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Salvatore Filippone
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HREF="node3.html">General Overview</A>
<LI><A NAME="tex2html29"
HREF="node4.html">Code Distribution</A>
<UL>
<LI><A NAME="tex2html30"
HREF="node5.html">Configuring and Building MLD2P4</A>
<UL>
HREF="node5.html">Contributors</A>
</UL>
<BR>
<LI><A NAME="tex2html31"
HREF="node6.html">Prerequisites</A>
HREF="node6.html">Configuring and Building MLD2P4</A>
<UL>
<LI><A NAME="tex2html32"
HREF="node7.html">Optional third party libraries</A>
HREF="node7.html">Prerequisites</A>
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HREF="node8.html">Configuration options</A>
HREF="node8.html">Optional third party libraries</A>
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HREF="node9.html">Bug reporting</A>
HREF="node9.html">Configuration options</A>
<LI><A NAME="tex2html35"
HREF="node10.html">Example and test programs</A>
</UL>
<BR>
HREF="node10.html">Bug reporting</A>
<LI><A NAME="tex2html36"
HREF="node11.html">Multigrid Background</A>
<UL>
HREF="node11.html">Example and test programs</A>
</UL>
<BR>
<LI><A NAME="tex2html37"
HREF="node12.html">AMG preconditioners</A>
HREF="node12.html">Multigrid Background</A>
<UL>
<LI><A NAME="tex2html38"
HREF="node13.html">Smoothed Aggregation</A>
HREF="node13.html">AMG preconditioners</A>
<LI><A NAME="tex2html39"
HREF="node14.html">Smoothers and coarsest-level solvers</A>
</UL>
<BR>
HREF="node14.html">Smoothed Aggregation</A>
<LI><A NAME="tex2html40"
HREF="node15.html">Getting Started</A>
<UL>
HREF="node15.html">Smoothers and coarsest-level solvers</A>
</UL>
<BR>
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HREF="node16.html">Examples</A>
</UL>
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HREF="node17.html">User Interface</A>
HREF="node16.html">Getting Started</A>
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<LI><A NAME="tex2html51"
HREF="node26.html">Adding new smoother and solver objects to MLD2P4</A>
HREF="node26.html">Subroutine descr</A>
</UL>
<BR>
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HREF="node27.html">Error Handling</A>
HREF="node27.html">Adding new smoother and solver objects to MLD2P4</A>
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<H2><A NAME="SECTION00055000000000000000"></A><A NAME="sec:ex_and_test"></A>
<BR>
Example and test programs
<H2><A NAME="SECTION00054000000000000000">
Bug reporting</A>
</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
The package contains the <code>examples</code> and <code>tests</code> directories;
both of them are further divided into <code>fileread</code> and
<code>pdegen</code> subdirectories. Their purpose is as follows:
</FONT></FONT></FONT><DL>
<DT><STRONG><TT>examples</TT></STRONG></DT>
<DD>contains a set of simple example programs with a
predefined choice of preconditioners, selectable via integer
values. These are intended to get an acquaintance with the
multi-level preconditioners available in MLD2P4.
</DD>
<DT><STRONG><TT>tests</TT></STRONG></DT>
<DD>contains a set of more sophisticated examples that
will allow the user, via the input files in the <code>runs</code>
subdirectories, to experiment with the full range of preconditioners
implemented in the package.
</DD>
</DL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
The <code>fileread</code> directories contain sample programs that read
sparse matrices from files, according to the Matrix Market or the
Harwell-Boeing storage format; the <code>pdegen</code> programs generate
matrices in full parallel mode from the discretization of a sample partial
differential equation.
If you find any bugs in our codes, please send an email to <BR><TT>pasqua.dambra@cnr.it</TT>
<BR><TT>daniela.diserafino@unicampania.it</TT>
<BR><TT>salvatore.filippone@cranfield.ac.uk</TT> <BR>
You should be aware that the amount of information needed to reproduce a problem
in a parallel program may vary quite a lot.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><HR>
</BODY>
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<H1><A NAME="SECTION00060000000000000000"></A><A NAME="sec:background"></A>
<H2><A NAME="SECTION00055000000000000000"></A><A NAME="sec:ex_and_test"></A>
<BR>
Multigrid Background
</H1><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
Example and test programs
</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
The package contains the <code>examples</code> and <code>tests</code> directories;
both of them are further divided into <code>fileread</code> and
<code>pdegen</code> subdirectories. Their purpose is as follows:
</FONT></FONT></FONT><DL>
<DT><STRONG><TT>examples</TT></STRONG></DT>
<DD>contains a set of simple example programs with a
predefined choice of preconditioners, selectable via integer
values. These are intended to get an acquaintance with the
multi-level preconditioners available in MLD2P4.
</DD>
<DT><STRONG><TT>tests</TT></STRONG></DT>
<DD>contains a set of more sophisticated examples that
will allow the user, via the input files in the <code>runs</code>
subdirectories, to experiment with the full range of preconditioners
implemented in the package.
</DD>
</DL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
The <code>fileread</code> directories contain sample programs that read
sparse matrices from files, according to the Matrix Market or the
Harwell-Boeing storage format; the <code>pdegen</code> programs generate
matrices in full parallel mode from the discretization of a sample partial
differential equation.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Multigrid preconditioners, coupled with Krylov iterative
solvers, are widely used in the parallel solution of large and sparse linear systems,
because of their optimality in the solution of linear systems arising from the
discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids.
Optimality, also known as algorithmic scalability, is the property
of having a computational cost per iteration that depends linearly on
the problem size, and a convergence rate that is independent of the problem size.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Multigrid preconditioners are based on a recursive application of a two-grid process
consisting of smoother iterations and a coarse-space (or coarse-level) correction.
The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones,
or more complex subspace-correction methods, such as the Schwarz ones.
The coarse-space correction consists of solving, in an appropriately chosen
coarse space, the residual equation associated with the approximate solution computed
by the smoother, and of using the solution of this equation to correct the
previous approximation. The transfer of information between the original
(fine) space and the coarse one is performed by using suitable restriction and
prolongation operators. The construction of the coarse space and the corresponding
transfer operators is carried out by applying a so-called coarsening algorithm to the system
matrix. Two main approaches can be used to perform coarsening: the geometric approach,
which exploits the knowledge of some physical grid associated with the matrix
and requires the user to define transfer operators from the fine
to the coarse level and vice versa, and the algebraic approach, which builds
the coarse-space correction and the associate transfer operators using only matrix
information. The first approach may be difficult when the system comes from
discretizations on complex geometries;
furthermore, ad hoc one-level smoothers may be required to get an efficient
interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients
are considered. The second approach performs a fully automatic coarsening and enforces the
interplay between fine and coarse level by suitably choosing the coarse space and
the coarse-to-fine interpolation (see, e.g., [<A
HREF="node29.html#Briggs2000">3</A>,<A
HREF="node29.html#Stuben_01">23</A>,<A
HREF="node29.html#dd2_96">21</A>] for details.)
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">MLD2P4 uses a pure algebraic approach, based on the smoothed
aggregation algorithm [<A
HREF="node29.html#BREZINA_VANEK">2</A>,<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>],
for building the sequence of coarse matrices and transfer operators,
starting from the original one.
A decoupled version of this algorithm is implemented, where the smoothed
aggregation is applied locally to each submatrix [<A
HREF="node29.html#TUMINARO_TONG">24</A>].
A brief description of the AMG preconditioners implemented in MLD2P4 is given in
Sections&nbsp;<A HREF="node12.html#sec:multilevel">4.1</A>-<A HREF="node14.html#sec:smoothers">4.3</A>. For further details the reader
is referred to [<A
HREF="node29.html#para_04">4</A>,<A
HREF="node29.html#aaecc_07">5</A>,<A
HREF="node29.html#apnum_07">7</A>,<A
HREF="node29.html#MLD2P4_TOMS">8</A>].
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">We note that optimal multigrid preconditioners do not necessarily correspond
to minimum execution times in a parallel setting. Indeed, to obtain effective parallel
multigrid preconditioners, a tradeoff between the optimality and the cost of building and
applying the smoothers and the coarse-space corrections must be achieved. Effective
parallel preconditioners require algorithmic scalability to be coupled with implementation
scalability, i.e., a computational cost per iteration which remains (almost) constant as
the number of parallel processors increases.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><HR>
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<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
<H1><A NAME="SECTION00060000000000000000"></A><A NAME="sec:background"></A>
<BR>
AMG preconditioners
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to describe the AMG preconditioners available in MLD2P4, we consider a
linear system
</FONT></FONT></FONT>
<BR>
<DIV ALIGN="RIGHT">
Multigrid Background
</H1><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<!-- MATH
\begin{equation}
Ax=b,
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:system"></A><IMG
WIDTH="58" HEIGHT="30" BORDER="0"
SRC="img2.png"
ALT="\begin{displaymath}
Ax=b,
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
where <!-- MATH
$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
-->
<IMG
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img5.png"
ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;
for ease of presentation we assume <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$"> is real, but the
results are valid for the complex case as well.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Let us assume as finest index space the set of row (column) indices of <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">, i.e.,
<!-- MATH
$\Omega = \{1, 2, \ldots, n\}$
-->
<IMG
WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img6.png"
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,
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Multigrid preconditioners, coupled with Krylov iterative
solvers, are widely used in the parallel solution of large and sparse linear systems,
because of their optimality in the solution of linear systems arising from the
discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids.
Optimality, also known as algorithmic scalability, is the property
of having a computational cost per iteration that depends linearly on
the problem size, and a convergence rate that is independent of the problem size.
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},
\quad A^1 \equiv A, A^2, \ldots, A^{nlev},
\end{displaymath}
-->
<IMG
WIDTH="398" HEIGHT="30" BORDER="0"
SRC="img7.png"
ALT="\begin{displaymath}\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},␍\quad A^1 \equiv A, A^2, \ldots, A^{nlev}, \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
by using the information contained in <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">, without assuming any
knowledge of the geometry of the problem from which <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$"> originates.
A vector space <!-- MATH
$\mathbb{R}^{n_{k}}$
-->
<IMG
WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img8.png"
ALT="$\mathbb{R}^{n_{k}}$"> is associated with <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">,
where <IMG
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img10.png"
ALT="$n_k$"> is the size of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">.
For all <IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img11.png"
ALT="$k &lt; nlev$">, a restriction operator and a prolongation one are built,
which connect two levels <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img12.png"
ALT="$k$"> and <IMG
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img13.png"
ALT="$k+1$">:
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Multigrid preconditioners are based on a recursive application of a two-grid process
consisting of smoother iterations and a coarse-space (or coarse-level) correction.
The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones,
or more complex subspace-correction methods, such as the Schwarz ones.
The coarse-space correction consists of solving, in an appropriately chosen
coarse space, the residual equation associated with the approximate solution computed
by the smoother, and of using the solution of this equation to correct the
previous approximation. The transfer of information between the original
(fine) space and the coarse one is performed by using suitable restriction and
prolongation operators. The construction of the coarse space and the corresponding
transfer operators is carried out by applying a so-called coarsening algorithm to the system
matrix. Two main approaches can be used to perform coarsening: the geometric approach,
which exploits the knowledge of some physical grid associated with the matrix
and requires the user to define transfer operators from the fine
to the coarse level and vice versa, and the algebraic approach, which builds
the coarse-space correction and the associate transfer operators using only matrix
information. The first approach may be difficult when the system comes from
discretizations on complex geometries;
furthermore, ad hoc one-level smoothers may be required to get an efficient
interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients
are considered. The second approach performs a fully automatic coarsening and enforces the
interplay between fine and coarse level by suitably choosing the coarse space and
the coarse-to-fine interpolation (see, e.g., [<A
HREF="node30.html#Briggs2000">3</A>,<A
HREF="node30.html#Stuben_01">23</A>,<A
HREF="node30.html#dd2_96">21</A>] for details.)
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
R^k \in \mathbb{R}^{n_{k+1}\times n_k};
\end{displaymath}
-->
<IMG
WIDTH="254" HEIGHT="30" BORDER="0"
SRC="img14.png"
ALT="\begin{displaymath}␍ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad ␍ R^k \in \mathbb{R}^{n_{k+1}\times n_k};␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
the matrix <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$"> is computed by using the previous operators according
to the Galerkin approach, i.e.,
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">MLD2P4 uses a pure algebraic approach, based on the smoothed
aggregation algorithm [<A
HREF="node30.html#BREZINA_VANEK">2</A>,<A
HREF="node30.html#VANEK_MANDEL_BREZINA">25</A>],
for building the sequence of coarse matrices and transfer operators,
starting from the original one.
A decoupled version of this algorithm is implemented, where the smoothed
aggregation is applied locally to each submatrix [<A
HREF="node30.html#TUMINARO_TONG">24</A>].
A brief description of the AMG preconditioners implemented in MLD2P4 is given in
Sections&nbsp;<A HREF="node13.html#sec:multilevel">4.1</A>-<A HREF="node15.html#sec:smoothers">4.3</A>. For further details the reader
is referred to [<A
HREF="node30.html#para_04">4</A>,<A
HREF="node30.html#aaecc_07">5</A>,<A
HREF="node30.html#apnum_07">7</A>,<A
HREF="node30.html#MLD2P4_TOMS">8</A>].
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A^{k+1}=R^kA^kP^k.
\end{displaymath}
-->
<IMG
WIDTH="131" HEIGHT="27" BORDER="0"
SRC="img16.png"
ALT="\begin{displaymath}␍ A^{k+1}=R^kA^kP^k.␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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="$k &lt; nlev$">, 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.
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">We note that optimal multigrid preconditioners do not necessarily correspond
to minimum execution times in a parallel setting. Indeed, to obtain effective parallel
multigrid preconditioners, a tradeoff between the optimality and the cost of building and
applying the smoothers and the coarse-space corrections must be achieved. Effective
parallel preconditioners require algorithmic scalability to be coupled with implementation
scalability, i.e., a computational cost per iteration which remains (almost) constant as
the number of parallel processors increases.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:application_alg"></A><A NAME="517"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
Application phase of a V-cycle preconditioner.</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<!-- MATH
$\framebox{
\begin{minipage}{.85\textwidth}
\begin{tabbing}
\quad \=\quad \=\quad \=\quad \\[-3mm]
procedure V-cycle$\left(k,A^k,b^k,u^k\right)$\ \\[2mm]
\>if $\left(k \ne nlev \right)$\ then \\[1mm]
\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
\>\> $b^{k+1} = R^{k+1}\left(b^k - A^k u^k\right)$\ \\[1mm]
\>\> $u^{k+1} =$\ V-cycle$\left(k+1,A^{k+1},b^{k+1},0\right)$\ \\[1mm]
\>\> $u^k = u^k + P^{k+1} u^{k+1}$\ \\[1mm]
\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
\>else \\[1mm]
\>\> $u^k = \left(A^k\right)^{-1} b^k$\\[1mm]
\>endif \\[1mm]
\>return $u^k$\ \\[1mm]
end
\end{tabbing}
\end{minipage}
}$
-->
<IMG
WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="\framebox{␍\begin{minipage}{.85\textwidth}␍\begin{tabbing}␍\quad \=\quad \=\quad...
...mm]␍\&gt;endif \ [1mm]␍\&gt;return $u^k$ \ [1mm]␍end␍\end{tabbing}␍\end{minipage}␍}">
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">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&nbsp;<A HREF="#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&nbsp;[<A
HREF="node29.html#Briggs2000">3</A>], and a version of the K-cycle described
in&nbsp;[<A
HREF="node29.html#Notay2008">19</A>].
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
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<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
<BR>
Smoothed Aggregation
AMG preconditioners
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to define the prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$">, used to compute
the coarse-level matrix <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation
algorithm described in [<A
HREF="node29.html#BREZINA_VANEK">2</A>,<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>].
The basic idea of this algorithm is to build a coarse set of indices
<IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img26.png"
ALT="$\Omega^{k+1}$"> by suitably grouping the indices of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"> into disjoint
subsets (aggregates), and to define the coarse-to-fine space transfer operator
<IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> by applying a suitable smoother to a simple piecewise constant
prolongation operator, with the aim of improving the quality of the coarse-space correction.
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to describe the AMG preconditioners available in MLD2P4, we consider a
linear system
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Three main steps can be identified in the smoothed aggregation procedure:
</FONT></FONT></FONT>
<OL>
<LI>aggregation of the indices of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"> to obtain <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img26.png"
ALT="$\Omega^{k+1}$">;
</LI>
<LI>construction of the prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$">;
</LI>
<LI>application of <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> and <IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
ALT="$R^k=(P^k)^T$"> to build <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$">.
</LI>
</OL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to perform the coarsening step, the smoothed aggregation algorithm
described in&nbsp;[<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
each index <!-- MATH
$j \in \Omega^{k+1}$
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
Ax=b,
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:system"></A><IMG
WIDTH="58" HEIGHT="30" BORDER="0"
SRC="img2.png"
ALT="\begin{displaymath}
Ax=b,
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
where <!-- MATH
$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
-->
<IMG
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.png"
ALT="$j \in \Omega^{k+1}$"> corresponds to an aggregate <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img28.png"
ALT="$\Omega^k_j$"> of <IMG
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img5.png"
ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;
for ease of presentation we assume <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$"> is real, but the
results are valid for the complex case as well.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Let us assume as finest index space the set of row (column) indices of <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">, i.e.,
<!-- MATH
$\Omega = \{1, 2, \ldots, n\}$
-->
<IMG
WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img6.png"
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,
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},
\quad A^1 \equiv A, A^2, \ldots, A^{nlev},
\end{displaymath}
-->
<IMG
WIDTH="398" HEIGHT="30" BORDER="0"
SRC="img7.png"
ALT="\begin{displaymath}\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},␍\quad A^1 \equiv A, A^2, \ldots, A^{nlev}, \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
by using the information contained in <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">, without assuming any
knowledge of the geometry of the problem from which <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$"> originates.
A vector space <!-- MATH
$\mathbb{R}^{n_{k}}$
-->
<IMG
WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img8.png"
ALT="$\mathbb{R}^{n_{k}}$"> is associated with <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">,
consisting of a suitably chosen index <!-- MATH
$i \in \Omega^k$
-->
<IMG
WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img29.png"
ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a
strongly-coupled neighborood of <IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img30.png"
ALT="$i$">, i.e.,
</FONT></FONT></FONT>
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
\left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
WIDTH="387" HEIGHT="72" BORDER="0"
SRC="img31.png"
ALT="\begin{displaymath}
\Omega^k_j \subset \mathcal{N}_i^k(\theta) = ␍ \left\{ r ...
...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},␍\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
for a given threshold <!-- MATH
$\theta \in [0,1]$
-->
<IMG
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img32.png"
ALT="$\theta \in [0,1]$"> (see&nbsp;[<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
Since this algorithm has a sequential nature, a decoupled
version of it is applied, where each processor independently executes
the algorithm on the set of indices assigned to it in the initial data
distribution. This version is embarrassingly parallel, since it does not require any data
communication. On the other hand, it may produce some nonuniform aggregates
and is strongly dependent on the number of processors and on the initial partitioning
of the matrix <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">. Nevertheless, this parallel algorithm has been chosen for
MLD2P4, since it has been shown to produce good results in practice
[<A
HREF="node29.html#aaecc_07">5</A>,<A
HREF="node29.html#apnum_07">7</A>,<A
HREF="node29.html#TUMINARO_TONG">24</A>].
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> is built starting from a tentative prolongator
<!-- MATH
$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$
-->
<IMG
WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img33.png"
ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$">, defined as
</FONT></FONT></FONT>
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
\left\{ \begin{array}{ll}
1 & \quad \mbox{if} \; i \in \Omega^k_j, \\
0 & \quad \mbox{otherwise},
\end{array} \right.
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
WIDTH="287" HEIGHT="51" BORDER="0"
SRC="img34.png"
ALT="\begin{displaymath}␍\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k = ␍\left\{...
...ega^k_j, \ ␍0 &amp; \quad \mbox{otherwise},␍\end{array} \right.
