tfrac {1}{1-t}, &  \tfrac {1}{2} < t < 1. \end{cases} \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p><div class="lemma_thmwrapper " id="a0000000019">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">5</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Let </p>
<div class="displaymath" id="a0000000020">
  \[  s_m(t)=t^m+\sum _{k=0}^mt^k~ ,\  t\in (0,1),\  m=1,2,\ldots ~ .  \]
</div>
<p> Then \(s_m(t){\lt}g(t)\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000021">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Proof is elementary. <div class="proof_wrapper" id="a0000000022">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Teorema31Petkovic">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let the iterative method <em>(<a href="#MetodaCorectii">5</a>)</em> have the iterative correction of the form <em>(<a href="#FormaCorectiilor">4</a>)</em> and let \(z^{(0)}_1\), \(z^{(0)}_2\), …\(z^{(0)}_n\) be distinct initial approximations of zeros for the polynomial \(P\). If there exists a real number \(\beta \) such that the following two inequalities: </p>
<ol class="enumerate">
  <li><p>\(\left\vert C^{(m+1)}_k\right\vert \le \beta \left\vert C^{(m)}_k\right\vert ~ ,\  \textnormal{for}\  m=0,1,\ldots \), </p>
</li>
  <li><p>\(\left\vert z^{(0)}_k-z^{(0)}_j\right\vert {\gt}g(\beta )\left(\left\vert C^{(0)}_k\right\vert +\left\vert C^{(0)}_j\right\vert \right),\  \textnormal{for}\  k\ne j,\  k,j\in I_n\), </p>
</li>
</ol>
<p> are valid, then the iterative method <em>(<a href="#MetodaCorectii">5</a>)</em> is convergent. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000023">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	T 3.1
	]
</span>. <div class="proof_wrapper" id="a0000000024">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>All the simultaneous inclusion methods convergence theorems try to determine the values of \(a\) and \(b\) such that the initial condition (<a href="#ConditiaInitiala">6</a>) assures the convergence for the method. We try to find the optimum values for \(a\) and \(b\), namely an <em>optimum i-factor</em> for every simultaneous method with corrections, <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	pp. 80-81
	]
</span>, <span class="cite">
	[
	<a href="#CiraOsiC-MinIEEE" >30</a>
	]
</span>. </p>
<p>Let us consider the nonlinear general problem with restrictions for simultaneous inclusion methods with the disks </p>
<div class="equation" id="DiscuriD">
<p>
  <div class="equation_content">
    \begin{equation} \label{DiscuriD} D_k=D\big(\widehat{z}_k;\left\vert C_k\right\vert \big),\  \  k\in I_n, \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<div class="equation" id="PNR">
<p>
  <div class="equation_content">
    \begin{equation} \label{PNR} \left\{ \begin{array}{ll} \max c(a,b,n) &  \textnormal{maximization},\\ a>1,\  b\ge 0,\  n\in \mathbf{d}_5 &  \textnormal{defining \emph{c}},\\ c(a,b,n)\le \phi (n) &  \textnormal{asymptotic condition to}\  \phi (n),\\ 0<\lambda (a,b,n)<\tfrac {1}{2} &  \Rightarrow \left\vert C_k\right\vert \le \tfrac {\lambda (a,b,n)}{c(a,b,n)}\left\vert W_k\right\vert ,\\ 0<\delta (a,b,n)<1 &  \Rightarrow \  \displaystyle \lim _{m\to \infty }\left\vert W_k^{(m)}\right\vert =0,\\ 0<\beta (a,b,n)<1 &  \Rightarrow \  \displaystyle \lim _{m\to \infty }\left\vert C_k^{(m)}\right\vert =0,\\ \theta (a,b,n)\le 1 &  \Rightarrow \  if\  w\le c(a,b,n)d\Rightarrow \widehat{w}\le c(a,b,n)\widehat{d},\\ \eta (a,b,n)<0 &  \Rightarrow \  D_k\cap D_j=\emptyset , \end{array}\right. \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> for \(k,j\in I_n\). The function \(\phi (n)\) is given by (<a href="#FunctiaPhi">9</a>). Functions \(\delta \), \(\beta \), \(\theta \) and \(\eta \) are defined by </p>
<div class="equation" id="FunctiilePNR">
<p>
  <div class="equation_content">
    \begin{equation} \label{FunctiilePNR} \left\{ \begin{array}{lcl} \lambda & =&  \lambda (c,n)~ ,\\ \Pi (\lambda ,n) & =&  \left(1+\tfrac {\lambda }{1-2\lambda }\right)^{n-1}~ , \\ \delta (\lambda ,c,n) & =&  \mu _\delta (\lambda ,c,n)\cdot \Pi (\lambda ,n)~ , \\ \beta (\lambda ,c,n) & =&  \mu _\beta (\lambda ,c,n)\cdot \delta (\lambda ,c,n)~ , \\ \mu _\theta (\lambda ) & =&  \tfrac {1}{1-2\lambda }~ , \\ \theta (\lambda ,c,n) & =&  \mu _\theta (\lambda )\cdot \beta (\lambda ,c,n)~ , \\ \eta (\lambda ,c,n) & =&  2\lambda -\tfrac {1}{g\big(\beta (\lambda ,c,n)\big)}~ . \end{array}\right. \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p> The functions \(\lambda \), \(\mu _\delta \), \(\mu _\beta \), will be defined for every inclusion method from the lemmas and convergence theorems. The functions \(\lambda \) depend on \(c\) and \(n\) while \(c\) depends on \(a\) \(b\) and \(n\). The function \(g\) is given in (<a href="#Functiag">11</a>). </p>
<p>Let us consider the nonlinear general problem with restrictions for simultaneous inclusion methods with the disks \(D^*_k\) given by (<a href="#DiscuriD*">10</a>): </p>
<div class="equation" id="PNRpentruD*">
<p>
  <div class="equation_content">
    \begin{equation} \label{PNRpentruD*} \left\{ \begin{array}{ll} \max c(a,b,n) &  \textnormal{maximization},\\ a\ge 2,\  b\ge 0,\  n\in \mathbf{d}_5 &  \textnormal{defining \emph{c}},\\ c(a,b,n)\le \psi (n) &  \textnormal{asymptotic condition to}\  \psi (n),\\ 0<\lambda (a,b,n)<\tfrac {1}{2}, &  \Rightarrow \left\vert C_k\right\vert \le \tfrac {\lambda (a,b,n)}{c(a,b,n)}\left\vert W_k\right\vert ,\\ 0<\delta (a,b,n)<1 &  \Rightarrow \  \displaystyle \lim _{m\to \infty }\left\vert W_k^{(m)}\right\vert =0,\\ 0<\beta (a,b,n)<1 &  \Rightarrow \  \displaystyle \lim _{m\to \infty }\left\vert C_k^{(m)}\right\vert =0,\\ \theta (a,b,n)\le 1 &  \Rightarrow \  if\  w\le c(a,b,n)d\Rightarrow \widehat{w}\le c(a,b,n)\widehat{d}, \end{array}\right. \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p> for \(k,j\in I_n\), where functions are defined by (<a href="#FunctiilePNR">16</a>). </p>
<p><div class="proposition_thmwrapper " id="Propozitiaf">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">7</span>
  </div>
  <div class="proposition_thmcontent">
  <p> If the functions \(f_1,f_2:I\subset \mathbb {R}\to \mathbb {R}_+\) are monotonically increasing (monotonically decreasing), then the function \(f:I\subset \mathbb {R}\to \mathbb {R}_+\), \(f(t)=f_1(t)\cdot f_2(t)\) is monotonically increasing (monotonically decreasing). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000025">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> If the functions \(f_1,f_2\) are positive and increasing (strictly increasing), then \(f'_1(t)\), \(f'_2(t)\ge 0,\  ({\gt}0)\), and \(f'(t)=f'_1(t)\cdot f_2(t)+f_1(t)\cdot f'_2(t)\ge 0,\  ({\gt}0)\). Then it follows that the function \(f(t)\) is increasing (strictly increasing). <div class="proof_wrapper" id="a0000000026">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000027">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">8</span>
  </div>
  <div class="remark_thmcontent">
  <ol class="enumerate">
  <li><p>If the functions \(\mu _\beta (a,b,n)\) and \(\mu _\theta (a,b,n)\), are monotonically and \(\ge 1\), then we have \(\delta (a,b,n)\le \beta (a,b,n)\le \theta (a,b,n)\). </p>
</li>
  <li><p>If the function \(\Pi =\Pi (a,b,n)\nearrow \) it is necessary for the functions \(\delta =\delta (a,b,n)\nearrow \), \(\beta =\beta (a,b,n)\nearrow \) and \(\theta =\theta (a,b,n)\nearrow \). These involve the followings: </p>
<ul class="itemize">
  <li><p>If \(\Pi \nearrow \), then is necessary that : \(\mu _\delta \nearrow \) for as \(\delta =\mu _\delta \Pi \nearrow \). </p>
</li>
  <li><p>If \(\delta \nearrow \), then is necessary that: \(\mu _\beta \nearrow \) for as \(\beta =\mu _\beta \delta \nearrow \) or if \(\mu _\beta \searrow \), then \(\mu _{\beta \delta }=\mu _\beta \mu _\delta \nearrow \) for as \(\beta =\mu _{\beta \delta }\Pi \nearrow \). </p>
</li>
  <li><p>If \(\beta \nearrow \), then is necessary that: \(\mu _\theta \nearrow \) for as \(\theta =\mu _\theta \beta \nearrow \) or if \(\mu _\theta \searrow \), then: </p>
<ul class="itemize">
  <li><p>\(\mu _{\beta \delta }=\mu _\beta \mu _\delta \nearrow \) for as \(\theta =\mu _{\beta \delta }\delta \nearrow \) or if \(\mu _{\beta \delta }\searrow \), then: </p>
</li>
  <li><p>\(\mu _{\theta \beta \delta }=\mu _\theta \mu _\beta \mu _\delta \nearrow \) for as \(\theta =\mu _{\theta \beta \delta }\Pi \nearrow \). </p>
</li>
</ul>
</li>
</ul>
</li>
</ol>
<p> The following notations where used: \(\mu _\delta =\mu _\delta (a,b,n)\), \(\mu _\beta =\mu _\beta (a,b,n)\), \(\mu _\theta =\mu _\theta (a,b,n)\). </p>
<p>If \(\Pi =\Pi (a,b,n)\searrow \) then we have conditions similar to the case \(\nearrow \).<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="MonotoniaLuiPi">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">9</span>
  </div>
  <div class="lemma_thmcontent">
