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<head><title>AMG preconditioners</title>
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</head><body
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<!--l. 56--><div class="crosslinks"><p class="noindent"><span
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class="cmbx-12">[</span><a
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href="userhtmlsu7.html" ><span
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class="cmbx-12">] </span></p></div>
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<h4 class="subsectionHead"><span class="titlemark"><span
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class="cmbx-12">4.1 </span></span> <a
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id="x14-130004.1"></a><span
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class="cmbx-12">AMG preconditioners</span></h4>
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<!--l. 58--><p class="noindent" ><span
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class="cmbx-12">In order to describe the AMG preconditioners available in MLD2P4, we</span>
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<span
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class="cmbx-12">consider a linear system</span>
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<table
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class="equation"><tr><td>
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<center class="math-display" >
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<img
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src="userhtml2x.png" alt="Ax = b,
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" class="math-display" ><a
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id="x14-13001r2"></a></center></td><td class="equation-label"><span
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class="cmr-12">(2)</span></td></tr></table>
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<!--l. 62--><p class="nopar" >
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<span
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class="cmbx-12">where </span><span
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class="cmmi-12">A </span><span
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class="cmr-12">= (</span><span
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class="cmmi-12">a</span><sub><span
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class="cmmi-8">ij</span></sub><span
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class="cmr-12">) </span><span
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class="cmsy-10x-x-120">∈ </span><span
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class="msbm-10x-x-120">ℝ</span><sup><span
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class="cmmi-8">n</span><span
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class="cmsy-8">×</span><span
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class="cmmi-8">n</span></sup> <span
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class="cmbx-12">is a nonsingular sparse matrix; for ease of</span>
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<span
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class="cmbx-12">presentation we assume </span><span
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class="cmmi-12">A </span><span
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class="cmbx-12">has a symmetric sparsity pattern.</span>
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<!--l. 67--><p class="indent" > <span
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class="cmbx-12">Let us consider as finest index space the set of row (column) indices of </span><span
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class="cmmi-12">A</span><span
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class="cmbx-12">,</span>
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<span
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class="cmbx-12">i.e., </span><span
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class="cmr-12">Ω = </span><span
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class="cmsy-10x-x-120">{</span><span
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class="cmr-12">1</span><span
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class="cmmi-12">, </span><span
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class="cmr-12">2</span><span
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class="cmmi-12">,</span><span
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class="cmmi-12">…</span><span
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class="cmmi-12">,n</span><span
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class="cmsy-10x-x-120">}</span><span
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class="cmbx-12">. Any algebraic multilevel preconditioners implemented in</span>
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<span
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class="cmbx-12">MLD2P4 generates a hierarchy of index spaces and a corresponding</span>
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<span
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class="cmbx-12">hierarchy of matrices,</span>
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<center class="math-display" >
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<img
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src="userhtml3x.png" alt="Ω1 ≡ Ω ⊃ Ω2 ⊃ ...⊃ Ωnlev, A1 ≡ A, A2, ...,Anlev,
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" class="math-display" ></center>
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<!--l. 72--><p class="nopar" > <span
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class="cmbx-12">by using the information contained in </span><span
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class="cmmi-12">A</span><span
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class="cmbx-12">, without assuming any knowledge</span>
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<span
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class="cmbx-12">of the geometry of the problem from which </span><span
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class="cmmi-12">A </span><span
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class="cmbx-12">originates. A vector space </span><span
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class="msbm-10x-x-120">ℝ</span><sup><span
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class="cmmi-8">n</span><sub><span
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class="cmmi-6">k</span></sub></sup>
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<span
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class="cmbx-12">is associated with </span><span
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class="cmr-12">Ω</span><sup><span
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class="cmmi-8">k</span></sup><span
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class="cmbx-12">, where </span><span
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class="cmmi-12">n</span><sub>
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<span
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class="cmmi-8">k</span></sub> <span
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class="cmbx-12">is the size of </span><span
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class="cmr-12">Ω</span><sup><span
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class="cmmi-8">k</span></sup><span
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class="cmbx-12">. For all </span><span
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class="cmmi-12">k < nlev</span><span
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class="cmbx-12">, a</span>
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<span
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class="cmbx-12">restriction operator and a prolongation one are built, which connect two</span>
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<span
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class="cmbx-12">levels </span><span
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class="cmmi-12">k </span><span
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class="cmbx-12">and </span><span
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class="cmmi-12">k </span><span
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class="cmr-12">+ 1</span><span
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class="cmbx-12">:</span>
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<center class="math-display" >
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<img
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src="userhtml4x.png" alt=" k nk×nk+1 k nk+1×nk
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P ∈ ℝ , R ∈ ℝ ;
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" class="math-display" ></center>
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<!--l. 82--><p class="nopar" > <span
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class="cmbx-12">the matrix </span><span
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class="cmmi-12">A</span><sup><span
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class="cmmi-8">k</span><span
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class="cmr-8">+1</span></sup> <span
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class="cmbx-12">is computed by using the previous operators according to</span>
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<span
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class="cmbx-12">the Galerkin approach, i.e.,</span>
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<center class="math-display" >
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<img
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src="userhtml5x.png" alt="Ak+1 = RkAkP k.
