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<head><title>Getting Started</title>
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<!--l. 1--><div class="crosslinks"><p class="noindent"><span
class="cmbx-12">[</span><a
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<h3 class="sectionHead"><span class="titlemark"><span
class="cmbx-12">5 </span></span> <a
id="x17-160005"></a><span
class="cmbx-12">Getting Started</span></h3>
<!--l. 5--><p class="noindent" ><span
class="cmbx-12">We describe the basics for building and applying AMG4PSBLAS one-level</span>
<span
class="cmbx-12">and multilevel (i.e., AMG) preconditioners with the Krylov solvers included</span>
<span
class="cmbx-12">in PSBLAS </span><span class="cite"><span
class="cmbx-12">[</span><a
href="userhtmlli4.html#XPSBLASGUIDE"><span
class="cmbx-12">15</span></a><span
class="cmbx-12">]</span></span><span
class="cmbx-12">. The following steps are required:</span>
<ol class="enumerate1" >
<li
class="enumerate" id="x17-16002x1"><span
class="cmbxti-10x-x-120">Declare the preconditioner data structure</span><span
class="cmbx-12">. It is a derived data</span>
<span
class="cmbx-12">type, </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">amg_</span></span></span><span
class="cmbxti-10x-x-120">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="cmbx-12">, where </span><span
class="cmbxti-10x-x-120">x </span><span
class="cmbx-12">may be </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">s</span></span></span><span
class="cmbx-12">, </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">d</span></span></span><span
class="cmbx-12">, </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">c</span></span></span> <span
class="cmbx-12">or </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">z</span></span></span><span
class="cmbx-12">, according to the</span>
<span
class="cmbx-12">basic data type of the sparse matrix (</span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">s</span></span></span> <span
class="cmbx-12">= real single precision; </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">d</span></span></span> <span
class="cmbx-12">=</span>
<span
class="cmbx-12">real double precision; </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">c</span></span></span> <span
class="cmbx-12">= complex single precision; </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">z</span></span></span> <span
class="cmbx-12">= complex</span>
<span
class="cmbx-12">double precision). This data structure is accessed by the user only</span>
<span
class="cmbx-12">through the AMG4PSBLAS routines, following an object-oriented</span>
<span
class="cmbx-12">approach.</span>
</li>
<li
class="enumerate" id="x17-16004x2"><span
class="cmbxti-10x-x-120">Allocate and initialize the preconditioner data structure,</span>
<span
class="cmbxti-10x-x-120">according to a preconditioner type chosen by the user</span><span
class="cmbx-12">. This is</span>
<span
class="cmbx-12">performed by the routine </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">init</span></span></span><span
class="cmbx-12">, which also sets defaults for each</span>
<span
class="cmbx-12">preconditioner type selected by the user. The preconditioner types</span>
<span
class="cmbx-12">and the defaults associated with them are given in Table</span><span
class="cmbx-12">&#x00A0;</span><a
href="#x17-160151"><span
class="cmbx-12">1</span><!--tex4ht:ref: tab:precinit --></a><span
class="cmbx-12">, where</span>
<span
class="cmbx-12">the strings used by </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">init</span></span></span> <span
class="cmbx-12">to identify the preconditioner types are</span>
<span
class="cmbx-12">also given. Note that these strings are valid also if uppercase letters</span>
<span
class="cmbx-12">are substituted by corresponding lowercase ones.</span>
</li>
<li
class="enumerate" id="x17-16006x3"><span
class="cmbxti-10x-x-120">Modify the selected preconditioner type, by properly setting</span>
<span
class="cmbxti-10x-x-120">preconditioner parameters. </span><span
class="cmbx-12">This is performed by the routine </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">set</span></span></span><span
class="cmbx-12">.</span>
<span
class="cmbx-12">This routine must be called only if the user wants to modify</span>
<span
class="cmbx-12">the default values of the parameters associated with the selected</span>