  <p> The function \(\Pi (\lambda ,n)\) is strictly increasing if \(\lambda (n){\lt}\Lambda (n)\) and strictly decreasing if \(\lambda (n){\gt}\Lambda (n)\), where </p>
<div class="equation" id="SolutiaLambda">
<p>
  <div class="equation_content">
    \begin{equation} \label{SolutiaLambda} \Lambda (n)=\tfrac {1-\sqrt[n-1]{e}}{1-2\sqrt[n-1]{e}}~ . \end{equation}
  </div>
  <span class="equation_label">18</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> If we derive the function \(\Pi (\lambda ,n)\) in respect to the variable \(n\) we obtain </p>
<div class="displaymath" id="a0000000029">
  \begin{equation*}  \tfrac {\partial }{\partial n}\Pi (\lambda ,n) =\left[(n-1)\lambda ’(n)\! +\! \big(1-2\lambda (n)\big)\big(1\! -\! \lambda (n)\big) \ln \left(\tfrac {1\! -\! \lambda (n)}{1-2\lambda (n)}\right)\right] \times \left(\tfrac {1-\lambda (n)}{1-2\lambda (n)}\right)^n. \end{equation*}
</div>
<p> Since \(\lambda (n)\in (0,\tfrac {1}{2})\), under the constraints of nonlinear problems with restrictions (<a href="#PNR">15</a>) and (<a href="#PNRpentruD*">17</a>), we have that </p>
<div class="displaymath" id="a0000000030">
  \[  \left(\tfrac {1-\lambda (n)}{1-2\lambda (n)}\right)^n =\left(1+\tfrac {\lambda (n)}{1-2\lambda (n)}\right)^n{\gt}0.  \]
</div>
<p> The solution of differential equation </p>
<div class="displaymath" id="a0000000031">
  \[  \lambda '(n)=-\tfrac {1}{n-1}\big(1-2\lambda (n)\big)\big(1-\lambda (n)\big) \ln \left(\tfrac {1-\lambda (n)}{1-2\lambda (n)}\right),  \]
</div>
<p> is \(\Lambda (n)\), given by (<a href="#SolutiaLambda">18</a>). Then: </p>
<ul class="itemize">
  <li><p>if \(\lambda (n){\lt}\Lambda (n)\), for \(n\in \mathbf{d}_5\), then \(\Pi (\lambda ,n)\) is increasing, </p>
</li>
  <li><p>if \(\lambda (n){\gt}\Lambda (n)\), for \(n\in \mathbf{d}_5\), then \(\Pi (\lambda ,n)\) is decreasing. </p>
</li>
</ul>
<p> <div class="proof_wrapper" id="a0000000032">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="a0000000033">3 The Durand–Kerner’s method</h1>
<p>Durand Kerner method, <span class="cite">
	[
	<a href="#Durand1960" >3</a>
	, 
	<a href="#Kerner1966" >6</a>
	]
</span>, or Weierstrass-Docev method, <span class="cite">
	[
	<a href="#Weierstrass1903" >2</a>
	, 
	<a href="#Docev1962" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#Petkovic1995" >15</a>
	]
</span> is defined by </p>
<div class="equation" id="MD-KsauMW-D">
<p>
  <div class="equation_content">
    \begin{equation} \label{MD-KsauMW-D} z^{(m+1)}_k=z^{(m)}_k-W^{(m)}_k\  \textnormal{for}\  k\in I_n\  \textnormal{and}\  m=0,1,\ldots ~ . \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p>This method is part of the correction methods of form \(C_k=W_k=\tfrac {P(z_k)}{F_k(z)}\) where </p>
<div class="displaymath" id="a0000000034">
  \[  F_k(z)=F_k(z_1,z_2,\ldots ,z_n)=\prod _{j\ne k}(z_k-z_j)~ ,\  k\in I_n.  \]
</div>
<h2 id="a0000000035">3.1 Durand–Kerner’s method convergence</h2>
<p>Proof of Durand–Kerner’s method convergence requires the next lemma and theorem. <div class="lemma_thmwrapper " id="Lema1PentruConvMDK">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">10</span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let \(z_1\), \(z_2\), …, \(z_n\) be distinct approximations and let initial conditions </p>
<div class="displaymath" id="a0000000036">
  \begin{equation*}  w\le c(a,b,n)d, \end{equation*}
</div>
<div class="equation" id="PsiDinMDK">
<p>
  <div class="equation_content">
    \begin{equation} \label{PsiDinMDK} 0<c(a,b,n)<\tfrac {1}{2}, \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<div class="equation" id="ThetaDinMDK">
<p>
  <div class="equation_content">
    \begin{equation} \label{ThetaDinMDK} \theta (a,b,n)=\tfrac {\delta (a,b,n)}{1-2c(a,b,n)}\le 1, \end{equation}
  </div>
  <span class="equation_label">21</span>
</p>
</div>
<p> where </p>
<div class="equation" id="DeltaDinMDK">
<p>
  <div class="equation_content">
    \begin{equation} \label{DeltaDinMDK} \delta (a,b,n)=\tfrac {(n-1)c(a,b,n)}{1-c(a,b,n)} \left(1+\tfrac {c(a,b,n)}{1-2c(a,b,n)}\right)^{n-1}. \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p> hold. Then: </p>
<ol class="enumerate">
  <li><p>\(\left\vert \widehat{W}_k\right\vert \le \delta (a,b,n)\left\vert W_k\right\vert \), </p>
</li>
  <li><p>\(\widehat{w}\le c(a,b,n)\widehat{d}\). </p>
</li>
</ol>

  </div>
</div> <div class="proof_wrapper" id="a0000000037">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	Lemma 3.3.
	]
</span>. <div class="proof_wrapper" id="a0000000038">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <div class="theorem_thmwrapper " id="TeoremaConvMDK">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">11</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let the i-factor </p>
<div class="equation" id="I-FactorPetkovicMDK">
<p>
  <div class="equation_content">
    \begin{equation} \label{I-FactorPetkovicMDK} c_P(n)=\tfrac {1}{1.76325n+0.8689425}, \end{equation}
  </div>
  <span class="equation_label">23</span>
</p>
</div>
<p> then the Durnad-Kerner method is convergent if the initial condition is satisfied, namely if </p>
<div class="displaymath" id="a0000000039">
  \begin{equation*}  w^{(0)}{\lt}c_P(n)d^{(0)}. \end{equation*}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000040">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	T 3.3.
	]
</span>. <div class="proof_wrapper" id="a0000000041">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Petković in <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	pp. 80
	]
</span> stated that the <em>i–factor</em> is “<em>almost optimum</em>". </p>
<h2 id="a0000000042">3.2 The analytically optimum i-factor for Durand–Kerner’s method with inclusion disks \(D_k\)</h2>
<p> These inclusion disks are \(D_k=D(\widehat{z}_k,\left\vert C_k\right\vert )\), where for Durand-Kerner method the corrections \(C_k\) are Weierstrass factors \(W_k\). </p>
<p>The solution for the Lambert equation, \(x {\rm e}^x=1\), <span class="cite">
	[
	<a href="#Mathworld" >32</a>
	, 
	LambertW-Function
	]
</span>, <span class="cite">
	[
	<a href="#Corless1996" >19</a>
	]
</span>, is denoted by \(W(1)\). The constant \(1/W(1)\), is called <em>omega</em> <em>constant</em> <span class="cite">
	[
	<a href="#Mathworld" >32</a>
	, 
	OmegaConstant
	]
</span>, <span class="cite">
	[
	<a href="#Mathworld" >32</a>
	, 
	A030178
	]
</span> prints 59 decimals for the constant \(W(1)\), </p>
<div class="displaymath" id="a0000000043">
  \[  W(1)\approx 0.56714329040978387299996866221\ldots .  \]
</div>
<p> Next \(1/W(1)\) will be denoted by \(\omega \), </p>
<div class="displaymath" id="a0000000044">
  \[  \omega \approx 1.76322283435189671022520177695\ldots .  \]
</div>
<p>Let the constant </p>
<div class="equation" id="ConstantaTau">
<p>
  <div class="equation_content">
    \begin{equation} \label{ConstantaTau} \tau \approx 0.88049674007368891\ldots , \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p> be the real solution of </p>
<div class="equation" id="EcuatiaPentruTau">
<p>
  <div class="equation_content">
    \begin{equation} \label{EcuatiaPentruTau} b^3+(11\omega -3)b^2+(35\omega ^2-30\omega +2)b+25\omega ^3-75\omega ^2+18\omega =0. \end{equation}
  </div>
  <span class="equation_label">25</span>
</p>
</div>
<p>For Durand–Kerner’s method we have the function \(c(a,b,n)\), given by (<a href="#I-FactorPetkovic">7</a>), and the functions: </p>
<div class="equation" id="FunctiileMDK">
<p>
  <div class="equation_content">
    \begin{equation} \label{FunctiileMDK} \left\{ \begin{array}{l} \lambda (c)=c,\\ \mu _\delta (\lambda ,n)=\tfrac {(n-1)\lambda }{1-\lambda },\\ \mu _\beta =1, \end{array}\right. \end{equation}
  </div>
  <span class="equation_label">26</span>
</p>
</div>
<p> resulting from Lemma <a href="#Lema1PentruConvMDK">10</a> relations (<a href="#ThetaDinMDK">21</a>) and (<a href="#DeltaDinMDK">22</a>), and the functions \(\delta \), \(\beta \), \(\mu _\theta \), \(\theta \) and \(\eta \) are defined in (<a href="#FunctiilePNR">16</a>). </p>
<p><div class="proposition_thmwrapper " id="PropozitiaLambdaMDK">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">12</span>
  </div>
  <div class="proposition_thmcontent">
  <p> If \(b{\gt}h(a,5)\), then the function \(\lambda (a,b,n)=c(a,b,n){\lt}\Lambda (n)\), where </p>
<div class="displaymath" id="a0000000045">
  \[  h(a,n)=\tfrac {1}{\Lambda (n)}-an  \]
</div>
<p> and \(\Lambda \) is given by <em>(<a href="#SolutiaLambda">18</a>)</em>. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000046">
  <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 function </p>
<div class="equation" id="Functiah(a,n)MDK">
<p>
  <div class="equation_content">
    \begin{equation} \label{Functiah(a,n)MDK} h(a,n)=\tfrac {1-\sqrt[n-1]{e}}{1-2\sqrt[n-1]{e}}-an, \end{equation}
  </div>
  <span class="equation_label">27</span>
</p>
</div>
<p> is decreasing, because </p>
<div class="displaymath" id="a0000000047">
  \[  h'(a,n)=\tfrac {\partial }{\partial n}h(a,n) =\tfrac {1}{\left(2(n-1)\sinh \left(\tfrac {1}{2(n-1)}\right)\right)^2}-a{\lt}0,  \]
