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<title>On General Fixed Point Method <br />Based On Matrix Splitting <br />For Solving Linear Complementarity Problem: On General Fixed Point Method <br />Based On Matrix Splitting <br />For Solving Linear Complementarity Problem</title>
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<h1>On General Fixed Point Method <br />Based On Matrix Splitting <br />For Solving Linear Complementarity Problem</h1>
<p class="authors">
<span class="author">Bharat Kumar\(^1\), Deepmala\(^2\) A.K. Das\(^3\)</span>
</p>
<p class="date">November 11, 2022; accepted: December 18, 2022; published online: December 31, 2022.</p>
</div>
<div class="abstract"><p> In this article, we introduce a modified fixed point method to process the large and sparse linear complementarity problem (LCP) and formulate an equivalent fixed point equation for the LCP and show the equivalence. Also, we provide convergence conditions when the system matrix is a \(P\)-matrix and two sufficient convergence conditions when the system matrix is an \(H_+\)-matrix. To show the efficiency of our proposed method, we illustrate two numerical examples for different parameters. </p>
<p><b class="bf">MSC.</b> 65F10, 65F50. </p>
<p><b class="bf">Keywords.</b> Linear complementarity problem, \(H_{+}\)-matrix, \(P\)-matrix, Matrix splitting, Convergence. </p>
</div>
<p>\(^1\)The first author is thankful to the University Grants Commission (UGC), Government of India under the SRF fellowship Program No. 1068/(CSIR-UGC NET DEC.2017). </p>
<p>\(^1\)Mathematics Discipline, PDPM-Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, M.P., India, e-mail: <span class="tt">bharatnishad.kanpu@gmail.com</span> </p>
<p>\(^2\)Mathematics Discipline, PDPM-Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, M.P., India, e-mail: <span class="tt">dmrai23@gmail.com</span> </p>
<p>\(^3\)Indian Statistical Institute, 203 B.T. Road, Kolkata-700108, India, e-mail: <span class="tt">akdas@isical.ac.in</span> </p>
<h1 id="sec.1">1 Introduction</h1>

<p>Given \(A_1\in {\mathbb R}^{n\times n}\) and a vector \( q \in \, {\mathbb R}^{n},\) the linear complementarity problem denoted as LCP\((q,A_{1})\) is to find the solution \(z\; \in {\mathbb R}^{n}\; \) to the following system </p>
<div class="displaymath" id="eq1">
  \begin{eqnarray} \label{eq1} z\geq 0, ~ ~ ~ ~  A_{1}z +q \geq 0,~ ~ ~ ~  z^T(A_{1}z +q)=0. \end{eqnarray}
</div>
<p>The LCP has many applications, including operations research, control theory, mathematical economics, optimization theory, stochastic optimal control, the American option pricing problem, economics, elasticity theory, the free boundary problem, and the Nash equilibrium point of the bimatrix game, which has been extensively studied in the literature on mathematical programming. For details see <span class="cite">
	[
	<a href="#Cottle1992" >14</a>
	]
</span>, <span class="cite">
	[
	<a href="#Neogy2008" >10</a>
	]
</span>, <span class="cite">
	[
	<a href="#Murty1988" >21</a>
	]
</span>. For recent works on this problem see <span class="cite">
	[
	<a href="#Neogy2016" >8</a>
	]
</span>, <span class="cite">
	[
	<a href="#Neogy2021" >9</a>
	]
</span>. </p>
<p>The methods available for solving the LCP may be classified into two groups namely pivoting method <span class="cite">
	[
	<a href="#Das2014" >2</a>
	]
</span>, <span class="cite">
	[
	<a href="#Dutta2022" >5</a>
	]
</span>, <span class="cite">
	[
	<a href="#Neogy2012" >23</a>
	]
</span> and iterative method <span class="cite">
	[
	<a href="#Najafi2013i" >22</a>
	]
</span>, <span class="cite">
	[
	<a href="#Hadjidimos2018" >18</a>
	]
</span>. The basic idea of the pivotal method is to obtain a basic feasible complementary vector through a series of pivot steps, whereas the iterative method generates a series of iterations that converge to a solution <span class="cite">
	[
	<a href="#Das2016" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#Jana2019" >7</a>
	]
</span>. Lemke and Howson <span class="cite">
	[
	<a href="#lmke" >20</a>
	]
</span> introduced the complementary pivot method. Following this method, Lemke introduced a technique known as Lemke’s algorithm, which is well known for finding the solution to the LCP. </p>
<p>In order to develop effective iteration methods, we commonly use matrix splittings to find a numerical solution of the large and sparse LCP\((q, A_{1})\), such as the projected methods <span class="cite">
	[
	<a href="#Dehghan2009" >16</a>
	]
</span>, <span class="cite">
	[
	<a href="#Han2011" >19</a>
	]
</span>, <span class="cite">
	[
	<a href="#Xu2007" >25</a>
	]
</span>, the modulus algorithm <span class="cite">
	[
	<a href="#Bai2010" >11</a>
	]
</span> and the modulus based matrix splitting iterative methods <span class="cite">
	[
	<a href="#Zhang2013" >26</a>
	]
</span>, <span class="cite">
	[
	<a href="#Zheng2017" >27</a>
	]
</span>. </p>
<p>A general fixed point method (GFP) is proposed by Fang <span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span> assuming the case where \(\Omega = \omega D_{1}^{-1}\) with \(\omega \) \(\textgreater \) \( 0\) and \(D_{1}\) is the diagonal matrix of \(A_{1}\). The GFP approach takes less iterations than the modulus based successive over relaxation method (MSOR) <span class="cite">
	[
	<a href="#Bai2010" >11</a>
	]
</span>. We present a modified form of GFP <span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span> that incorporates projected type iteration techniques by including two positive diagonal parameter matrices \(\Omega _1\), \(\Omega _2\) and \(\phi \) is a strictly lower triangular matrix in this article. We also show that the fixed point equation and the LCP is equivalent and discuss the convergence conditions and along with several convergence domains for our method. </p>
<p>The paper is organised as follows: we review some notation, definitions and lemmas in <a href="#Preli">section 2</a> in order to establish our key findings. The iterative fixed point approach for solving LCP\((q,A_{1})\) with convergence analysis is proposed in <a href="#sec.3">section 3</a>. We present two numerical examples in <a href="#sec.4">section 4</a> to demonstrate the efficiency of the proposed methods. <a href="#sec.5">section 5</a> ends the paper with some conclusions. </p>
<h1 id="Preli">2 Preliminaries</h1>