<span
class="cmbx-12">preconditioner type, to obtain a variant of that preconditioner.</span>
<span
class="cmbx-12">Examples of use of </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">set</span></span></span> <span
class="cmbx-12">are given in Section</span><span
class="cmbx-12">&#x00A0;</span><a
href="userhtmlsu9.html#x18-170005.1"><span
class="cmbx-12">5.1</span><!--tex4ht:ref: sec:examples --></a><span
class="cmbx-12">; a complete list</span>
<span
class="cmbx-12">of all the preconditioner parameters and their allowed and default</span>
<span
class="cmbx-12">values is provided in Section</span><span
class="cmbx-12">&#x00A0;</span><a
href="userhtmlse6.html#x19-180006"><span
class="cmbx-12">6</span><!--tex4ht:ref: sec:userinterface --></a><span
class="cmbx-12">, Tables</span><span
class="cmbx-12">&#x00A0;</span><a
href="userhtmlsu11.html#x21-200092"><span
class="cmbx-12">2</span><!--tex4ht:ref: tab:p_cycle --></a><span
class="cmbx-12">-</span><a
href="userhtmlsu11.html#x21-200158"><span
class="cmbx-12">8</span><!--tex4ht:ref: tab:p_smoother_1 --></a><span
class="cmbx-12">.</span>
</li>
<li
class="enumerate" id="x17-16008x4"><span
class="cmbxti-10x-x-120">Build the preconditioner for a given matrix</span><span
class="cmbx-12">. If the selected</span>
<span
class="cmbx-12">preconditioner is multilevel, then two steps must be performed, as</span>
<span
class="cmbx-12">specified next.</span>
<ol class="enumerate2" >
<li
class="enumerate" id="x17-16009x0"><span
class="cmbxti-10x-x-120">Build the aggregation hierarchy for a given matrix. </span><span
class="cmbx-12">This is</span>
<span
class="cmbx-12">performed by the routine </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">hierarchy_build</span></span></span><span
class="cmbx-12">.</span>
</li>
<li
class="enumerate" id="x17-16010x0"><span
class="cmbxti-10x-x-120">Build the preconditioner for a given matrix. </span><span
class="cmbx-12">This is</span>
<span
class="cmbx-12">performed by the routine </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">smoothers_build</span></span></span><span
class="cmbx-12">.</span></li></ol>
<!--l. 41--><p class="noindent" ><span
class="cmbx-12">If the selected preconditioner is one-level, it is built in a single step,</span>
<span
class="cmbx-12">performed by the routine </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">bld</span></span></span><span
class="cmbx-12">.</span>
</li>
<li
class="enumerate" id="x17-16012x5"><span
class="cmbxti-10x-x-120">Apply the preconditioner at each iteration of a Krylov solver. </span><span
class="cmbx-12">This is</span>
<span
class="cmbx-12">performed by the method </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">apply</span></span></span><span
class="cmbx-12">. When using the PSBLAS Krylov</span>
<span
class="cmbx-12">solvers, this step is completely transparent to the user, since </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">apply</span></span></span> <span
class="cmbx-12">is</span>
<span
class="cmbx-12">called by the PSBLAS routine implementing the Krylov solver</span>
<span
class="cmbx-12">(</span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">psb_krylov</span></span></span><span
class="cmbx-12">).</span>
</li>
<li
class="enumerate" id="x17-16014x6"><span
class="cmbxti-10x-x-120">Free the preconditioner data structure</span><span
class="cmbx-12">. This is performed by the</span>
<span
class="cmbx-12">routine </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">free</span></span></span><span
class="cmbx-12">. This step is complementary to step 1 and should be</span>
<span
class="cmbx-12">performed when the preconditioner is no more used.</span></li></ol>
<!--l. 52--><p class="indent" > <span
class="cmbx-12">All the previous routines are available as methods of the preconditioner</span>
<span
class="cmbx-12">object. A detailed description of them is given in Section</span><span
class="cmbx-12">&#x00A0;</span><a
href="userhtmlse6.html#x19-180006"><span
class="cmbx-12">6</span><!--tex4ht:ref: sec:userinterface --></a><span
class="cmbx-12">. Examples</span>
<span
class="cmbx-12">showing the basic use of AMG4PSBLAS are reported in Section</span><span
class="cmbx-12">&#x00A0;</span><a
href="userhtmlsu9.html#x18-170005.1"><span
class="cmbx-12">5.1</span><!--tex4ht:ref: sec:examples --></a><span
class="cmbx-12">.</span>
