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   <!--l. 1--><div class="crosslinks"><p class="noindent"><span 
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   <h3 class="sectionHead"><span class="titlemark"><span 
class="cmr-12">4   </span></span> <a 
 id="x14-130004"></a><span 
class="cmr-12">Getting Started</span></h3>
<!--l. 5--><p class="noindent" ><span 
class="cmr-12">This section describes the basics for building and applying AMG4PSBLAS one-level</span>
<span 
class="cmr-12">and multilevel (i.e., AMG) preconditioners with the Krylov solvers included in</span>
<span 
class="cmr-12">PSBLAS</span><span 
class="cmr-12">&#x00A0;</span><span class="cite"><span 
class="cmr-12">[</span><a 
href="userhtmlli5.html#XPSBLASGUIDE"><span 
class="cmr-12">20</span></a><span 
class="cmr-12">]</span></span><span 
class="cmr-12">.</span>
<!--l. 9--><p class="indent" >   <span 
class="cmr-12">The following steps are required:</span>
     <ol  class="enumerate1" >
     <li 
  class="enumerate" id="x14-13002x1"><span 
class="cmti-12">Declare  the  preconditioner  data  structure</span><span 
class="cmr-12">.  It  is  a  derived  data  type,</span>
     <span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">amg_</span></span></span><span 
class="cmti-12">x</span><span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">prec_</span></span></span> <span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">type</span></span></span><span 
class="cmr-12">, where </span><span 
class="cmti-12">x </span><span 
class="cmr-12">may be </span><span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">s</span></span></span><span 
class="cmr-12">, </span><span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">d</span></span></span><span 
class="cmr-12">, </span><span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">c</span></span></span> <span 
class="cmr-12">or </span><span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">z</span></span></span><span 
class="cmr-12">, according to the basic data</span>
     <span 
class="cmr-12">type of the sparse matrix (</span><span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">s</span></span></span> <span 
class="cmr-12">= real single precision; </span><span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">d</span></span></span> <span 
class="cmr-12">= real double precision;</span>
     <span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">c</span></span></span> <span 
class="cmr-12">= complex single precision; </span><span class="obeylines-h"><span class="verb"><span 
class="cmtt-12">z</span></span></span> <span 
class="cmr-12">= complex double precision). This data</span>
     <span 
class="cmr-12">structure is accessed by the user only through the AMG4PSBLAS routines,</span>
     <span 
class="cmr-12">following an object-oriented approach.</span>
     </li>
     <li 
  class="enumerate" id="x14-13004x2"><span 
class="cmti-12">Allocate  and  initialize  the  preconditioner  data  structure,  according  to  a</span>
     <span 
class="cmti-12">preconditioner type chosen by the user</span><span 
class="cmr-12">. This is performed by the routine</span>
     <span class="lstinline"></span><span 
class="cmtt-12">init</span><span 
class="cmr-12">,  which  also  sets  defaults  for  each  preconditioner  type  selected  by</span>
     <span 
class="cmr-12">the user. The preconditioner types and the defaults associated with them</span>
     <span 
class="cmr-12">are  given  in  Table</span><span 
class="cmr-12">&#x00A0;</span><a 
href="#x14-13015r1"><span 
class="cmr-12">1</span><!--tex4ht:ref: tab:precinit --></a><span 
class="cmr-12">,  where  the  strings  used  by  </span><span class="lstinline"></span><span 
class="cmtt-12">init</span> <span 
class="cmr-12">to  identify  the</span>
     <span 
class="cmr-12">preconditioner types are also given. Note that these strings are valid also if</span>
     <span 
class="cmr-12">uppercase letters are substituted by corresponding lowercase ones.</span>
     </li>
     <li 
  class="enumerate" id="x14-13006x3"><span 
class="cmti-12">Modify the selected preconditioner type, by properly setting preconditioner</span>
     <span 
class="cmti-12">parameters. </span><span 
class="cmr-12">This is performed by the routine </span><span class="lstinline"></span><span 
class="cmtt-12">set</span><span 
class="cmr-12">. This routine must be</span>
     <span 
class="cmr-12">called  if  the  user  wants  to  modify  the  default  values  of  the  parameters</span>
     <span 
class="cmr-12">associated with the selected preconditioner type, to obtain a variant of that</span>
     <span 
class="cmr-12">preconditioner. Examples of use of </span><span class="lstinline"></span><span 
class="cmtt-12">set</span> <span 
class="cmr-12">are given in Section</span><span 
class="cmr-12">&#x00A0;</span><a 
href="userhtmlsu6.html#x15-140004.1"><span 
class="cmr-12">4.1</span><!--tex4ht:ref: sec:examples --></a><span 