<p>Some notation, preliminary definitions, and required lemmas are reviewed. </p>
<p>Here \( A_{1}=( \bar{a}_{ij}) \in {{\mathbb R}}^{n\times n}\) and \( B_{1}=( \bar{b}_{ij}) \in { {\mathbb R}}^{n\times n}\) are square matrices. For \(A_{1}=(\bar{a}_{ij}) \in {{\mathbb R}}^{n\times n}\) and \(B_1=(\bar{b}_{ij}) \in {{\mathbb R}}^{n\times n}\), \(A_{1} \geq \) \((\textgreater )\) \( B_{1}\) means \(\bar{a}_{ij}\geq (\textgreater )\) \( \bar{b}_{ij}\) for all \( i,j \in \{ 1,2,\ldots ,n\} \). </p>
<p><div class="definition_thmwrapper " id="a0000000002">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> Let \( A_{1}=(\bar{a}_{ij})\in { {\mathbb R}}^{n\times n}\) be a square matrix, then \(\lvert A_{1}\rvert =(\bar{b}_{ij})\) is defined by \( \bar{b}_{ij} = \lvert \bar{a}_{ij}\rvert \) \(\forall ~ i,j\) and \(|A_{1}| \) represent that \( \bar{a}_{ij} \geq 0\) \(\forall ~ i,j \). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="defn0">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> Let \(A_{1}, B_{1} \in {{\mathbb R}}^{n \times n}\) be two square matrices. Then \(|A_{1}+B_{1}|\leq |A_{1}|+|B_{1}|\) and \(|A_{1}B_{1}|\leq |A_{1}|\cdot |B_{1}|\). Moreover, when \(a_{1}, b_{1} \in {{\mathbb R}}^{n}\) then \(|a_{1}+b_{1}|\leq |a_{1}|+|b_{1}|\) and \(||a_{1}|-|b_{1}||\leq |a_{1}-b_{1}|\). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="a0000000003">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">3</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="#Das2016" >4</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> Let \(A_{1}\in {{\mathbb R}}^{n\times n}\) be a square matrix. \(A_{1}\) is said to be a \(P\)-matrix if all its principle minors are positive such that \(det({A_1}_{\alpha _1 \alpha _1}) ~ \textgreater ~  0\) for all \(\alpha _1 \subseteq \{ 1,2,\ldots , n\} \). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="a0000000004">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">4</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> Suppose \( A_{1}\in {{\mathbb R}}^{n\times n}\) is a square matrix, then its comparison matrix is defined as \(\langle \bar{a}_{ij}\rangle =|\bar{a}_{ij}| \) if \(i=j\) and \(\langle \bar{a}_{ij}\rangle =-|\bar{a}_{ij}| \) if \(i \neq j\). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="def1">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">5</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="#Frommer1992" >6</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> Suppose \( A_{1} \in {{\mathbb R}}^{n\times n}\) is a square matrix. It is said to be a \(Z\)-matrix if all of its non diagonal elements are less than or equal to zero; an \(M\)-matrix if \(A_1^{-1}\geq 0\) as well as \(Z\)-matrix. The matrix \( A_{1}\) is said to be an \(H\)-matrix if \(\langle A_1 \rangle \) is an \(M\)-matrix. \( A_{1}\) is said to be an \(H_+\)-matrix if \( A_{1}\) is an \(H_+\)-matrix with \(\bar{a}_{ii} ~ \textgreater ~  0 ~ \forall ~ i \in \{ 1,2,\ldots ,n\} \). </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="def2">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">6</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="#Frommer1992" >6</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> The splitting \(A_{1} = M_{1}-N_{1} \) is called an \(M\)-splitting if \(M_{1}\) is a nonsingular \(M\)-matrix and \(N_{1} \geq 0\); an \(H\)-splitting if \(\langle M_{1} \rangle -|N_1| \) is an \(M\)-matrix; an \(H\)-compatible splitting if \(\langle A_{1} \rangle = \langle M_1 \rangle - \lvert N_1\rvert \). </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lem0">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">7</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#rashid2022" >1</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let \(a_{1}, b_1 \in {\mathbb R}^{n} \). Then \(a_{1}\geq 0\), \(b_{1}\geq 0\), \(a_1^{T}b_{1}=0\) if and only if \(a_{1}+b_{1}=|a_{1}-b_{1}|\). </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lem1">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">8</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#Frommer1992" >6</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> Suppose \(A_{1}, B_{1} \in {{\mathbb R}}^{n\times n}\). If \(A_{1}\) and \(B_{1}\) are \(M\) and \(Z\)-matrices respectively with \(A_{1} \leq B_{1}\), then \(B_{1}\) is an \(M\)-matrix. If \(A_{1}\) is an \(H\)-matrix, then \(|A_{1}^{-1}|\leq \langle A_{1}\rangle ^{-1}\). If \(A_{1} \leq B_{1}\), then \(\rho (A_{1}) \leq \rho (B_{1})\). </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lem2">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">9</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> Let \(A_{1}\in {{\mathbb R}}^{n\times n}\) be an \(M\)-matrix and \(A_{1}=M_{1}-N_{1}\) be an \(M\)-splitting. Let \(\rho \) be the spectral radius, then &#160;\(\rho (M_{1}^{-1}N_{1})\) \(\textless \) \(1\). </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lem3">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">10</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#Berman1994" >13</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> If splitting is an \(H\)-compatible of an \(H\)-matrix, then it is an \(H\)-splitting but converse is not true. </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lem4">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">11</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#Frommer1992" >6</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> Suppose \(A_{1} \geq 0 \). If there exist \(v ~ \textgreater ~  0 \in {{\mathbb R}}^{n}\) and a scalar \(\alpha _{1}~ \textgreater ~  0\) such that \(A_{1}v \leq \alpha _{1} v\), then \(\rho (A_{1}) \leq \alpha _{1} \). Moreover, if \( A_{1}v \textless \alpha _{1} v\), then \(\rho (A_{1})\textless \alpha _{1}\). </p>