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" class="math-display" ></center>
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<!--l. 87--><p class="nopar" > <span
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class="cmbx-12">In the current implementation of MLD2P4 we have </span><span
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class="cmmi-12">R</span><sup><span
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class="cmmi-8">k</span></sup> <span
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class="cmr-12">= (</span><span
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class="cmmi-12">P</span><sup><span
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class="cmmi-8">k</span></sup><span
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class="cmr-12">)</span><sup><span
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class="cmmi-8">T</span> </sup> <span
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class="cmbx-12">A smoother</span>
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<span
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class="cmbx-12">with iteration matrix </span><span
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class="cmmi-12">M</span><sup><span
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class="cmmi-8">k</span></sup> <span
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class="cmbx-12">is set up at each level </span><span
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class="cmmi-12">k < nlev</span><span
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class="cmbx-12">, and a solver is set</span>
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<span
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class="cmbx-12">up at the coarsest level, so that they are ready for application (for example,</span>
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<span
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class="cmbx-12">setting up a solver based on the </span><span
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class="cmmi-12">LU </span><span
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class="cmbx-12">factorization means computing and</span>
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<span
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class="cmbx-12">storing the </span><span
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class="cmmi-12">L </span><span
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class="cmbx-12">and </span><span
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class="cmmi-12">U </span><span
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class="cmbx-12">factors). The construction of the hierarchy of AMG</span>
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<span
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class="cmbx-12">components described so far corresponds to the so-called build phase of the</span>
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<span
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class="cmbx-12">preconditioner.</span>
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<!--l. 95--><p class="indent" > <hr class="figure"><div class="figure"
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>
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<a
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id="x14-130021"></a>
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<div class="center"
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>
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<!--l. 96--><p class="noindent" >
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<div class="fbox"> <div class="minipage"><table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td style="width:16;"
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class="tabbing"> </td><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing"></td></tr></table>
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<!--l. 115--><p class="noindent" ><table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td
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class="tabbing">procedure V-cycle<img
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src="userhtml6x.png" alt="( k k k)
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k,A ,b ,u" class="left" align="middle">
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</td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing">if <img
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src="userhtml7x.png" alt="(k ⁄= nlev)" class="left" align="middle"> then</td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing"><span
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class="cmmi-10x-x-109">u</span><sup><span
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class="cmmi-8">k</span></sup> = <span
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class="cmmi-10x-x-109">u</span><sup><span
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class="cmmi-8">k</span></sup> + <span
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class="cmmi-10x-x-109">M</span><sup><span
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class="cmmi-8">k</span></sup><img
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src="userhtml8x.png" alt="( )
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bk - Akuk" class="left" align="middle"></td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing"><span
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class="cmmi-10x-x-109">b</span><sup><span
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class="cmmi-8">k</span><span
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class="cmr-8">+1</span></sup> = <span
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class="cmmi-10x-x-109">R</span><sup><span
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class="cmmi-8">k</span><span
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class="cmr-8">+1</span></sup><img
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src="userhtml9x.png" alt="(bk - Akuk )" class="left" align="middle"></td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing"><span
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class="cmmi-10x-x-109">u</span><sup><span
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class="cmmi-8">k</span><span
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class="cmr-8">+1</span></sup> = V-cycle<img
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src="userhtml10x.png" alt="( )
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k + 1,Ak+1,bk+1,0" class="left" align="middle"></td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing"><span
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class="cmmi-10x-x-109">u</span><sup><span
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class="cmmi-8">k</span></sup> = <span
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class="cmmi-10x-x-109">u</span><sup><span
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class="cmmi-8">k</span></sup> + <span
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class="cmmi-10x-x-109">P</span><sup><span
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class="cmmi-8">k</span><span
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class="cmr-8">+1</span></sup><span
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class="cmmi-10x-x-109">u</span><sup><span
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class="cmmi-8">k</span><span
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class="cmr-8">+1</span></sup></td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing"><span
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class="cmmi-10x-x-109">u</span><sup><span
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class="cmmi-8">k</span></sup> = <span
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class="cmmi-10x-x-109">u</span><sup><span
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class="cmmi-8">k</span></sup> + <span
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class="cmmi-10x-x-109">M</span><sup><span
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class="cmmi-8">k</span></sup><img