</div>
<p> for \(a{\gt}1\) and \(n\in \mathbf{d}_5\). Then, if \(b{\gt}h(a,5)\) resulting that \(\lambda (a,b,n){\lt}\Lambda (n)\). <div class="proof_wrapper" id="a0000000048">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="Lema1MDK">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">13</span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let the <em>i-factor</em> </p>
<div class="equation" id="I-FactorOptimMDK">
<p>
  <div class="equation_content">
    \begin{equation} \label{I-FactorOptimMDK} c(n)=c(\omega ,\tau ,n)=\tfrac {1}{\omega n+\tau }, \end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<p> then: </p>
<ol class="enumerate">
  <li><p>\(0{\lt}\lambda (\omega ,\tau ,n){\lt}0.10313\), </p>
</li>
  <li><p>\(0.707{\lt}\delta (\omega ,\tau ,n){\lt}1\), </p>
</li>
  <li><p>\(0.944{\lt}\theta (\omega ,\tau ,n){\lt}1\) </p>
</li>
  <li><p>\(\eta (\omega ,\tau ,n){\lt}0\), </p>
</li>
</ol>
<p> for \(n\in \mathbf{d}_5\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000049">
  <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 derivative function \(\lambda (a,b,n)=c(a,b,n)\) with respect to \(n\) is </p>
<div class="displaymath" id="a0000000050">
  \[  -\tfrac {a}{(an+b)^2}{\lt}0,  \]
</div>
<p> which implies that the functions \(c\) and \(\lambda \) are strictly decreasing. For </p>
<div class="displaymath" id="a0000000051">
  \[  \lambda (\omega ,\tau ,n)=c(\omega ,\tau ,n)  \]
</div>
<p> we have </p>
<div class="displaymath" id="a0000000052">
  \[  \lim _{n\to \infty }\lambda (\omega ,\tau ,n)=0\  \  \textnormal{and}\  \   \lambda (5)=\tfrac {1}{5\omega +\tau }\approx 0.10312881573707708\ldots {\lt}\tfrac {1}{2},  \]
</div>
<p> then we have (1) of lemma. </p>
<p>If the function \(\lambda (a,b,n)\to 0\), then the function \(\Pi (\lambda (a,b,n),n)\) converges and we have </p>
<div class="displaymath" id="a0000000053">
  \[  \lim _{n\to \infty }\Pi \big(\lambda (a,b,n),n\big)= \lim _{n\to \infty }\Pi (a,b,n)={\rm e}^{\tfrac {1}{a}},  \]
</div>
<p> for \(a{\gt}1\). In these conditions we have that </p>
<div class="displaymath" id="a0000000054">
  \[  \lim _{n\to \infty }\delta (a,b,n)=\tfrac {1}{a}{\rm e}^{\tfrac {1}{a}}\  \  \textnormal{and}\   \lim _{n\to \infty }\theta (a,b,n)=\tfrac {1}{a}{\rm e}^{\tfrac {1}{a}}.  \]
</div>
<p> Since \(\mu _\theta (a,b,n)=(an+b)/(an+b-2){\gt}1\) it follows that \(\theta (a,b,n){\gt}\beta (a,b,n)=\delta (a,b,n)\). Imposing the inequality \(\theta (a,b,n)\le 1\) it follows that the equation </p>
<div class="displaymath" id="a0000000055">
  \[  \tfrac {{\rm e}^{\tfrac {1}{a}}}{a}=1.  \]
</div>
<p> This equation is a Lambert type equation, <span class="cite">
	[
	<a href="#Corless1996" >19</a>
	]
</span>, whose solution is the constant \(\omega \). </p>
<p>The function \(\mu _\delta (\omega ,b,n)=(n-1)/(an+b-1)\) is increasing, because </p>
<div class="displaymath" id="a0000000056">
  \[  \tfrac {\partial }{\partial n}\mu _\delta (\omega ,b,n)=\tfrac {\omega +b-1}{(\omega n+b-1)^2}{\gt}0,  \]
</div>
<p> for \(b{\gt}1-\omega \approx -0.7632228\ldots \) and \(n\in \mathbf{d}_5\). </p>
<p>Under Lemma <a href="#MonotoniaLuiPi">9</a> and Proposition <a href="#PropozitiaLambdaMDK">12</a>, for function \(\Pi (\omega ,b,n)\) to be increasing it is necessary that \(b{\gt}h(\omega ,5)\approx -3.295\ldots \), where \(h\) is given by (<a href="#Functiah(a,n)MDK">27</a>). Then, it follows that \(\delta (\omega ,b,n)\nearrow 1\), for \(b\ge 0\) when \(n\to \infty \). </p>
<p>The derivative for the function \(\mu _\theta (\omega ,b,n)\) is </p>
<div class="displaymath" id="a0000000057">
  \[  \tfrac {\partial }{\partial n}\mu _\theta (\omega ,b,n)=\tfrac {-2\omega }{(\omega n+b-2)^2}{\lt}0.  \]
</div>
<p> Therefore we can not say that the function \(\theta (\omega ,b,n)=\mu _\theta (\omega ,b,n)\delta (\omega ,b,n)\) is increasing on Proposition <a href="#Propozitiaf">7</a>. But the function \(\theta (a,b,n)\) can be expressed as </p>
<div class="equation" id="Theta">
<p>
  <div class="equation_content">
    \begin{equation} \label{Theta} \theta (a,b,n)=\mu _\theta (a,b,n)\delta (a,b,n)=\mu _\theta (a,b,n)\mu _\delta (a,b,n)\Pi (a,b,n). \end{equation}
  </div>
  <span class="equation_label">29</span>
</p>
</div>
<p> Using the notation \(\mu _{\theta \delta }(a,b,n)=\mu _\theta (a,b,n)\mu _\delta (a,b,n)\), it follows that </p>
<div class="equation" id="Mu">
<p>
  <div class="equation_content">
    \begin{equation} \label{Mu} \mu _{\theta \delta }(a,b,n)=(n-1)\tfrac {an+b}{(an+b-1)(an+b-2)} \end{equation}
  </div>
  <span class="equation_label">30</span>
</p>
</div>
<p> and </p>
<div class="equation" id="MuPrim">
<p>
  <div class="equation_content">
    \begin{equation} \label{MuPrim} \mu '_{\theta \delta }(a,b,n)=\tfrac {\partial }{\partial n}\mu _{\theta \delta }(a,b,n) =\tfrac {P_3(a,b,n)}{(an+b-1)^2(an+b-2)^2}. \end{equation}
  </div>
  <span class="equation_label">31</span>
</p>
</div>
<p> Thus we have the polynomial </p>
<div class="equation" id="PolinomulMuPrim">
<p>
  <div class="equation_content">
    \begin{equation} \label{PolinomulMuPrim} P_3(a,b,n)=b^3+\alpha _2(a,n)b^2+\alpha _1(a,n)b+\alpha _0(a,n), \end{equation}
  </div>
  <span class="equation_label">32</span>
</p>
</div>
<p> with coefficients </p>
<div class="equation" id="CoefPolinomMuPrim">
<p>
  <div class="equation_content">
    \begin{equation} \label{CoefPolinomMuPrim} \begin{array}{lcl} \alpha _2(a,n) & =&  (2n+1)a-3, \\ \alpha _1(a,n) & =&  n(n+2)a^2-6na+2, \\ \alpha _0(a,n) & =&  a(n^2a^2-3n^2a+4n-2) , \end{array} \end{equation}
  </div>
  <span class="equation_label">33</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000058">
  \begin{eqnarray*}  \alpha _2(a,n) & \ge 0&  for\  a\ge \tfrac {3}{11}\  \textnormal{and}\  n\in \mathbf{d}_5, \\ \alpha _1(a,n) & \ge 0&  for\  a\notin \left(\tfrac {3}{7}-\sqrt{\tfrac {31}{7}}, \tfrac {3}{7}+\sqrt{\tfrac {31}{7}}\right)\approx (0.073,0.748)\  \textnormal{and}\  n\in \mathbf{d}_5,\\ \alpha _0(a,n) & \le 0&  for\  a\in \left(\tfrac {3}{2}-\tfrac {3\sqrt{17}}{10}, \tfrac {3}{2}+\tfrac {3\sqrt{17}}{10}\right)\approx (0.263,2.737)\  \textnormal{and}\  n\in \mathbf{d}_5. \end{eqnarray*}
</div>
<p> Then, according to Cauchy’s theorem, <span class="cite">
	[
	<a href="#CiraO2005" >29</a>
	, 
	pp. 3
	]
</span>, the equation \(P_3(a,b,n)=0\) has a unique positive real root, in relation to \(b\). The greatest root results for \(n=5\). For \(a=\omega \) and \(n=5\) we have the equation (<a href="#EcuatiaPentruTau">25</a>). For \(a=\omega \) and \(n=6\) we have the solution \(\approx 0.92989\ldots \), for \(a=\omega \) and \(n=7\) we have the solution \(\approx 0.96706\ldots \) and so on. </p>
<p>Then it follows that \(\mu _{\theta \delta }(\omega ,b,n)\) is strictly increasing if \(\mu '_{\theta \delta }(\omega ,b,5){\gt}0\). We have this inequality if \(b{\gt}\tau \approx 0.8804967400681223\ldots \) is true. The function \(\theta (\omega ,b,n)\) is the product of two positive and increasing functions, \(\Pi (\omega ,b,n)\) and \(\mu _{\theta \delta }(\omega ,b,n)\). If \(b{\gt}\tau \), then, according to Proposition <a href="#Propozitiaf">7</a>, the function \(\theta (\omega ,b,n)\) is strictly increasing. </p>
<p>We compute the function values \(\delta \) and \(\theta \) for \(n=5\), </p>
<div class="displaymath" id="a0000000059">
  \begin{eqnarray*}  \delta (\omega ,\tau ,5) & \approx &  0.7497422461897223\ldots , \\ \theta (\omega ,\tau ,5) & \approx &  0.9445662420442020\ldots . \end{eqnarray*}
</div>
<p> Because functions \(\delta \) and \(\theta \) are strictly increasing resulting that claims the second and the third from the lemma are true. </p>
<p>For Durand–Kerner’s method we have \(\delta (a,b,n)=\beta (a,b,n)\), when taking into account (2) of the lemma and the definition of function g, given by (<a href="#Functiag">11</a>), we have </p>
<div class="displaymath" id="a0000000060">
  \[  \eta (\omega ,\tau ,n)=2\lambda (\omega ,\tau ,n)+\beta (\omega ,\tau ,n)-1 =2\lambda (\omega ,\tau ,n)+\delta (\omega ,\tau ,n)-1.  \]
</div>
<p> To demonstrate that \(\eta (\omega ,\tau ,n){\lt}0\), for \(n\in \mathbf{d}_5\), but this relation is equivalent to \(\theta (\omega ,\tau ,n){\lt}1\), for \(n\in \mathbf{d}_5\), relation that has already been demonstrated. <div class="proof_wrapper" id="a0000000061">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="TeoremaI-FactorOptimMDK">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">14</span>
  </div>
  <div class="theorem_thmcontent">
  <p> The function \(c(n)\) given by <em>(<a href="#I-FactorOptimMDK">28</a>)</em> is the optimum <em>i-factor</em> for Durand–Kerner’s method with inclusion disks \(D_k\), given by <em>(<a href="#DiscuriD">14</a>)</em>. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000062">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Since \(\omega \) was computed from the limit condition </p>