  </div>
</div> </p>
<h1 id="sec.3">3 Main results</h1>

<p>For a given vector \(s\in {\mathbb R}^{n }\), we consider the vectors \(s_{+}=\max \{ 0,s\} \), \(s_{-}=\max \{ 0,-s\} \) and \(A_{1}=(D_1 +\phi )-(L_1+U_1+\phi )\), where \(\phi \) is a strictly lower triangular matrix, \(U_{1}\) is a strictly upper triangular matrix of \(A_{1}\). \(U^T_1\) denotes the transpose of \(U_{1}\), \(L_1\) is strictly lower triangular matrix of \(A_1\) and \(\alpha \) is a positive real number. In the following theorem we convert the LCP\((q, A_{1})\) into a fixed point equation. </p>
<p><div class="theorem_thmwrapper " id="thm1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">12</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(A\in {\mathbb R}^{n\times n} \) with the splitting \(A_{1}=(D_1 +\phi )-(L_1+U_1+\phi )\). If \(z=\Omega _{1}s_+\) and \(\omega =\Omega _2s_{-}\), then the equivalent formulation of the LCP\((q,A_{1})\) in the form of fixed point equation is </p>
<div class="displaymath" id="eq2">
  \begin{eqnarray} \label{eq2} s=(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})s_{+}+\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}s_{+}-\Omega ^{-1}_{2}q. \end{eqnarray}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000005">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let \(z=\Omega _{1}s_{+}\) and \(w=\Omega _{2}s_{-}\), and \(s=s_{+}-s_{-}\). From LCP\((q,A_{1})\) </p>
<div class="displaymath" id="a0000000006">
  \begin{align*}  \Omega _{2}s_{-} & =A_{1}\Omega _{1}s_{+} + q, \\ s& =s_{+}-\Omega ^{-1}_{2}(A_{1}\Omega _{1}s_{+}+ q), \\ s& =(I_{1}-\Omega ^{-1}_{2}A_{1}\Omega _{1})s_{+}-\Omega ^{-1}_{2}q, \\ s& =(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})s_{+}+\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}s_{+}-\Omega ^{-1}_{2}q. \end{align*}
</div>
<p> <div class="proof_wrapper" id="a0000000007">
  <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>Based on <a href="#eq2" class="eqref">2</a> we propose the following iteration method which is referred to as modified general fixed point method (MGFP) for solving the LCP\((q, A_{1})\), </p>
<div class="equation" id="mthd1">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq3} s^{(k+1)}=(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})s^{(k)}_{+}+\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}s^{(k+1)}_{+}-\Omega ^{-1}_{2}q. \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p>Let \(\operatorname {Res}\) be the Euclidean norm of the error vector, which is defined <span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span> as follows, </p>
<div class="displaymath" id="a0000000008">
  \[  \operatorname {Res}(z^{(k)})=\| \min (z^{(k)}, A_{1}z^{(k)}+q) \| _{2}.  \]
</div>
<p> Consider the nonnegative initial vector \(z^{(0)}\in {\mathbb R}^n\). The iteration process continues until the iteration sequence \(\{ z^{(k)}\} _{k=0}^{+\infty } \subset {\mathbb R}^n\) converges. The iteration process stop if \(\operatorname {Res}(z^{(k)})\) \(\textless \) \( 10^{-5} \). For computing \(z^{(k+1)}\in {\mathbb R}^{n}\) we use the following algorithm. </p>
<p> <figcaption>
  <span class="caption_title"></span> 
  <span class="caption_ref">None</span> 
  <span class="caption_text"></span> 
</figcaption><b class="bfseries">(Modified General Fixed Point Method)</b> </p>
<ul class="itemize">
  <li><p>Given any initial vector \(s^{(0)} \in {\mathbb R}^{n}\), \(\epsilon ~ \textgreater ~  0 \) and set \( k=0 \).<br /></p>
</li>
  <li><p><b class="bfseries">for</b> \(k=0,1,2,\ldots \) do<br /></p>
</li>
  <li><p>\(s^{(k)}_{+}=\max \{ 0,s^{(k)}\} \)<br /></p>
</li>
  <li><p>compute \(\operatorname {Res}= norm (\min (s^{(k)}_+, A_1s^{(k)}_++q\)))<br /></p>
</li>
  <li><p><b class="bfseries">if</b> \(\operatorname {Res}{\lt} \epsilon \) then <br /></p>
</li>
  <li><p>\(s=s^{(k)}\)<br /></p>
</li>
  <li><p><b class="bfseries">break</b><br /></p>
</li>
  <li><p><b class="bfseries">else</b> <br /></p>
</li>
  <li><p>Using the following scheme, create the sequence \(s^{(k)}\): </p>
<div class="displaymath" id="a0000000009">
  \begin{equation*}  s_1^{(k+1)}=((I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})s^{(k)}_{+}-\Omega ^{-1}_{2}q)_1, \end{equation*}
</div>
</li>
  <li><p><b class="bfseries"> for</b> \(i= 2,3. \ldots , n\) do<br />\(s_i^{(k+1)}=((I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})s^{(k)}_{+}-\Omega ^{-1}_{2}q)_i\) <br />&#8195;&#8195;&#8195;&#8195;\(+(\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}s_{+}^{{(k+1)}})_{i}\) <br />and set \(z^{(k+1)}=\Omega _{1}s_{+}^{(k+1)}\).<br /></p>
</li>
  <li><p><b class="bfseries">end for</b><br /></p>
</li>
  <li><p><b class="bfseries">end if</b> <br /></p>
</li>
  <li><p><b class="bfseries">end for</b>. </p>
</li>
</ul>