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src="userhtml11x.png" alt="( )
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bk - Akuk" class="left" align="middle"></td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing">else</td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing"><span
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class="cmmi-10x-x-109">u</span><sup><span
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class="cmmi-8">k</span></sup> = <img
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src="userhtml12x.png" alt="( )
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Ak" class="left" align="middle"><sup><span
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class="cmsy-8">-</span><span
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class="cmr-8">1</span></sup><span
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class="cmmi-10x-x-109">b</span><sup><span
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class="cmmi-8">k</span></sup></td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing">endif</td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
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class="tabbing"> </td><td
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class="tabbing">return <span
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class="cmmi-10x-x-109">u</span><sup><span
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class="cmmi-8">k</span></sup></td></tr></table>
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<!--l. 115--><p class="noindent" >
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<table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td
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class="tabbing">end</td></tr></table>
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<!--l. 115--><p class="noindent" > </div> </div>
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<br /> <div class="caption"
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><span class="id">Figure 1: </span><span
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class="content">Application phase of a V-cycle preconditioner.</span></div><!--tex4ht:label?: x14-130021 -->
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</div>
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<!--l. 118--><p class="indent" > </div><hr class="endfigure">
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<!--l. 120--><p class="indent" > <span
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class="cmbx-12">The components produced in the build phase may be combined in</span>
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<span
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|
class="cmbx-12">several ways to obtain different multilevel preconditioners; this is done in</span>
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<span
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|
class="cmbx-12">the application phase, i.e., in the computation of a vector of type </span><span
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class="cmmi-12">w </span><span
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|
class="cmr-12">= </span><span
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class="cmmi-12">B</span><sup><span
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|
class="cmsy-8">-</span><span
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|
class="cmr-8">1</span></sup><span
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|
class="cmmi-12">v</span><span
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class="cmbx-12">,</span>
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<span
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class="cmbx-12">where </span><span
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class="cmmi-12">B </span><span
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|
class="cmbx-12">denotes the preconditioner, usually within an iteration of a Krylov</span>
|
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|
<span
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class="cmbx-12">solver </span><span class="cite"><span
|
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|
class="cmbx-12">[</span><a
|
|
|
|
|
href="userhtmlli4.html#XSaad_book"><span
|
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class="cmbx-12">23</span></a><span
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|
class="cmbx-12">]</span></span><span
|
|
|
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|
class="cmbx-12">. An example of such a combination, known as V-cycle, is given in</span>
|
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|
<span
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|
class="cmbx-12">Figure</span><span
|
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|
class="cmbx-12"> </span><a
|
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|
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|
href="#x14-130021"><span
|
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|
class="cmbx-12">1</span><!--tex4ht:ref: fig:application_alg --></a><span
|
|
|
|
|
class="cmbx-12">. In this case, a single iteration of the same smoother is used before</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmbx-12">and after the the recursive call to the V-cycle (i.e., in the pre-smoothing</span>
|
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|
|
<span
|
|
|
|
|
class="cmbx-12">and post-smoothing phases); however, different choices can be performed.</span>
|
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|
|
<span
|
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|
|
|
class="cmbx-12">Other cycles can be defined; in MLD2P4, we implemented the standard</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmbx-12">V-cycle and W-cycle</span><span
|
|
|
|
|
class="cmbx-12"> </span><span class="cite"><span
|
|
|
|
|
class="cmbx-12">[</span><a
|
|
|
|
|
href="userhtmlli4.html#XBriggs2000"><span
|
|
|
|
|
class="cmbx-12">3</span></a><span
|
|
|
|
|
class="cmbx-12">]</span></span><span
|
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|
|
class="cmbx-12">, and a version of the K-cycle described</span>
|
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|
<span
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|
class="cmbx-12">in</span><span
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class="cmbx-12"> </span><span class="cite"><span
|
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class="cmbx-12">[</span><a
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|
href="userhtmlli4.html#XNotay2008"><span
|
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|
class="cmbx-12">22</span></a><span
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class="cmbx-12">]</span></span><span
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class="cmbx-12">.</span>
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<!--l. 133--><div class="crosslinks"><p class="noindent"><span
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class="cmbx-12">[</span><a
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href="userhtmlsu7.html" ><span
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class="cmbx-12">next</span></a><span
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class="cmbx-12">] [</span><a
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href="userhtmlsu6.html" ><span
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class="cmbx-12">front</span></a><span
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class="cmbx-12">] [</span><a
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href="userhtmlse4.html#userhtmlsu6.html" ><span
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class="cmbx-12">up</span></a><span
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class="cmbx-12">] </span></p></div>
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<!--l. 133--><p class="indent" > <a
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id="tailuserhtmlsu6.html"></a>
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</body></html>
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