</H2><FONTSIZE="+1"><FONTSIZE="+1"><FONTSIZE="+1">
In order to describe the AMG preconditioners available in MLD2P4, we consider a
linear system
</FONT></FONT></FONT>
ALT="$A$"> is real, but the
results are valid for the complex case as well.
Let us assume as finest index space the set of row (column) indices of <IMG
ALT="$\Omega = \{1, 2, \ldots, n\}$">.
Any algebraic multilevel preconditioners implemented in MLD2P4 generates
a hierarchy of index spaces and a corresponding hierarchy of matrices,
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In the current implementation of MLD2P4 we have <IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
ALT="$R^k=(P^k)^T$">
A smoother with iteration matrix <IMG
WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img18.png"
ALT="$M^k$"> is set up at each level <IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img11.png"
ALT="$k < nlev$">, and a solver
is set up at the coarsest level, so that they are ready for application
(for example, setting up a solver based on the <IMG
WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img19.png"
ALT="$LU$"> factorization means computing
and storing the <IMG
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$L$"> and <IMG
WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$U$"> factors). The construction of the hierarchy of AMG components
described so far corresponds to the so-called build phase of the preconditioner.
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The components produced in the build phase may be combined in several ways
to obtain different multilevel preconditioners;
this is done in the application phase, i.e., in the computation of a vector
of type <IMG
WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$w=B^{-1}v$">, where <IMG
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img24.png"
ALT="$B$"> denotes the preconditioner, usually within an iteration
of a Krylov solver [<A
HREF="node29.html#Saad_book">20</A>]. An example of such a combination, known as
V-cycle, is given in Figure <AHREF="#fig:application_alg">1</A>. In this case, a single iteration
of the same smoother is used before and after the the recursive call to the V-cycle (i.e.,
in the pre-smoothing and post-smoothing phases); however, different choices can be
performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle
and W-cycle [<A
HREF="node29.html#Briggs2000">3</A>], and a version of the K-cycle described
in [<A