<p>Moreover, the MGFP provides a general structure for solving LCP\((q, A_{1})\). Using some particular values of the parameter matrices \(\Omega _{1}, \Omega _2\) and we obtain an iterative method. In particular, </p>
<ul class="itemize">
  <li><p>When \(\Omega _{1}=I\), \(\Omega _2=\Omega ^{-1}\) and \(\phi =0 \), from <a href="#mthd1" class="eqref">3</a> we have, </p>
<div class="displaymath" id="eq4">
  \begin{eqnarray} \label{eq4} s^{(k+1)}=(I_{1}-\Omega (D_{1}-U_{1}))s^{(k)}_{+}+\Omega L_1 s^{(k+1)}_{+}-\Omega q, \end{eqnarray}
</div>
<p> this is a GFP <span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span>. </p>
</li>
</ul>
<p><div class="theorem_thmwrapper " id="th00">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">13</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Suppose \(A_{1}=(D_1 +\phi )-(L_1+U_1+\phi ) \in {\mathbb R}^{n \times n}\) and \(q\in {\mathbb R}^n\). Then \(s^{*}\) is a solution of <a href="#eq2" class="eqref">2</a> if and only if \(z^{*}=\Omega _{1}s^{*}_{+}\) is a solution of LCP\((q,A_{1})\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000010">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let \(s^{*} \) be a solution of <a href="#eq2" class="eqref">2</a>. Then </p>
<div class="displaymath" id="a0000000011">
  \begin{align*}  s^*& =(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})s^*_{+}+\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}s^*_{+}-\Omega ^{-1}_{2}q, \\ s^{*}& =(I_{1}-\Omega ^{-1}_{2}A_{1}\Omega _{1})s^{*}_{+}-\Omega ^{-1}_{2}q, \\ \Omega _{2}s^{*}_{-} & =A_{1}\Omega _{1}s^{*}_{+} + q. \end{align*}
</div>
<p>Since \(\Omega _{2}s^{*}_{-} \geq 0\), </p>
<div class="displaymath" id="a0000000012">
  \[ A_{1}\Omega _{1}s^{*}_{+} + q \geq 0. \]
</div>
<p>Moreover, </p>
<div class="displaymath" id="a0000000013">
  \[  (\Omega _{1}s^{*}_{+})^T(A_{1}\Omega _{1}s^{*}_{+}+ q) = (\Omega _{1}s^{*}_{+})^T(\Omega _{2}s^{*}_{-}) = 0,  \]
</div>
<p> and \(\Omega _{1}s^{*}_{+}\geq 0\). Therefore \(z^{*}=\Omega _{1}s^{*}_{+}\) is a solution of LCP\((q,A_{1})\). </p>
<p>Let \(z^{*}=\Omega _{1}s^{*}_{+}\), \(w^{*}=\Omega _{2}s^{*}_{-}\) and \(s^{*}=s^{*}_{+}-s^{*}_{-}\). From LCP\((q,A_{1})\) </p>
<div class="displaymath" id="a0000000014">
  \begin{align*}  \Omega _{2}s^{*}_{-} & =A_{1}\Omega _{1}s^{*}_{+} + q, \\ s^{*}& =s^{*}_{+}-\Omega ^{-1}_{2}(A_{1}\Omega _{1}s^{*}_{+}+ q), \\ s^{*}& =(I_{1}-\Omega ^{-1}_{2}A_{1}\Omega _{1})s^{*}_{+}-\Omega ^{-1}_{2}q, \\ s^*& =(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})s^*_{+}+\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}s^*_{+}-\Omega ^{-1}_{2}q. \end{align*}
</div>
<p> Thus, \(s^{*}\) is a solution of <a href="#eq2" class="eqref">2</a>. <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> </p>
<p>In the following theorem, we show that the solution of <a href="#eq2" class="eqref">2</a> is unique when the system matrix \(A _1\) of LCP\((q,A_1)\) is a \(P\)-matrix. </p>
<p><div class="theorem_thmwrapper " id="thm14">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">14</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(A_1\) be a \(P\)-matrix and \(A_{1}=(D_1 +\phi )-(L_1+U_1+\phi ) \in {\mathbb R}^{n \times n}\) and \(q\in {\mathbb R}^n\). Then for any positive diagonal matrices \(\Omega _1\) and \(\Omega _2\), <a href="#eq2" class="eqref">2</a> has a unique solution. </p>