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(4)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
where <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img28.png"
ALT="$\Omega^k_j$"> is the aggregate of <IMG
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img10.png"
ALT="$n_k$"> is the size of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">
corresponding to the index <!-- MATH
$j \in \Omega^{k+1}$
-->
<IMG
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.png"
ALT="$j \in \Omega^{k+1}$">.
<IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> is obtained by applying to <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img35.png"
ALT="$\bar{P}^k$"> a smoother
<!-- MATH
$S^k \in \mathbb{R}^{n_k \times n_k}$
-->
<IMG
WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img36.png"
ALT="$S^k \in \mathbb{R}^{n_k \times n_k}$">:
ALT="$\Omega^k$">.
For all <IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img11.png"
ALT="$k &lt; nlev$">, a restriction operator and a prolongation one are built,
which connect two levels <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img12.png"
ALT="$k$"> and <IMG
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img13.png"
ALT="$k+1$">:
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
P^k = S^k \bar{P}^k,
P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
R^k \in \mathbb{R}^{n_{k+1}\times n_k};
\end{displaymath}
-->
<IMG
WIDTH="90" HEIGHT="30" BORDER="0"
SRC="img37.png"
ALT="\begin{displaymath}␍P^k = S^k \bar{P}^k,␍\end{displaymath}">
WIDTH="254" HEIGHT="30" BORDER="0"
SRC="img14.png"
ALT="\begin{displaymath}␍ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad ␍ R^k \in \mathbb{R}^{n_{k+1}\times n_k};␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
in order to remove nonsmooth components from the range of the prolongator,
and hence to improve the convergence properties of the multi-level
method&nbsp;[<A
HREF="node29.html#BREZINA_VANEK">2</A>,<A
HREF="node29.html#Stuben_01">23</A>].
A simple choice for <IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img38.png"
ALT="$S^k$"> is the damped Jacobi smoother:
the matrix <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$"> is computed by using the previous operators according
to the Galerkin approach, i.e.,
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
A^{k+1}=R^kA^kP^k.
\end{displaymath}
-->
<IMG
WIDTH="175" HEIGHT="31" BORDER="0"
SRC="img39.png"
ALT="\begin{displaymath}␍S^k = I - \omega^k (D^k)^{-1} A^k_F , ␍\end{displaymath}">
WIDTH="131" HEIGHT="27" BORDER="0"
SRC="img16.png"
ALT="\begin{displaymath}␍ A^{k+1}=R^kA^kP^k.␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
where <IMG
WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img40.png"
ALT="$D^k$"> is the diagonal matrix with the same diagonal entries as <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$">,
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="$k &lt; nlev$">, 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.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:application_alg"></A><A NAME="524"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
Application phase of a V-cycle preconditioner.</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<!-- MATH
$A^k_F = (\bar{a}_{ij}^k)$
$\framebox{
\begin{minipage}{.85\textwidth}
\begin{tabbing}
\quad \=\quad \=\quad \=\quad \\[-3mm]
procedure V-cycle$\left(k,A^k,b^k,u^k\right)$\ \\[2mm]
\>if $\left(k \ne nlev \right)$\ then \\[1mm]
\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
\>\> $b^{k+1} = R^{k+1}\left(b^k - A^k u^k\right)$\ \\[1mm]
\>\> $u^{k+1} =$\ V-cycle$\left(k+1,A^{k+1},b^{k+1},0\right)$\ \\[1mm]
\>\> $u^k = u^k + P^{k+1} u^{k+1}$\ \\[1mm]
\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
\>else \\[1mm]
\>\> $u^k = \left(A^k\right)^{-1} b^k$\\[1mm]
\>endif \\[1mm]
\>return $u^k$\ \\[1mm]
end
\end{tabbing}
\end{minipage}
}$
-->
<IMG
WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$A^k_F = (\bar{a}_{ij}^k)$"> is the filtered matrix defined as
</FONT></FONT></FONT>
<BR>
<DIV ALIGN="RIGHT">
WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="\framebox{␍\begin{minipage}{.85\textwidth}␍\begin{tabbing}␍\quad \=\quad \=\quad...
...mm]␍\&gt;endif [1mm]␍\&gt;return $u^k$ [1mm]␍end␍\end{tabbing}␍\end{minipage}␍}">
<!-- MATH
\begin{equation}
\bar{a}_{ij}^k =
\left \{ \begin{array}{ll}
a_{ij}^k & \mbox{if } j \in \mathcal{N}_i^k(\theta), \\
0 & \mbox{otherwise},
\end{array} \right.
\; (j \ne i),
\qquad
\bar{a}_{ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
WIDTH="514" HEIGHT="74" BORDER="0"
SRC="img43.png"
ALT="\begin{displaymath}
\bar{a}_{ij}^k =␍ \left \{ \begin{array}{ll}␍ a_{ij}^k &amp; ...
...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),␍\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5)</TD></TR>
</DIV></TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
and <IMG
WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img44.png"
ALT="$\omega^k$"> is an approximation of <IMG
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img45.png"
ALT="$4/(3\rho^k)$">, where
<IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img46.png"
ALT="$\rho^k$"> is the spectral radius of <!-- MATH
$(D^k)^{-1}A^k_F$
-->
<IMG
WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img47.png"
ALT="$(D^k)^{-1}A^k_F$"> [<A
HREF="node29.html#BREZINA_VANEK">2</A>].
In MLD2P4 this approximation is obtained by using <!-- MATH
$\| A^k_F \|_\infty$
-->
<IMG
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img48.png"
ALT="$\Vert A^k_F \Vert _\infty$"> as an estimate
of <IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img46.png"
ALT="$\rho^k$">. Note that for systems coming from uniformly elliptic
problems, filtering the matrix <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"> has little or no effect, and
<IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"> can be used instead of <IMG
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img49.png"
ALT="$A^k_F$">. The latter choice is the default in MLD2P4.
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">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="node30.html#Saad_book">20</A>]. An example of such a combination, known as
V-cycle, is given in Figure&nbsp;<A HREF="#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&nbsp;[<A
HREF="node30.html#Briggs2000">3</A>], and a version of the K-cycle described
in&nbsp;[<A
HREF="node30.html#Notay2008">19</A>].
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
@@ -378,7 +289,7 @@ problems, filtering the matrix <IMG
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<H2><A NAME="SECTION00063000000000000000"></A><A NAME="sec:smoothers"></A>
<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
<BR>
Smoothers and coarsest-level solvers
Smoothed Aggregation
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,
a hybrid version of the forward and backward Gauss-Seidel methods, and the
additive Schwarz (AS) ones (see, e.g., [<A
HREF="node29.html#Saad_book">20</A>,<A
HREF="node29.html#dd2_96">21</A>]).
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The hybrid Gauss-Seidel
version is considered because the original Gauss-Seidel method is inherently sequential.
At each iteration of the hybrid version, each parallel process uses the most recent values
of its own local variables and the values of the non-local variables computed at the
previous iteration, obtained by exchanging data with other processes before
the beginning of the current iteration.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In the AS methods, the index space <IMG
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to define the prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$">, used to compute
the coarse-level matrix <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation
algorithm described in [<A
HREF="node30.html#BREZINA_VANEK">2</A>,<A
HREF="node30.html#VANEK_MANDEL_BREZINA">25</A>].
The basic idea of this algorithm is to build a coarse set of indices
<IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img26.png"
ALT="$\Omega^{k+1}$"> by suitably grouping the indices of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"> is divided into <IMG
WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
ALT="$m_k$">
subsets <IMG
ALT="$\Omega^k$"> into disjoint
subsets (aggregates), and to define the coarse-to-fine space transfer operator
<IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> by applying a suitable smoother to a simple piecewise constant
prolongation operator, with the aim of improving the quality of the coarse-space correction.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Three main steps can be identified in the smoothed aggregation procedure:
</FONT></FONT></FONT>
<OL>
<LI>aggregation of the indices of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"> to obtain <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img26.png"
ALT="$\Omega^{k+1}$">;
</LI>
<LI>construction of the prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$">;
</LI>
<LI>application of <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> and <IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
ALT="$R^k=(P^k)^T$"> to build <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$">.
</LI>
</OL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to perform the coarsening step, the smoothed aggregation algorithm
described in&nbsp;[<A
HREF="node30.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
each index <!-- MATH
$j \in \Omega^{k+1}$
-->
<IMG
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.png"
ALT="$j \in \Omega^{k+1}$"> corresponds to an aggregate <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$"> of size <IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img52.png"
ALT="$n_{k,i}$">, possibly
overlapping. For each <IMG
SRC="img28.png"
ALT="$\Omega^k_j$"> of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">,
consisting of a suitably chosen index <!-- MATH
$i \in \Omega^k$
-->
<IMG
WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img29.png"
ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a
strongly-coupled neighborood of <IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img30.png"
ALT="$i$"> we consider the restriction
operator <!-- MATH
$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$
-->
<IMG
WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img53.png"
ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$">
that maps a vector <IMG
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img54.png"
ALT="$x^k$"> to the vector <IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img55.png"
ALT="$x_i^k$"> made of the components of <IMG
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img54.png"
ALT="$x^k$">
with indices in <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$">, and the prolongation operator
ALT="$i$">, i.e.,
</FONT></FONT></FONT>
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
$P^k_i = (R_i^k)^T$
\begin{equation}
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
\left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
WIDTH="387" HEIGHT="72" BORDER="0"
SRC="img31.png"
ALT="\begin{displaymath}
\Omega^k_j \subset \mathcal{N}_i^k(\theta) = ␍ \left\{ r ...
...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},␍\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
for a given threshold <!-- MATH
$\theta \in [0,1]$
-->
<IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img56.png"
ALT="$P^k_i = (R_i^k)^T$">. These operators are then used to build
<!-- MATH
$A_i^k=R_i^kA^kP_i^k$
-->
<IMG
WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img57.png"
ALT="$A_i^k=R_i^kA^kP_i^k$">, which is the restriction of <IMG
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img32.png"
ALT="$\theta \in [0,1]$"> (see&nbsp;[<A
HREF="node30.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
Since this algorithm has a sequential nature, a decoupled
version of it is applied, where each processor independently executes
the algorithm on the set of indices assigned to it in the initial data
distribution. This version is embarrassingly parallel, since it does not require any data
communication. On the other hand, it may produce some nonuniform aggregates
and is strongly dependent on the number of processors and on the initial partitioning
of the matrix <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">. Nevertheless, this parallel algorithm has been chosen for
MLD2P4, since it has been shown to produce good results in practice
[<A
HREF="node30.html#aaecc_07">5</A>,<A
HREF="node30.html#apnum_07">7</A>,<A
HREF="node30.html#TUMINARO_TONG">24</A>].
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The prolongator <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"> to the index
space <IMG
SRC="img25.png"
ALT="$P^k$"> is built starting from a tentative prolongator
<!-- MATH
$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$
-->
<IMG
WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img33.png"
ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$">, defined as
</FONT></FONT></FONT>
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
\left\{ \begin{array}{ll}
1 & \quad \mbox{if} \; i \in \Omega^k_j, \\
0 & \quad \mbox{otherwise},
\end{array} \right.
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
WIDTH="287" HEIGHT="51" BORDER="0"
SRC="img34.png"
ALT="\begin{displaymath}␍\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k = ␍\left\{...
...ega^k_j, ␍0 &amp; \quad \mbox{otherwise},␍\end{array} \right.
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(4)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
where <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$">.
The classical AS preconditioner <IMG
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img58.png"
ALT="$M^k_{AS}$"> is defined as
SRC="img28.png"
ALT="$\Omega^k_j$"> is the aggregate of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">
corresponding to the index <!-- MATH
$j \in \Omega^{k+1}$
-->
<IMG
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.png"
ALT="$j \in \Omega^{k+1}$">.
<IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"> is obtained by applying to <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img35.png"
ALT="$\bar{P}^k$"> a smoother
<!-- MATH
$S^k \in \mathbb{R}^{n_k \times n_k}$
-->
<IMG
WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img36.png"
ALT="$S^k \in \mathbb{R}^{n_k \times n_k}$">:
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},
P^k = S^k \bar{P}^k,
\end{displaymath}
-->
<IMG
WIDTH="219" HEIGHT="59" BORDER="0"
SRC="img59.png"
ALT="\begin{displaymath}␍ ( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},␍\end{displaymath}">
WIDTH="90" HEIGHT="30" BORDER="0"
SRC="img37.png"
ALT="\begin{displaymath}␍P^k = S^k \bar{P}^k,␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
in order to remove nonsmooth components from the range of the prolongator,
and hence to improve the convergence properties of the multi-level
method&nbsp;[<A
HREF="node30.html#BREZINA_VANEK">2</A>,<A
HREF="node30.html#Stuben_01">23</A>].
A simple choice for <IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img38.png"
ALT="$S^k$"> is the damped Jacobi smoother:
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
\end{displaymath}
-->
<IMG
WIDTH="175" HEIGHT="31" BORDER="0"
SRC="img39.png"
ALT="\begin{displaymath}␍S^k = I - \omega^k (D^k)^{-1} A^k_F , ␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
where <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$"> is supposed to be nonsingular. We observe that an approximate
inverse of <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$"> is usually considered instead of <IMG
WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img61.png"
ALT="$(A_i^k)^{-1}$">.
The setup of <IMG
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img58.png"
ALT="$M^k_{AS}$"> during the multilevel build phase
involves
WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img40.png"
ALT="$D^k$"> is the diagonal matrix with the same diagonal entries as <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$">,
<!-- MATH
$A^k_F = (\bar{a}_{ij}^k)$
-->
<IMG
WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$A^k_F = (\bar{a}_{ij}^k)$"> is the filtered matrix defined as
</FONT></FONT></FONT>
<UL>
<LI>the definition of the index subspaces <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img62.png"
ALT="$\Omega_i^k$"> and of the corresponding
operators <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img63.png"
ALT="$R_i^k$"> (and <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img64.png"
ALT="$P_i^k$">);
</LI>
<LI>the computation of the submatrices <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$">;
</LI>
<LI>the computation of their inverses (usually approximated
through some form of incomplete factorization).
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
The computation of <!-- MATH
$z^k=M^k_{AS}w^k$
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
\bar{a}_{ij}^k =
\left \{ \begin{array}{ll}
a_{ij}^k & \mbox{if } j \in \mathcal{N}_i^k(\theta), \\
0 & \mbox{otherwise},
\end{array} \right.
\; (j \ne i),
\qquad
\bar{a}_{ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
WIDTH="514" HEIGHT="74" BORDER="0"
SRC="img43.png"
ALT="\begin{displaymath}
\bar{a}_{ij}^k =␍ \left \{ \begin{array}{ll}␍ a_{ij}^k &amp; ...
...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),␍\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
and <IMG
WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img44.png"
ALT="$\omega^k$"> is an approximation of <IMG
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img45.png"
ALT="$4/(3\rho^k)$">, where
<IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img46.png"
ALT="$\rho^k$"> is the spectral radius of <!-- MATH
$(D^k)^{-1}A^k_F$
-->
<IMG
WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img65.png"
ALT="$z^k=M^k_{AS}w^k$">, with <!-- MATH
$w^k \in \mathbb{R}^{n_k}$
WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img47.png"
ALT="$(D^k)^{-1}A^k_F$"> [<A
HREF="node30.html#BREZINA_VANEK">2</A>].
In MLD2P4 this approximation is obtained by using <!-- MATH
$\| A^k_F \|_\infty$
-->
<IMG
WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img66.png"
ALT="$w^k \in \mathbb{R}^{n_k}$">, during the
multilevel application phase, requires
</FONT></FONT></FONT>
<UL>
<LI>the restriction of <IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img67.png"
ALT="$w^k$"> to the subspaces <!-- MATH
$\mathbb{R}^{n_{k,i}}$
-->
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img48.png"
ALT="$\Vert A^k_F \Vert _\infty$"> as an estimate
of <IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img46.png"
ALT="$\rho^k$">. Note that for systems coming from uniformly elliptic
problems, filtering the matrix <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"> has little or no effect, and
<IMG
WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img68.png"
ALT="$\mathbb{R}^{n_{k,i}}$">,
i.e. <!-- MATH
$w_i^k = R_i^{k} w^k$
-->
<IMG
WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img69.png"
ALT="$w_i^k = R_i^{k} w^k$">;
</LI>
<LI>the computation of the vectors <!-- MATH
$z_i^k=(A_i^k)^{-1} w_i^k$
-->
<IMG
WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img70.png"
ALT="$z_i^k=(A_i^k)^{-1} w_i^k$">;
</LI>
<LI>the prolongation and the sum of the previous vectors,
i.e. <!-- MATH
$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$
-->
<IMG
WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img71.png"
ALT="$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$">.
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
Variants of the classical AS method, which use modifications of the
restriction and prolongation operators, are also implemented in MLD2P4.
Among them, the Restricted AS (RAS) preconditioner usually
outperforms the classical AS preconditioner in terms of convergence
rate and of computation and communication time on parallel distributed-memory
computers, and is therefore the most widely used among the AS
preconditioners&nbsp;[<A
HREF="node29.html#CAI_SARKIS">6</A>].
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"> can be used instead of <IMG
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img49.png"
ALT="$A^k_F$">. The latter choice is the default in MLD2P4.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Direct solvers based on sparse LU factorizations, implemented in the
third-party libraries reported in Section&nbsp;<A HREF="node7.html#sec:third-party">3.2</A>, can be applied
as coarsest-level solvers by MLD2P4. Native inexact solvers based on
incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel,
and block Jacobi preconditioners are also available. Direct solvers usually
lead to more effective preconditioners in terms of algorithmic scalability;
however, this does not guarantee parallel efficiency.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT></FONT><HR>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
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<H1><A NAME="SECTION00070000000000000000"></A><A NAME="sec:started"></A>
<H2><A NAME="SECTION00063000000000000000"></A><A NAME="sec:smoothers"></A>
<BR>
Getting Started
</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
Smoothers and coarsest-level solvers
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">We describe the basics for building and applying MLD2P4 one-level and multi-level
(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [<A
HREF="node29.html#PSBLASGUIDE">13</A>].
The following steps are required:
</FONT></FONT></FONT>
<OL>
<LI><I>Declare the preconditioner data structure</I>. It is a derived data type,
<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>
or <code>z</code>, according to the basic data type of the sparse matrix
(<code>s</code> = real single precision; <code>d</code> = real double precision;
<code>c</code> = complex single precision; <code>z</code> = complex double precision).
This data structure is accessed by the user only through the MLD2P4 routines,
following an object-oriented approach.
</LI>
<LI><I>Allocate and initialize the preconditioner data structure, according to
a preconditioner type chosen by the user</I>. This is performed by the routine
<code>init</code>, which also sets defaults for each preconditioner
type selected by the user. The preconditioner types and the defaults associated
with them are given in Table&nbsp;<A HREF="#tab:precinit">1</A>, where the strings used by
<code>init</code> to identify the preconditioner types are also given.
Note that these strings are valid also if uppercase letters are substituted by
corresponding lowercase ones.
</LI>
<LI><I>Modify the selected preconditioner type, by properly setting
preconditioner parameters.</I> This is performed by the routine <code>set</code>.
This routine must be called only if the user wants to modify the default values
of the parameters associated with the selected preconditioner type, to obtain a variant
of that preconditioner. Examples of use of <code>set</code> are given in
Section&nbsp;<A HREF="node16.html#sec:examples">5.1</A>; a complete list of all the
preconditioner parameters and their allowed and default values is provided in
Section&nbsp;<A HREF="node17.html#sec:userinterface">6</A>, Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.
</LI>
<LI><I>Build the preconditioner for a given matrix</I>. If the selected preconditioner
is multi-level, then two steps must be performed, as specified next.
<DL COMPACT>
<DT>4.1</DT>
<DD><I>Build the aggregation hierarchy for a given matrix.</I> This is
performed by the routine <code>hierarchy_build</code>.
</DD>
<DT>4.2</DT>
<DD><I>Build the preconditioner for a given matrix.</I> This is performed
by the routine <code>smoothers_build</code>.
</DD>
</DL>
If the selected preconditioner is one-level, it is built in a single step,
performed by the routine <code>bld</code>.