<div class="table">
<!--l. 56--><p class="indent" > <a
id="x17-160151"></a><hr class="float"><div class="float"
>
<div class="center"
>
<!--l. 57--><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. 61--><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. 61--><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. 62--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">&#8217;NONE&#8217;</span></span></span> </td><td style="white-space:wrap; text-align:left;" id="TBL-1-2-3"
class="td11"><!--l. 62--><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. 64--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">&#8217;DIAG&#8217;</span></span></span> or
<span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">&#8217;JACOBI&#8217;</span></span></span> </td><td style="white-space:wrap; text-align:left;" id="TBL-1-3-3"
class="td11"><!--l. 64--><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. 67--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">&#8217;GS&#8217;</span></span></span> </td><td style="white-space:wrap; text-align:left;" id="TBL-1-4-3"
class="td11"><!--l. 67--><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. 70--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">&#8217;FBGS&#8217;</span></span></span> </td><td style="white-space:wrap; text-align:left;" id="TBL-1-5-3"
class="td11"><!--l. 70--><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. 73--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">&#8217;BJAC&#8217;</span></span></span> </td><td style="white-space:wrap; text-align:left;" id="TBL-1-6-3"
class="td11"><!--l. 73--><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. 74--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">&#8217;AS&#8217;</span></span></span> </td><td style="white-space:wrap; text-align:left;" id="TBL-1-7-3"
class="td11"><!--l. 74--><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. 76--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">&#8217;ML&#8217;</span></span></span> </td><td style="white-space:wrap; text-align:left;" id="TBL-1-8-3"
class="td11"><!--l. 76--><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, basic
smoothed aggregation as coarsening algorithm,
and LU (plus triangular solve) as coarsest-level
solver. See the default values in Tables&#x00A0;<a
href="userhtmlsu11.html#x21-200092">2<!--tex4ht:ref: tab:p_cycle --></a>-<a
href="userhtmlsu11.html#x21-200158">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?: x17-160151 -->
</div>
</div><hr class="endfloat" />
</div>
<!--l. 91--><p class="indent" > <span
class="cmbx-12">Note that the module </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">amg_prec_mod</span></span></span><span
class="cmbx-12">, containing the definition of</span>
<span
class="cmbx-12">the preconditioner data type and the interfaces to the routines of</span>
<span
class="cmbx-12">AMG4PSBLAS, must be used in any program calling such routines. The</span>
<span
class="cmbx-12">modules </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">psb_base_mod</span></span></span><span
class="cmbx-12">, for the sparse matrix and communication descriptor</span>
<span
class="cmbx-12">data types, and </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">psb_krylov_mod</span></span></span><span
class="cmbx-12">, for interfacing with the Krylov solvers,</span>
<span
class="cmbx-12">must be also used (see Section</span><span
class="cmbx-12">&#x00A0;</span><a
href="userhtmlsu9.html#x18-170005.1"><span
class="cmbx-12">5.1</span><!--tex4ht:ref: sec:examples --></a><span
class="cmbx-12">). </span><br
class="newline" />
<!--l. 98--><p class="indent" > <span
class="cmbx-12">Remark 1. Coarsest-level solvers based on the LU factorization, such as</span>
<span
class="cmbx-12">those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU</span><span
class="cmbx-12">_Dist,</span>
<span
class="cmbx-12">usually lead to smaller numbers of preconditioned Krylov iterations</span>
<span
class="cmbx-12">than inexact solvers, when the linear system comes from a standard</span>
<span
class="cmbx-12">discretization of basic scalar elliptic PDE problems. However, this does not</span>
<span
class="cmbx-12">necessarily correspond to the smallest execution time on parallel</span>
<span
class="cmbx-12">computers.</span>
<!--l. 107--><p class="indent" > <span
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