  </div>
</div> </p>
<p><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>Since \(A_{1}\) is a \(P\)-matrix, for any \(q \in {\mathbb R}^{n}\) LCP\((q, A_{1})\) has a unique solution. Let \( y^{*}\) and \( u^{*}\) be the solutions of <a href="#eq2" class="eqref">2</a>. Then </p>
<div class="displaymath" id="a0000000017">
  \begin{align*}  s^*& =(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})s^*_{+}+\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}s^*_{+}-\Omega ^{-1}_{2}q, \\ u^*& =(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})u^*_{+}+\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}u^*_{+}-\Omega ^{-1}_{2}q. \end{align*}
</div>
<p>Since \(\Omega _{1}y^{*}_{+} =\Omega _{1}u^{*}_{+}\) \(\implies y^{*}_{+} =u^{*}_{+}\), therefore </p>
<div class="displaymath" id="a0000000018">
  \[  y^{*}=u^{*}.  \]
</div>
<p>In the following, we prove the convergence conditions when \(A_{1}\) is a \(P\)-matrix. </p>
<p><div class="theorem_thmwrapper " id="thm2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">15</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(A_{1}\) be a \(P\)-matrix with \(A_{1}=(D_1 +\phi )-(L_1+U_1+\phi ) \in {\mathbb R}^{n \times n}\) and \(q\in {\mathbb R}^n\). Assume </p>
<div class="displaymath" id="a0000000019">
  \[  \rho \Big(\Big(I-|\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}|\Big)^{-1}\Big|I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1}\Big|\Big){\lt}1  \]
</div>
<p> and \(s^{*}\) is the solution of <a href="#eq2" class="eqref">2</a>. Then the sequence \(\{ z^{(k)}\} ^{+\infty }_{k=1}\) generated by \(\cref{mthd1} \) converges to \(z^{*}\) for any initial vector \(s^{(0)}\in {\mathbb R}^{n}\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000020">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Suppose \(A_{1}\) is a \(P\)-matrix, then \(s^{*}\) is a unique solution of <a href="#eq2" class="eqref">2</a>. Thus </p>
<div class="displaymath" id="a0000000021">
  \[  s^*=(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})s^*_{+}+\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}s^*_{+}-\Omega ^{-1}_{2}q.  \]
</div>
<p> From <a href="#mthd1" class="eqref">3</a>, this implies </p>
<div class="displaymath" id="a0000000022">
  \begin{equation*} \begin{split}  s^{(k+1)}-s^{*}& =(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})(s_{+}^{(k)}-s^*_{+}) \\ & \quad +\Omega ^{-1}_{2} (L_1+ \phi )\Omega _{1}(s_{+}^{(k+1)}-s^*_{+}). \end{split}\end{equation*}
</div>
<p> It follows that </p>
<div class="displaymath" id="a0000000023">
  \begin{equation*} \begin{split} & |s^{(k+1)}-s^{*}|= \\ & =|(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})(s_+^{(k)}-s^*_{+})+\Omega ^{-1}_{2} (L_1 +\phi )\Omega _{1}(s_+^{(k+1)}-s^*_{+})| \\ & \leq |(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})(s_{+}^{(k)}-s^*_{+})|+|\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}(s_+^{(k+1)}-s^*_{+})| \\ & \leq |(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})||s^{(k)}-s^*|+|\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}||s^{(k+1)}-s^*| \end{split}\end{equation*}
</div>
<div class="displaymath" id="a0000000024">
  \[  |s^{(k+1)}-s^{*}|-|\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}||s^{(k+1)}-s^*| \leq |(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})||s^{(k)}-s^*|  \]
</div>
<div class="displaymath" id="a0000000025">
  \[  (I-|\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}|)|s^{(k+1)}-s^*| \leq |(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})||s^{(k)}-s^*|  \]
</div>
<p> and </p>
<div class="displaymath" id="a0000000026">
  \[  |s^{(k+1)}-s^*| \leq \big(I-|\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}|\big)^{-1}\big|(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})\big|\cdot |s^{(k)}-s^*|.  \]
</div>
<p> Therefore, if \(\rho \Big(\big(I-|\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}|\big)^{-1}\big|(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})\big|\Big){\lt}1, \) for any initial vector \(s^{(0)}\in {\mathbb R}^{n}\) the sequence \(\{ z^{(k)}\} ^{+\infty }_{k=1}\) converges to the \(z^{*}\). <div class="proof_wrapper" id="a0000000027">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Now, when \(A _1\) is an \(H _+\)-matrix, we analyze the convergence domains for parameter matrices \(\Omega _1\) and \(\Omega _2\) for MGFP. </p>
<p><div class="theorem_thmwrapper " id="thm6">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">16</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(A_1\) be a \(H_{+ }\)-matrix with \(A_{1}=(D_1 +\phi )-(L_1+U_1+\phi ) \in {\mathbb R}^{n \times n}\) and either one of the following is true: </p>
<p><b class="bfseries">\((1)\)</b> \( \Omega ^{-1}_{2}\Omega _{1} \) \( \textgreater \) \( (D_{1}+\phi )^{-1} \) and \((2\Omega ^{-1}_{2}\Omega _{1}-(D_{1}+\phi )-|B+\phi |)\), where \(B=L_{1}+U_1\). </p>
<p><b class="bfseries">\((2)\)</b> \( 0 \) \(\textless \) \( \Omega ^{-1}_{2}\Omega _{1}\) \(\leq \) \((D_{1}+\phi )^{-1}.\) </p>
<p>Then the sequence \(\{ z^{(k)}\} ^{+\infty }_{k=1}\) generated by \(\cref{mthd1}\) converges to \(z^{*}\) for any initial vector \(s^{(0)}\in {\mathbb R}^{n}\). </p>