</LI>
<LI><I>Apply the preconditioner at each iteration of a Krylov solver.</I>
This is performed by the routine <code>aply</code>. When using the PSBLAS Krylov solvers,
this step is completely transparent to the user, since <code>aply</code> is called
by the PSBLAS routine implementing the Krylov solver (<code>psb_krylov</code>).
</LI>
<LI><I>Free the preconditioner data structure</I>. This is performed by
the routine <code>free</code>. This step is complementary to step 1 and should
be performed when the preconditioner is no more used.
</LI>
</OL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">All the previous routines are available as methods of the preconditioner object.
A detailed description of them is given in Section&nbsp;<A HREF="node17.html#sec:userinterface">6</A>.
Examples showing the basic use of MLD2P4 are reported in Section&nbsp;<A HREF="node16.html#sec:examples">5.1</A>.
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,
a hybrid version of the forward and backward Gauss-Seidel methods, and the
additive Schwarz (AS) ones (see, e.g., [<A
HREF="node30.html#Saad_book">20</A>,<A
HREF="node30.html#dd2_96">21</A>]).
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The hybrid Gauss-Seidel
version is considered because the original Gauss-Seidel method is inherently sequential.
At each iteration of the hybrid version, each parallel process uses the most recent values
of its own local variables and the values of the non-local variables computed at the
previous iteration, obtained by exchanging data with other processes before
the beginning of the current iteration.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In the AS methods, the index space <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"> is divided into <IMG
WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
ALT="$m_k$">
subsets <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$"> of size <IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img52.png"
ALT="$n_{k,i}$">, possibly
overlapping. For each <IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img30.png"
ALT="$i$"> we consider the restriction
operator <!-- MATH
$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$
-->
<IMG
WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img53.png"
ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$">
that maps a vector <IMG
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img54.png"
ALT="$x^k$"> to the vector <IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img55.png"
ALT="$x_i^k$"> made of the components of <IMG
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img54.png"
ALT="$x^k$">
with indices in <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$">, and the prolongation operator
<!-- MATH
$P^k_i = (R_i^k)^T$
-->
<IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img56.png"
ALT="$P^k_i = (R_i^k)^T$">. These operators are then used to build
<!-- MATH
$A_i^k=R_i^kA^kP_i^k$
-->
<IMG
WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img57.png"
ALT="$A_i^k=R_i^kA^kP_i^k$">, which is the restriction of <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"> to the index
space <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$">.
The classical AS preconditioner <IMG
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img58.png"
ALT="$M^k_{AS}$"> is defined as
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="897"></A>
<TABLE>
<CAPTION><STRONG>Table 1:</STRONG>
Preconditioner types, corresponding strings and default choices.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT"><SMALL>TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><SMALL>STRING</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232><SMALL>DEFAULT PRECONDITIONER</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT">No preconditioner</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'NOPREC'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Considered only to use the PSBLAS
Krylov solvers with no preconditioner.</TD>
</TR>
<TR><TD ALIGN="LEFT">Diagonal</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'DIAG'</code> or <code>'JACOBI'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Diagonal preconditioner.
For any zero diagonal entry of the matrix to be preconditioned,
the corresponding entry of the preconditioner is set to&nbsp;1.</TD>
</TR>
<TR><TD ALIGN="LEFT">Block Jacobi</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'BJAC'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Block-Jacobi with ILU(0) on the local blocks.</TD>
</TR>
<TR><TD ALIGN="LEFT">Additive Schwarz</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'AS'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Restricted Additive Schwarz (RAS),
with overlap&nbsp;1 and ILU(0) on the local blocks.</TD>
</TR>
<TR><TD ALIGN="LEFT">Multilevel</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'ML'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>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&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>
for further details of the preconditioner.</TD>
</TR>
</TABLE>
</DIV></TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Note that the module <code>mld_prec_mod</code>, containing the definition of the
preconditioner data type and the interfaces to the routines of MLD2P4,
must be used in any program calling such routines.
The modules <code>psb_base_mod</code>, for the sparse matrix and communication descriptor
data types, and <code>psb_krylov_mod</code>, for interfacing with the
Krylov solvers, must be also used (see Section&nbsp;<A HREF="node16.html#sec:examples">5.1</A>).
<BR></FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Remark 1.</B> Coarsest-level solvers based on the LU factorization,
such as those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU_Dist,
usually lead to smaller numbers of preconditioned Krylov
iterations than inexact solvers, when the linear system comes from
a standard discretization of basic scalar elliptic PDE problems. However,
this does not necessarily correspond to the smallest execution time
on parallel computers. </FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
<!-- MATH
\begin{displaymath}
( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},
\end{displaymath}
-->
<IMG
WIDTH="219" HEIGHT="59" BORDER="0"
SRC="img59.png"
ALT="\begin{displaymath}␍ ( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
where <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$"> is supposed to be nonsingular. We observe that an approximate
inverse of <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$"> is usually considered instead of <IMG
WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img61.png"
ALT="$(A_i^k)^{-1}$">.
The setup of <IMG
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img58.png"
ALT="$M^k_{AS}$"> during the multilevel build phase
involves
</FONT></FONT></FONT>
<UL>
<LI><A NAME="tex2html265"
HREF="node16.html">Examples</A>
</UL>
<!--End of Table of Child-Links-->
<HR>
<LI>the definition of the index subspaces <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img62.png"
ALT="$\Omega_i^k$"> and of the corresponding
operators <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img63.png"
ALT="$R_i^k$"> (and <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img64.png"
ALT="$P_i^k$">);
</LI>
<LI>the computation of the submatrices <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$">;
</LI>
<LI>the computation of their inverses (usually approximated
through some form of incomplete factorization).
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
The computation of <!-- MATH
$z^k=M^k_{AS}w^k$
-->
<IMG
WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img65.png"
ALT="$z^k=M^k_{AS}w^k$">, with <!-- MATH
$w^k \in \mathbb{R}^{n_k}$
-->
<IMG
WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img66.png"
ALT="$w^k \in \mathbb{R}^{n_k}$">, during the
multilevel application phase, requires
</FONT></FONT></FONT>
<UL>
<LI>the restriction of <IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img67.png"
ALT="$w^k$"> to the subspaces <!-- MATH
$\mathbb{R}^{n_{k,i}}$
-->
<IMG
WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img68.png"
ALT="$\mathbb{R}^{n_{k,i}}$">,
i.e. <!-- MATH
$w_i^k = R_i^{k} w^k$
-->
<IMG
WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img69.png"
ALT="$w_i^k = R_i^{k} w^k$">;
</LI>
<LI>the computation of the vectors <!-- MATH
$z_i^k=(A_i^k)^{-1} w_i^k$
-->
<IMG
WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img70.png"
ALT="$z_i^k=(A_i^k)^{-1} w_i^k$">;
</LI>
<LI>the prolongation and the sum of the previous vectors,
i.e. <!-- MATH
$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$
-->
<IMG
WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img71.png"
ALT="$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$">.
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
Variants of the classical AS method, which use modifications of the
restriction and prolongation operators, are also implemented in MLD2P4.
Among them, the Restricted AS (RAS) preconditioner usually
outperforms the classical AS preconditioner in terms of convergence
rate and of computation and communication time on parallel distributed-memory
computers, and is therefore the most widely used among the AS
preconditioners&nbsp;[<A
HREF="node30.html#CAI_SARKIS">6</A>].
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Direct solvers based on sparse LU factorizations, implemented in the
third-party libraries reported in Section&nbsp;<A HREF="node8.html#sec:third-party">3.2</A>, can be applied
as coarsest-level solvers by MLD2P4. Native inexact solvers based on
incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel,
and block Jacobi preconditioners are also available. Direct solvers usually
lead to more effective preconditioners in terms of algorithmic scalability;
however, this does not guarantee parallel efficiency.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT></FONT><HR>
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<H2><A NAME="SECTION00071000000000000000"></A><A NAME="sec:examples"></A>
<H1><A NAME="SECTION00070000000000000000"></A><A NAME="sec:started"></A>
<BR>
Examples
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
Getting Started
</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The code reported in Figure&nbsp;<A HREF="#fig:ex1">2</A> shows how to set and apply the default
multi-level preconditioner available in the real double precision version
of MLD2P4 (see Table&nbsp;<A HREF="#tab:precinit">1</A>). This preconditioner is chosen
by simply specifying <code>'ML'</code> as the second argument of <code>P%init</code>
(a call to <code>P%set</code> is not needed) and is applied with the CG
solver provided by PSBLAS (the matrix of the system to be solved is
assumed to be positive definite). As previously observed, the modules
<code>psb_base_mod</code>, <code>mld_prec_mod</code> and <code>psb_krylov_mod</code>
must be used by the example program.
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">We describe the basics for building and applying MLD2P4 one-level and multi-level
(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [<A
HREF="node30.html#PSBLASGUIDE">13</A>].
The following steps are required:
</FONT></FONT></FONT>
<OL>
<LI><I>Declare the preconditioner data structure</I>. It is a derived data type,
<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>
or <code>z</code>, according to the basic data type of the sparse matrix
(<code>s</code> = real single precision; <code>d</code> = real double precision;
<code>c</code> = complex single precision; <code>z</code> = complex double precision).
This data structure is accessed by the user only through the MLD2P4 routines,
following an object-oriented approach.
</LI>
<LI><I>Allocate and initialize the preconditioner data structure, according to
a preconditioner type chosen by the user</I>. This is performed by the routine
<code>init</code>, which also sets defaults for each preconditioner
type selected by the user. The preconditioner types and the defaults associated
with them are given in Table&nbsp;<A HREF="#tab:precinit">1</A>, where the strings used by
<code>init</code> to identify the preconditioner types are also given.
Note that these strings are valid also if uppercase letters are substituted by
corresponding lowercase ones.
</LI>
<LI><I>Modify the selected preconditioner type, by properly setting
preconditioner parameters.</I> This is performed by the routine <code>set</code>.
This routine must be called only if the user wants to modify the default values
of the parameters associated with the selected preconditioner type, to obtain a variant
of that preconditioner. Examples of use of <code>set</code> are given in
Section&nbsp;<A HREF="node17.html#sec:examples">5.1</A>; a complete list of all the
preconditioner parameters and their allowed and default values is provided in
Section&nbsp;<A HREF="node18.html#sec:userinterface">6</A>, Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.
</LI>
<LI><I>Build the preconditioner for a given matrix</I>. If the selected preconditioner
is multi-level, then two steps must be performed, as specified next.
<DL COMPACT>
<DT>4.1</DT>
<DD><I>Build the aggregation hierarchy for a given matrix.</I> This is
performed by the routine <code>hierarchy_build</code>.
</DD>
<DT>4.2</DT>
<DD><I>Build the preconditioner for a given matrix.</I> This is performed
by the routine <code>smoothers_build</code>.
</DD>
</DL>
If the selected preconditioner is one-level, it is built in a single step,
performed by the routine <code>bld</code>.
</LI>
<LI><I>Apply the preconditioner at each iteration of a Krylov solver.</I>
This is performed by the routine <code>aply</code>. When using the PSBLAS Krylov solvers,
this step is completely transparent to the user, since <code>aply</code> is called
by the PSBLAS routine implementing the Krylov solver (<code>psb_krylov</code>).
</LI>
<LI><I>Free the preconditioner data structure</I>. This is performed by
the routine <code>free</code>. This step is complementary to step 1 and should
be performed when the preconditioner is no more used.
</LI>
</OL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The part of the code concerning the
reading and assembling of the sparse matrix and the right-hand side vector, performed
through the PSBLAS routines for sparse matrix and vector management, is not reported
here for brevity; the statements concerning the deallocation of the PSBLAS
data structure are neglected too.
The complete code can be found in the example program file <code>mld_dexample_ml.f90</code>,
in the directory <code>examples/fileread</code> of the MLD2P4 implementation (see
Section&nbsp;<A HREF="node10.html#sec:ex_and_test">3.5</A>). A sample test problem along with the relevant
input data is available in <code>examples/fileread/runs</code>.
For details on the use of the PSBLAS routines, see the PSBLAS User's
Guide&nbsp;[<A
HREF="node29.html#PSBLASGUIDE">13</A>].
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The setup and application of the default multi-level preconditioner
for the real single precision and the complex, single and double
precision, versions are obtained with straightforward modifications of the previous
example (see Section&nbsp;<A HREF="node17.html#sec:userinterface">6</A> for details). If these versions are installed,
the corresponding codes are available in <code>examples/fileread/</code>.
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">All the previous routines are available as methods of the preconditioner object.
A detailed description of them is given in Section&nbsp;<A HREF="node18.html#sec:userinterface">6</A>.
Examples showing the basic use of MLD2P4 are reported in Section&nbsp;<A HREF="node17.html#sec:examples">5.1</A>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex1"></A><A NAME="900"></A>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="904"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
setup and application of the default multi-level preconditioner (example 1).
<CAPTION><STRONG>Table 1:</STRONG>
Preconditioner types, corresponding strings and default choices.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
use psb_base_mod
use mld_prec_mod
use psb_krylov_mod
... ...
!
! sparse matrix
type(psb_dspmat_type) :: A
! sparse matrix descriptor
type(psb_desc_type) :: desc_A
! preconditioner
type(mld_dprec_type) :: P
! right-hand side and solution vectors
type(psb_d_vect_type) :: b, x
... ...
!
! initialize the parallel environment
call psb_init(ictxt)
call psb_info(ictxt,iam,np)
... ...
!
! read and assemble the spd matrix A and the right-hand side b
! using PSBLAS routines for sparse matrix / vector management
... ...
!
! initialize the default multi-level preconditioner, i.e. V-cycle
! with basic smoothed aggregation, 1 hybrid forward/backward
! GS sweep as pre/post-smoother and UMFPACK as coarsest-level
! solver
call P%init('ML',info)
!
! build the preconditioner
call P%hierarchy_build(A,desc_A,info)
call P%smoothers_build(A,desc_A,info)
!
! set the solver parameters and the initial guess
... ...
!
! solve Ax=b with preconditioned CG
call psb_krylov('CG',A,P,b,x,tol,desc_A,info)
... ...
!
! deallocate the preconditioner
call P%free(info)
!
! deallocate other data structures
... ...
!
! exit the parallel environment
call psb_exit(ictxt)
stop
</PRE>
</TD></TR>
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT"><SMALL>TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><SMALL>STRING</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232><SMALL>DEFAULT PRECONDITIONER</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT">No preconditioner</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'NOPREC'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Considered only to use the PSBLAS
Krylov solvers with no preconditioner.</TD>
</TR>
<TR><TD ALIGN="LEFT">Diagonal</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'DIAG'</code> or <code>'JACOBI'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Diagonal preconditioner.
For any zero diagonal entry of the matrix to be preconditioned,
the corresponding entry of the preconditioner is set to&nbsp;1.</TD>
</TR>
<TR><TD ALIGN="LEFT">Block Jacobi</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'BJAC'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Block-Jacobi with ILU(0) on the local blocks.</TD>
</TR>
<TR><TD ALIGN="LEFT">Additive Schwarz</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'AS'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>Restricted Additive Schwarz (RAS),
with overlap&nbsp;1 and ILU(0) on the local blocks.</TD>
</TR>
<TR><TD ALIGN="LEFT">Multilevel</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=51><code>'ML'</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=232>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&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>
for further details of the preconditioner.</TD>
</TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Different versions of the multi-level preconditioner can be obtained by changing
the default values of the preconditioner parameters. The code reported in
Figure&nbsp;<A HREF="#fig:ex2">3</A> shows how to set a V-cycle preconditioner
which applies 1 block-Jacobi sweep as pre- and post-smoother,
and solves the coarsest-level system with 8 block-Jacobi sweeps.
Note that the ILU(0) factorization (plus triangular solve) is used as
local solver for the block-Jacobi sweeps, since this is the default associated
with block-Jacobi and set by&nbsp;<code>P%init</code>.
Furthermore, specifying block-Jacobi as coarsest-level
solver implies that the coarsest-level matrix is distributed
among the processes.
Figure&nbsp;<A HREF="#fig:ex3">4</A> shows how to set a W-cycle preconditioner which
applies 2 hybrid Gauss-Seidel sweeps as pre- and post-smoother,
and solves the coarsest-level system with the multifrontal LU factorization
implemented in MUMPS. It is specified that the coarsest-level
matrix is distributed, since MUMPS can be used on both
replicated and distributed matrices, and by default
it is used on replicated ones. The code fragments shown in Figures&nbsp;<A HREF="#fig:ex2">3</A> and <A HREF="#fig:ex3">4</A> are
included in the example program file <code>mld_dexample_ml.f90</code> too.
</FONT></FONT></FONT>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Note that the module <code>mld_prec_mod</code>, containing the definition of the
preconditioner data type and the interfaces to the routines of MLD2P4,
must be used in any program calling such routines.
The modules <code>psb_base_mod</code>, for the sparse matrix and communication descriptor
data types, and <code>psb_krylov_mod</code>, for interfacing with the
Krylov solvers, must be also used (see Section&nbsp;<A HREF="node17.html#sec:examples">5.1</A>).
<BR></FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Finally, Figure&nbsp;<A HREF="#fig:ex4">5</A> shows the setup of a one-level
additive Schwarz preconditioner, i.e., RAS with overlap 2.
Note also that a Krylov method different from CG must be used to solve
the preconditioned system, since the preconditione in nonsymmetric.
The corresponding example program is available in the file
<code>mld_dexample_1lev.f90</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">For all the previous preconditioners, example programs where the sparse matrix and
the right-hand side are generated by discretizing a PDE with Dirichlet
boundary conditions are also available in the directory <code>examples/pdegen</code>.
</FONT></FONT></FONT>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Remark 1.</B> Coarsest-level solvers based on the LU factorization,
such as those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU_Dist,
usually lead to smaller numbers of preconditioned Krylov
iterations than inexact solvers, when the linear system comes from
a standard discretization of basic scalar elliptic PDE problems. However,
this does not necessarily correspond to the smallest execution time
on parallel computers. </FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex2"></A><A NAME="902"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
setup of a multi-level preconditioner</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
... ...
! build a V-cycle preconditioner with 1 block-Jacobi sweep (with
! ILU(0) on the blocks) as pre- and post-smoother, and 8 block-Jacobi
! sweeps (with ILU(0) on the blocks) as coarsest-level solver
call P%init('ML',info)
call_P%set('SMOOTHER_TYPE','BJAC',info)
call P%set('COARSE_SOLVE','BJAC',info)
call P%set('COARSE_SWEEPS',8,info)
call P%hierarchy_build(A,desc_A,info)
call P%smoothers_build(A,desc_A,info)
... ...
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV>
<P>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex3"></A><A NAME="904"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4:</STRONG>
setup of a multi-level preconditioner</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
... ...
! build a W-cycle preconditioner with 2 hybrid Gauss-Seidel sweeps
! as pre- and post-smoother, a distributed coarsest
! matrix, and MUMPS as coarsest-level solver
call P%init('ML',info)
call P%set('ML_CYCLE','WCYCLE',info)
call P%set('SMOOTHER_TYPE','FBGS',info)
call P%set('SMOOTHER_SWEEPS',2,info)
call P%set('COARSE_SOLVE','MUMPS',info)
call P%set('COARSE_MAT','DIST',info)
call P%hierarchy_build(A,desc_A,info)
call P%smoothers_build(A,desc_A,info)
... ...
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex4"></A><A NAME="906"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5:</STRONG>
setup of a one-level Schwarz preconditioner.</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
... ...
! set RAS with overlap 2 and ILU(0) on the local blocks
call P%init('AS',info)
call P%set('SUB_OVR',2,info)
call P%bld(A,desc_A,info)
... ...
! solve Ax=b with preconditioned BiCGSTAB
call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info)
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT></FONT><HR>
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<H1><A NAME="SECTION00080000000000000000"></A><A NAME="sec:userinterface"></A>
<H2><A NAME="SECTION00071000000000000000"></A><A NAME="sec:examples"></A>
<BR>
User Interface
</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
Examples
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The basic user interface of MLD2P4 consists of eight routines. The six
routines <code>init</code>, <code>set</code>,
<code>hierarchy_build</code>, <code>smoothers_build</code>,
<code>bld</code>, and <code>apply</code> encapsulate all the
functionalities for the setup and the application of any multi-level and one-level
preconditioner implemented in the package.