<div class="displaymath" id="a0000000063">
  \[  \tfrac {1}{\omega }{\rm e}^{\tfrac {1}{\omega }}=1,  \]
</div>
<p> and \(\tau \) from the extreme condition, namely \(\tau \) is the real solution for the equation, it implies that \(\omega \) and \(\tau \) are the best values for the given conditions. We can then state that \(c(n)\) given by (<a href="#I-FactorOptimMDK">28</a>) is the optimum <em>i-factor</em> for Durand–Kerner’s method. <div class="proof_wrapper" id="a0000000064">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="TeoremaCiraConvMDK">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">15</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If the initial distinct approximations \(z^{(0)}_1,z^{(0)}_2,\ldots ,z^{(0)}_n\) satisfy the initial condition </p>
<div class="equation" id="CondInitialaMDK">
<p>
  <div class="equation_content">
    \begin{equation} \label{CondInitialaMDK} w^{(0)}<c(n)d^{(0)}~ , \end{equation}
  </div>
  <span class="equation_label">34</span>
</p>
</div>
<p> for \(n\in \mathbf{d}_5\), where \(c\) given by <em>(<a href="#I-FactorOptimMDK">28</a>)</em>, then the Durand-Kerner method with inclusion disks \(D_k\), given by <em>(<a href="#DiscuriD">14</a>)</em>, is convergent. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000065">
  <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 in the Lemma <a href="#Lema1MDK">13</a> assure that the conclusions of Lemma <a href="#Lema1PentruConvMDK">10</a> are satisfied, which, in turn assure the convergence of the Durand-Kerner method if the initial condition is verified (<a href="#CondInitialaMDK">34</a>). <div class="proof_wrapper" id="a0000000066">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000067">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">16</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \(n\in \mathbf{d}_3\), case considered by Petković et al., \(a=\omega \) and \(b=\tau _3\), where \(\tau _3\) is the solution for the equation \(\mu '_{\theta \delta }(\omega ,b,3)=0\), and the function \(\mu '_{\theta \delta }\) is given by (<a href="#MuPrim">31</a>). So we have </p>
<div class="displaymath" id="a0000000068">
  \[  b^3+(7\omega -3)b^2+(15\omega ^2-18\omega +2)b+9\omega ^3-27\omega ^2+10\omega =0,  \]
</div>
<p> and the optimum <em>i-factor</em> </p>
<div class="displaymath" id="a0000000069">
  \[  c(n)=\tfrac {1}{\omega n+\tau _3},\  \  \textnormal{with}\  \  \tau _3=0.7071447767242046\ldots ,  \]
</div>
<p> for \(n\in \mathbf{d}_3\).<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h2 id="a0000000070">3.3 The analytically optimum i-factor for Durand–Kerner’s method<br />with inclusion disks \(D^*_k\)</h2>
<p> The inclusion disks \(D^*_k\) are given by (<a href="#DiscuriD*">10</a>). </p>
<p><div class="lemma_thmwrapper " id="Lema2MDK">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">17</span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let the <em>i-factor</em> be </p>
<div class="equation" id="I-FactorOptimMDKD*">
<p>
  <div class="equation_content">
    \begin{equation} \label{I-FactorOptimMDKD*} c(n)=c(2,\tau _*,n)=\tfrac {1}{2n+\tau _*},\  \  \tau _*\approx 0.67211423631036255\ldots , \end{equation}
  </div>
  <span class="equation_label">35</span>
</p>
</div>
<p> where \(\tau _*\) is a root of equation </p>
<div class="equation" id="EcuatiaPentruTau*">
<p>
  <div class="equation_content">
    \begin{equation} \label{EcuatiaPentruTau*} b^3+19b^2+82b-64=0, \end{equation}
  </div>
  <span class="equation_label">36</span>
</p>
</div>
<p> then: </p>
<ol class="enumerate">
  <li><p>\(0{\lt}\lambda (2,\tau _*,n){\lt}0.094\), </p>
</li>
  <li><p>\(0.639{\lt}\delta (2,\tau _*,n)\le \tfrac {\sqrt{e}}{2}\approx 0.824\), </p>
</li>
  <li><p>\(0.787{\lt}\theta (2,\tau _*,n)\le \tfrac {\sqrt{e}}{2}\approx 0.824\), </p>
</li>
</ol>
<p> for \(n\in \mathbf{d}_5\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000071">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> According to corollary <a href="#Corolarul11Petkovic">4</a> if we consider the inclusion disks \(D^*_k\) given by (<a href="#DiscuriD*">10</a>), then \(a\ge 2\) and \(b{\gt}(2-a)n\). Let \(a=2\), the lowest value of \(a\), then it follows that \(b{\gt}0\). </p>
<p>The function \(\lambda (2,b,n)\searrow 0\) when \(n\to \infty \), because the derivative function \(\lambda \) in relation to the variable \(n\) is \(-2/(2n+b)^2{\lt}0\). Then, \(\Pi (2,b,n)\to \sqrt{{\rm e}}\) when \(n\to \infty \). According to Proposition <a href="#Propozitiaf">7</a>, the function \(\delta (2,b,n)\) is increasing if the functions \(\mu _\delta (2,b,n)\) and \(\Pi (2,b,n)\) are increasing. The derivative of the function \(\mu _\delta (2,b,n)\) is </p>
<div class="displaymath" id="a0000000072">
  \[  \tfrac {\partial }{\partial n}\mu _\delta (2,b,n)=\tfrac {b+1}{(b+2n-1)^2}~ ,  \]
</div>
<p> and is positive if \(b{\gt}-1\). The function \(\Pi (2,b,n)\) is increasing if \(b{\gt}h(2,5)\approx -4.479\), where \(h\) is given by (<a href="#Functiah(a,n)MDK">27</a>). Therefore it follows that \(\delta (2,b,n)\) is increasing for \(b\ge 0\) and \(n\in \mathbf{d}_5\). In these circumstances we have </p>
<div class="displaymath" id="a0000000073">
  \[  \lim _{n\to \infty }\delta (2,b,n)=\lim _{n\to \infty }\theta (2,b,n)=\tfrac {\sqrt{{\rm e}}}{2}\approx 0.824{\lt}1.  \]
</div>
<p>The derivative of the function \(\mu _\theta (2,b,n)\) is </p>
<div class="displaymath" id="a0000000074">
  \[  \tfrac {\partial }{\partial n}\mu _\theta (2,b,n)=\tfrac {-4}{(2n+b-2)^2}{\lt}0.  \]
</div>
<p> We can not say that the function \(\theta (2,b,n)=\mu _\theta (2,b,n)\delta (2,b,n)\) is increasing in the Proposition <a href="#Propozitiaf">7</a>. But the function \(\theta (a,b,n)\) can be expressed as (<a href="#Theta">29</a>). We denote by \(\mu _{\theta \delta }(a,b,n)=\mu _\theta (a,b,n)\mu _\delta (a,b,n)\) and consider the polynomial (<a href="#PolinomulMuPrim">32</a>) with coefficients (<a href="#CoefPolinomMuPrim">33</a>). If \(0.748{\lt}a{\lt}2.737\), then, according to Cauchy’s theorem, <span class="cite">
	[
	<a href="#CiraO2005" >29</a>
	, 
	pp. 3
	]
</span>, the equation \(P_3(a,b,n)=0\), in relation to the variable \(b\), has a unique real positive solution. The largest positive real root results for \(n=5\). For \(a=2\) and \(n=5\) resulting the equation (<a href="#EcuatiaPentruTau*">36</a>), with real positive solution \(\tau _*\approx 0.67211423631036255\ldots \). For \(a=2\) and \(n=6\) we have the solution \(0.71912\ldots \), for \(a=2\) and \(n=7\) we have the solution \(0.75419\ldots \) and so on. </p>
<p>Then it follows that the function \(\mu _{\theta \delta }(2,b,n)\) is increasing if \(b{\gt}\tau _*\). The function \(\theta (2,b,n)\) is the product of two positive and increasing functions, namely \(\Pi (2,b,n)\) and \(\mu _{\theta \delta }(2,b,n)\), if \(b{\gt}\tau _*\), then, according to Proposition <a href="#Propozitiaf">7</a>, the function \(\theta (2,b,n)\) is increasing. </p>
<p>Monotonous function values \(\theta \), \(\delta \) and \(\lambda \) for \(n=5\) are: </p>
<div class="displaymath" id="a0000000075">
  \begin{eqnarray*}  \theta (2,\tau _*,5) & \approx &  0.787498541207337\ldots , \\ \delta (2,\tau _*,5) & \approx &  0.6399179355710173\ldots , \\ \lambda (2,\tau _*,5) & \approx &  0.09370214540972971. \end{eqnarray*}
</div>
<p> <div class="proof_wrapper" id="a0000000076">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="TeoremaI-FactorOptimMDKD*">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">18</span>
  </div>
  <div class="theorem_thmcontent">
  <p> The function \(c(n)\) given by <em>(<a href="#I-FactorOptimMDKD*">35</a>)</em> is the optimum <em>i-factor</em> for Durand–Kerner’s method with the inclusion disks \(D^*_k\) given by <em>(<a href="#DiscuriD*">10</a>)</em>. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000077">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Since \(a=2\) and \(b=\tau _*\) are the best values under the circumstances, then we say that \(c(n)\) given by (<a href="#I-FactorOptimMDKD*">35</a>) is optimum <em>i-factor</em> for Durand–Kerner’s method with the inclusion disks \(D^*_k\), given by (<a href="#DiscuriD*">10</a>). <div class="proof_wrapper" id="a0000000078">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="TeoremaCiraConvMDKD*">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">19</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If the initial distinct approximations \(z^{(0)}_1,z^{(0)}_2,\ldots ,z^{(0)}_n\) satisfy the initial condition </p>
<div class="equation" id="CondInitialaMDKD*">
<p>
  <div class="equation_content">
    \begin{equation} \label{CondInitialaMDKD*} w^{(0)}<c(n)d^{(0)}, \end{equation}
  </div>
  <span class="equation_label">37</span>
</p>
</div>
<p> for \(n\in \mathbf{d}_5\), where \(c\) is given by <em>(<a href="#I-FactorOptimMDKD*">35</a>)</em>, then Durand–Kerner’s method with inclusion disks \(D^*_k\), given by <em>(<a href="#DiscuriD*">10</a>)</em>, is convergent. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000079">