  </div>
</div> </p>
<p><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> Since \(A_{1}\) is an \( H _+\)-matrix. there LCP\((q,A_{1})\) has unique solution <span class="cite">
	[
	<a href="#Bai2010" >11</a>
	]
</span>. Now we will look at the splitting, </p>
<div class="displaymath" id="a0000000029">
  \begin{equation*} \begin{split} & (I_{1}-|\Omega ^{-1}_{2}(L_1+\phi )\Omega _{1}|)-|I_{1}-\Omega ^{-1}_{2}(D_1+\phi -U_{1})\Omega _{1}|= \\ & =(I_{1}-|I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi )\Omega _{1}|)-\Omega ^{-1}_{2}|B+\phi |\Omega _{1}. \end{split}\end{equation*}
</div>
<p> <b class="bfseries">(1)</b> If \( \Omega ^{-1}_{2}\Omega _{1} \textgreater (D_{1}+\phi )^{-1}\) then, </p>
<div class="displaymath" id="a0000000030">
  \begin{equation*} \begin{split} & (I_{1}-|I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi )\Omega _{1}|)-\Omega ^{-1}_{2}|B+\phi |\Omega _{1}= \\ & =2I_{1}-\Omega ^{-1}_{2}(D_1+\phi )\Omega _{1}-\Omega ^{-1}_{2}|B+\phi |\Omega _{1}\\ & =\Omega ^{-1}_{2}(2\Omega ^{-1}_{2}\Omega _{1}-(D_{1}+\phi )-|B+\phi |)\Omega _{1}. \end{split}\end{equation*}
</div>
<p>Since \((2\Omega ^{-1}_{2}\Omega _{1}-(D_{1}+\phi )-|B+\phi |)\) is an \(M\)-matrix. Then the splitting \((I_{1}-|\Omega ^{-1}_{2}(L_{1}+\phi )\Omega _{1}|)-|I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1}|\) represent an \(M\)-splitting of the \(M\)-matrix \(\Omega ^{-1}_{2}(2\Omega ^{-1}_{2}\Omega _{1}-(D_{1}+\phi )-|B+\phi |)\Omega _{1}\), hence \(\rho ((I-|\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}|)^{-1}|(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})|)\) \(\textless \) \( 1 \). </p>
<p><b class="bfseries">(2)</b> If \(\Omega ^{-1}_{2}\Omega _{1}\) \(\leq \) \( (D_{1}+\phi )^{-1}\) then, </p>
<div class="displaymath" id="a0000000031">
  \begin{align*}  (I_{1}-|I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi )\Omega _{1}|)-\Omega ^{-1}_{2}|B+\phi |\Omega _{1}& =\Omega ^{-1}_{2}(D_{1}+\phi -|B+\phi |)\Omega _{1}\\ & =\Omega ^{-1}_{2}\langle A_{1} \rangle \Omega _{1}. \end{align*}
</div>
<p> Therefore, \((I_{1}-|\Omega ^{-1}_{2}(L_{1}+\phi )\Omega _{1}|)-|I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1}|\) represents an \(M\)-splitting of \(M\)-matrix \(\Omega ^{-1}_{2}\langle A_{1} \rangle \Omega _{1}\) <span class="cite">
	[
	<a href="#Berman1994" >13</a>
	]
</span>. Therefore, from \(\cref{lem1}\) \(\rho ((I-|\Omega ^{-1}_{2} (L_1 + \phi )\Omega _{1}|)^{-1}|(I_{1}-\Omega ^{-1}_{2}(D_{1}+\phi -U_{1})\Omega _{1})|)\) \(\textless \) \( 1 \). <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="sec.4">4 Numerical examples</h1>
<p>  In this section, two numerical examples are provided to show the effectiveness of our new method. We use some notation as, IT=number of iteration steps, CPU = CPU time in seconds. The system matrix \(A_{1} \) is generated by </p>
<div class="displaymath" id="a0000000033">
  \[ A_{1}(p_{1}, p_{2}, p_{3})= Q+p_{1}I_{1}+p_{2}G+p_{3}H, \]
</div>
<p> where \(p_{1}, p_{2}\) and \(p_{3}\) are given constants, \(I_{1}\) is the identity matrix of order \(n\) and \(G\) = tridiag \((0, 0, 1)= \begin{bmatrix}  0 

&  1 

& 0 

&  \ldots 

& 0 

\\ 0 

&  0 

&  1 

& \ldots 

& 0 

\\ 0

&  0 

&  0 

& 1 

& \vdots 

\\ \vdots 

&  \ldots 

&  0 

&  \ddots 

& 1

\\ 0

&  \ldots 

&  0 

& 0 

& 0

\\ \end{bmatrix} \in {\mathbb R}^{n \times n} \) and <br /> <br /> <br />\(H\)=diag([1, 2, 1, 2, …])\(= \begin{bmatrix}  1 

& 0 

&  0

& \ldots 

& 0 

\\ 0

&  2 

&  0 

&  \ldots 

&  0 

\\ 0

&  0

&  1 

& 0 

&  0

\\ \vdots 

&  \ldots 

& 0 

&  2 

&  \vdots 

\\ 0

&  \ldots 

&  \ldots 

& 0 

& \ddots 

\\ \end{bmatrix} \in {\mathbb R}^{n \times n}\). <br /> <br />Let \(s^{(0)}=(0,0,0,0,\ldots 0,0,\ldots )^T\in {\mathbb R}^{n}\) be an initial vector. We consider the LCP\((q, A_1)\) which has always a unique solution, where \( q= (1 \;  -1 \) \(1 \;  -1 \;  \ldots 1 \;  -1 \ldots )^T \) \(\in {\mathbb R}^{n}\). We set \(\Omega _{1}= I_{1}\) and \(\Omega _{2}= \omega ^{-1} D_{1}\) in the MGFP. The suggested method is compared with GFP <span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span>, which is effective in solving LCP\((q,A_{1})\). </p>
<p>Matlab version 2021a was used for all calculations. <a href="#tab1">example 17</a> and <a href="#tab2">example 18</a> show the numerical results for GFP <span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span> and MGFP respectively. </p>
<p><div class="example_thmwrapper " id="tab1">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">17</span>
  </div>
  <div class="example_thmcontent">
  <p> The system matrix \(A_{1}, B_{1}\in {{\mathbb R}}^{n\times n}\) are generated by <br />\(A_{1}(p_{1}, p_{2}, p_{3})= Q+p_{1}I_{1}+p_{2}G+p_{3}H,\) where \(p_{1}, p_{2}\) and \(p_{3}\) are given constants and </p>
<div class="displaymath" id="a0000000034">
  \[  Q=\operatorname {tridiag} (-I_{2},L_{1}, -I_{2})= \begin{bmatrix}  L_{1} 