The routine <code>free</code> deallocates the preconditioner data structure, while
<code>descr</code> prints a description of the preconditioner setup by the user.
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The code reported in Figure&nbsp;<A HREF="#fig:ex1">2</A> shows how to set and apply the default
multi-level preconditioner available in the real double precision version
of MLD2P4 (see Table&nbsp;<A HREF="#tab:precinit">1</A>). This preconditioner is chosen
by simply specifying <code>'ML'</code> as the second argument of <code>P%init</code>
(a call to <code>P%set</code> is not needed) and is applied with the CG
solver provided by PSBLAS (the matrix of the system to be solved is
assumed to be positive definite). As previously observed, the modules
<code>psb_base_mod</code>, <code>mld_prec_mod</code> and <code>psb_krylov_mod</code>
must be used by the example program.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">All the routines are available as methods of the preconditioner object.
For each routine, the same user interface is overloaded with
respect to the real/ complex case and the single/double precision;
arguments with appropriate data types must be passed to the routine,
i.e.,
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The part of the code concerning the
reading and assembling of the sparse matrix and the right-hand side vector, performed
through the PSBLAS routines for sparse matrix and vector management, is not reported
here for brevity; the statements concerning the deallocation of the PSBLAS
data structure are neglected too.
The complete code can be found in the example program file <code>mld_dexample_ml.f90</code>,
in the directory <code>examples/fileread</code> of the MLD2P4 implementation (see
Section&nbsp;<A HREF="node11.html#sec:ex_and_test">3.5</A>). A sample test problem along with the relevant
input data is available in <code>examples/fileread/runs</code>.
For details on the use of the PSBLAS routines, see the PSBLAS User's
Guide&nbsp;[<A
HREF="node30.html#PSBLASGUIDE">13</A>].
</FONT></FONT></FONT>
<UL>
<LI>the sparse matrix data structure, containing the matrix to be
preconditioned, must be of type <code>psb_</code><I>x</I><code>spmat_type</code>
with <I>x</I> = <code>s</code> for real single precision, <I>x</I> = <code>d</code>
for real double precision, <I>x</I> = <code>c</code> for complex single precision,
<I>x</I> = <code>z</code> for complex double precision;
</LI>
<LI>the preconditioner data structure must be of type
<code>mld_</code><I>x</I><code>prec_type</code>, with <I>x</I> =
<code>s</code>, <code>d</code>, <code>c</code>, <code>z</code>, according to the sparse
matrix data structure;
</LI>
<LI>the arrays containing the vectors <IMG
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img72.png"
ALT="$v$"> and <IMG
WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img73.png"
ALT="$w$"> involved in
the preconditioner application <IMG
WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$w=B^{-1}v$"> must be of type
<code>psb_</code><I>x</I><code>vect_type</code> with <I>x</I> =
<code>s</code>, <code>d</code>, <code>c</code>, <code>z</code>, in a manner completely
analogous to the sparse matrix type;
</LI>
<LI>real parameters defining the preconditioner must be declared
according to the precision of the sparse matrix and preconditioner
data structures (see Section&nbsp;<A HREF="node19.html#sec:precset">6.2</A>).
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
A description of each routine is given in the remainder of this section.
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The setup and application of the default multi-level preconditioner
for the real single precision and the complex, single and double
precision, versions are obtained with straightforward modifications of the previous
example (see Section&nbsp;<A HREF="node18.html#sec:userinterface">6</A> for details). If these versions are installed,
the corresponding codes are available in <code>examples/fileread/</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
<DIV ALIGN="CENTER"><A NAME="fig:ex1"></A><A NAME="907"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
setup and application of the default multi-level preconditioner (example 1).
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
use psb_base_mod
use mld_prec_mod
use psb_krylov_mod
... ...
!
! sparse matrix
type(psb_dspmat_type) :: A
! sparse matrix descriptor
type(psb_desc_type) :: desc_A
! preconditioner
type(mld_dprec_type) :: P
! right-hand side and solution vectors
type(psb_d_vect_type) :: b, x
... ...
!
! initialize the parallel environment
call psb_init(ictxt)
call psb_info(ictxt,iam,np)
... ...
!
! read and assemble the spd matrix A and the right-hand side b
! using PSBLAS routines for sparse matrix / vector management
... ...
!
! initialize the default multi-level preconditioner, i.e. V-cycle
! with basic smoothed aggregation, 1 hybrid forward/backward
! GS sweep as pre/post-smoother and UMFPACK as coarsest-level
! solver
call P%init('ML',info)
!
! build the preconditioner
call P%hierarchy_build(A,desc_A,info)
call P%smoothers_build(A,desc_A,info)
<UL>
<LI><A NAME="tex2html288"
HREF="node18.html">Subroutine init</A>
<LI><A NAME="tex2html289"
HREF="node19.html">Subroutine set</A>
<LI><A NAME="tex2html290"
HREF="node20.html">Subroutine build</A>
<LI><A NAME="tex2html291"
HREF="node21.html">Subroutine hierarchy_build</A>
<LI><A NAME="tex2html292"
HREF="node22.html">Subroutine smoothers_build</A>
<LI><A NAME="tex2html293"
HREF="node23.html">Subroutine apply</A>
<LI><A NAME="tex2html294"
HREF="node24.html">Subroutine free</A>
<LI><A NAME="tex2html295"
HREF="node25.html">Subroutine descr</A>
</UL>
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<HR>
!
! set the solver parameters and the initial guess
... ...
!
! solve Ax=b with preconditioned CG
call psb_krylov('CG',A,P,b,x,tol,desc_A,info)
... ...
!
! deallocate the preconditioner
call P%free(info)
!
! deallocate other data structures
... ...
!
! exit the parallel environment
call psb_exit(ictxt)
stop
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Different versions of the multi-level preconditioner can be obtained by changing
the default values of the preconditioner parameters. The code reported in
Figure&nbsp;<A HREF="#fig:ex2">3</A> shows how to set a V-cycle preconditioner
which applies 1 block-Jacobi sweep as pre- and post-smoother,
and solves the coarsest-level system with 8 block-Jacobi sweeps.
Note that the ILU(0) factorization (plus triangular solve) is used as
local solver for the block-Jacobi sweeps, since this is the default associated
with block-Jacobi and set by&nbsp;<code>P%init</code>.
Furthermore, specifying block-Jacobi as coarsest-level
solver implies that the coarsest-level matrix is distributed
among the processes.
Figure&nbsp;<A HREF="#fig:ex3">4</A> shows how to set a W-cycle preconditioner which
applies 2 hybrid Gauss-Seidel sweeps as pre- and post-smoother,
and solves the coarsest-level system with the multifrontal LU factorization
implemented in MUMPS. It is specified that the coarsest-level
matrix is distributed, since MUMPS can be used on both
replicated and distributed matrices, and by default
it is used on replicated ones. The code fragments shown in Figures&nbsp;<A HREF="#fig:ex2">3</A> and <A HREF="#fig:ex3">4</A> are
included in the example program file <code>mld_dexample_ml.f90</code> too.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Finally, Figure&nbsp;<A HREF="#fig:ex4">5</A> shows the setup of a one-level
additive Schwarz preconditioner, i.e., RAS with overlap 2.
Note also that a Krylov method different from CG must be used to solve
the preconditioned system, since the preconditione in nonsymmetric.
The corresponding example program is available in the file
<code>mld_dexample_1lev.f90</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">For all the previous preconditioners, example programs where the sparse matrix and
the right-hand side are generated by discretizing a PDE with Dirichlet
boundary conditions are also available in the directory <code>examples/pdegen</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex2"></A><A NAME="909"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
setup of a multi-level preconditioner</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
... ...
! build a V-cycle preconditioner with 1 block-Jacobi sweep (with
! ILU(0) on the blocks) as pre- and post-smoother, and 8 block-Jacobi
! sweeps (with ILU(0) on the blocks) as coarsest-level solver
call P%init('ML',info)
call_P%set('SMOOTHER_TYPE','BJAC',info)
call P%set('COARSE_SOLVE','BJAC',info)
call P%set('COARSE_SWEEPS',8,info)
call P%hierarchy_build(A,desc_A,info)
call P%smoothers_build(A,desc_A,info)
... ...
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV>
<P>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex3"></A><A NAME="911"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4:</STRONG>
setup of a multi-level preconditioner</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
... ...
! build a W-cycle preconditioner with 2 hybrid Gauss-Seidel sweeps
! as pre- and post-smoother, a distributed coarsest
! matrix, and MUMPS as coarsest-level solver
call P%init('ML',info)
call P%set('ML_CYCLE','WCYCLE',info)
call P%set('SMOOTHER_TYPE','FBGS',info)
call P%set('SMOOTHER_SWEEPS',2,info)
call P%set('COARSE_SOLVE','MUMPS',info)
call P%set('COARSE_MAT','DIST',info)
call P%hierarchy_build(A,desc_A,info)
call P%smoothers_build(A,desc_A,info)
... ...
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex4"></A><A NAME="913"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5:</STRONG>
setup of a one-level Schwarz preconditioner.</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
... ...
! set RAS with overlap 2 and ILU(0) on the local blocks
call P%init('AS',info)
call P%set('SUB_OVR',2,info)
call P%bld(A,desc_A,info)
... ...
! solve Ax=b with preconditioned BiCGSTAB
call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info)
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
</DIV>
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<P>
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<H2><A NAME="SECTION00081000000000000000"></A><A NAME="sec:precinit"></A>
<H1><A NAME="SECTION00080000000000000000"></A><A NAME="sec:userinterface"></A>
<BR>
Subroutine init
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
User Interface
</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%init(ptype,info)</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
This routine allocates and initializes the preconditioner
<code>p</code>, according to the preconditioner type chosen by the user.
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The basic user interface of MLD2P4 consists of eight routines. The six
routines <code>init</code>, <code>set</code>,
<code>hierarchy_build</code>, <code>smoothers_build</code>,
<code>bld</code>, and <code>apply</code> encapsulate all the
functionalities for the setup and the application of any multi-level and one-level
preconditioner implemented in the package.
The routine <code>free</code> deallocates the preconditioner data structure, while
<code>descr</code> prints a description of the preconditioner setup by the user.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Arguments</B></FONT> </FONT></FONT></FONT>
<P></P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>ptype</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>character(len=*), intent(in)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The type of preconditioner. Its values are specified
in Table&nbsp;<A HREF="#tab:precinit">1</A>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Note that the strings are case insensitive.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>info</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(out)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node27.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
</TR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">For compatibility with the previous versions of MLD2P4, this routine can be also invoked
as follows:
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">All the routines are available as methods of the preconditioner object.
For each routine, the same user interface is overloaded with
respect to the real/ complex case and the single/double precision;
arguments with appropriate data types must be passed to the routine,
i.e.,
</FONT></FONT></FONT>
<UL>
<LI>the sparse matrix data structure, containing the matrix to be
preconditioned, must be of type <code>psb_</code><I>x</I><code>spmat_type</code>
with <I>x</I> = <code>s</code> for real single precision, <I>x</I> = <code>d</code>
for real double precision, <I>x</I> = <code>c</code> for complex single precision,
<I>x</I> = <code>z</code> for complex double precision;
</LI>
<LI>the preconditioner data structure must be of type
<code>mld_</code><I>x</I><code>prec_type</code>, with <I>x</I> =
<code>s</code>, <code>d</code>, <code>c</code>, <code>z</code>, according to the sparse
matrix data structure;
</LI>
<LI>the arrays containing the vectors <IMG
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img72.png"
ALT="$v$"> and <IMG
WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img73.png"
ALT="$w$"> involved in
the preconditioner application <IMG
WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$w=B^{-1}v$"> must be of type
<code>psb_</code><I>x</I><code>vect_type</code> with <I>x</I> =
<code>s</code>, <code>d</code>, <code>c</code>, <code>z</code>, in a manner completely
analogous to the sparse matrix type;
</LI>
<LI>real parameters defining the preconditioner must be declared
according to the precision of the sparse matrix and preconditioner
data structures (see Section&nbsp;<A HREF="node20.html#sec:precset">6.2</A>).
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
A description of each routine is given in the remainder of this section.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precinit(p,ptype,info)</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
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<LI><A NAME="tex2html300"
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<LI><A NAME="tex2html301"
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<LI><A NAME="tex2html303"
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<LI><A NAME="tex2html304"
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<LI><A NAME="tex2html306"
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<H2><A NAME="SECTION00082000000000000000"></A><A NAME="sec:precset"></A>
<H2><A NAME="SECTION00081000000000000000"></A><A NAME="sec:precinit"></A>
<BR>
Subroutine set
Subroutine init
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%set(what,val,info [,ilev, ilmax, pos])</code>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%init(ptype,info)</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
This routine sets the parameters defining the preconditioner <code>p</code>. More
precisely, the parameter identified by <code>what</code> is assigned the value
contained in <code>val</code>.
This routine allocates and initializes the preconditioner
<code>p</code>, according to the preconditioner type chosen by the user.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Arguments</B></FONT> </FONT></FONT></FONT>
@@ -71,28 +70,17 @@ contained in <code>val</code>.
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>what</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>character(len=*)</code>. </FONT></FONT></FONT></TD>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>ptype</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>character(len=*), intent(in)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The parameter to be set. It can be specified through its name;
the string is case-insensitive. See
Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>val </code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer</code> <I>or</I> <code>character(len=*)</code> <I>or</I>
<code>real(psb_spk_)</code> <I>or</I> <code>real(psb_dpk_)</code>,
<code>intent(in)</code>.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The type of preconditioner. Its values are specified
in Table&nbsp;<A HREF="#tab:precinit">1</A>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The value of the parameter to be set. The list of allowed
values and the corresponding data types is given in
Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.
When the value is of type <code>character(len=*)</code>,
it is also treated as case insensitive.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Note that the strings are case insensitive.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>info</code> </FONT></FONT></FONT></TD>
@@ -100,49 +88,7 @@ contained in <code>val</code>.
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node27.html#sec:errors">8</A>
for details.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>ilev</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, optional, intent(in)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> For the multi-level preconditioner, the level at which the
preconditioner parameter has to be set.
The levels are numbered in increasing
order starting from the finest one, i.e., level 1 is the finest level.
If <code>ilev</code> is not present, the parameter identified by <code>what</code>
is set at all the appropriate levels (see
Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>).</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>ilmax</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, optional, intent(in)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> For the multi-level preconditioner, when both
<code>ilev</code> and <code>ilmax</code> are present, the settings
are applied at all levels <code>ilev:ilmax</code>. When
<code>ilev</code> is present but <code>ilmax</code> is not, then
the default is <code>ilmax=ilev</code>.
The levels are numbered in increasing
order starting from the finest one, i.e., level 1 is the finest level. </FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>pos</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>charater(len=*), optional, intent(in)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Whether the other arguments apply only to the pre-smoother (<code>'PRE'</code>)
or to the post-smoother (<code>'POST'</code>). If <code>pos</code> is not present,
the other arguments are applied to both smoothers.
If the preconditioner is one-level or the parameter identified by <code>what</code>
does not concern the smoothers, <code>pos</code> is ignored.
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node28.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
</TR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
@@ -151,731 +97,9 @@ as follows:
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precset(p,what,val,info)</code>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precinit(p,ptype,info)</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
However, in this case the optional arguments <code>ilev</code>, <code>ilmax</code>, and <code>pos</code>
cannot be used.
<BR></FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">A variety of preconditioners can be obtained
by a suitable setting of the preconditioner parameters. These parameters
can be logically divided into four groups, i.e., parameters defining
</FONT></FONT></FONT>
<OL>
<LI>the type of multi-level cycle and how many cycles must be applied;
</LI>
<LI>the aggregation algorithm;
</LI>
<LI>the coarse-space correction at the coarsest level (for multi-level
preconditioners only);
</LI>
<LI>the smoother of the multi-level preconditioners, or the one-level
preconditioner.
<P>
</LI>
</OL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
A list of the parameters that can be set, along with their allowed and
default values, is given in Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.
For a description of the meaning of the parameters, please
refer also to Section&nbsp;<A HREF="node11.html#sec:background">4</A>.
<BR></FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Remark 2.</B> A smoother is usually obtained by combining two objects:
a smoother (<code>SMOOTHER_TYPE</code>) and a local solver (<code>SUB_SOLVE</code>),
as specified in Tables&nbsp;<A HREF="#tab:p_smoother">7</A>-<A HREF="#tab:p_smoother_1">8</A>.
For example, the block-Jacobi smoother using
ILU(0) on the blocks is obtained by combining the block-Jacobi smoother
object with the ILU(0) solver object. Similarly,
the hybrid Gauss-Seidel smoother (see Note in Table&nbsp;<A HREF="#tab:p_smoother">7</A>)
is obtained by combining the block-Jacobi smoother object with a single sweep
of the Gauss-Seidel solver object, while the point-Jacobi smoother is the
result of combining the block-Jacobi smoother object with a single sweep
of the pointwise-Jacobi solver object. However, for simplicity, shortcuts are
provided to set point-Jacobi, hybrid (forward) Gauss-Seidel, and
hybrid backward Gauss-Seidel, i.e., the previous smoothers can be defined
by setting only <code>SMOOTHER_TYPE</code> to appropriate values (see
Tables&nbsp;<A HREF="#tab:p_smoother">7</A>), i.e., without setting
<code>SUB_SOLVE</code> too.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The smoother and solver objects are arranged in a
hierarchical manner. When specifying a smoother object, its parameters,
including the local solver, are set to their default values, and when a solver
object is specified, its defaults are also set, overriding in both
cases any previous settings even if explicitly specified. Therefore if
the user sets a smoother, and wishes to use a solver
different from the default one, the call to set the solver must come
<I>after</I> the call to set the smoother.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Similar considerations apply to the point-Jacobi, Gauss-Seidel and block-Jacobi
coarsest-level solvers, and shortcuts are available
in this case too (see Table&nbsp;<A HREF="#tab:p_coarse">5</A>).
<BR></FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Remark 3.</B> In general, a coarsest-level solver cannot be used with
both the replicated and distributed coarsest-matrix layout;
therefore, setting the solver after the layout may change the layout.
Similarly, setting the layout after the solver may change the solver.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">More precisely, UMFPACK and SuperLU require the coarsest-level
matrix to be replicated, while SuperLU_Dist requires it to be distributed.
In these cases, setting the coarsest-level solver implies that
the layout is redefined according to the solver, ovverriding any
previous settings. MUMPS, point-Jacobi,
hybrid Gauss-Seidel and block-Jacobi can be applied to
replicated and distributed matrices, thus their choice
does not modify any previously specified layout.
It is worth noting that, when the matrix is replicated,
the point-Jacobi, hybrid Gauss-Seidel and block-Jacobi solvers
reduce to the corresponding local solver objects (see Remark&nbsp;2).
For the point-Jacobi and Gauss-Seidel solvers, these objects
correspond to a <I>single</I> point-Jacobi sweep and a <I>single</I>
Gauss-Seidel sweep, respectively, which are very poor solvers.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">On the other hand, the distributed layout can be used with any solver
but UMFPACK and SuperLU; therefore, if any of these two solvers has already
been selected, the coarsest-level solver is changed to block-Jacobi,
with the previously chosen solver applied to the local blocks.
Likewise, the replicated layout can be used with any solver but SuperLu_Dist;
therefore, if SuperLu_Dist has been previously set, the coarsest-level
solver is changed to the default sequential solver.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1330"></A>
<TABLE>
<CAPTION><STRONG>Table 2:</STRONG>
Parameters defining the multi-level cycle and the number of cycles to
be applied.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68><code>val</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=204><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><code>'ML_CYCLE'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68><TT>'VCYCLE'</TT>
<P>
<TT>'WCYCLE'</TT>
<P>
<TT>'KCYCLE'</TT>
<P>
<TT>'MULT'</TT>
<P>
<TT>'ADD'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68><TT>'VCYCLE'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=204>Multi-level cycle: V-cycle, W-cycle, K-cycle, hybrid Multiplicative Schwarz,
and Additive Schwarz.