class="cmr-12">; a complete</span>
     <span 
class="cmr-12">list of all the preconditioner parameters and their allowed and default values</span>
     <span 
class="cmr-12">is provided in Section</span><span 
class="cmr-12">&#x00A0;</span><a 
href="userhtmlse5.html#x17-160005"><span 
class="cmr-12">5</span><!--tex4ht:ref: sec:userinterface --></a><span 
class="cmr-12">, Tables</span><span 
class="cmr-12">&#x00A0;</span><a 
href="userhtmlsu9.html#x19-18009r2"><span 
class="cmr-12">2</span><!--tex4ht:ref: tab:p_cycle --></a><span 
class="cmr-12">-</span><a 
href="userhtmlsu9.html#x19-18015r8"><span 
class="cmr-12">8</span><!--tex4ht:ref: tab:p_smoother_1 --></a><span 
class="cmr-12">.</span>
     </li>
     <li 
  class="enumerate" id="x14-13008x4"><span 
class="cmti-12">Build the preconditioner for a given matrix</span><span 
class="cmr-12">. If the selected preconditioner is</span>
     <span 
class="cmr-12">multilevel, then two steps must be performed, as specified next.</span>
                                                                               

                                                                               
          <ol  class="enumerate2" >
          <li 
  class="enumerate" id="x14-13009x0"><span 
class="cmti-12">Build the AMG hierarchy for a given matrix. </span><span 
class="cmr-12">This is performed by the</span>
          <span 
class="cmr-12">routine </span><span class="lstinline"></span><span 
class="cmtt-12">hierarchy_build</span><span 
class="cmr-12">.</span>
          </li>
          <li 
  class="enumerate" id="x14-13010x0"><span 
class="cmti-12">Build the preconditioner for a given matrix. </span><span 
class="cmr-12">This is performed by the</span>
          <span 
class="cmr-12">routine </span><span class="lstinline"></span><span 
class="cmtt-12">smoothers_build</span><span 
class="cmr-12">.</span></li></ol>
     <!--l. 48--><p class="noindent" ><span 
class="cmr-12">If the selected preconditioner is one-level, it is built in a single step, performed by</span>
     <span 
class="cmr-12">the routine </span><span class="lstinline"></span><span 
class="cmtt-12">bld</span><span 
class="cmr-12">.</span>
     </li>
     <li 
  class="enumerate" id="x14-13012x5"><span 
class="cmti-12">Apply the preconditioner at each iteration of a Krylov solver. </span><span 
class="cmr-12">This is performed by</span>
     <span 
class="cmr-12">the method </span><span class="lstinline"></span><span 
class="cmtt-12">apply</span><span 
class="cmr-12">. When using the PSBLAS Krylov solvers, this step is</span>
     <span 
class="cmr-12">completely transparent to the user, since </span><span class="lstinline"></span><span 
class="cmtt-12">apply</span> <span 
class="cmr-12">is called by the PSBLAS routine</span>
     <span 
class="cmr-12">implementing the Krylov solver (</span><span class="lstinline"></span><span 
class="cmtt-12">psb_krylov</span><span 
class="cmr-12">).</span>
     </li>
     <li 
  class="enumerate" id="x14-13014x6"><span 
class="cmti-12">Free the preconditioner data structure</span><span 
class="cmr-12">. This is performed by the routine </span><span class="lstinline"></span><span 
class="cmtt-12">free</span><span 
class="cmr-12">.</span>
     <span 
class="cmr-12">This step is complementary to step 1 and should be performed when the</span>
     <span 
class="cmr-12">preconditioner is no more used.</span></li></ol>
<!--l. 59--><p class="indent" >   <span 
class="cmr-12">All the previous routines are available as methods of the preconditioner object. A</span>
<span 
class="cmr-12">detailed description of them is given in Section</span><span 
class="cmr-12">&#x00A0;</span><a 
href="userhtmlse5.html#x17-160005"><span 
class="cmr-12">5</span><!--tex4ht:ref: sec:userinterface --></a><span 
class="cmr-12">. Examples showing the basic use of</span>
<span 
class="cmr-12">AMG4PSBLAS are reported in Section</span><span 
class="cmr-12">&#x00A0;</span><a 
href="userhtmlsu6.html#x15-140004.1"><span 
class="cmr-12">4.1</span><!--tex4ht:ref: sec:examples --></a><span 
class="cmr-12">.</span>
   <div class="table">
                                                                               

                                                                               
<!--l. 63--><p class="indent" >   <a 
 id="x14-13015r1"></a><hr class="float"><div class="float" 
>
                                                                               

                                                                               
<div class="center" 
>
<!--l. 64--><p class="noindent" >
<div class="tabular"> <table id="TBL-1" class="tabular" 
cellspacing="0" cellpadding="0"  
><colgroup id="TBL-1-1g"><col 
id="TBL-1-1"></colgroup><colgroup id="TBL-1-2g"><col 
id="TBL-1-2"></colgroup><colgroup id="TBL-1-3g"><col 
id="TBL-1-3"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-1-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-1-1-1"  
class="td11"><span 
class="cmcsc-10x-x-109"><span 