&  -I_{2} 

& 0 

&  \ldots 

& 0 

\\ -I_{2} 

&  L_{1} 

&  -I_{2} 

& \ldots 

& 0 

\\ \vdots 

&  -I_{2} 

&  L_{1} 

& -I_{2} 

& \vdots 

\\ 0

&  \ldots 

&  -I_{2} 

&  \ddots 

&  -I_{2}

\\ 0

&  \ldots 

&  0 

& -I_{2} 

& L_{1}

\\ \end{bmatrix} \in {{\mathbb R}}^{n\times n},  \]
</div>
<div class="displaymath" id="a0000000035">
  \[  L_{1}=\operatorname {tridiag} (-1,4, -1)= \begin{bmatrix}  4 

& -1 

&  \ldots 

& \ldots 

& 0 

\\ -1

&  4 

&  -1 

&  \ldots 

&  0 

\\ \vdots 

&  -1

&  4 

& -1 

&  \vdots 

\\ 0

&  \ldots 

& -1 

&  \ddots 

&  -1

\\ 0

&  \ldots 

&  \ldots 

& -1 

& 4

\\ \end{bmatrix} \in {{\mathbb R}}^{m\times m},  \]
</div>
<p> where \(I_{2}\) is the identity matrix of order \(m\), where \(n=m^{2}\) with \(m\) being a positive integer. </p>

  </div>
</div> </p>
<div class="table"  id="a0000000036">
   <div class="centered"><small class="footnotesize"> <table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\mathbf{A_1(1, 1, -1)}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p><b class="bfseries">n</b> </p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \({400}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({900}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1600}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2500}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\textbf{GFP}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT} \)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>18 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 21 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 22 </p>

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

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

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0058 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1267 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.8356 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>12.8507</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>76.8588 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.8e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.5e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>7.3e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>5.3e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>7.0e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\textbf{MGFP}\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT}\)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 15 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>18 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>19 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>20</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>20</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0050 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.1028 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.6105 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>11.2363</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>67.0586 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\alpha =0.1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>6.4e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.8e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>6.8e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>4.4e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>5.9e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\mathbf{A_1(0, 1, 0)}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p><b class="bfseries">n</b> </p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \({400}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({900}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1600}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2500}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\textbf{GFP}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT} \)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>13 </p>

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

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

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

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

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0045 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0582 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3728 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>5.4135</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>31.8341 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.3e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.6e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9.3e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>2.3e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>2.6e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\textbf{MGFP}\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT}\)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 12 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 13</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>13 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>13</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>13</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0041 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0817 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.4127</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>7.2064</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>42.3931 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\alpha =0.1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>3.1e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>3.2e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>5.3e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>7.4e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9.5e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\mathbf{A_1(1, 1, 1)}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p><b class="bfseries">n</b> </p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \({400}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({900}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1600}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2500}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\textbf{GFP}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT} \)</p>

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

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

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

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

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

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0046 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0580 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.447</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>5.2868</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>31.3160 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>2.1e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>6.7e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>1.8e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>2.5e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3.2e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\textbf{MGFP}\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT}\)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 9 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 9</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>10</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>10</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0050 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0530 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3177 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>5.3091</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>31.5628 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\alpha =0.02\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>1.6e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.0e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>8.3e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>1.8e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>2.3e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\mathbf{A_1(1, 0, 1)}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p><b class="bfseries">n</b> </p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \({400}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({900}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1600}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2500}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\textbf{GFP}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT} \)</p>

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

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

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

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

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

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1.1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0045 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0582 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3728 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>5.4135</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>31.8341 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.3e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.6e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9.3e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>2.3e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>2.6e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\textbf{MGFP}\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT}\)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 9 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 9</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>10</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1.1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0040 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0507 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3671 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>5.4161</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>31.9112 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\alpha =0.05\)</p>

    </td>
    <td  style="border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>4.7e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>6.6e-06 </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>8.1e-06 </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>9.3e-06</p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>2.1e-06</p>

    </td>
  </tr>
</table> </small> <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">1</span> 
  <span class="caption_text">Results for GFP <span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span> and MGFP with \(\phi =\alpha (L_{1}+U^T_{1})\)</span> 
</figcaption> </div>
</div>
<p><div class="example_thmwrapper " id="tab2">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">18</span>
  </div>
  <div class="example_thmcontent">
  <p>The system matrix \(A_{1}\in {{\mathbb R}}^{n\times n}\) is generated by </p>
<div class="displaymath" id="a0000000037">
  \[ A_{1}(p_{1}, p_{2}, p_{3})= Q+p_{1}I_{1}+p_{2}G+p_{3}H, \]
</div>
<p> where \(p_{1}, p_{2}\) and \(p_{3}\) are given constants, \(I_{1}\) is the identity matrix of order \(n\) and </p>
<div class="displaymath" id="a0000000038">
  \begin{align*}  Q=& \operatorname {tridiag} (-1.5I_2,L_1, -0.5I_{2}) \\ = & \begin{bmatrix}  L_{1} 

&  -0.5I_{2} 

& 0 

&  \ldots 

& 0 

\\ -1.5I_{2} 

&  L_{1} 

&  -0.5I_{2} 

& \ldots 

& 0 

\\ \vdots 

&  -1.5I_{2} 

&  L_{1} 

& -0.5I_{2} 

& \vdots 

\\ 0

&  \ldots 

&  -1.5I_{2} 

&  \ddots 

&  -0.5I_{2}

\\ 0

&  \ldots 

&  0 

& -1.5I_{2} 

& L_{1}

\\ \end{bmatrix} \in {{\mathbb R}}^{n\times n}, \end{align*}
</div>
<div class="displaymath" id="a0000000039">
  \[  L_{1}=\operatorname {tridiag} (-1.5,4, -0.5)= \begin{bmatrix}  4 

& -0.5 

&  \ldots 

& \ldots 

& 0 

\\ -1.5

&  4 

&  -0.5 

&  \ldots 

&  0 

\\ \vdots 

&  -1.5

&  4 

& -0.5 

&  \vdots 

\\ 0

&  \ldots 

& -1.5 

&  \ddots 

&  -0.5

\\ 0

&  \ldots 

&  \ldots 

& -1.5 

& 4

\\ \end{bmatrix} \in {{\mathbb R}}^{m\times m},  \]
</div>
<p> where \(I_{2}\) is the identity matrix of order \(m\). </p>