<P>
Note that hybrid Multiplicative Schwarz is equivalent to V-cycle and
is included for compatibility with previous versions of MLD2P4.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><code>'OUTER_SWEEPS'</code></TD>
<TD ALIGN="LEFT"><TT>integer</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68>Any integer
<P>
number <IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img74.png"
ALT="$\ge 1$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68>1</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=204>Number of multi-level cycles.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1335"></A>
<TABLE>
<CAPTION><STRONG>Table 3:</STRONG>
Parameters defining the aggregation algorithm.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><code>val</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'MIN_COARSE_SIZE'</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any number
<P>
<IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img75.png"
ALT="$&gt; 0$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82><!-- MATH
$\lfloor 40 \sqrt[3]{n} \rfloor$
-->
<IMG
WIDTH="63" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img76.png"
ALT="$\lfloor 40 \sqrt[3]{n} \rfloor$">, where <IMG
WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img77.png"
ALT="$n$"> is the dimension
of the matrix at the finest level</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Coarse size threshold. The aggregation stops
if the global number of variables of the
computed coarsest matrix
is lower than or equal to this threshold
(see Note).</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'MIN_CR_RATIO'</code></TD>
<TD ALIGN="LEFT"><code>real</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any number
<P>
<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img78.png"
ALT="$&gt; 1$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82>1.5</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Minimum coarsening ratio. The aggregation stops
if the ratio between the matrix dimensions
at two consecutive levels is lower than or equal to this
threshold (see Note).</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'MAX_LEVS'</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any integer
<P>
number <IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img78.png"
ALT="$&gt; 1$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82>20</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Maximum number of levels. The aggregation stops
if the number of levels reaches this value (see Note).</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'PAR_AGGR'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'DEC'</TT>, <TT>'SYMDEC'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82><TT>'DEC'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Parallel aggregation algorithm.
<P>
Currently, only the
decoupled aggregation (<code>DEC</code>) is available; the
<code>SYMDEC</code> option applies decoupled
aggregation to the sparsity pattern
of <IMG
WIDTH="63" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img79.png"
ALT="$A+A^T$">.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'AGGR_TYPE'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><B><TT>'VMB'</TT></B></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82><B><TT>'VMB'</TT></B></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Type of aggregation algorithm: currently, the scalar aggregation
algorithm by Vanek, Mandel and Brezina is implemented
[<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>].</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'AGGR_PROL'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'SMOOTHED'</TT>, <TT>'UNSMOOTHED'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82><TT>'SMOOTHED'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Prolongator used by the aggregation algorithm: smoothed or unsmoothed
(i.e., tentative prolongator).</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5><B>Note.</B> The aggregation algorithm stops when
at least one of the following criteria is met:
the coarse size threshold, the</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5>maximum coarsening ratio, or the maximum number
of levels is reached. Therefore, the actual number of levels may be</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5>smaller than the specified maximum number
of levels.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1339"></A>
<TABLE>
<CAPTION><STRONG>Table 4:</STRONG>
Parameters defining the aggregation algorithm (continued).
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=108><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><code>val</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=187><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=108><code>'AGGR_ORD'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><TT>'NATURAL'</TT>
<P>
<TT>'DEGREE'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'NATURAL'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=187>Initial ordering of indices for the aggregation
algorithm: either natural ordering or sorted by
descending degrees of the nodes in the
matrix graph.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=108><code>'AGGR_THRESH'</code></TD>
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71>Any&nbsp;real
<P>
number&nbsp;<IMG
WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img80.png"
ALT="$\in [0, 1]$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>0.05</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=187>The threshold <IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img81.png"
ALT="$\theta$"> in the aggregation algorithm,
see (<A HREF="node13.html#eq:strongly_coup">3</A>) in Section&nbsp;<A HREF="node13.html#sec:aggregation">4.2</A>.
See also the note at the bottom of this table.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=108><code>'AGGR_FILTER'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><TT>'FILTER'</TT>
<P>
<TT>'NOFILTER'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'NOFILTER'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=187>Matrix used in computing the smoothed
prolongator: filtered or unfiltered (see&nbsp;(<A HREF="node13.html#eq:filtered">5</A>) in Section&nbsp;<A HREF="node13.html#sec:aggregation">4.2</A>).</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5><B>Note.</B> Different thresholds at different levels, such as
those used in [<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>, Section&nbsp;5.1], can be easily set by
invoking the rou-</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5>tine <TT>set</TT> with
the parameter <TT>ilev</TT>.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1344"></A>
<TABLE>
<CAPTION><STRONG>Table 5:</STRONG>
Parameters defining the coarse-space correction at the coarsest
level.</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><code>val</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=244><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_MAT'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><TT>'DIST'</TT>
<P>
<TT>'REPL'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><TT>'REPL'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=244>Coarsest matrix layout: distributed among the processes or
replicated on each of them.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_SOLVE'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><TT>'MUMPS'</TT>
<P>
<TT>'UMF'</TT>
<P>
<TT>'SLU'</TT>
<P>
<TT>'SLUDIST'</TT>
<P>
<TT>'JACOBI'</TT>
<P>
<TT>'GS'</TT>
<P>
<TT>'BJAC'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48>See&nbsp;Note.</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=244>Solver used at the coarsest level: sequential
LU from MUMPS, UMFPACK, or SuperLU
(plus triangular solve);
distributed LU from MUMPS or SuperLU_Dist
(plus triangular solve);
point-Jacobi, hybrid Gauss-Seidel or block-Jacobi.
<P>
Note that <TT>UMF</TT> and <TT>SLU</TT> require the coarsest
matrix to be replicated, <TT>SLUDIST</TT>, <TT>JACOBI</TT>,
<TT>GS</TT> and <TT>BJAC</TT> require it to be
distributed, <TT>MUMPS</TT> can be used with either
a replicated or a distributed matrix. When any of the previous
solvers is specified, the matrix layout is set to a default
value
which allows the use
value UMFPACK and SuperLU_Dist
are available only in double precision.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_SUBSOLVE'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><TT>'ILU'</TT>
<P>
<TT>'ILUT'</TT>
<P>
<TT>'MILU'</TT>
<P>
<TT>'MUMPS'</TT>
<P>
<TT>'SLU'</TT>
<P>
<TT>'UMF'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48>See&nbsp;Note.</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=244>Solver for the diagonal blocks of the coarse matrix,
in case the block Jacobi solver
is chosen as coarsest-level solver: ILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$">), ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img83.png"
ALT="$p,t$">),
MILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$">), LU from MUMPS, SuperLU or UMFPACK
(plus triangular solve).
Note that UMFPACK and SuperLU_Dist
are available only in double precision.</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5><B>Note.</B> Defaults for <TT>COARSE_SOLVE</TT> and
<TT>COARSE_SUBSOLVE</TT> are chosen in the following order:</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5>single precision version - <TT>MUMPS</TT> if installed,
then <TT>SLU</TT> if installed,
<TT>ILU</TT> otherwise;</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5>double precision version - <TT>UMF</TT> if installed,
then <TT>MUMPS</TT> if installed, then <TT>SLU</TT> if
installed, <TT>ILU</TT> otherwise.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1346"></A>
<TABLE>
<CAPTION><STRONG>Table 6:</STRONG>
Parameters defining the coarse-space correction at the coarsest
level (continued).</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57><code>val</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=43><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=213><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_SWEEPS'</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57>Any integer
<P>
number <IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img75.png"
ALT="$&gt; 0$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=43>10</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=213>Number of sweeps when <code>JACOBI</code>, <code>GS</code> or <code>BJAC</code>
is chosen as coarsest-level solver.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_FILLIN'</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57>Any integer
<P>
number <IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=43>0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=213>Fill-in level <IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$"> of the ILU factorizations.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_ILUTHRS'</code></TD>
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57>Any real
<P>
number <IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=43>0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=213>Drop tolerance <IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img85.png"
ALT="$t$"> in the ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img83.png"
ALT="$p,t$">) factorization.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1348"></A>
<TABLE>
<CAPTION><STRONG>Table 7:</STRONG>
Parameters defining the smoother or the details of the one-level preconditioner.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1">
<code>what</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <SMALL>DATA TYPE</SMALL> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54><FONT SIZE="-1"> <code>val</code> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <SMALL>DEFAULT</SMALL> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=184><FONT SIZE="-1">
<SMALL>COMMENTS</SMALL> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <code>'SMOOTHER_TYPE'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>character(len=*)</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54><FONT SIZE="-1"> <code>'JACOBI'</code> </FONT>
<P>
<FONT SIZE="-1"><code>'GS'</code> </FONT>
<P>
<FONT SIZE="-1"><code>'BGS'</code> </FONT>
<P>
<FONT SIZE="-1"><code>'BJAC'</code>
</FONT>
<P>
<FONT SIZE="-1"><code>'AS'</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <code>'FBGS'</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=184><FONT SIZE="-1"> Type of smoother used in the multi-level preconditioner:
point-Jacobi, hybrid (forward) Gauss-Seidel,
hybrid backward Gauss-Seidel, block-Jacobi, and
Additive Schwarz. </FONT>
<P>
<FONT SIZE="-1">It is ignored by one-level preconditioners. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <code>'SUB_SOLVE'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>character(len=*)</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54><FONT SIZE="-1"> <TT>'JACOBI'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'GS'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'BGS'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'ILU'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'ILUT'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'MILU'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'MUMPS'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'SLU'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'UMF'</TT>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <TT>GS</TT> and <TT>BGS</TT> for pre- and post-smoothers
of multi-level preconditioners, respectively </FONT>
<P>
<FONT SIZE="-1"><TT>ILU</TT> for block-Jacobi and Additive Schwarz
one-level preconditioners
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=184><FONT SIZE="-1"> The local solver to be used with the smoother or one-level
preconditioner (see Remark&nbsp;2, page&nbsp;24): point-Jacobi,
hybrid (forward) Gauss-Seidel, hybrid backward
Gauss-Seidel, ILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$">), ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img83.png"
ALT="$p,t$">), MILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$">),
LU from MUMPS, SuperLU or UMFPACK
(plus triangular solve). See Note for details on hybrid
Gauss-Seidel. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <code>'SMOOTHER_SWEEPS'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>integer</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54><FONT SIZE="-1"> Any integer </FONT>
<P>
<FONT SIZE="-1">number&nbsp;<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$">
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> 1
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=184><FONT SIZE="-1"> Number of sweeps of the smoother or one-level preconditioner.
In the multi-level case, no pre-smother or
post-smoother is used if this parameter is set to 0
together with <code>pos='PRE'</code> or <code>pos='POST</code>,
respectively. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <code>'SUB_OVR'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>integer</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54><FONT SIZE="-1"> Any integer </FONT>
<P>
<FONT SIZE="-1">number&nbsp;<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$">
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> 1
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=184><FONT SIZE="-1"> Number of overlap layers, for Additive Schwarz only. </FONT></TD>
</TR>
</TABLE></DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1350"></A>
<TABLE>
<CAPTION><STRONG>Table 8:</STRONG>
Parameters defining the smoother or the details of the one-level preconditioner
(continued).</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1">
<code>what</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <SMALL>DATA TYPE</SMALL> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><FONT SIZE="-1"> <code>val</code> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><FONT SIZE="-1"> <SMALL>DEFAULT</SMALL> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><FONT SIZE="-1">
<SMALL>COMMENTS</SMALL> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1"> <code>'SUB_RESTR'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>character(len=*)</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><FONT SIZE="-1"> <TT>'HALO'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'NONE'</TT>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><FONT SIZE="-1"> <TT>'HALO'</TT>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><FONT SIZE="-1"> Type of restriction operator, for Additive Schwarz only:
<TT>HALO</TT> for taking into account the overlap, <TT>NONE</TT>
for neglecting it. </FONT>
<P>
<FONT SIZE="-1">Note that <TT>HALO</TT> must be chosen for
the classical Addditive Schwarz smoother and its RAS variant.</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1"> <code>'SUB_PROL'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>character(len=*)</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><FONT SIZE="-1"> <TT>'SUM'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'NONE'</TT>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><FONT SIZE="-1"> <TT>'NONE'</TT>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><FONT SIZE="-1"> Type of prolongation operator, for Additive Schwarz only:
<TT>SUM</TT> for adding the contributions from the overlap, <TT>NONE</TT>
for neglecting them. </FONT>
<P>
<FONT SIZE="-1">Note that <TT>SUM</TT> must be chosen for the classical Additive
Schwarz smoother, and <TT>NONE</TT> for its RAS variant. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1"> <code>'SUB_FILLIN'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>integer</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><FONT SIZE="-1"> Any integer </FONT>
<P>
<FONT SIZE="-1">number&nbsp;<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$">
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><FONT SIZE="-1"> 0
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><FONT SIZE="-1"> Fill-in level <IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$"> of the incomplete LU factorizations. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1"> <code>'SUB_ILUTHRS'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>real(</code><I>kind_parameter</I><code>)</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><FONT SIZE="-1"> Any real number&nbsp;<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$">
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><FONT SIZE="-1"> 0
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><FONT SIZE="-1"> Drop tolerance <IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img85.png"
ALT="$t$"> in the ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img83.png"
ALT="$p,t$">) factorization. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1"> </FONT></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD></TD>
</TR>
</TABLE></DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
@@ -884,7 +108,7 @@ Parameters defining the smoother or the details of the one-level preconditioner
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<H2><A NAME="SECTION00083000000000000000"></A><A NAME="sec:precbld"></A>
<H2><A NAME="SECTION00082000000000000000"></A><A NAME="sec:precset"></A>
<BR>
Subroutine build
Subroutine set
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%build(a,desc_a,info)</code>
<BR></FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%set(what,val,info [,ilev, ilmax, pos])</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
This routine builds the one-level preconditioner <code>p</code> according to the requirements
made by the user through the routines <code>init</code> and <code>set</code>
(see Sections&nbsp;<A HREF="node21.html#sec:hier_bld">6.4</A> and&nbsp;<A HREF="node22.html#sec:smooth_bld">6.5</A> for multi-level preconditioners).
This routine sets the parameters defining the preconditioner <code>p</code>. More
precisely, the parameter identified by <code>what</code> is assigned the value
contained in <code>val</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Arguments</B></FONT> </FONT></FONT></FONT>
@@ -71,34 +71,78 @@ made by the user through the routines <code>init</code> and <code>set</code>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>a</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>type(psb_</code><I>x</I><code>spmat_type), intent(in)</code>. </FONT></FONT></FONT></TD>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>what</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>character(len=*)</code>. </FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The sparse matrix structure containing the local part of the
matrix to be preconditioned. Note that <I>x</I> must be chosen according
to the real/complex, single/double precision version of MLD2P4 under use.
See the PSBLAS User's Guide for details [<A
HREF="node29.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The parameter to be set. It can be specified through its name;
the string is case-insensitive. See
Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>desc_a</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>type(psb_desc_type), intent(in)</code>. </FONT></FONT></FONT></TD>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>val </code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer</code> <I>or</I> <code>character(len=*)</code> <I>or</I>
<code>real(psb_spk_)</code> <I>or</I> <code>real(psb_dpk_)</code>,
<code>intent(in)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The communication descriptor of <code>a</code>. See the PSBLAS User's Guide for
details [<A
HREF="node29.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The value of the parameter to be set. The list of allowed
values and the corresponding data types is given in
Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.
When the value is of type <code>character(len=*)</code>,
it is also treated as case insensitive.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>info</code> </FONT></FONT></FONT></TD>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>info</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(out)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node27.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node28.html#sec:errors">8</A>
for details.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>ilev</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, optional, intent(in)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> For the multi-level preconditioner, the level at which the
preconditioner parameter has to be set.
The levels are numbered in increasing
order starting from the finest one, i.e., level 1 is the finest level.
If <code>ilev</code> is not present, the parameter identified by <code>what</code>
is set at all the appropriate levels (see
Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>).</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>ilmax</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, optional, intent(in)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> For the multi-level preconditioner, when both
<code>ilev</code> and <code>ilmax</code> are present, the settings
are applied at all levels <code>ilev:ilmax</code>. When
<code>ilev</code> is present but <code>ilmax</code> is not, then
the default is <code>ilmax=ilev</code>.
The levels are numbered in increasing
order starting from the finest one, i.e., level 1 is the finest level. </FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>pos</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>charater(len=*), optional, intent(in)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Whether the other arguments apply only to the pre-smoother (<code>'PRE'</code>)
or to the post-smoother (<code>'POST'</code>). If <code>pos</code> is not present,
the other arguments are applied to both smoothers.
If the preconditioner is one-level or the parameter identified by <code>what</code>
does not concern the smoothers, <code>pos</code> is ignored.
</FONT></FONT></FONT></TD>
</TR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
@@ -107,12 +151,730 @@ as follows:
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precbld(p,what,val,info)</code>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precset(p,what,val,info)</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
In this case, the routine can be used to build multi-level preconditioners too.
However, in this case the optional arguments <code>ilev</code>, <code>ilmax</code>, and <code>pos</code>
cannot be used.
<BR></FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">A variety of preconditioners can be obtained
by a suitable setting of the preconditioner parameters. These parameters
can be logically divided into four groups, i.e., parameters defining
</FONT></FONT></FONT>
<OL>
<LI>the type of multi-level cycle and how many cycles must be applied;
</LI>
<LI>the aggregation algorithm;
</LI>
<LI>the coarse-space correction at the coarsest level (for multi-level
preconditioners only);
</LI>
<LI>the smoother of the multi-level preconditioners, or the one-level
preconditioner.
<P>
</LI>
</OL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
A list of the parameters that can be set, along with their allowed and
default values, is given in Tables&nbsp;<A HREF="#tab:p_cycle">2</A>-<A HREF="#tab:p_smoother_1">8</A>.
For a description of the meaning of the parameters, please
refer also to Section&nbsp;<A HREF="node12.html#sec:background">4</A>.
<BR></FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Remark 2.</B> A smoother is usually obtained by combining two objects:
a smoother (<code>SMOOTHER_TYPE</code>) and a local solver (<code>SUB_SOLVE</code>),
as specified in Tables&nbsp;<A HREF="#tab:p_smoother">7</A>-<A HREF="#tab:p_smoother_1">8</A>.
For example, the block-Jacobi smoother using
ILU(0) on the blocks is obtained by combining the block-Jacobi smoother
object with the ILU(0) solver object. Similarly,
the hybrid Gauss-Seidel smoother (see Note in Table&nbsp;<A HREF="#tab:p_smoother">7</A>)
is obtained by combining the block-Jacobi smoother object with a single sweep
of the Gauss-Seidel solver object, while the point-Jacobi smoother is the
result of combining the block-Jacobi smoother object with a single sweep
of the pointwise-Jacobi solver object. However, for simplicity, shortcuts are
provided to set point-Jacobi, hybrid (forward) Gauss-Seidel, and
hybrid backward Gauss-Seidel, i.e., the previous smoothers can be defined
by setting only <code>SMOOTHER_TYPE</code> to appropriate values (see
Tables&nbsp;<A HREF="#tab:p_smoother">7</A>), i.e., without setting
<code>SUB_SOLVE</code> too.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The smoother and solver objects are arranged in a
hierarchical manner. When specifying a smoother object, its parameters,
including the local solver, are set to their default values, and when a solver
object is specified, its defaults are also set, overriding in both
cases any previous settings even if explicitly specified. Therefore if
the user sets a smoother, and wishes to use a solver
different from the default one, the call to set the solver must come
<I>after</I> the call to set the smoother.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Similar considerations apply to the point-Jacobi, Gauss-Seidel and block-Jacobi
coarsest-level solvers, and shortcuts are available
in this case too (see Table&nbsp;<A HREF="#tab:p_coarse">5</A>).
<BR></FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Remark 3.</B> In general, a coarsest-level solver cannot be used with
both the replicated and distributed coarsest-matrix layout;
therefore, setting the solver after the layout may change the layout.
Similarly, setting the layout after the solver may change the solver.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">More precisely, UMFPACK and SuperLU require the coarsest-level
matrix to be replicated, while SuperLU_Dist requires it to be distributed.
In these cases, setting the coarsest-level solver implies that
the layout is redefined according to the solver, ovverriding any
previous settings. MUMPS, point-Jacobi,
hybrid Gauss-Seidel and block-Jacobi can be applied to
replicated and distributed matrices, thus their choice
does not modify any previously specified layout.