class="small-caps">t</span><span 
class="small-caps">y</span><span 
class="small-caps">p</span><span 
class="small-caps">e</span>                        </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-1-1-2"  
class="td11"><!--l. 68--><p class="noindent" ><span 
class="cmcsc-10x-x-109"><span 
class="small-caps">s</span><span 
class="small-caps">t</span><span 
class="small-caps">r</span><span 
class="small-caps">i</span><span 
class="small-caps">n</span><span 
class="small-caps">g</span></span>     </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-1-3"  
class="td11"><!--l. 68--><p class="noindent" ><span 
class="cmcsc-10x-x-109"><span 
class="small-caps">d</span><span 
class="small-caps">e</span><span 
class="small-caps">f</span><span 
class="small-caps">a</span><span 
class="small-caps">u</span><span 
class="small-caps">l</span><span 
class="small-caps">t</span> <span 
class="small-caps">p</span><span 
class="small-caps">r</span><span 
class="small-caps">e</span><span 
class="small-caps">c</span><span 
class="small-caps">o</span><span 
class="small-caps">n</span><span 
class="small-caps">d</span><span 
class="small-caps">i</span><span 
class="small-caps">t</span><span 
class="small-caps">i</span><span 
class="small-caps">o</span><span 
class="small-caps">n</span><span 
class="small-caps">e</span><span 
class="small-caps">r</span></span>               </td></tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-1-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-1-2-1"  
class="td11">No preconditioner </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-2-2"  
class="td11"><!--l. 69--><p class="noindent" ><span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">NONE</span><span 
class="cmtt-10x-x-109">&#8217;</span> </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-2-3"  
class="td11"><!--l. 69--><p class="noindent" >Considered to use the PSBLAS Krylov
solvers with no preconditioner.             </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-1-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-1-3-1"  
class="td11">Diagonal                       </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-3-2"  
class="td11"><!--l. 71--><p class="noindent" ><span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">DIAG</span><span 
class="cmtt-10x-x-109">&#8217;</span>,
<span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">JACOBI</span><span 
class="cmtt-10x-x-109">&#8217;</span>,
<span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">L1</span><span 
class="cmtt-10x-x-109">-</span><span 
class="cmtt-10x-x-109">JACOBI</span><span 
class="cmtt-10x-x-109">&#8217;</span> </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-3-3"  
class="td11"><!--l. 71--><p class="noindent" >Diagonal  preconditioner.  For  any  zero
diagonal  entry  of  the  matrix  to  be
preconditioned, the corresponding entry
of the preconditioner is set to&#x00A0;1.          </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-1-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-1-4-1"  
class="td11">Gauss-Seidel                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-4-2"  
class="td11"><!--l. 74--><p class="noindent" ><span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">GS</span><span 
class="cmtt-10x-x-109">&#8217;</span>,
<span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">L1</span><span 
class="cmtt-10x-x-109">-</span><span 
class="cmtt-10x-x-109">GS</span><span 
class="cmtt-10x-x-109">&#8217;</span>   </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-4-3"  
class="td11"><!--l. 74--><p class="noindent" >Hybrid Gauss-Seidel (forward), that is,
global block Jacobi with Gauss-Seidel as
local solver.                                      </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-1-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-1-5-1"  
class="td11">Symmetrized Gauss-Seidel</td><td  style="white-space:wrap; text-align:left;" id="TBL-1-5-2"  
class="td11"><!--l. 77--><p class="noindent" ><span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">FBGS</span><span 
class="cmtt-10x-x-109">&#8217;</span>,
<span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">L1</span><span 
class="cmtt-10x-x-109">-</span><span 
class="cmtt-10x-x-109">FBGS</span><span 
class="cmtt-10x-x-109">&#8217;</span> </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-5-3"  
class="td11"><!--l. 77--><p class="noindent" >Symmetrized  hybrid  Gauss-Seidel,  that
is,  forward  Gauss-Seidel  followed  by
backward Gauss-Seidel.                      </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-1-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-1-6-1"  
class="td11">Block Jacobi                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-6-2"  
class="td11"><!--l. 80--><p class="noindent" ><span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">BJAC</span><span 
class="cmtt-10x-x-109">&#8217;</span>,