  </div>
</div> </p>
<div class="table"  id="a0000000040">
   <div class="centered"><small class="footnotesize"> <table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\mathbf{A_1(1, 1, -1)}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p><b class="bfseries">n</b> </p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \({400}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({900}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1600}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2500}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\textbf{GFP}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT} \)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>13 </p>

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

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

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

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

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0061 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0807 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.5871 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>8.3912</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>48.8538 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.0e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>6.2e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3.5e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>5.0e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>6.5e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\textbf{MGFP}\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT}\)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 11 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 12</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>13 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>15</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>15</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0051 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0607 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.5392 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>8.44191</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>49.4862 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\alpha =0.1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>3.0e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>4.8e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>7.9e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3.1e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>8.7e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\mathbf{A_1(0, 1, 0)}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p><b class="bfseries">n</b> </p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \({400}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({900}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1600}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2500}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\textbf{GFP}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT} \)</p>

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

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

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 11 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 11</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0041 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0548 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3621 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>5.999</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>35.2935 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>1.9e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>5.8e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9.4e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>2.6e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3.3e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\textbf{MGFP}\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT}\)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 9</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0045 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0.0507 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3412 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>4.5795</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>28.4975 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\alpha =0.1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7.8e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>2.05e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>2.7e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3.4e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>4.0e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\mathbf{A_1(1, 1, 1)}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p><b class="bfseries">n</b> </p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \({400}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({900}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1600}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2500}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\textbf{GFP}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT} \)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>7 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 7 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 8 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 8 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 8</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0041 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0407 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.3266 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>4.1892</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>24.6431 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>2.7e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>6.8e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>9.7e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>1.3e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>1.7e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\textbf{MGFP}\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT}\)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 6 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 7</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>7 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>7</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>7</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0043 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0474 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2796 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3.4536</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>22.6106 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\alpha =0.1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>9.7e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>1.3e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>1.6e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>2.0e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>2.2e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\mathbf{A_1(1, 0, 1)}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p><b class="bfseries">n</b> </p>

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

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \({400}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({900}\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({1600}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> \({2500}\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px" 
        rowspan=""
        colspan="">
      <p> \(\textbf{GFP}\)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT} \)</p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>8 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 8 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 8 </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left" 
        rowspan=""
        colspan="">
      <p> 8</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1.1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0041 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.0407 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2699 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>4.4741</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>24.6650 </p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">&nbsp;</td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>1.9e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>2.7e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3.2e-06 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>3.8e-06</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>4.2e-06</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\textbf{MGFP}\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(\textbf{IT}\)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 8</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>8 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>8</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>8</p>

    </td>
  </tr>
  <tr>
    <td  style="None" 
        rowspan=""
        colspan="">
      <p>\(\omega =1.1\)</p>

    </td>
    <td  style="border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(\textbf{CPU}\) </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> 0.0047 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p>0.05303 </p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p> 0.2554</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>4.0929</p>

    </td>
    <td  style="text-align:left" 
        rowspan=""
        colspan="">
      <p>24.3851 </p>

    </td>
  </tr>
  <tr>
    <td  style="border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\alpha =0.1\)</p>

    </td>
    <td  style="border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(\textbf{Res} \)</p>

    </td>
    <td  style="text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>1.7e-06 </p>

    </td>
    <td  style="text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>2.4e-06 </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>2.9e-06 </p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>3.4e-06</p>

    </td>
    <td  style="text-align:left; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>3.8e-06</p>

    </td>
  </tr>
</table> </small> <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">2</span> 
  <span class="caption_text">Results for GFP <span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span> and MGFP with \(\phi =\alpha (L_{1}+U^T_{1})\)</span> 
</figcaption> </div>
</div>
<p>From <a href="#tab1">example 17</a> and <a href="#tab2">example 18</a>, we see that our proposed MGFP have requires less iteration steps than the GFP <span class="cite">
	[
	<a href="#Xi2021" >17</a>
	]
</span> respectively. </p>
<h1 id="sec.5">5 Conclusion</h1>
<p>  In this article, we introduced a modified general fixed point method based on new matrix splitting for solving the LCP\((q, A _1)\) with parameter matrices \(\Omega _1\) and \(\Omega _2\). Also, we showed how the iterative form is linked to the new matrix splitting and the parameter matrices \(\Omega _1\) and \(\Omega _2\) . This iterative form preserves the big and sparse structure of \(A_1\) during the iteration process. Moreover, we showed the convergence condition for \(P\)-matrix and presented sufficient convergence domains for \(\Omega _1\) and \(\Omega _2\) when system matrix \(A_{1}\) is \(H_{+}\)-matrix. At the end, two examples are discussed to show the efficiency of our proposed method. </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000041">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors are grateful to the editor associated with this paper for the excellent cooperation. Also, the authors thank the anonymous referees who spent some precious time reviewing this paper. The authors would like to acknowledge their contribution, due to which there is a significant improvement in the paper. </p>

  </div>
</div> </p>
<p><br /></p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
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</a> </p>
</dd>
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</a> </p>
</dd>
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</a> </p>
</dd>
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</a> </p>
</dd>
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</dd>
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  <dd><p><a href ="https://doi.org/10.1142/9789812813220_0025"> <i class="sc">S.K. Neogy, A.K. Das, S. Sinha, A. Gupta</i>, <i class="it">On a mixture class of stochastic game with ordered field property</i>, in: <i class="it">Mathematical programming and game theory for decision making</i>, pp. 451–477, World Scientific, 2008. <a href="https://doi.org/10.1142/9789812813220_0025">https://doi.org/10.1142/9789812813220_0025</a> <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
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</dl>


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