  <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 in the Lemma <a href="#Lema2MDK">17</a> assure that the conclusions of Lemma <a href="#Lema1PentruConvMDK">10</a> are satisfied, which, in turn assure the convergence of the Durand-Kerner method if the initial condition is verified (<a href="#CondInitialaMDKD*">37</a>). <div class="proof_wrapper" id="a0000000080">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="a0000000081">4 The Tanabe’s method</h1>
<p> The Tanabe’s method, <span class="cite">
	[
	<a href="#Tanabe1983" >10</a>
	]
</span>, is given by the formula: </p>
<div class="equation" id="MetodaTanabe">
<p>
  <div class="equation_content">
    \begin{equation} \label{MetodaTanabe} z^{(m+1)}_k=z^{(m)}_k-W^{(m)}_k\left(1-\sum _{j\ne k}\tfrac {W^{(m)}_j}{z^{(m)}_k-z^{(m)}_j}\right),\  k\in I_n\  m=0,1,\ldots . \end{equation}
  </div>
  <span class="equation_label">38</span>
</p>
</div>
<p> If we denote by </p>
<div class="displaymath" id="a0000000082">
  \[  t_k=\sum _{j\ne k}\tfrac {W_j}{z_k-z_j},\  k\in I_n,  \]
</div>
<p> then for reasonably small values of \(t_k\), we can state that \(1/(1+t_k)=1-t_k+O(t_k^2)\). It is well known that Tanabe’s method results form Börsch-Supan’s method based on this observation. Tanabe’s method is one with corrections, where </p>
<div class="displaymath" id="a0000000083">
  \[  C_k(z_1,z_2,\ldots ,z_n)=\tfrac {P(z_k)}{F_k(z_1,z_2,\ldots ,z_n)},\  k\in I_n,  \]
</div>
<p> but </p>
<div class="displaymath" id="a0000000084">
  \[  F_k(z_1,z_2,\ldots ,z_n) =\tfrac {1}{1-\displaystyle \sum _{j\ne k}\tfrac {W^{(m)}_k}{z^{(m)}_k-z^{(m)}_j}} \cdot \prod _{j\ne k}(z_k-z_j).  \]
</div>
<h2 id="a0000000085">4.1 Tanabe’s method convergence</h2>
<p> In order to prove the convergence we state the following theorem and 4 lemmas. <div class="lemma_thmwrapper " id="Lema1MT">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">20</span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let \(z_1,z_2,\ldots ,z_n\in \mathbb {C}\) be distinct numbers and the <em>i-factor</em> \(c(a,b,n)\) that satisfies the conditions </p>
<div class="equation" id="PsiDinMT">
<p>
  <div class="equation_content">
    \begin{equation} \label{PsiDinMT} 0<c(a,b,n)<\psi (n)=\tfrac {1}{1+\sqrt{2n-1}} \end{equation}
  </div>
  <span class="equation_label">39</span>
</p>
</div>
<p> and </p>
<div class="equation" id="CondInitialaMT">
<p>
  <div class="equation_content">
    \begin{equation} \label{CondInitialaMT} w\le c(a,b,n)d. \end{equation}
  </div>
  <span class="equation_label">40</span>
</p>
</div>
<p> Then: </p>
<div class="displaymath" id="a0000000086">
  \begin{eqnarray*}  (1) & &  \tfrac {\lambda (a,b,n)}{c(a,b,n)}\ge \left\vert 1-\sum _{j\ne k}\tfrac {W_j}{z_k-z_j}\right\vert \ge 2-\tfrac {\lambda (a,b,n)}{c(a,b,n)},\\ (2) & &  \left\vert \widehat{z}_k-z_k\right\vert \le \tfrac {\lambda (a,b,n)}{c(a,b,n)}\left\vert W_k\right\vert  \le \lambda (a,b,n)d,\\ (3) & &  \left\vert \widehat{z}_k-z_j\right\vert \ge \big(1-\lambda (a,b,n)\big)d,\\ (4) & &  \left\vert \widehat{z}_k-\widehat{z}_j\right\vert \ge \big(1-2\lambda (a,b,n)\big)d,\\ (5) & &  \left\vert 1+\sum \tfrac {W_j}{\widehat{z}_k-z_j}\right\vert \le \tfrac {(n-1)\big(\lambda (a,b,n)+ (n-1)c(a,b,n)\big)c(a,b,n)^2}{\big(2c(a,b,n)-\lambda (a,b,n)\big)\big(1-\lambda (a,b,n)\big)},\\ (6) & &  \left\vert \prod _{j\ne k}\tfrac {\widehat{z}_k-z_j}{\widehat{z}_k-\widehat{z}_j}\right\vert \le \left(1+\tfrac {\lambda (a,b,n)}{1-2\lambda (a,b,n)}\right)^{n-1},\\ \end{eqnarray*}
</div>
<p> where </p>
<div class="equation" id="LambdaDinMT">
<p>
  <div class="equation_content">
    \begin{equation} \label{LambdaDinMT} \lambda (a,b,n)=\big(1+(n-1)c(a,b,n)\big)c(a,b,n). \end{equation}
  </div>
  <span class="equation_label">41</span>
</p>
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000087">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	Lemma 3.8.
	]
</span>. <div class="proof_wrapper" id="a0000000088">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="Lema2MT">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">21</span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let \(z_1\), \(z_2\), …, \(z_n\) be distinct approximations for the roots \(\zeta _1\), \(\zeta _2\), …, \( \zeta _n\) of the polynomial \(P\) and let us assume the conditions <em>(<a href="#PsiDinMT">39</a>)</em> and <em>(<a href="#CondInitialaMT">40</a>)</em> from Lemma <em><a href="#Lema1MT">20</a></em> to be true. Let us also consider true the following inequality </p>
<div class="equation" id="ThetaDinMT">
<p>
  <div class="equation_content">
    \begin{equation} \label{ThetaDinMT} \theta (a,b,n)=\tfrac {\delta (a,b,n)}{1-2\lambda (a,b,n)}\le 1, \end{equation}
  </div>
  <span class="equation_label">42</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="DeltaDinMT">
  \begin{multline} \label{DeltaDinMT} \delta (a,b,n)=\tfrac {(n-1)c(a,b,n)\lambda (a,b,n)\big(\lambda (a,b,n)+(n-1)c(a,b,n)\big)} {\big(2c(a,b,n)-\lambda (a,b,n)\big)\big(1-\lambda (a,b,n)\big)} \times \left(1+\tfrac {\lambda (a,b,n)}{1-2\lambda (a,b,n)}\right)^{n-1}. \end{multline}
</div>
<p> Then: </p>
<div class="displaymath" id="a0000000089">
  \begin{eqnarray*}  (1) & &  \left\vert \widehat{W}_k\right\vert \le \delta (a,b,n)\left\vert W_k\right\vert ,\  k\in I_n \\ (2) & &  \widehat{w}\le c(a,b,n)w. \end{eqnarray*}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000090">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	Lemma 3.9.
	]
</span>. <div class="proof_wrapper" id="a0000000091">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="Lema3MT">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">22</span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let us consider all the conditions form Lemma <em><a href="#Lema1MT">20</a></em> and <em><a href="#Lema2MT">21</a></em> and the next two ones to be true </p>
<div class="equation" id="BetaDinMT">
<p>
  <div class="equation_content">
    \begin{equation} \label{BetaDinMT} \beta (a,b,n)=\tfrac {\lambda (a,b,n)\delta (a,b,n)}{2c(a,b,n)-\lambda (a,b,n)}<1 \end{equation}
  </div>
  <span class="equation_label">44</span>
</p>
</div>
<p> and </p>
<div class="equation" id="EtaDinMT">
<p>
  <div class="equation_content">
    \begin{equation} \label{EtaDinMT} \eta (a,b,n)=2\lambda (a,b,n)-\tfrac {1}{g\big(\beta (a,b,n)\big)}<0~ . \end{equation}
  </div>
  <span class="equation_label">45</span>
</p>
</div>
<p> If the initial distinct approximations \(z^{(0)}_1,z^{(0)}_2,\ldots ,z^{(0)}_n\) satisfy the initial condition </p>
<div class="equation" id="CIMT">
<p>
  <div class="equation_content">
    \begin{equation} \label{CIMT} w^{(0)}\le c(a,b,n)d^{(0)}, \end{equation}
  </div>
  <span class="equation_label">46</span>
</p>
</div>
<p> then Tanabe’s method given by <em>(<a href="#MetodaTanabe">38</a>)</em> is convergent. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000092">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	T 3.6.
	]
</span>. <div class="proof_wrapper" id="a0000000093">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="Lema4MT">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">23</span>
  </div>
  <div class="lemma_thmcontent">
  <p> The <em>i-factor</em> function </p>
<div class="equation" id="I-FactorMTPetkovic">
<p>
  <div class="equation_content">
    \begin{equation} \label{I-FactorMTPetkovic} c_P(n)=\tfrac {1}{3n}. \end{equation}
  </div>
  <span class="equation_label">47</span>
</p>
</div>
<p> satisfies then conditions <em>(<a href="#PsiDinMT">39</a></em>), <em>(<a href="#ThetaDinMT">42</a>)</em>, <em>(<a href="#BetaDinMT">44</a>)</em> and <em>(<a href="#EtaDinMT">45</a>)</em>. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000094">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> See <span class="cite">
	[
	<a href="#Petkovic2008" >31</a>
	, 
	Lemma 3.10.
	]
</span>. <div class="proof_wrapper" id="a0000000095">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="TeoremaConvMT">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">24</span>
  </div>
  <div class="theorem_thmcontent">
  <p> The Tanabe’s method, given by <em>(<a href="#MetodaTanabe">38</a>)</em>, is convergent if the initial condition <em>(<a href="#CondInitialaMT">40</a>)</em> is true for the initial distinct approximations \(z^{(0)}_1\), \(z^{(0)}_2\), …\(z^{(0)}_n\), where the <em>i-factor</em> is given by <em>(<a href="#I-FactorMTPetkovic">47</a>)</em>. </p>

  </div>
</div> </p>
<h2 id="a0000000096">4.2 The optimum analytic i-factor</h2>
<p>For Tanabe’s method we have the function \(c(a,b,n)\), given by (<a href="#I-FactorPetkovic">7</a>) and the functions: </p>
<div class="equation" id="FunctiileMT">
<p>
  <div class="equation_content">