<span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">L1</span><span 
class="cmtt-10x-x-109">-</span><span 
class="cmtt-10x-x-109">BJAC</span><span 
class="cmtt-10x-x-109">&#8217;</span> </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-6-3"  
class="td11"><!--l. 80--><p class="noindent" >Block-Jacobi  with  ILU(0)  on  the  local
blocks.                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-1-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-1-7-1"  
class="td11">Additive Schwarz            </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-7-2"  
class="td11"><!--l. 81--><p class="noindent" ><span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">AS</span><span 
class="cmtt-10x-x-109">&#8217;</span>      </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-7-3"  
class="td11"><!--l. 81--><p class="noindent" >Additive Schwarz (AS), with overlap&#x00A0;1
and ILU(0) on the local blocks.            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-1-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-1-8-1"  
class="td11">Multilevel                      </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-8-2"  
class="td11"><!--l. 83--><p class="noindent" ><span class="lstinline"></span><span 
class="cmtt-10x-x-109">&#8217;</span><span 
class="cmtt-10x-x-109">ML</span><span 
class="cmtt-10x-x-109">&#8217;</span>      </td><td  style="white-space:wrap; text-align:left;" id="TBL-1-8-3"  
class="td11"><!--l. 83--><p class="noindent" >V-cycle        with        one        hybrid
forward  Gauss-Seidel  (GS)  sweep  as
pre-smoother and one hybrid backward
GS  sweep  as  post-smoother,  decoupled
smoothed   aggregation   as   coarsening
algorithm, and LU (plus triangular solve)
as coarsest-level solver. See the default
values in Tables&#x00A0;<a 
href="userhtmlsu9.html#x19-18009r2">2<!--tex4ht:ref: tab:p_cycle --></a>-<a 
href="userhtmlsu9.html#x19-18015r8">8<!--tex4ht:ref: tab:p_smoother_1 --></a> for further details of
the preconditioner.                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-1-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-1-9-1"  
class="td11">                      </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;1: </span><span  
class="content">Preconditioner types, corresponding strings and default choices. </span></div><!--tex4ht:label?: x14-13015r1 -->
</div>
                                                                               

                                                                               
   </div><hr class="endfloat" />
   </div>
<!--l. 98--><p class="indent" >   <span 
class="cmr-12">Note that the module </span><span class="lstinline"></span><span 
class="cmtt-12">amg_prec_mod</span><span 
class="cmr-12">, containing the definition of the preconditioner</span>
<span 
class="cmr-12">data type and the interfaces to the routines of AMG4PSBLAS, must be used</span>
<span 
class="cmr-12">in any program calling such routines. The modules </span><span class="lstinline"></span><span 
class="cmtt-12">psb_base_mod</span><span 
class="cmr-12">, for the</span>
<span 
class="cmr-12">sparse matrix and communication descriptor data types, and </span><span class="lstinline"></span><span 
class="cmtt-12">psb_krylov_mod</span><span 
class="cmr-12">,</span>
<span 
class="cmr-12">for interfacing with the Krylov solvers, must be also used (see Section</span><span 
class="cmr-12">&#x00A0;</span><a 
href="userhtmlsu6.html#x15-140004.1"><span 
class="cmr-12">4.1</span><!--tex4ht:ref: sec:examples --></a><span 
class="cmr-12">).</span>
<br 
class="newline" />
<!--l. 105--><p class="indent" >   <span 
class="cmbx-12">Remark 1. </span><span 
class="cmr-12">Coarsest-level solvers based on the LU factorization, such as those</span>
<span 
class="cmr-12">implemented in UMFPACK, MUMPS, SuperLU, and SuperLU</span><span 
class="cmr-12">_Dist, usually lead to</span>
<span 
class="cmr-12">smaller numbers of preconditioned Krylov iterations than inexact solvers, when the</span>
<span 
class="cmr-12">linear system comes from a standard discretization of basic scalar elliptic PDE</span>
<span 
class="cmr-12">problems. However, this does not necessarily correspond to the shortest execution time</span>
<span 
class="cmr-12">on parallel</span><span 
class="cmr-12">&#x00A0;computers.</span>
   <div class="subsectionTOCS">
   <span 
class="cmr-12">&#x00A0;</span><span class="subsectionToc" ><span 
class="cmr-12">4.1 </span><a 
href="userhtmlsu6.html#x15-140004.1"><span 
class="cmr-12">Examples</span></a></span>
<br />   <span 
class="cmr-12">&#x00A0;</span><span class="subsectionToc" ><span 
class="cmr-12">4.2 </span><a 
href="userhtmlsu7.html#x16-150004.2"><span 
class="cmr-12">GPU example</span></a></span>
   </div>

                                                                               

                                                                               
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