It is worth noting that, when the matrix is replicated,
the point-Jacobi, hybrid Gauss-Seidel and block-Jacobi solvers
reduce to the corresponding local solver objects (see Remark&nbsp;2).
For the point-Jacobi and Gauss-Seidel solvers, these objects
correspond to a <I>single</I> point-Jacobi sweep and a <I>single</I>
Gauss-Seidel sweep, respectively, which are very poor solvers.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">On the other hand, the distributed layout can be used with any solver
but UMFPACK and SuperLU; therefore, if any of these two solvers has already
been selected, the coarsest-level solver is changed to block-Jacobi,
with the previously chosen solver applied to the local blocks.
Likewise, the replicated layout can be used with any solver but SuperLu_Dist;
therefore, if SuperLu_Dist has been previously set, the coarsest-level
solver is changed to the default sequential solver.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1337"></A>
<TABLE>
<CAPTION><STRONG>Table 2:</STRONG>
Parameters defining the multi-level cycle and the number of cycles to
be applied.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68><code>val</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=204><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><code>'ML_CYCLE'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68><TT>'VCYCLE'</TT>
<P>
<TT>'WCYCLE'</TT>
<P>
<TT>'KCYCLE'</TT>
<P>
<TT>'MULT'</TT>
<P>
<TT>'ADD'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68><TT>'VCYCLE'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=204>Multi-level cycle: V-cycle, W-cycle, K-cycle, hybrid Multiplicative Schwarz,
and Additive Schwarz.
<P>
Note that hybrid Multiplicative Schwarz is equivalent to V-cycle and
is included for compatibility with previous versions of MLD2P4.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><code>'OUTER_SWEEPS'</code></TD>
<TD ALIGN="LEFT"><TT>integer</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68>Any integer
<P>
number <IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img74.png"
ALT="$\ge 1$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=68>1</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=204>Number of multi-level cycles.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1342"></A>
<TABLE>
<CAPTION><STRONG>Table 3:</STRONG>
Parameters defining the aggregation algorithm.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><code>val</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'MIN_COARSE_SIZE'</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any number
<P>
<IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img75.png"
ALT="$&gt; 0$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82><!-- MATH
$\lfloor 40 \sqrt[3]{n} \rfloor$
-->
<IMG
WIDTH="63" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img76.png"
ALT="$\lfloor 40 \sqrt[3]{n} \rfloor$">, where <IMG
WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img77.png"
ALT="$n$"> is the dimension
of the matrix at the finest level</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Coarse size threshold. The aggregation stops
if the global number of variables of the
computed coarsest matrix
is lower than or equal to this threshold
(see Note).</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'MIN_CR_RATIO'</code></TD>
<TD ALIGN="LEFT"><code>real</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any number
<P>
<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img78.png"
ALT="$&gt; 1$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82>1.5</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Minimum coarsening ratio. The aggregation stops
if the ratio between the matrix dimensions
at two consecutive levels is lower than or equal to this
threshold (see Note).</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'MAX_LEVS'</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any integer
<P>
number <IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img78.png"
ALT="$&gt; 1$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82>20</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Maximum number of levels. The aggregation stops
if the number of levels reaches this value (see Note).</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'PAR_AGGR'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'DEC'</TT>, <TT>'SYMDEC'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82><TT>'DEC'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Parallel aggregation algorithm.
<P>
Currently, only the
decoupled aggregation (<code>DEC</code>) is available; the
<code>SYMDEC</code> option applies decoupled
aggregation to the sparsity pattern
of <IMG
WIDTH="63" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img79.png"
ALT="$A+A^T$">.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'AGGR_TYPE'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><B><TT>'VMB'</TT></B></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82><B><TT>'VMB'</TT></B></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Type of aggregation algorithm: currently, the scalar aggregation
algorithm by Vanek, Mandel and Brezina is implemented
[<A
HREF="node30.html#VANEK_MANDEL_BREZINA">25</A>].</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'AGGR_PROL'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'SMOOTHED'</TT>, <TT>'UNSMOOTHED'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=82><TT>'SMOOTHED'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=196>Prolongator used by the aggregation algorithm: smoothed or unsmoothed
(i.e., tentative prolongator).</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5><B>Note.</B> The aggregation algorithm stops when
at least one of the following criteria is met:
the coarse size threshold, the</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5>maximum coarsening ratio, or the maximum number
of levels is reached. Therefore, the actual number of levels may be</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5>smaller than the specified maximum number
of levels.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1346"></A>
<TABLE>
<CAPTION><STRONG>Table 4:</STRONG>
Parameters defining the aggregation algorithm (continued).
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=108><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><code>val</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=187><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=108><code>'AGGR_ORD'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><TT>'NATURAL'</TT>
<P>
<TT>'DEGREE'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'NATURAL'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=187>Initial ordering of indices for the aggregation
algorithm: either natural ordering or sorted by
descending degrees of the nodes in the
matrix graph.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=108><code>'AGGR_THRESH'</code></TD>
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71>Any&nbsp;real
<P>
number&nbsp;<IMG
WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img80.png"
ALT="$\in [0, 1]$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>0.05</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=187>The threshold <IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img81.png"
ALT="$\theta$"> in the aggregation algorithm,
see (<A HREF="node14.html#eq:strongly_coup">3</A>) in Section&nbsp;<A HREF="node14.html#sec:aggregation">4.2</A>.
See also the note at the bottom of this table.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=108><code>'AGGR_FILTER'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><TT>'FILTER'</TT>
<P>
<TT>'NOFILTER'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'NOFILTER'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=187>Matrix used in computing the smoothed
prolongator: filtered or unfiltered (see&nbsp;(<A HREF="node14.html#eq:filtered">5</A>) in Section&nbsp;<A HREF="node14.html#sec:aggregation">4.2</A>).</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5><B>Note.</B> Different thresholds at different levels, such as
those used in [<A
HREF="node30.html#VANEK_MANDEL_BREZINA">25</A>, Section&nbsp;5.1], can be easily set by
invoking the rou-</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5>tine <TT>set</TT> with
the parameter <TT>ilev</TT>.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1351"></A>
<TABLE>
<CAPTION><STRONG>Table 5:</STRONG>
Parameters defining the coarse-space correction at the coarsest
level.</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><code>val</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=244><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_MAT'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><TT>'DIST'</TT>
<P>
<TT>'REPL'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><TT>'REPL'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=244>Coarsest matrix layout: distributed among the processes or
replicated on each of them.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_SOLVE'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><TT>'MUMPS'</TT>
<P>
<TT>'UMF'</TT>
<P>
<TT>'SLU'</TT>
<P>
<TT>'SLUDIST'</TT>
<P>
<TT>'JACOBI'</TT>
<P>
<TT>'GS'</TT>
<P>
<TT>'BJAC'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48>See&nbsp;Note.</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=244>Solver used at the coarsest level: sequential
LU from MUMPS, UMFPACK, or SuperLU
(plus triangular solve);
distributed LU from MUMPS or SuperLU_Dist
(plus triangular solve);
point-Jacobi, hybrid Gauss-Seidel or block-Jacobi.
<P>
Note that <TT>UMF</TT> and <TT>SLU</TT> require the coarsest
matrix to be replicated, <TT>SLUDIST</TT>, <TT>JACOBI</TT>,
<TT>GS</TT> and <TT>BJAC</TT> require it to be
distributed, <TT>MUMPS</TT> can be used with either
a replicated or a distributed matrix. When any of the previous
solvers is specified, the matrix layout is set to a default
value
which allows the use
value UMFPACK and SuperLU_Dist
are available only in double precision.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_SUBSOLVE'</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48><TT>'ILU'</TT>
<P>
<TT>'ILUT'</TT>
<P>
<TT>'MILU'</TT>
<P>
<TT>'MUMPS'</TT>
<P>
<TT>'SLU'</TT>
<P>
<TT>'UMF'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=48>See&nbsp;Note.</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=244>Solver for the diagonal blocks of the coarse matrix,
in case the block Jacobi solver
is chosen as coarsest-level solver: ILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$">), ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img83.png"
ALT="$p,t$">),
MILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$">), LU from MUMPS, SuperLU or UMFPACK
(plus triangular solve).
Note that UMFPACK and SuperLU_Dist
are available only in double precision.</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5><B>Note.</B> Defaults for <TT>COARSE_SOLVE</TT> and
<TT>COARSE_SUBSOLVE</TT> are chosen in the following order:</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5>single precision version - <TT>MUMPS</TT> if installed,
then <TT>SLU</TT> if installed,
<TT>ILU</TT> otherwise;</TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=5>double precision version - <TT>UMF</TT> if installed,
then <TT>MUMPS</TT> if installed, then <TT>SLU</TT> if
installed, <TT>ILU</TT> otherwise.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1353"></A>
<TABLE>
<CAPTION><STRONG>Table 6:</STRONG>
Parameters defining the coarse-space correction at the coarsest
level (continued).</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57><code>val</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=43><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=213><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_SWEEPS'</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57>Any integer
<P>
number <IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img75.png"
ALT="$&gt; 0$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=43>10</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=213>Number of sweeps when <code>JACOBI</code>, <code>GS</code> or <code>BJAC</code>
is chosen as coarsest-level solver.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_FILLIN'</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57>Any integer
<P>
number <IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=43>0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=213>Fill-in level <IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$"> of the ILU factorizations.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=111><code>'COARSE_ILUTHRS'</code></TD>
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57>Any real
<P>
number <IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=43>0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=213>Drop tolerance <IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img85.png"
ALT="$t$"> in the ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img83.png"
ALT="$p,t$">) factorization.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1355"></A>
<TABLE>
<CAPTION><STRONG>Table 7:</STRONG>
Parameters defining the smoother or the details of the one-level preconditioner.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1">
<code>what</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <SMALL>DATA TYPE</SMALL> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54><FONT SIZE="-1"> <code>val</code> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <SMALL>DEFAULT</SMALL> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=184><FONT SIZE="-1">
<SMALL>COMMENTS</SMALL> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <code>'SMOOTHER_TYPE'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>character(len=*)</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54><FONT SIZE="-1"> <code>'JACOBI'</code> </FONT>
<P>
<FONT SIZE="-1"><code>'GS'</code> </FONT>
<P>
<FONT SIZE="-1"><code>'BGS'</code> </FONT>
<P>
<FONT SIZE="-1"><code>'BJAC'</code>
</FONT>
<P>
<FONT SIZE="-1"><code>'AS'</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <code>'FBGS'</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=184><FONT SIZE="-1"> Type of smoother used in the multi-level preconditioner:
point-Jacobi, hybrid (forward) Gauss-Seidel,
hybrid backward Gauss-Seidel, block-Jacobi, and
Additive Schwarz. </FONT>
<P>
<FONT SIZE="-1">It is ignored by one-level preconditioners. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <code>'SUB_SOLVE'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>character(len=*)</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54><FONT SIZE="-1"> <TT>'JACOBI'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'GS'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'BGS'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'ILU'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'ILUT'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'MILU'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'MUMPS'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'SLU'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'UMF'</TT>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <TT>GS</TT> and <TT>BGS</TT> for pre- and post-smoothers
of multi-level preconditioners, respectively </FONT>
<P>
<FONT SIZE="-1"><TT>ILU</TT> for block-Jacobi and Additive Schwarz
one-level preconditioners
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=184><FONT SIZE="-1"> The local solver to be used with the smoother or one-level
preconditioner (see Remark&nbsp;2, page&nbsp;24): point-Jacobi,
hybrid (forward) Gauss-Seidel, hybrid backward
Gauss-Seidel, ILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$">), ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img83.png"
ALT="$p,t$">), MILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$">),
LU from MUMPS, SuperLU or UMFPACK
(plus triangular solve). See Note for details on hybrid
Gauss-Seidel. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <code>'SMOOTHER_SWEEPS'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>integer</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54><FONT SIZE="-1"> Any integer </FONT>
<P>
<FONT SIZE="-1">number&nbsp;<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$">
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> 1
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=184><FONT SIZE="-1"> Number of sweeps of the smoother or one-level preconditioner.
In the multi-level case, no pre-smother or
post-smoother is used if this parameter is set to 0
together with <code>pos='PRE'</code> or <code>pos='POST</code>,
respectively. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> <code>'SUB_OVR'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>integer</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54><FONT SIZE="-1"> Any integer </FONT>
<P>
<FONT SIZE="-1">number&nbsp;<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$">
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=102><FONT SIZE="-1"> 1
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=184><FONT SIZE="-1"> Number of overlap layers, for Additive Schwarz only. </FONT></TD>
</TR>
</TABLE></DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1357"></A>
<TABLE>
<CAPTION><STRONG>Table 8:</STRONG>
Parameters defining the smoother or the details of the one-level preconditioner
(continued).</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1">
<code>what</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <SMALL>DATA TYPE</SMALL> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><FONT SIZE="-1"> <code>val</code> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><FONT SIZE="-1"> <SMALL>DEFAULT</SMALL> </FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><FONT SIZE="-1">
<SMALL>COMMENTS</SMALL> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1"> <code>'SUB_RESTR'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>character(len=*)</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><FONT SIZE="-1"> <TT>'HALO'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'NONE'</TT>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><FONT SIZE="-1"> <TT>'HALO'</TT>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><FONT SIZE="-1"> Type of restriction operator, for Additive Schwarz only:
<TT>HALO</TT> for taking into account the overlap, <TT>NONE</TT>
for neglecting it. </FONT>
<P>
<FONT SIZE="-1">Note that <TT>HALO</TT> must be chosen for
the classical Addditive Schwarz smoother and its RAS variant.</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1"> <code>'SUB_PROL'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>character(len=*)</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><FONT SIZE="-1"> <TT>'SUM'</TT> </FONT>
<P>
<FONT SIZE="-1"><TT>'NONE'</TT>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><FONT SIZE="-1"> <TT>'NONE'</TT>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><FONT SIZE="-1"> Type of prolongation operator, for Additive Schwarz only:
<TT>SUM</TT> for adding the contributions from the overlap, <TT>NONE</TT>
for neglecting them. </FONT>
<P>
<FONT SIZE="-1">Note that <TT>SUM</TT> must be chosen for the classical Additive
Schwarz smoother, and <TT>NONE</TT> for its RAS variant. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1"> <code>'SUB_FILLIN'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>integer</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><FONT SIZE="-1"> Any integer </FONT>
<P>
<FONT SIZE="-1">number&nbsp;<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$">
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><FONT SIZE="-1"> 0
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><FONT SIZE="-1"> Fill-in level <IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img82.png"
ALT="$p$"> of the incomplete LU factorizations. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1"> <code>'SUB_ILUTHRS'</code> </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>real(</code><I>kind_parameter</I><code>)</code>
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71><FONT SIZE="-1"> Any real number&nbsp;<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img84.png"
ALT="$\ge 0$">
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><FONT SIZE="-1"> 0
</FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><FONT SIZE="-1"> Drop tolerance <IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img85.png"
ALT="$t$"> in the ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img83.png"
ALT="$p,t$">) factorization. </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=85><FONT SIZE="-1"> </FONT></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD></TD>
</TR>
</TABLE></DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
@@ -122,7 +884,7 @@ In this case, the routine can be used to build multi-level preconditioners too.
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<H2><A NAME="SECTION00084000000000000000"></A><A NAME="sec:hier_bld"></A>
<H2><A NAME="SECTION00083000000000000000"></A><A NAME="sec:precbld"></A>
<BR>
Subroutine hierarchy_build
Subroutine build
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%hierarchy_build(a,desc_a,info)</code>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%build(a,desc_a,info)</code>
<BR></FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
This routine builds the hierarchy of matrices and restriction/prolongation
operators for the multi-level preconditioner <code>p</code>, according to the requirements
made by the user through the routines <code>init</code> and <code>set</code>.
This routine builds the one-level preconditioner <code>p</code> according to the requirements
made by the user through the routines <code>init</code> and <code>set</code>
(see Sections&nbsp;<A HREF="node22.html#sec:hier_bld">6.4</A> and&nbsp;<A HREF="node23.html#sec:smooth_bld">6.5</A> for multi-level preconditioners).
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Arguments</B></FONT> </FONT></FONT></FONT>
@@ -72,17 +72,16 @@ made by the user through the routines <code>init</code> and <code>set</code>.
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>a</code> </FONT></FONT></FONT></TD>
<code>a</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>type(psb_</code><I>x</I><code>spmat_type), intent(in)</code>. </FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The sparse matrix structure containing the local part of the
matrix to be preconditioned. Note that <I>x</I> must be chosen according
to the real/complex,
single/double precision version of MLD2P4 under use.
to the real/complex, single/double precision version of MLD2P4 under use.
See the PSBLAS User's Guide for details [<A
HREF="node29.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
HREF="node30.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>desc_a</code> </FONT></FONT></FONT></TD>
@@ -92,17 +91,29 @@ single/double precision version of MLD2P4 under use.
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The communication descriptor of <code>a</code>. See the PSBLAS User's Guide for
details [<A
HREF="node29.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
HREF="node30.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>info</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(out)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node27.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node28.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
</TR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">For compatibility with the previous versions of MLD2P4, this routine can be also invoked
as follows:
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precbld(p,what,val,info)</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
In this case, the routine can be used to build multi-level preconditioners too.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
@@ -111,7 +122,7 @@ single/double precision version of MLD2P4 under use.
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@@ -121,11 +132,11 @@ single/double precision version of MLD2P4 under use.
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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION00085000000000000000"></A><A NAME="sec:smooth_bld"></A>
<H2><A NAME="SECTION00084000000000000000"></A><A NAME="sec:hier_bld"></A>
<BR>
Subroutine smoothers_build
Subroutine hierarchy_build
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%smoothers_build(a,desc_a,p,info)</code>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%hierarchy_build(a,desc_a,info)</code>
<BR></FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
This routine builds the smoothers and the coarsest-level solvers for the
multi-level preconditioner <code>p</code>, according to the requirements made by
the user through the routines <code>init</code> and <code>set</code>, and based on the aggregation
hierarchy produced by a previous call to <code>hierarchy_build</code>
(see Section&nbsp;<A HREF="node21.html#sec:hier_bld">6.4</A>).
This routine builds the hierarchy of matrices and restriction/prolongation
operators for the multi-level preconditioner <code>p</code>, according to the requirements
made by the user through the routines <code>init</code> and <code>set</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Arguments</B></FONT> </FONT></FONT></FONT>
@@ -81,9 +79,10 @@ hierarchy produced by a previous call to <code>hierarchy_build</code>
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The sparse matrix structure containing the local part of the
matrix to be preconditioned. Note that <I>x</I> must be chosen according
to the real/complex, single/double precision version of MLD2P4 under use.
to the real/complex,
single/double precision version of MLD2P4 under use.
See the PSBLAS User's Guide for details [<A
HREF="node29.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
HREF="node30.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>desc_a</code> </FONT></FONT></FONT></TD>
@@ -93,24 +92,26 @@ hierarchy produced by a previous call to <code>hierarchy_build</code>
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The communication descriptor of <code>a</code>. See the PSBLAS User's Guide for
details [<A
HREF="node29.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
HREF="node30.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>info</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(out)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node27.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node28.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
</TR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
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@@ -120,11 +121,11 @@ hierarchy produced by a previous call to <code>hierarchy_build</code>
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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -20,7 +20,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -30,7 +30,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION00086000000000000000"></A><A NAME="sec:precapply"></A>
<H2><A NAME="SECTION00085000000000000000"></A><A NAME="sec:smooth_bld"></A>
<BR>
Subroutine apply
Subroutine smoothers_build
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%apply(x,y,desc_a,info [,trans,work])</code>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%smoothers_build(a,desc_a,p,info)</code>
<BR></FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
This routine computes <!-- MATH
$y = op(B^{-1})\, x$
-->
<IMG
WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img86.png"
ALT="$y = op(B^{-1}) x$">, where <IMG
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img24.png"
ALT="$B$"> is a previously built
preconditioner, stored into <code>p</code>, and <IMG
WIDTH="21" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img87.png"
ALT="$op$">
denotes the preconditioner itself or its transpose, according to
the value of <code>trans</code>.