    \begin{equation} \label{FunctiileMT} \left\{ \begin{array}{lcl} \lambda (c,n) & =&  c+(n-1)c^2 \\ \mu _\delta (\lambda ,c,n) & =&  \tfrac {(n-1)^2c^2\lambda +(n-1)c\lambda ^2}{(2c-\lambda )(1-\lambda )}, \\ \mu _\beta (\lambda ,c) & =& \tfrac {\lambda }{2c-\lambda }, \end{array}\right. \end{equation}
  </div>
  <span class="equation_label">48</span>
</p>
</div>
<p> resulting from the Lemmas <a href="#Lema1MT">20</a>, <a href="#Lema2MT">21</a> and <a href="#Lema3MT">22</a>, respectively from the relations (<a href="#LambdaDinMT">41</a>), (<a href="#DeltaDinMT">43</a>) and (<a href="#BetaDinMT">44</a>) and the functions \(\delta \), \(\beta \), \(\mu _\theta \) and \(\theta \) defined in (<a href="#FunctiilePNR">16</a>). Let the constant \(\omega _{_\textrm {T}}\) be the solution for the Lambert type equation, <span class="cite">
	[
	<a href="#Corless1996" >19</a>
	]
</span>, </p>
<div class="equation" id="EcuatiaLambertOmegaT">
<p>
  <div class="equation_content">
    \begin{equation} \label{EcuatiaLambertOmegaT} \tfrac {x+1}{x^2}\exp \left(\tfrac {x+1}{x^2}\right) =\tfrac {(x-1)^2}{x+1}~ . \end{equation}
  </div>
  <span class="equation_label">49</span>
</p>
</div>
<p> The approximative value \(\omega _{_\textrm {T}}\) is \(2.7480500253477966212\ldots \). </p>
<p><div class="proposition_thmwrapper " id="PropozitiaLambdaMT">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">25</span>
  </div>
  <div class="proposition_thmcontent">
  <p> If \(b{\gt}h_\emph {T}(a,5)\), then \(\lambda (a,b,n){\lt}\Lambda (n)\), where \(\lambda (a,b,n)=c(a,b,n)+(n-1)c(a,b,n)^2\), </p>
<div class="equation" id="Functiah(a,n)MT">
<p>
  <div class="equation_content">
    \begin{equation} \label{Functiah(a,n)MT} h_\emph {T}(a,n)=\tfrac {-2(a-1)\sqrt[n-1]{e}+2an-1}{2(\sqrt[n-1]{e}-1)} +\tfrac {\sqrt{4(2n-1)\sqrt[n-1]{e^2}-4(3n-2)\sqrt[n-1]{e}+4n-3}}{2(\sqrt[n-1]{e}-1)} \end{equation}
  </div>
  <span class="equation_label">50</span>
</p>
</div>
<p> and \(\Lambda \) is given by <em>(<a href="#SolutiaLambda">18</a>)</em>. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000097">
  <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 equation \(\lambda (a,b,n)=\Lambda (n)\) has solutions: </p>
<div class="displaymath" id="a0000000098">
  \[  \tfrac {-2(a-1)\sqrt[n-1]{e}+2an-1}{2(\sqrt[n-1]{e}-1)} \pm \tfrac {\sqrt{4(2n-1)\sqrt[n-1]{e^2}-4(3n-2)\sqrt[n-1]{e}+4n-3}}{2(\sqrt[n-1]{e}-1)}.  \]
</div>
<p> The function \(h_\textrm {T}(a,n)\), given by (<a href="#Functiah(a,n)MT">50</a>), for \(a{\gt}1\) and \(n\in \mathbf{d}_5\). Then, if \(b{\gt}h_\textrm {T}(a,5)\) resulting that \(\lambda (a,b,n){\lt}\Lambda (n)\). <div class="proof_wrapper" id="a0000000099">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="Lema1PentruConvMT">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">26</span>
  </div>
  <div class="lemma_thmcontent">
  <p> If \(a=\omega _\emph {T}\) and \(b=0\), then for the <em>i-factor</em> \(c(n)=c(\omega _\emph {T},0,n)\) the following inequalities must be true for \(n\in \mathbf{d}_5\), </p>
<ol class="enumerate">
  <li><p>\(0{\lt}c(\omega _\emph {T},0,n){\lt}\psi (n)\), </p>
</li>
  <li><p>\(0{\lt}\lambda (\omega _\emph {T},0,n){\lt}0.094\), </p>
</li>
  <li><p>\(0.349{\lt}\delta (\omega _\emph {T},0,n){\lt}0.4664\), </p>
</li>
  <li><p>\(0.636{\lt}\beta (\omega _\emph {T},0,n){\lt}1\), </p>
</li>
  <li><p>\(0.783{\lt}\theta (\omega _\emph {T},0,n){\lt}1\), </p>
</li>
</ol>
<p> where \(\psi (n)\) is given by <em>(<a href="#PsiDinMT">39</a>)</em>. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000100">
  <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 function \(c(\omega _{_\textrm {T}},0,n)\) is clearly greater than \(0\) and verifies the inequality \(c(\omega _{_\textrm {T}},0,n){\lt}\psi (n)\), for \(n\in \mathbf{d}_5\). </p>
<p>For the function </p>
<div class="displaymath" id="a0000000101">
  \[  \lambda (a,b,n)=\lambda \big(c(a,b,n),n\big)=\tfrac {1}{an+b}+\tfrac {n-1}{(an+b)^2},  \]
</div>
<p> we have </p>
<div class="displaymath" id="a0000000102">
  \[  \lim _{n\to \infty }\lambda (a,b,n)=0,\  \  \textnormal{for}\  \  a{\gt}0.  \]
</div>
<p>The function \(\Pi (a,b,n)\) is </p>
<div class="displaymath" id="a0000000103">
  \[  \Pi (a,b,n)=\left(1+\tfrac {\lambda (a,b,n)}{1-2\lambda (a,b,n)}\right)^{n-1}.  \]
</div>
<p> Since the limit the function \(\lambda \) is \(0\), then we move to limit the function \(\Pi (a,b,n)\) and we have </p>
<div class="displaymath" id="a0000000104">
  \[  L_\Pi (a)=\lim _{n\to \infty }\Pi (a,b,n)={\rm e}^{\tfrac {a+1}{a^2}}.  \]
</div>
<p>The function \(\mu _\delta (a,b,n)\) is </p>
<div class="equation" id="FunctiaMuDelta">
<p>
  <div class="equation_content">
    \begin{equation} \label{FunctiaMuDelta} \mu _\delta (a,b,n) =\tfrac {(n-1)[(a+1)n+b-1][an^2+(b+1)n-1]}{(an+b)[(a-1)n+b+1][a^2n^2+(2ab-a-1)n+b^2-b+1]}, \end{equation}
  </div>
  <span class="equation_label">51</span>
</p>
</div>
<p> and its limit is </p>
<div class="displaymath" id="a0000000105">
  \[  L_{\mu _\delta }(a)=\lim _{n\to \infty }\mu _\delta (a,b,n)=\tfrac {a+1}{a^2(a-1)}.  \]
</div>
<p>Then the function \(\delta (a,b,n)=\mu _\delta (a,b,n)\Pi (a,b,n)\), has the limit </p>
<div class="displaymath" id="a0000000106">
  \[  L_\delta (a)=\lim _{n\to \infty }\delta (a,b,n)=\tfrac {a+1}{a^2(a-1)}{\rm e}^{\tfrac {a+1}{a^2}}.  \]
</div>
<p>The function \(\mu _\beta (a,b,n)\) is </p>
<div class="displaymath" id="a0000000107">
  \[  \mu _\beta (a,b,n)=\tfrac {(a+1)n+b-1}{(a-1)n+b+1},  \]
</div>
<p> and its limit is </p>
<div class="displaymath" id="a0000000108">
  \[  L_{\mu _\beta }(a)=\lim _{n\to \infty }\mu _\beta (a,b,n)=\tfrac {a+1}{a-1}.  \]
</div>
<p>Then the function \(\beta (a,b,n)=\mu _\beta (a,b,n)\delta (a,b,n)\), has the limit </p>
<div class="displaymath" id="a0000000109">
  \[  L_\beta (a)=\lim _{n\to \infty }\beta (a,b,n)=\tfrac {(a+1)^2}{a^2(a-1)^2}{\rm e}^{\tfrac {a+1}{a^2}}.  \]
</div>
<p>The function \(\mu _\theta (a,b,n)\) is </p>
<div class="displaymath" id="a0000000110">
  \[  \mu _\theta (a,b,n)=\tfrac {(an+b)^2}{a^2n^2-2(a+1-ab)n+b^2-2b+2},  \]
</div>
<p> and its limit is \(1\) when \(n\to \infty \). Then it follows that the function </p>
<div class="displaymath" id="a0000000111">
  \[  \theta (a,b,n)=\mu _\theta (a,b,n) \beta (a,b,n),  \]
</div>
<p> has the limit </p>
<div class="displaymath" id="a0000000112">
  \[  L_\theta (a)=\tfrac {(a+1)^2}{a^2(a-1)^2}{\rm e}^{\tfrac {a+1}{a^2}}.  \]
</div>
<p>Impose conditions \(L_\delta (a)\le 1\) and \(L_\beta (a)=L_\theta (a)\le 1\). Lambert type equations, \(L_\delta (a)=1\) has solution \(\approx 2.236180389652745005\) and \(L_\theta (a)=1\) has solution \(\omega _{_\textrm {T}}\approx 2.7480500253477966212\ldots \). Let \(a=\omega _{_\textrm {T}}\), sine for any \(a{\gt}\omega _{_\textrm {T}}\) we have \(L_\delta (a){\lt}1\). </p>
<p>To prove that the function \(\Pi (\omega _{_\textrm {T}},b,n)\) is monotonically increasing to its limit. According to Lemma <a href="#MonotoniaLuiPi">9</a> and Proposition <a href="#PropozitiaLambdaMT">25</a>, the function \(\Pi (\omega _{_\textrm {T}},b,n)\) is increasing for \(b{\gt}h_\textrm {T}(\omega _{_\textrm {T}},5)\approx -5.529\ldots \). It follows that for \(b\ge 0\) the function \(\Pi (\omega _{_\textrm {T}},0,n)\) is monotonically increasing to its limit. Let further \(b=0\). </p>
<p>To prove that the function \(\delta (\omega _{_\textrm {T}},0,n)\) is monotonically increasing to its limit, we derivative the function \(\mu _\delta (\omega _{_\textrm {T}},0,n)\), where \(\mu _\delta (a,b,n)\) is given by (<a href="#FunctiaMuDelta">51</a>), so </p>
<div class="displaymath" id="a0000000113">
  \[  \mu '_\delta (\omega _{_\textrm {T}},0,n) =\tfrac {\alpha _6n^6 +\alpha _5n^5+\alpha _4n^4+\alpha _3n^3+\alpha _2n^2+\alpha _1n+\alpha _0} {\omega _{_\textrm {T}}^2n^2[(\omega _{_\textrm {T}}-1)n+1]^2[\omega _{_\textrm {T}}^2n^2-(\omega _{_\textrm {T}}+1)n+1]^2},  \]
</div>
<p> where </p>
<div class="displaymath" id="a0000000114">
  \begin{eqnarray*}  \alpha _6 & =&  \omega _{_\textrm {T}}^5-2\omega _{_\textrm {T}}^3+2\omega _{_\textrm {T}}^2+\omega _{_\textrm {T}}\approx 133.06582372905672183, \\ \alpha _5 & =&  2\omega _{_\textrm {T}}^4+4\omega _{_\textrm {T}}^3-10\omega _{_\textrm {T}}^2-4\omega _{_\textrm {T}}\approx 110.55940610241204683, \\ \alpha _4 & =&  -3\omega _{_\textrm {T}}^4-6\omega _{_\textrm {T}}^3+14\omega _{_\textrm {T}}^2+6\omega _{_\textrm {T}}+1 \approx -172.39088809543209589, \\ \alpha _3 & =&  4\omega _{_\textrm {T}}^3-6\omega _{_\textrm {T}}^2-4\omega _{_\textrm {T}}-4 \approx 22.707791497616619358, \\ \alpha _2 & =&  \omega _{_\textrm {T}}+6\approx 8.7480500253477966212, \\ \alpha _1 & =&  -4, \\ \alpha _0 & =&  1. \end{eqnarray*}
</div>
<p>The largest real root of the nominator polynomial function \(\mu '_\delta (\omega _{_\textrm {T}},0,n)\) is \(\approx 0.63786\ldots \), hence for \(n\in \mathbf{d}_5\), \(\mu '_\delta (\omega _{_\textrm {T}},0,n){\gt}0\). Then, according to Proposition <a href="#Propozitiaf">7</a>, the function \(\delta (\omega _{_\textrm {T}},0,n)=\mu _\delta (\omega _{_\textrm {T}},0,n)\Pi (\omega _{_\textrm {T}},0,n)\) is monotonically increasing. </p>
