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original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node15.html">Smoothers and coarsest-level solvers</A>
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<H1><A NAME="SECTION00070000000000000000"></A><A NAME="sec:started"></A>
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<BR>
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Getting Started
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</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">We describe the basics for building and applying MLD2P4 one-level and multilevel
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(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [<A
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HREF="node30.html#PSBLASGUIDE">13</A>].
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The following steps are required:
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</FONT></FONT></FONT>
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<OL>
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<LI><I>Declare the preconditioner data structure</I>. It is a derived data type,
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<code>mld_</code><I>x</I><code>prec_</code> <code>type</code>, where <I>x</I> may be <code>s</code>, <code>d</code>, <code>c</code>
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or <code>z</code>, according to the basic data type of the sparse matrix
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(<code>s</code> = real single precision; <code>d</code> = real double precision;
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<code>c</code> = complex single precision; <code>z</code> = complex double precision).
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This data structure is accessed by the user only through the MLD2P4 routines,
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following an object-oriented approach.
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</LI>
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<LI><I>Allocate and initialize the preconditioner data structure, according to
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a preconditioner type chosen by the user</I>. This is performed by the routine
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<code>init</code>, which also sets defaults for each preconditioner
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type selected by the user. The preconditioner types and the defaults associated
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with them are given in Table <A HREF="#tab:precinit">1</A>, where the strings used by
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<code>init</code> to identify the preconditioner types are also given.
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Note that these strings are valid also if uppercase letters are substituted by
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corresponding lowercase ones.
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</LI>
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<LI><I>Modify the selected preconditioner type, by properly setting
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preconditioner parameters.</I> This is performed by the routine <code>set</code>.
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This routine must be called only if the user wants to modify the default values
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of the parameters associated with the selected preconditioner type, to obtain a variant
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of that preconditioner. Examples of use of <code>set</code> are given in
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Section <A HREF="node17.html#sec:examples">5.1</A>; a complete list of all the
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preconditioner parameters and their allowed and default values is provided in
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Section <A HREF="node18.html#sec:userinterface">6</A>, Tables <A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.
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</LI>
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<LI><I>Build the preconditioner for a given matrix</I>. If the selected preconditioner
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is multilevel, then two steps must be performed, as specified next.
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<DL COMPACT>
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<DT>4.1</DT>
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<DD><I>Build the aggregation hierarchy for a given matrix.</I> This is
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performed by the routine <code>hierarchy_build</code>.
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</DD>
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<DT>4.2</DT>
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<DD><I>Build the preconditioner for a given matrix.</I> This is performed
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by the routine <code>smoothers_build</code>.
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</DD>
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</DL>
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If the selected preconditioner is one-level, it is built in a single step,
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performed by the routine <code>bld</code>.
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</LI>
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<LI><I>Apply the preconditioner at each iteration of a Krylov solver.</I>
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This is performed by the routine <code>aply</code>. When using the PSBLAS Krylov solvers,
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this step is completely transparent to the user, since <code>aply</code> is called
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by the PSBLAS routine implementing the Krylov solver (<code>psb_krylov</code>).
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</LI>
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<LI><I>Free the preconditioner data structure</I>. This is performed by
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the routine <code>free</code>. This step is complementary to step 1 and should
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be performed when the preconditioner is no more used.
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</LI>
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</OL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">All the previous routines are available as methods of the preconditioner object.
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A detailed description of them is given in Section <A HREF="node18.html#sec:userinterface">6</A>.
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Examples showing the basic use of MLD2P4 are reported in Section <A HREF="node17.html#sec:examples">5.1</A>.
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<BR><P></P>
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<DIV ALIGN="CENTER"><A NAME="904"></A>
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<TABLE>
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<CAPTION><STRONG>Table 1:</STRONG>
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Preconditioner types, corresponding strings and default choices.
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</CAPTION>
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<TR><TD>
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<DIV ALIGN="CENTER">
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<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
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<TR><TD ALIGN="LEFT"><SMALL>TYPE</SMALL></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><SMALL>STRING</SMALL></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232><SMALL>DEFAULT PRECONDITIONER</SMALL></TD>
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</TR>
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<TR><TD ALIGN="LEFT">No preconditioner</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'NOPREC'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Considered only to use the PSBLAS
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Krylov solvers with no preconditioner.</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Diagonal</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'DIAG'</code> or <code>'JACOBI'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Diagonal preconditioner.
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For any zero diagonal entry of the matrix to be preconditioned,
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the corresponding entry of the preconditioner is set to 1.</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Gauss-Seidel</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'GS'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Hybrid Gauss-Seidel (forward), that is,
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global block Jacobi with
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Gauss-Seidel as local solver.</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Symmetrized Gauss-Seidel</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'FBGS'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Symmetrized hybrid Gauss-Seidel,that is,
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forward Gauss-Seidel followed by
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backward Gauss-Seidel.</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Block Jacobi</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'BJAC'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Block-Jacobi with ILU(0) on the local blocks.</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Additive Schwarz</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'AS'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Additive Schwarz (AS),
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with overlap 1 and ILU(0) on the local blocks.</TD>
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</TR>
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<TR><TD ALIGN="LEFT">Multilevel</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'ML'</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>V-cycle with one hybrid forward Gauss-Seidel
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(GS) sweep as pre-smoother and one hybrid backward
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GS sweep as post-smoother, basic smoothed aggregation
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as coarsening algorithm, and LU (plus triangular solve)
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as coarsest-level solver. See the default values in
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Tables <A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>
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for further details of the preconditioner.</TD>
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</TR>
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</TABLE>
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</DIV></TD></TR>
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</TABLE>
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</DIV><P></P>
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<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Note that the module <code>mld_prec_mod</code>, containing the definition of the
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preconditioner data type and the interfaces to the routines of MLD2P4,
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must be used in any program calling such routines.
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The modules <code>psb_base_mod</code>, for the sparse matrix and communication descriptor
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data types, and <code>psb_krylov_mod</code>, for interfacing with the
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Krylov solvers, must be also used (see Section <A HREF="node17.html#sec:examples">5.1</A>).
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<BR></FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Remark 1.</B> Coarsest-level solvers based on the LU factorization,
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such as those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU_Dist,
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usually lead to smaller numbers of preconditioned Krylov
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iterations than inexact solvers, when the linear system comes from
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a standard discretization of basic scalar elliptic PDE problems. However,
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this does not necessarily correspond to the smallest execution time
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on parallel computers. </FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<BR><HR>
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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