Note that, when MLD2P4 is used with a Krylov solver from PSBLAS,
<code>p%apply</code> is called within the PSBLAS routine <code>psb_krylov</code>
and hence it is completely transparent to the user.
This routine builds the smoothers and the coarsest-level solvers for the
multi-level preconditioner <code>p</code>, according to the requirements made by
the user through the routines <code>init</code> and <code>set</code>, and based on the aggregation
hierarchy produced by a previous call to <code>hierarchy_build</code>
(see Section&nbsp;<A HREF="node22.html#sec:hier_bld">6.4</A>).
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Arguments</B></FONT> </FONT></FONT></FONT>
@@ -87,30 +73,17 @@ and hence it is completely transparent to the user.
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>x</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <I>type</I><code>(</code><I>kind_parameter</I><code>), dimension(:), intent(in)</code>.</FONT></FONT></FONT></TD>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>a</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>type(psb_</code><I>x</I><code>spmat_type), intent(in)</code>. </FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The local part of the vector <IMG
WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img88.png"
ALT="$x$">. Note that <I>type</I> and
<I>kind_parameter</I> must be chosen according
to the real/complex, single/double precision version of MLD2P4 under use.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>y</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <I>type</I><code>(</code><I>kind_parameter</I><code>), dimension(:), intent(out)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The local part of the vector <IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img89.png"
ALT="$y$">. Note that <I>type</I> and
<I>kind_parameter</I> must be chosen according
to the real/complex, single/double precision version of MLD2P4 under use.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The sparse matrix structure containing the local part of the
matrix to be preconditioned. Note that <I>x</I> must be chosen according
to the real/complex, single/double precision version of MLD2P4 under use.
See the PSBLAS User's Guide for details [<A
HREF="node30.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>desc_a</code> </FONT></FONT></FONT></TD>
@@ -118,82 +91,26 @@ and hence it is completely transparent to the user.
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The communication descriptor associated to the matrix to be
preconditioned.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The communication descriptor of <code>a</code>. See the PSBLAS User's Guide for
details [<A
HREF="node30.html#PSBLASGUIDE">13</A>].</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>info</code> </FONT></FONT></FONT></TD>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>info</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(out)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node27.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>trans</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>character(len=1), optional, intent(in).</code></FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> If <code>trans</code> = <code>'N','n'</code> then <!-- MATH
$op(B^{-1}) = B^{-1}$
-->
<IMG
WIDTH="123" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img90.png"
ALT="$op(B^{-1}) = B^{-1}$">;
if <code>trans</code> = <code>'T','t'</code> then <!-- MATH
$op(B^{-1}) = B^{-T}$
-->
<IMG
WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img91.png"
ALT="$op(B^{-1}) = B^{-T}$">
(transpose of <IMG
WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img92.png"
ALT="$B^{-1})$">; if <code>trans</code> = <code>'C','c'</code> then <!-- MATH
$op(B^{-1}) = B^{-C}$
-->
<IMG
WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img93.png"
ALT="$op(B^{-1}) = B^{-C}$">
(conjugate transpose of <IMG
WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img92.png"
ALT="$B^{-1})$">.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>work</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <I>type</I><code>(</code><I>kind_parameter</I><code>), dimension(:), optional, target</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Workspace. Its size should be at
least <code>4 * psb_cd_get_local_</code> <code>cols(desc_a)</code> (see the PSBLAS User's Guide).
Note that <I>type</I> and <I>kind_parameter</I> must be chosen according
to the real/complex, single/double precision version of MLD2P4 under use.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node28.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
</TR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">For compatibility with the previous versions of MLD2P4, this routine can be also invoked
as follows:
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precaply(p,what,val,info)</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
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<H2><A NAME="SECTION00087000000000000000"></A><A NAME="sec:precfree"></A>
<H2><A NAME="SECTION00086000000000000000"></A><A NAME="sec:precapply"></A>
<BR>
Subroutine free
Subroutine apply
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%free(p,info)</code>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%apply(x,y,desc_a,info [,trans,work])</code>
<BR></FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
This routine deallocates the preconditioner data structure <code>p</code>.
This routine computes <!-- MATH
$y = op(B^{-1})\, x$
-->
<IMG
WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img86.png"
ALT="$y = op(B^{-1}) x$">, where <IMG
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img24.png"
ALT="$B$"> is a previously built
preconditioner, stored into <code>p</code>, and <IMG
WIDTH="21" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img87.png"
ALT="$op$">
denotes the preconditioner itself or its transpose, according to
the value of <code>trans</code>.
Note that, when MLD2P4 is used with a Krylov solver from PSBLAS,
<code>p%apply</code> is called within the PSBLAS routine <code>psb_krylov</code>
and hence it is completely transparent to the user.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Arguments</B></FONT> </FONT></FONT></FONT>
@@ -69,12 +87,93 @@ This routine deallocates the preconditioner data structure <code>p</code>.
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>info</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(out)</code>.</FONT></FONT></FONT></TD>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>x</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <I>type</I><code>(</code><I>kind_parameter</I><code>), dimension(:), intent(in)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node27.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The local part of the vector <IMG
WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img88.png"
ALT="$x$">. Note that <I>type</I> and
<I>kind_parameter</I> must be chosen according
to the real/complex, single/double precision version of MLD2P4 under use.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>y</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <I>type</I><code>(</code><I>kind_parameter</I><code>), dimension(:), intent(out)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The local part of the vector <IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img89.png"
ALT="$y$">. Note that <I>type</I> and
<I>kind_parameter</I> must be chosen according
to the real/complex, single/double precision version of MLD2P4 under use.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>desc_a</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>type(psb_desc_type), intent(in)</code>. </FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The communication descriptor associated to the matrix to be
preconditioned.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>info</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(out)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node28.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>trans</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>character(len=1), optional, intent(in).</code></FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> If <code>trans</code> = <code>'N','n'</code> then <!-- MATH
$op(B^{-1}) = B^{-1}$
-->
<IMG
WIDTH="123" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img90.png"
ALT="$op(B^{-1}) = B^{-1}$">;
if <code>trans</code> = <code>'T','t'</code> then <!-- MATH
$op(B^{-1}) = B^{-T}$
-->
<IMG
WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img91.png"
ALT="$op(B^{-1}) = B^{-T}$">
(transpose of <IMG
WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img92.png"
ALT="$B^{-1})$">; if <code>trans</code> = <code>'C','c'</code> then <!-- MATH
$op(B^{-1}) = B^{-C}$
-->
<IMG
WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img93.png"
ALT="$op(B^{-1}) = B^{-C}$">
(conjugate transpose of <IMG
WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img92.png"
ALT="$B^{-1})$">.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>work</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <I>type</I><code>(</code><I>kind_parameter</I><code>), dimension(:), optional, target</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Workspace. Its size should be at
least <code>4 * psb_cd_get_local_</code> <code>cols(desc_a)</code> (see the PSBLAS User's Guide).
Note that <I>type</I> and <I>kind_parameter</I> must be chosen according
to the real/complex, single/double precision version of MLD2P4 under use.</FONT></FONT></FONT></TD>
</TR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
@@ -83,13 +182,35 @@ as follows:
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precfree(p,info)</code>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precaply(p,what,val,info)</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<H2><A NAME="SECTION00088000000000000000"></A><A NAME="sec:precdescr"></A>
<H2><A NAME="SECTION00087000000000000000"></A><A NAME="sec:precfree"></A>
<BR>
Subroutine descr
Subroutine free
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%descr(info, [iout])</code>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%free(p,info)</code>
<BR></FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
This routine prints a description of the preconditioner <code>p</code> to the standard output or
to a file. It must be called after <code>hierachy_build</code> and <code>smoothers_build</code>,
or <code>build</code>, have been called.
This routine deallocates the preconditioner data structure <code>p</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Arguments</B></FONT> </FONT></FONT></FONT>
@@ -71,20 +70,11 @@ or <code>build</code>, have been called.
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>info</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(out)</code>.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(out)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node27.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>iout</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(in), optional</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The id of the file where the preconditioner description
will be printed; the default is the standard output.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node28.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
</TR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
@@ -93,37 +83,13 @@ as follows:
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precdescr(p,info [,iout])</code>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precfree(p,info)</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<H1><A NAME="SECTION00090000000000000000"></A><A NAME="sec:adding"></A>
<H2><A NAME="SECTION00088000000000000000"></A><A NAME="sec:precdescr"></A>
<BR>
Adding new smoother and solver objects to MLD2P4
</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
Subroutine descr
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Developers can add completely new smoother and/or solver classes
derived from the base objects in the library (see Remark&nbsp;2 in Section&nbsp;<A HREF="node19.html#sec:precset">6.2</A>),
without recompiling the library itself.
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call p%descr(info, [iout])</code>
<BR></FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
This routine prints a description of the preconditioner <code>p</code> to the standard output or
to a file. It must be called after <code>hierachy_build</code> and <code>smoothers_build</code>,
or <code>build</code>, have been called.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">To do so, it is necessary first to select the base type to be extended.
In our experience, it is quite likely that the new application needs
only the definition of a ``solver'' object, which is almost
always acting only on the local part of the distributed matrix.
The parallel actions required to connect the various solver objects
are most often already provided by the block-Jacobi or the additive
Schwarz smoothers. To define a new solver, the developer will then
have to define its components and methods, perhaps taking one of the
predefined solvers as a starting point, if possible.
</FONT></FONT></FONT>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><B>Arguments</B></FONT> </FONT></FONT></FONT>
<P></P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Once the new smoother/solver class has been developed, to use it in
the context of the multilevel preconditioners it is necessary to:
</FONT></FONT></FONT>
<UL>
<LI>declare in the application program a variable of the new type;
</LI>
<LI>pass that variable as the argument to the <code>set</code> routine as in the
following:
<DIV ALIGN="CENTER">
<code>call p%set(smoother,info [,ilev,ilmax,pos])</code>
<BR><code>call p%set(solver,info [,ilev,ilmax,pos])</code>
</DIV>
</LI>
<LI>link the code implementing the various methods into the application executable.
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
The new solver object is then dynamically included in the
preconditioner structure, and acts as a <I>mold</I> to which the
preconditioner will conform, even though the MLD2P4 library has not
been modified to account for this new development.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">It is possible to define new values for the keyword <code>WHAT</code> in the
<code>set</code> routine; if the library code does not recognize a keyword,
it passes it down the composition hierarchy (levels containing
smoothers containing in turn solvers), so that it can be eventually caught by
the new solver.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">An example is provided in the source code distribution under the
folder <code>tests/newslv</code>. In this example we are implementing a new
incomplete factorization variant (which is simply the ILU(0)
factorization under a new name). Because of the specifics of this case, it is
possible to reuse the basic structure of the ILU solver, with its
L/D/U components and the methods needed to apply the solver; only a
few methods, such as the description and most importantly the build,
need to be ovverridden (rewritten).
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The interfaces for the calls shown above are defined using
</FONT></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>smoother</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>class(mld_x_base_smoother_type)</code> </FONT></FONT></FONT></TD>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>info</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(out)</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The user-defined new smoother to be employed in the
preconditioner.</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node28.html#sec:errors">8</A> for details.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>solver</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>class(mld_x_base_solver_type)</code> </FONT></FONT></FONT></TD>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>iout</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>integer, intent(in), optional</code>.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The user-defined new solver to be employed in the
preconditioner.
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The id of the file where the preconditioner description
will be printed; the default is the standard output.</FONT></FONT></FONT></TD>
</TR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
The other arguments are defined in the way described in
Sec.&nbsp;<A HREF="node19.html#sec:precset">6.2</A>. As an example, in the <code>tests/newslv</code>
code we define a new object of type <code>mld_d_tlu_solver_type</code>, and
we pass it as follows:
</FONT></FONT></FONT><PRE>
! sparse matrix and preconditioner
type(psb_dspmat_type) :: a
type(mld_dprec_type) :: prec
type(mld_d_tlu_solver_type) :: tlusv
......
!
! prepare the preconditioner: an ML with defaults, but with TLU solver at
! intermediate levels. All other parameters are at default values.
!
call prec%init('ML', info)
call prec%hierarchy_build(a,desc_a,info)
nlv = prec%get_nlevs()
call prec%set(tlusv, info,ilev=1,ilmax=max(1,nlv-1))
call prec%smoothers_build(a,desc_a,info)
</PRE><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT><HR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">For compatibility with the previous versions of MLD2P4, this routine can be also invoked
as follows:
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"><code>call mld_precdescr(p,info [,iout])</code>
</FONT></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
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<H1><A NAME="SECTION000100000000000000000"></A><A NAME="sec:errors"></A>
<H1><A NAME="SECTION00090000000000000000"></A><A NAME="sec:adding"></A>
<BR>
Error Handling
Adding new smoother and solver objects to MLD2P4
</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The error handling in MLD2P4 is based on the PSBLAS (version 2) error
handling. Error conditions are signaled via an integer argument
<code>info</code>; whenever an error condition is detected, an error trace
stack is built by the library up to the top-level, user-callable
routine. This routine will then decide, according to the user
preferences, whether the error should be handled by terminating the
program or by returning the error condition to the user code, which
will then take action, and whether
an error message should be printed. These options may be set by using
the PSBLAS error handling routines; for further details see the PSBLAS
User's Guide [<A
HREF="node29.html#PSBLASGUIDE">13</A>].
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Developers can add completely new smoother and/or solver classes
derived from the base objects in the library (see Remark&nbsp;2 in Section&nbsp;<A HREF="node20.html#sec:precset">6.2</A>),
without recompiling the library itself.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">To do so, it is necessary first to select the base type to be extended.
In our experience, it is quite likely that the new application needs
only the definition of a ``solver'' object, which is almost
always acting only on the local part of the distributed matrix.
The parallel actions required to connect the various solver objects
are most often already provided by the block-Jacobi or the additive
Schwarz smoothers. To define a new solver, the developer will then
have to define its components and methods, perhaps taking one of the
predefined solvers as a starting point, if possible.
</FONT></FONT></FONT>
<BR><HR>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Once the new smoother/solver class has been developed, to use it in
the context of the multilevel preconditioners it is necessary to:
</FONT></FONT></FONT>
<UL>
<LI>declare in the application program a variable of the new type;
</LI>
<LI>pass that variable as the argument to the <code>set</code> routine as in the
following:
<DIV ALIGN="CENTER">
<code>call p%set(smoother,info [,ilev,ilmax,pos])</code>
<BR><code>call p%set(solver,info [,ilev,ilmax,pos])</code>
</DIV>
</LI>
<LI>link the code implementing the various methods into the application executable.
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
The new solver object is then dynamically included in the
preconditioner structure, and acts as a <I>mold</I> to which the
preconditioner will conform, even though the MLD2P4 library has not
been modified to account for this new development.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">It is possible to define new values for the keyword <code>WHAT</code> in the
<code>set</code> routine; if the library code does not recognize a keyword,
it passes it down the composition hierarchy (levels containing
smoothers containing in turn solvers), so that it can be eventually caught by
the new solver.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">An example is provided in the source code distribution under the
folder <code>tests/newslv</code>. In this example we are implementing a new
incomplete factorization variant (which is simply the ILU(0)
factorization under a new name). Because of the specifics of this case, it is
possible to reuse the basic structure of the ILU solver, with its
L/D/U components and the methods needed to apply the solver; only a
few methods, such as the description and most importantly the build,
need to be ovverridden (rewritten).
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The interfaces for the calls shown above are defined using
</FONT></FONT></FONT>
<DIV ALIGN="CENTER"><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>smoother</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>class(mld_x_base_smoother_type)</code> </FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The user-defined new smoother to be employed in the
preconditioner.</FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<code>solver</code> </FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> <code>class(mld_x_base_solver_type)</code> </FONT></FONT></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=40><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"> The user-defined new solver to be employed in the
preconditioner.
</FONT></FONT></FONT></TD>
</TR>
</TABLE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT></DIV><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
The other arguments are defined in the way described in
Sec.&nbsp;<A HREF="node20.html#sec:precset">6.2</A>. As an example, in the <code>tests/newslv</code>
code we define a new object of type <code>mld_d_tlu_solver_type</code>, and
we pass it as follows:
</FONT></FONT></FONT><PRE>
! sparse matrix and preconditioner
type(psb_dspmat_type) :: a
type(mld_dprec_type) :: prec
type(mld_d_tlu_solver_type) :: tlusv
......
!
! prepare the preconditioner: an ML with defaults, but with TLU solver at
! intermediate levels. All other parameters are at default values.
!
call prec%init('ML', info)
call prec%hierarchy_build(a,desc_a,info)
nlv = prec%get_nlevs()
call prec%set(tlusv, info,ilev=1,ilmax=max(1,nlv-1))
call prec%smoothers_build(a,desc_a,info)
</PRE><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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License
Error Handling
</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The MLD2P4 is freely distributable under the following copyright
terms: </FONT></FONT></FONT><PRE>
MLD2P4 version 2.1
MultiLevel Domain Decomposition Parallel Preconditioners Package
based on PSBLAS (Parallel Sparse BLAS version 3.4)
(C) Copyright 2008, 2010, 2012, 2017
Salvatore Filippone Cranfield University, Cranfield, UK
Ambra Abdullahi Hassan University of Rome Tor Vergata, Rome, IT
Alfredo Buttari CNRS-IRIT, Toulouse, FR
Pasqua D'Ambra IAC-CNR, Naples, IT
Daniela di Serafino University of Campania L. Vanvitelli, Caserta, IT
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:
1. Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions, and the following disclaimer in the
documentation and/or other materials provided with the distribution.
3. The name of the MLD2P4 group or the names of its contributors may
not be used to endorse or promote products derived from this
software without specific written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE MLD2P4 GROUP OR ITS CONTRIBUTORS
BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
</PRE><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The error handling in MLD2P4 is based on the PSBLAS (version 2) error
handling. Error conditions are signaled via an integer argument
<code>info</code>; whenever an error condition is detected, an error trace
stack is built by the library up to the top-level, user-callable
routine. This routine will then decide, according to the user
preferences, whether the error should be handled by terminating the
program or by returning the error condition to the user code, which
will then take action, and whether
an error message should be printed. These options may be set by using
the PSBLAS error handling routines; for further details see the PSBLAS
User's Guide [<A
HREF="node30.html#PSBLASGUIDE">13</A>].
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT>
<BR><HR>
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@@ -40,140 +40,63 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION000120000000000000000">
Bibliography</A>
</H2><DL COMPACT><DD>
<H1><A NAME="SECTION000110000000000000000"></A><A NAME="sec:license"></A>
<BR>
License
</H1><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<P></P><DT><A NAME="MUMPS">1</A>
<DD>
P.&nbsp;R.&nbsp;Amestoy, C.&nbsp;Ashcraft, O.&nbsp;Boiteau, A.&nbsp;Buttari, J.&nbsp;L'Excellent, C.&nbsp;Weisbecker,
<EM>Improving multifrontal methods by means of block low-rank representations</EM>,
SIAM Journal on Scientific Computing, volume 37 (3), 2015, A1452-A1474.