<p>To prove that \(\beta (\omega _{_\textrm {T}},0,n)\) is monotone increasing function to its limit, we derivative the function \(\mu _\beta (\omega _{_\textrm {T}},0,n)\), is \(2\omega _{_\textrm {T}}/[(\omega _{_\textrm {T}}-1)n+1]^2{\gt}0\). Then, according to Proposition <a href="#Propozitiaf">7</a>, the function \(\beta (\omega _{_\textrm {T}},0,n)=\mu _\beta (\omega _{_\textrm {T}},0,n)\delta (\omega _{_\textrm {T}},0,n)\) is monotonically increasing. </p>
<p>To prove that \(\theta (\omega _{_\textrm {T}},0,n)\) is monotonically increasing function to its limit. The derivative function \(\mu _\theta (\omega _{_\textrm {T}},0,n)\) is </p>
<div class="displaymath" id="a0000000115">
  \[  -\tfrac {2\omega _{_\textrm {T}}^2n[(\omega _{_\textrm {T}}+1)n-2]} {[\omega _{_\textrm {T}}^2n^2-2(\omega _{_\textrm {T}}+1)+2]^2}{\lt}0~ .  \]
</div>
<p> Then we can say that the function \(\theta (\omega _{_\textrm {T}},0,n)=\mu _\theta (\omega _{_\textrm {T}},0,n)\beta (\omega _{_\textrm {T}},0,n)\) is monotonically increasing. Let the function </p>
<div class="displaymath" id="a0000000116">
  \begin{align*}  \mu _{\theta \beta \delta }(\omega _{_\textrm {T}},0,n) & =\mu _\theta (\omega _{_\textrm {T}},0,n)\mu _\beta (\omega _{_\textrm {T}},0,n)\mu _\delta (\omega _{_\textrm {T}},0,n)\\ & =\tfrac {\omega _{_\textrm {T}}n(n-1)[(\omega _{_\textrm {T}}+1)n-1]^2(\omega _{_\textrm {T}}n^2+n-1)} {[(\omega _{_\textrm {T}}-1)n+1]^2[\omega _{_\textrm {T}}^2n^2-2(\omega _{_\textrm {T}}+1)n+2] [\omega _{_\textrm {T}}^2n^2-(\omega _{_\textrm {T}}+1)n+1]}, \end{align*}
</div>
<p> its derivative is </p>
<div class="displaymath" id="a0000000117">
  \begin{align*}  \mu ’_{\theta \beta \delta }(\omega _{_\textrm {T}},0,n) =\tfrac {\omega _{_\textrm {T}}[(\omega _{_\textrm {T}}+1)n-1]P_8(\omega _{_\textrm {T}},n)} {[(\omega _{_\textrm {T}}-1)n+1]^3[\omega _{_\textrm {T}}^2n^2-2(\omega _{_\textrm {T}}+1)n+2]^2 [\omega _{_\textrm {T}}^2n^2-(\omega _{_\textrm {T}}+1)n+1]^2} \end{align*}
</div>
<p> where polynomial \(P_8(\omega _{_\textrm {T}},n)=\alpha _8n^8+\alpha _7n^7+\alpha _6n^6+\alpha _5n^5+\alpha _4n^4+\alpha _3n^3+\alpha _2n^2+\alpha _1n+\alpha _0\) has the following coefficients: </p>
<div class="displaymath" id="a0000000118">
  \begin{eqnarray*}  \alpha _8 & =&  \omega _{_\textrm {T}}^7-4\omega _{_\textrm {T}}^5+4\omega _{_\textrm {T}}^4+3\omega _{_\textrm {T}}^3 \approx 847.0086516, \\ \alpha _7 & =&  4\omega _{_\textrm {T}}^5-16\omega _{_\textrm {T}}^4-12\omega _{_\textrm {T}}^3-8\omega _{_\textrm {T}}^2 -4\omega _{_\textrm {T}}\approx -606.0300766, \\ \alpha _6 & =&  -3\omega _{_\textrm {T}}^6-8\omega _{_\textrm {T}}^5+18\omega _{_\textrm {T}}^4+28\omega _{_\textrm {T}}^3 +43\omega _{_\textrm {T}}^2+18\omega _{_\textrm {T}}-2\approx -565.9812397, \\ \alpha _5 & =&  10\omega _{_\textrm {T}}^5+8\omega _{_\textrm {T}}^4-18\omega _{_\textrm {T}}^3-72\omega _{_\textrm {T}}^2 -24\omega _{_\textrm {T}}+12\approx 1052.2011279, \\ \alpha _4 & =&  -14\omega _{_\textrm {T}}^4-15\omega _{_\textrm {T}}^3+38\omega _{_\textrm {T}}^2-4\omega _{_\textrm {T}}-30 \approx -863.72570760, \\ \alpha _3 & =&  14\omega _{_\textrm {T}}^3+8\omega _{_\textrm {T}}^2+36\omega _{_\textrm {T}}+40\approx 489.8813608, \\ \alpha _2 & =&  -9\omega _{_\textrm {T}}^2-30\omega _{_\textrm {T}}-30\approx -180.4075112, \\ \alpha _1 & =&  8\omega _{_\textrm {T}}+12\approx 33.9844002, \\ \alpha _0 & =&  -2. \end{eqnarray*}
</div>
<p> The monomials \((\omega _{_\textrm {T}}+1)n-1\) and \((\omega _{_\textrm {T}}-1)n+1\) are equal to 0 in \(0.2668054\ldots \) and \(-0.572066\ldots \). The largest positive real root of polynomial \(P_8(\omega _{_\textrm {T}},n)\) is \(\approx 0.600633\ldots \), hence \(\mu '_{\theta \beta \delta }(\omega _{_\textrm {T}},0,n){\gt}0\) for \(n\in \mathbf{d}_5\), i.e. \(\mu _{\theta \beta \delta }(\omega _{_\textrm {T}},0,n)\) is increasing, then and the function \(\theta (\omega _{_\textrm {T}},0,n)=\mu _{\theta \beta \delta }(\omega _{_\textrm {T}},0,n)\Pi (\omega _{_\textrm {T}},0,n)\) is increasing for \(n\in \mathbf{d}_5\). </p>
<p>Monotonous function values \(\lambda \), \(\delta \), \(\beta \) and \(\theta \) for \(n=5\) are: </p>
<div class="displaymath" id="a0000000119">
  \begin{eqnarray*}  \lambda (\omega _{_\textrm {T}},0,5) & \approx &  0.093965939752349623079, \\ \delta (\omega _{_\textrm {T}},0,5) & \approx &  0.349197650213356803400, \\ \beta (\omega _{_\textrm {T}},0,5) & \approx &  0.636005603331306147860, \\ \theta (\omega _{_\textrm {T}},0,5) & \approx &  0.783192428417692776398. \end{eqnarray*}
</div>
<p> For the function \(\delta \) we have </p>
<div class="displaymath" id="a0000000120">
  \[  \lim _{n\to \infty }\delta (\omega _{_\textrm {T}},0,n)=\tfrac {\omega _{_\textrm {T}}-1}{\omega _{_\textrm {T}}+1}\approx 0.46638919265374213777.  \]
</div>
<p> <div class="proof_wrapper" id="a0000000121">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="TeoremaI-FactorOptimMT">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">27</span>
  </div>
  <div class="theorem_thmcontent">
  <p> The optimum <em>i-factor</em> for Tanabe’s method is </p>
<div class="equation" id="I-FactorOptimMT">
<p>
  <div class="equation_content">
    \begin{equation} \label{I-FactorOptimMT} c(n)=\tfrac {1}{\omega _\emph {T}n}. \end{equation}
  </div>
  <span class="equation_label">52</span>
</p>
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000122">
  <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 constants \(a=\omega _{_\textrm {T}}\) and \(b=0\), are the best values in the given conditions, so the <em>i-factor</em> \(c(n)\) given by (<a href="#I-FactorOptimMT">52</a>), is optimal. <div class="proof_wrapper" id="a0000000123">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="TeoremaCiraConvMT">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">28</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If the initial distinct approximation \(z^{(0)}_1,z^{(0)}_2,\ldots ,z^{(0)}_n\) satisfy the initial condition </p>
<div class="equation" id="CondInitialaCuCOptimMT">
<p>
  <div class="equation_content">
    \begin{equation} \label{CondInitialaCuCOptimMT} w^{(0)}<c(n)d^{(0)}, \end{equation}
  </div>
  <span class="equation_label">53</span>
</p>
</div>
<p> for \(n\in \mathbf{d}_5\), where \(c\) given by <em>(<a href="#I-FactorOptimMT">52</a>)</em>, then Tanabe’s method with inclusion disks \(D^*_k\), given by <em>(<a href="#DiscuriD*">10</a>)</em>, is convergent. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000124">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Since \(\omega _{_\textrm {T}}{\gt}2\) and \(b=0\), then according to corollary <a href="#Corolarul11Petkovic">4</a> the disks \(D^*_k\), given by (<a href="#DiscuriD*">10</a>), are mutually disjoint and each of them contain exactly one zero of polynomial \(P\). </p>
<p>The conclusions on Lemma <a href="#Lema1PentruConvMT">26</a> assure the fulfillment of Lemmas <a href="#Lema1MT">20</a>, <a href="#Lema2MT">21</a> and <a href="#Lema3MT">22</a>, fact, that in turn, assures the convergence for Tanabe’s method if the initial condition (<a href="#CondInitialaCuCOptimMT">53</a>) is verified. <div class="proof_wrapper" id="a0000000125">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="a0000000126">5 Conclusions</h1>
<ol class="enumerate">
  <li><p>For Durand–Kerner’s method, given by (<a href="#MD-KsauMW-D">19</a>), with the inclusion disks \(D_k\), given by (<a href="#DiscuriD">14</a>), the optimum <em>i-factor</em> is </p>
<div class="displaymath" id="a0000000127">
  \[  c(n)=\tfrac {1}{\omega n+\tau }~ ,\  \  n\in \mathbf{d}_5,  \]
</div>
<p> where \(\omega \approx 1.7632228343518967\ldots \), \(\tau \approx 0.88049674007368891\ldots \). </p>
</li>
  <li><p>For Durand–Kerner’s method, given by (<a href="#MD-KsauMW-D">19</a>), with the inclusion disks \(D^*_k\), given by (<a href="#DiscuriD*">10</a>), the optimum <em>i-factor</em> is </p>
<div class="displaymath" id="a0000000128">
  \[  c(n)=\tfrac {1}{2n+\tau _*},\  \  n\in \mathbf{d}_5,  \]
</div>
<p> where \(\tau _*\approx 0.67211423631036255\ldots \). </p>
</li>
  <li><p>For Tanabe’s method, given by (<a href="#MetodaTanabe">38</a>), with the inclusion disks \(D^*_k\), given by (<a href="#DiscuriD*">10</a>), the optimum <em>i-factor</em> is </p>
<div class="displaymath" id="a0000000129">
  \[  c(n)=\tfrac {1}{\omega _{_\textrm {T}}n},\  \  n\in \mathbf{d}_5,  \]
</div>
<p> where \(\omega _{_\textrm {T}}\approx 2.7480500253477966212\ldots \). </p>
</li>
</ol>
<p> <small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="Abel1826">1</a></dt>
  <dd><p><i class="sc">N. H. Abel</i>, <i class="it">Beweis der Unmoglichkeit, algebraische Gleichungen von hoheren Graden als dem vierten allgemein aufzulosen</i>, J. Reine Angew. Math., <b class="bf">1</b>, Reprinted in Abel, N. H. Œ (Ed. L. Sylow and S. Lie), Christiania (Oslo), Norway, 1881. Reprinted in New York: Johnson Reprint Corp., pp. 66–87, 1988, pp.&#160;65–84, 1826. </p>
</dd>
  <dt><a name="Weierstrass1903">2</a></dt>
  <dd><p><i class="sc">K. Weierstrass</i>, <i class="it">Neuer Beweis des Satzes, dass jede ganze rationale Funktion einer Veränderlichen dargestellt werden kann als ein Product aus linearen Funktionen dertelben Verändelichen</i>, Ges. Werke, <b class="bf">3</b>, pp.&#160;251–269, 1903. </p>
</dd>
  <dt><a name="Durand1960">3</a></dt>
  <dd><p><i class="sc">I. E. Durand</i>, <i class="it">Solutions Numérique des Équations Algébriques. Équations du Type <em>F(x)=0</em>; Racines d’une Polynôme</i>, Masson, Paris, <b class="bf">1</b>, pp.&#160;279–281, 1960. </p>
</dd>
  <dt><a name="Docev1962">4</a></dt>
  <dd><p><i class="sc">K. Docev</i>, <i class="it">An Alternative Method of Newton for Simultaneous Calculation of All the Roots of a Given Algebraic Equation</i>, Phys. Math. J. Bulgar. Acad. Sci., <b class="bf">5</b>, No. 2, pp.&#160;136–139, 1962 (in bulgarian). </p>
</dd>
  <dt><a name="Borsch-Supan1963">5</a></dt>
  <dd><p><i class="sc">W. Börsch-Supan</i>, <i class="it">A posteriori error bounds for the zeros of polynomials</i>, Numer. Math., <b class="bf">5</b>, pp.&#160;380–398, 1963. </p>
</dd>
  <dt><a name="Kerner1966">6</a></dt>
  <dd><p><i class="sc">I. O. Kerner</i>, <i class="it">Simultaneous Displacement of Polynomial Roots if Real and Simple</i>, Comm. ACM, <b class="bf">9</b>, pp.&#160;273, 1966. </p>
</dd>
  <dt><a name="Borsch-Supan1970">7</a></dt>
  <dd><p><i class="sc">W.Börsch-Supan</i>, <i class="it">Residuenabschätzung für  Polynom-Nullstellen mittels Lagrange Interpolation</i>, Numer. Math. <b class="bf">14</b>, pp.&#160;287–296, 1970. </p>
</dd>
  <dt><a name="Braess+Hadeler1973">8</a></dt>
  <dd><p><i class="sc">D. Braess</i> and <i class="sc">K. P. Handeler</i>, <i class="it">Simultaneous inclusion of the zeros of polynomial</i>, Numer. Math., <b class="bf">21</b>, pp.&#160;161–165, 1973. </p>
</dd>
  <dt><a name="Nourein1975">9</a></dt>
  <dd><p><i class="sc">A. W. M. Nourein</i>, <i class="it">An iteration formula for the simultaneous determination of the zeroes of polynomial</i>, J. Comput. Appl. Math., <b class="bf">4</b>, pp.&#160;251–254, 1975. </p>
</dd>
  <dt><a name="Tanabe1983">10</a></dt>
  <dd><p><i class="sc">K. Tanabe</i>, <i class="it">Behavior of the sequences around multiple zeros generated by some simultaneous methods for solving algebraic equations</i>, Tech. Rep. Inf. Procces. Numer. Anal., <b class="bf">4-2</b>, pp.&#160;1–6, 1983 (in Japanese). </p>
</dd>
  <dt><a name="Wells1986">11</a></dt>
  <dd><p><i class="sc">D. Wells</i>, <i class="it">The Penguin Dictionary of Curious and Interesting Numbers</i>, Penguin Books, 1986, address: Middlesex, England. </p>
</dd>
  <dt><a name="Carstensen1991">12</a></dt>
  <dd><p><i class="sc">C. Carstensen</i>, <i class="it">Anwendungen von Begleitmatrizen</i>, Z. Angew. Math., <b class="bf">71</b>. pp.&#160;809–812, 1991. </p>
</dd>
  <dt><a name="Carstensen1991a">13</a></dt>
  <dd><p><i class="sc">C. Carstensen</i>, <i class="it">Inclusion of the roots of polynomial based on Gerschgorin’s theorem</i>, Numer. Math., <b class="bf">59</b>, pp.&#160;349–360, 1991. </p>
</dd>
  <dt><a name="Boyer+Merzbach1991">14</a></dt>
  <dd><p><i class="sc">C. B. Boyer</i> and <i class="sc">U. C. Merzbach </i>, <i class="it">A History of Mathematics</i>, CHAPTER: The Renaissance, Wiley, address: New York, edition: 2an, pp.&#160;282–287, 1991. </p>
</dd>
  <dt><a name="Petkovic1995">15</a></dt>
  <dd><p><i class="sc">M. S. Petković, C. Trajković</i> and <i class="sc">M. Carstensen</i>, <i class="it">Weierstrass’ formula and zero-finding methods</i>, Numer. Math., <b class="bf">69</b>, pp.&#160;353–372, 1995. </p>
</dd>
  <dt><a name="Petkovic1996">16</a></dt>
  <dd><p><i class="sc">M. S. Petković</i>, <i class="it">On initial conditions for the convergence of simultaneous root finding methods</i>, Computing, <b class="bf">57</b>, pp.&#160;163–177, 1996. </p>
</dd>
  <dt><a name="Petkovic+Herceg1996">17</a></dt>
  <dd><p><i class="sc">M. S. Petković</i> and <i class="sc">J. Herceg</i>, <i class="it">Point estimation and safe convergence of root-finding simultaneous methods</i>, Scientific Review, <b class="bf">21-22</b>, pp.&#160;117–130, 1996. </p>
</dd>
  <dt><a name="Birkhoff+MacLane1996">18</a></dt>
  <dd><p><i class="sc">G. Birkhoff</i> and <i class="sc">S. Mac Lane</i>, <i class="it">Galois Theory</i>, CHAPTER: 15 in A Survey of Modern Algebra, PUBLISHER: Macmillan, address: New York, edition: 5th, pp.&#160;395–421, 1996. </p>
</dd>
  <dt><a name="Corless1996">19</a></dt>
  <dd><p><i class="sc">R. M. Corless</i>, <i class="sc">G. H. Gonnet</i>, <i class="sc">D. E. G. Hare </i>, <i class="sc">D. J. Jeffrey</i> and <i class="sc">D. E. Knuth</i>, <i class="it">On the Lambert Function</i>, Adv. Comput. Math., <b class="bf">5</b>, pp.&#160;329–359, 1996. </p>
</dd>
  <dt><a name="Petkovic1997">20</a></dt>
  <dd><p><i class="sc">M. S. Petković</i>, <i class="sc">J. Herceg</i> and <i class="sc">S. Ilić</i>, <i class="it">Point Estimation Theory and its Applications</i>, Institute of Mathematics, address: Novi Sad, 1997. </p>
</dd>
  <dt><a name="Petkovic1997a">21</a></dt>
  <dd><p><i class="sc">M. S. Petković</i> and <i class="sc">J. Herceg</i>, <i class="it">Börsch-Supan-like methods: Point estimation and parallel implementation</i>, Inter. J. Comput. Math., <b class="bf">64</b>, pp.&#160;117–130, 1997. </p>
</dd>
  <dt><a name="Petkovic1997b">22</a></dt>
  <dd><p><i class="sc">M. S. Petković</i> and <i class="sc">S. Ilić</i>, <i class="it">Point estimation and the convergence of Ehrlich-Aberth metod</i>, Publ. Inst. Math., <b class="bf">62</b>, pp.&#160;141–149, 1997. </p>
</dd>
  <dt><a name="Batra1998">23</a></dt>
  <dd><p><i class="sc">P. Batra</i>, <i class="it">Improvement of a convergence condition for the Durand-Kerner iteration</i>, J. of Comp. and Appl. Math., <b class="bf">96</b> <b class="bf">2</b>, pp.&#160;117–125, 1998. </p>
</dd>
  <dt><a name="Petkovic1998">24</a></dt>
  <dd><p><i class="sc">M. S. Petković</i>, <i class="sc">J. Herceg</i> and <i class="sc">S. Ilić</i>, <i class="it">Safe Convergence of Simultaneous Methods for Polynomial Zeros</i>, Numerical Algorithms, <b class="bf">17</b>, pp.&#160;313–331, 1998. </p>
</dd>
  <dt><a name="Petkovic1998a">25</a></dt>
  <dd><p><i class="sc">M. S. Petković</i>, <i class="sc">J. Herceg</i> and <i class="sc">S. Ilić</i>, <i class="it">Point estimation and some applications to iterative methods</i>, BIT, <b class="bf">38</b>, pp.&#160;112–126, 1998. </p>
</dd>
  <dt><a name="Petkovic1998b">26</a></dt>
  <dd><p><i class="sc">M. S. Petković</i> and <i class="sc">J. Herceg</i>, <i class="it">On the convergence of Wang-Zheng’s metod</i>, J. Comput. Appl. Math., <b class="bf">91</b>, pp.&#160;123–135, 1998. </p>
</dd>
  <dt><a name="Dumit+Foote1998">27</a></dt>
  <dd><p><i class="sc">D. S. Dummit</i> and <i class="sc">R. M. Foote</i>, <i class="it">Galois Theory</i>, CHAPTER:14 in Abstract Algebra, PUBLISHER: Prentice-Hall, address: New York, edition: 2nd Englewood Cliffs, pp.&#160;471–570, 1998. </p>
</dd>
  <dt><a name="Petkovic+Herteg2001">28</a></dt>
  <dd><p><i class="sc">M. S. Petković</i> and <i class="sc">J. Herceg</i>, <i class="it">Point estimation of simultaneous methods for solving polynomial equations:a survey</i>, J. Comput. Appl. Math., <b class="bf">136</b>, pp.&#160;183–207, 2001. </p>
</dd>
  <dt><a name="CiraO2005">29</a></dt>
  <dd><p><i class="sc">O. Cira</i>, <i class="it">Metode numerice pentru rezolvarea ecuaţiilor algebrice</i>, Ed. Academiei Române, 2005, Bucureşti, (in romanian). </p>
</dd>
  <dt><a name="CiraOsiC-MinIEEE">30</a></dt>
  <dd><p><i class="sc">O. Cira</i> and <i class="sc">C. M. Cira</i>, <i class="it">The optimum convergence condition for the Durand-Kerner type simultaneous inclusion method</i>, BOOKTITLE: SYNASC 2006 – \(8^{th}\) International Symposium on Symbolic and Numeric Algorithms for Scientif<span class="mbox" style="width: "></span>ic Computing, pp.&#160;171–174, 2006, address: Timişoara, România, month: 26-29 September, publisher: IEEE Computer Society, Los Alamitos California, Washington, Tokyo. </p>
</dd>
  <dt><a name="Petkovic2008">31</a></dt>
  <dd><p><i class="sc">M. S. Petković</i>, <i class="it">Point estimation of root finding methods</i>, PUBLISHER: Springer, 2008, series: Lecture Notes in Mathematics 1933. </p>
</dd>
  <dt><a name="Mathworld">32</a></dt>
  <dd><p><i class="sc">Mathworld</i>, howpublished: <span class="ttfamily">http://mathworld.wolfram.com</span>, 30 Aug. 2010. </p>
</dd>
  <dt><a name="Wolfram">33</a></dt>
  <dd><p><i class="sc">Wolfram</i>, howpublished: <span class="ttfamily">http://www.wolfram.com/products/mathematica</span>, 30 Sept. 2010. </p>
</dd>
</dl>


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