See also <TT>http://mumps.enseeiht.fr</TT>. <P></P><DT><A NAME="BREZINA_VANEK">2</A>
<DD>
M.&nbsp;Brezina, P.&nbsp;Vanek,
<EM>A Black-Box Iterative Solver Based on a Two-Level Schwarz Method</EM>,
Computing, 63, 1999, 233-263. <P></P><DT><A NAME="Briggs2000">3</A>
<DD>
W.&nbsp;L.&nbsp;Briggs, V.&nbsp;E.&nbsp;Henson, S.&nbsp;F.&nbsp;McCormick,
<EM>A Multigrid Tutorial, Second Edition</EM>,
SIAM, 2000. <P></P><DT><A NAME="para_04">4</A>
<DD>
A.&nbsp;Buttari, P.&nbsp;D'Ambra, D.&nbsp;di Serafino, S.&nbsp;Filippone,
<EM>Extending PSBLAS to Build Parallel Schwarz Preconditioners</EM>,
in J.&nbsp;Dongarra, K.&nbsp;Madsen, J.&nbsp;Wasniewski, editors,
Proceedings of PARA&nbsp;04 Workshop on State of the Art
in Scientific Computing, Lecture Notes in Computer Science,
Springer, 2005, 593-602. <P></P><DT><A NAME="aaecc_07">5</A>
<DD>
A.&nbsp;Buttari, P.&nbsp;D'Ambra, D.&nbsp;di&nbsp;Serafino, S.&nbsp;Filippone,
<EM>2LEV-D2P4: a package of high-performance preconditioners
for scientific and engineering applications</EM>,
Applicable Algebra in Engineering, Communications and Computing,
18 (3) 2007, 223-239. <P></P><DT><A NAME="CAI_SARKIS">6</A>
<DD>
X.&nbsp;C.&nbsp;Cai, M.&nbsp;Sarkis,
<EM>A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems</EM>,
SIAM Journal on Scientific Computing, 21 (2), 1999, 792-797. <P></P><DT><A NAME="apnum_07">7</A>
<DD>
P.&nbsp;D'Ambra, S.&nbsp;Filippone, D.&nbsp;di&nbsp;Serafino,
<EM>On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners</EM>,
Applied Numerical Mathematics, Elsevier Science,
57 (11-12), 2007, 1181-1196. <P></P><DT><A NAME="MLD2P4_TOMS">8</A>
<DD>
P.&nbsp;D'Ambra, D.&nbsp;di&nbsp;Serafino, S.&nbsp;Filippone,
<I>MLD2P4: a Package of Parallel Multilevel
Algebraic Domain Decomposition Preconditioners
in Fortran 95</I>, ACM Trans. Math. Softw., 37(3), 2010, art. 30. <P></P><DT><A NAME="UMFPACK">9</A>
<DD>
T.&nbsp;A.&nbsp;Davis,
<EM>Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal
Method with a Column Pre-ordering Strategy</EM>,
ACM Transactions on Mathematical Software, 30, 2004, 196-199.
(See also <TT>http://www.cise.ufl.edu/&nbsp;davis/</TT>) <P></P><DT><A NAME="SUPERLU">10</A>
<DD>
J.&nbsp;W.&nbsp;Demmel, S.&nbsp;C.&nbsp;Eisenstat, J.&nbsp;R.&nbsp;Gilbert, X.&nbsp;S.&nbsp;Li, J.&nbsp;W.&nbsp;H.&nbsp;Liu,
A supernodal approach to sparse partial pivoting,
SIAM Journal on Matrix Analysis and Applications, 20 (3), 1999, 720-755. <P></P><DT><A NAME="blas3">11</A>
<DD>
J.&nbsp;J.&nbsp;Dongarra, J.&nbsp;Du Croz, I.&nbsp;S.&nbsp;Duff, S.&nbsp;Hammarling,
<I>A set of Level 3 Basic Linear Algebra Subprograms</I>,
ACM Transactions on Mathematical Software, 16 (1) 1990, 1-17. <P></P><DT><A NAME="blas2">12</A>
<DD>
J.&nbsp;J.&nbsp;Dongarra, J.&nbsp;Du Croz, S.&nbsp;Hammarling, R.&nbsp;J.&nbsp;Hanson,
<I>An extended set of FORTRAN Basic Linear Algebra Subprograms</I>,
ACM Transactions on Mathematical Software, 14 (1) 1988, 1-17. <P></P><DT><A NAME="PSBLASGUIDE">13</A>
<DD>
S.&nbsp;Filippone, A.&nbsp;Buttari,
<EM>PSBLAS-3.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library</EM>, 2012,
available from <TT>http://www.ce.uniroma2.it/psblas/</TT>. <P></P><DT><A NAME="PSBLAS3">14</A>
<DD>
S.&nbsp;Filippone, A.&nbsp;Buttari,
<EM>Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003</EM>.
ACM Transactions on on Mathematical Software, 38 (4), 2012, art.&nbsp;23. <P></P><DT><A NAME="psblas_00">15</A>
<DD>
S.&nbsp;Filippone, M.&nbsp;Colajanni,
<EM>PSBLAS: A Library for Parallel Linear Algebra
Computation on Sparse Matrices</EM>,
ACM Transactions on Mathematical Software, 26 (4), 2000, 527-550. <P></P><DT><A NAME="MPI2">16</A>
<DD>
W.&nbsp;Gropp, S.&nbsp;Huss-Lederman, A.&nbsp;Lumsdaine, E.&nbsp;Lusk, B.&nbsp;Nitzberg, W.&nbsp;Saphir, M.&nbsp;Snir,
<EM>MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions</EM>,
MIT Press, 1998. <P></P><DT><A NAME="blas1">17</A>
<DD>
C.&nbsp;L.&nbsp;Lawson, R.&nbsp;J.&nbsp;Hanson, D.&nbsp;Kincaid, F.&nbsp;T.&nbsp;Krogh,
<I>Basic Linear Algebra Subprograms for FORTRAN usage</I>,
ACM Transactions on Mathematical Software, 5 (3), 1979, 308-323. <P></P><DT><A NAME="SUPERLUDIST">18</A>
<DD>
X.&nbsp;S.&nbsp;Li, J.&nbsp;W.&nbsp;Demmel,
<EM>SuperLU_DIST: A Scalable Distributed-memory
Sparse Direct Solver for Unsymmetric Linear Systems</EM>,
ACM Transactions on Mathematical Software, 29 (2), 2003, 110-140. <P></P><DT><A NAME="Notay2008">19</A>
<DD>
Y.&nbsp;Notay, P.&nbsp;S.&nbsp;Vassilevski,
<EM>Recursive Krylov-based multigrid cycles</EM>,
Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487. <P></P><DT><A NAME="Saad_book">20</A>
<DD>
Y.&nbsp;Saad,
<EM>Iterative methods for sparse linear systems</EM>, 2nd edition, SIAM, 2003. <P></P><DT><A NAME="dd2_96">21</A>
<DD>
B.&nbsp;Smith, P.&nbsp;Bjorstad, W.&nbsp;Gropp,
<EM>Domain Decomposition: Parallel Multilevel Methods for Elliptic
Partial Differential Equations</EM>,
Cambridge University Press, 1996. <P></P><DT><A NAME="MPI1">22</A>
<DD>
M.&nbsp;Snir, S.&nbsp;Otto, S.&nbsp;Huss-Lederman, D.&nbsp;Walker, J.&nbsp;Dongarra,
<EM>MPI: The Complete Reference. Volume 1 - The MPI Core</EM>, second edition,
MIT Press, 1998. <P></P><DT><A NAME="Stuben_01">23</A>
<DD>
K.&nbsp;St&#252;ben,
<EM>An Introduction to Algebraic Multigrid</EM>,
in A.&nbsp;Sch&#252;ller, U.&nbsp;Trottenberg, C.&nbsp;Oosterlee, Multigrid,
Academic Press, 2001. <P></P><DT><A NAME="TUMINARO_TONG">24</A>
<DD>
R.&nbsp;S.&nbsp;Tuminaro, C.&nbsp;Tong,
<EM>Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines</EM>, in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000. <P></P><DT><A NAME="VANEK_MANDEL_BREZINA">25</A>
<DD>
P.&nbsp;Vanek, J.&nbsp;Mandel, M.&nbsp;Brezina,
<EM>Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems</EM>,
Computing, 56 (3) 1996, 179-196.
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The MLD2P4 is freely distributable under the following copyright
terms: </FONT></FONT></FONT><PRE>
MLD2P4 version 2.1
MultiLevel Domain Decomposition Parallel Preconditioners Package
based on PSBLAS (Parallel Sparse BLAS version 3.4)
(C) Copyright 2008, 2010, 2012, 2017
Salvatore Filippone Cranfield University, Cranfield, UK
Pasqua D'Ambra IAC-CNR, Naples, IT
Daniela di Serafino University of Campania L. Vanvitelli, Caserta, IT
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:
1. Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions, and the following disclaimer in the
documentation and/or other materials provided with the distribution.
3. The name of the MLD2P4 group or the names of its contributors may
not be used to endorse or promote products derived from this
software without specific written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE MLD2P4 GROUP OR ITS CONTRIBUTORS
BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
</PRE><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
<P>
</DL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT>
<P>
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@@ -59,9 +59,9 @@ General Overview
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The M<SMALL>ULTI-</SMALL>L<SMALL>EVEL </SMALL>D<SMALL>OMAIN </SMALL>D<SMALL>ECOMPOSITION </SMALL>P<SMALL>ARALLEL </SMALL>P<SMALL>RECONDITIONERS </SMALL>P<SMALL>ACKAGE BASED ON
</SMALL>PSBLAS (MLD2P4) provides parallel Algebraic MultiGrid (AMG) and Domain
Decomposition preconditioners (see, e.g., [<A
HREF="node29.html#Briggs2000">3</A>,<A
HREF="node29.html#Stuben_01">23</A>,<A
HREF="node29.html#dd2_96">21</A>]),
HREF="node30.html#Briggs2000">3</A>,<A
HREF="node30.html#Stuben_01">23</A>,<A
HREF="node30.html#dd2_96">21</A>]),
to be used in the iterative solution of linear systems,
</FONT></FONT></FONT>
<BR>
@@ -95,8 +95,8 @@ multi-level cycles and smoothers widely used in multigrid methods.
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The multi-level preconditioners implemented in MLD2P4 are obtained by combining
AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and
K-cycles&nbsp;[<A
HREF="node29.html#Briggs2000">3</A>,<A
HREF="node29.html#Notay2008">19</A>] are available, which allow to define
HREF="node30.html#Briggs2000">3</A>,<A
HREF="node30.html#Notay2008">19</A>] are available, which allow to define
almost all the preconditioners in the package, including the multi-level hybrid
Schwarz ones; a specific cycle is implemented to obtain multi-level additive
Schwarz preconditioners. The Jacobi, hybridforward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz methods
@@ -104,8 +104,8 @@ are available as smoothers. An algebraic approach is used to generate a hierarch
coarse-level matrices and operators, without explicitly using any information on the
geometry of the original problem, e.g., the discretization of a PDE. To this end,
the smoothed aggregation technique&nbsp;[<A
HREF="node29.html#BREZINA_VANEK">2</A>,<A
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>]
HREF="node30.html#BREZINA_VANEK">2</A>,<A
HREF="node30.html#VANEK_MANDEL_BREZINA">25</A>]
is applied. Either exact or approximate solvers can be used on the coarsest-level
system. Specifically, different sparse LU factorizations from external
packages, and native incomplete LU factorizations and Jacobi, hybrid Gauss-Seidel,
@@ -126,8 +126,8 @@ interface.
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">MLD2P4 has been designed to implement scalable and easy-to-use
multilevel preconditioners in the context of the PSBLAS (Parallel Sparse BLAS)
computational framework&nbsp;[<A
HREF="node29.html#psblas_00">15</A>,<A
HREF="node29.html#PSBLAS3">14</A>]. PSBLAS provides basic linear algebra
HREF="node30.html#psblas_00">15</A>,<A
HREF="node30.html#PSBLAS3">14</A>]. PSBLAS provides basic linear algebra
operators and data management facilities for distributed sparse matrices,
as well as parallel Krylov solvers which can be used with the MLD2P4 preconditioners.
The choice of PSBLAS has been mainly motivated by the need of having
@@ -150,55 +150,55 @@ few black-box routines at the upper layer allow all users to easily
build and apply any preconditioner available in MLD2P4;
facilities are also available allowing expert users to extend the set of smoothers
and solvers for building new versions of the preconditioners (see
Section&nbsp;<A HREF="node26.html#sec:adding">7</A>).
Section&nbsp;<A HREF="node27.html#sec:adding">7</A>).
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">We note that the user interface of MLD2P4 2.1 has been extended with respect to the
previous versions in order to separate the construction of the multi-level hierarchy from
the construction of the smoothers and solvers, and to allow for more flexibility
at each level. The software architecture described in&nbsp;[<A
HREF="node29.html#MLD2P4_TOMS">8</A>] has significantly
HREF="node30.html#MLD2P4_TOMS">8</A>] has significantly
evolved too, in order to fully exploit the Fortran&nbsp;2003 features implemented in PSBLAS 3.
However, compatibility with previous versions has been preserved.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">This guide is organized as follows. General information on the distribution of the source
code is reported in Section&nbsp;<A HREF="node4.html#sec:distribution">2</A>, while details on the configuration
and installation of the package are given in Section&nbsp;<A HREF="node5.html#sec:building">3</A>. A short description
of the preconditioners implemented in MLD2P4 is provided in Section&nbsp;<A HREF="node11.html#sec:background">4</A>,
and installation of the package are given in Section&nbsp;<A HREF="node6.html#sec:building">3</A>. A short description
of the preconditioners implemented in MLD2P4 is provided in Section&nbsp;<A HREF="node12.html#sec:background">4</A>,
to help the users in choosing among them. The basics for building and applying the
preconditioners with the Krylov solvers implemented in PSBLAS are reported
in&nbsp;Section&nbsp;<A HREF="node15.html#sec:started">5</A>, where the Fortran codes of a few sample programs
in&nbsp;Section&nbsp;<A HREF="node16.html#sec:started">5</A>, where the Fortran codes of a few sample programs
are also shown. A reference guide for the user interface routines is provided
in Section&nbsp;<A HREF="node17.html#sec:userinterface">6</A>. Information on the extension of the package
through the addition of new smoothers and solvers is reported in Section&nbsp;<A HREF="node26.html#sec:adding">7</A>.
in Section&nbsp;<A HREF="node18.html#sec:userinterface">6</A>. Information on the extension of the package
through the addition of new smoothers and solvers is reported in Section&nbsp;<A HREF="node27.html#sec:adding">7</A>.
The error handling mechanism used by the package
is briefly described in Section&nbsp;<A HREF="node27.html#sec:errors">8</A>. The copyright terms concerning the
distribution and modification of MLD2P4 are reported in Appendix&nbsp;<A HREF="node28.html#sec:license">A</A>.
is briefly described in Section&nbsp;<A HREF="node28.html#sec:errors">8</A>. The copyright terms concerning the
distribution and modification of MLD2P4 are reported in Appendix&nbsp;<A HREF="node29.html#sec:license">A</A>.
</FONT></FONT></FONT>
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<H2><A NAME="SECTION000120000000000000000">
Bibliography</A>
</H2><DL COMPACT><DD>
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About this document ...</A>
</H1><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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This document was generated using the
<A HREF="http://www.latex2html.org/"><STRONG>LaTeX</STRONG>2<tt>HTML</tt></A> translator Version 2012 (1.2)
<P>
Copyright &#169; 1993, 1994, 1995, 1996,
<A HREF="http://cbl.leeds.ac.uk/nikos/personal.html">Nikos Drakos</A>,
Computer Based Learning Unit, University of Leeds.
<BR>
Copyright &#169; 1997, 1998, 1999,
<A HREF="http://www.maths.mq.edu.au/~ross/">Ross Moore</A>,
Mathematics Department, Macquarie University, Sydney.
<P></P><DT><A NAME="MUMPS">1</A>
<DD>
P.&nbsp;R.&nbsp;Amestoy, C.&nbsp;Ashcraft, O.&nbsp;Boiteau, A.&nbsp;Buttari, J.&nbsp;L'Excellent, C.&nbsp;Weisbecker,
<EM>Improving multifrontal methods by means of block low-rank representations</EM>,
SIAM Journal on Scientific Computing, volume 37 (3), 2015, A1452-A1474.
See also <TT>http://mumps.enseeiht.fr</TT>. <P></P><DT><A NAME="BREZINA_VANEK">2</A>
<DD>
M.&nbsp;Brezina, P.&nbsp;Vanek,
<EM>A Black-Box Iterative Solver Based on a Two-Level Schwarz Method</EM>,
Computing, 63, 1999, 233-263. <P></P><DT><A NAME="Briggs2000">3</A>
<DD>
W.&nbsp;L.&nbsp;Briggs, V.&nbsp;E.&nbsp;Henson, S.&nbsp;F.&nbsp;McCormick,
<EM>A Multigrid Tutorial, Second Edition</EM>,
SIAM, 2000. <P></P><DT><A NAME="para_04">4</A>
<DD>
A.&nbsp;Buttari, P.&nbsp;D'Ambra, D.&nbsp;di Serafino, S.&nbsp;Filippone,
<EM>Extending PSBLAS to Build Parallel Schwarz Preconditioners</EM>,
in J.&nbsp;Dongarra, K.&nbsp;Madsen, J.&nbsp;Wasniewski, editors,
Proceedings of PARA&nbsp;04 Workshop on State of the Art
in Scientific Computing, Lecture Notes in Computer Science,
Springer, 2005, 593-602. <P></P><DT><A NAME="aaecc_07">5</A>
<DD>
A.&nbsp;Buttari, P.&nbsp;D'Ambra, D.&nbsp;di&nbsp;Serafino, S.&nbsp;Filippone,
<EM>2LEV-D2P4: a package of high-performance preconditioners
for scientific and engineering applications</EM>,
Applicable Algebra in Engineering, Communications and Computing,
18 (3) 2007, 223-239. <P></P><DT><A NAME="CAI_SARKIS">6</A>
<DD>
X.&nbsp;C.&nbsp;Cai, M.&nbsp;Sarkis,
<EM>A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems</EM>,
SIAM Journal on Scientific Computing, 21 (2), 1999, 792-797. <P></P><DT><A NAME="apnum_07">7</A>
<DD>
P.&nbsp;D'Ambra, S.&nbsp;Filippone, D.&nbsp;di&nbsp;Serafino,
<EM>On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners</EM>,
Applied Numerical Mathematics, Elsevier Science,
57 (11-12), 2007, 1181-1196. <P></P><DT><A NAME="MLD2P4_TOMS">8</A>
<DD>
P.&nbsp;D'Ambra, D.&nbsp;di&nbsp;Serafino, S.&nbsp;Filippone,
<I>MLD2P4: a Package of Parallel Multilevel
Algebraic Domain Decomposition Preconditioners
in Fortran 95</I>, ACM Trans. Math. Softw., 37(3), 2010, art. 30. <P></P><DT><A NAME="UMFPACK">9</A>
<DD>
T.&nbsp;A.&nbsp;Davis,
<EM>Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal
Method with a Column Pre-ordering Strategy</EM>,
ACM Transactions on Mathematical Software, 30, 2004, 196-199.
(See also <TT>http://www.cise.ufl.edu/&nbsp;davis/</TT>) <P></P><DT><A NAME="SUPERLU">10</A>
<DD>
J.&nbsp;W.&nbsp;Demmel, S.&nbsp;C.&nbsp;Eisenstat, J.&nbsp;R.&nbsp;Gilbert, X.&nbsp;S.&nbsp;Li, J.&nbsp;W.&nbsp;H.&nbsp;Liu,
A supernodal approach to sparse partial pivoting,
SIAM Journal on Matrix Analysis and Applications, 20 (3), 1999, 720-755. <P></P><DT><A NAME="blas3">11</A>
<DD>
J.&nbsp;J.&nbsp;Dongarra, J.&nbsp;Du Croz, I.&nbsp;S.&nbsp;Duff, S.&nbsp;Hammarling,
<I>A set of Level 3 Basic Linear Algebra Subprograms</I>,
ACM Transactions on Mathematical Software, 16 (1) 1990, 1-17. <P></P><DT><A NAME="blas2">12</A>
<DD>
J.&nbsp;J.&nbsp;Dongarra, J.&nbsp;Du Croz, S.&nbsp;Hammarling, R.&nbsp;J.&nbsp;Hanson,
<I>An extended set of FORTRAN Basic Linear Algebra Subprograms</I>,
ACM Transactions on Mathematical Software, 14 (1) 1988, 1-17. <P></P><DT><A NAME="PSBLASGUIDE">13</A>
<DD>
S.&nbsp;Filippone, A.&nbsp;Buttari,
<EM>PSBLAS-3.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library</EM>, 2012,
available from <TT>http://www.ce.uniroma2.it/psblas/</TT>. <P></P><DT><A NAME="PSBLAS3">14</A>
<DD>
S.&nbsp;Filippone, A.&nbsp;Buttari,
<EM>Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003</EM>.
ACM Transactions on on Mathematical Software, 38 (4), 2012, art.&nbsp;23. <P></P><DT><A NAME="psblas_00">15</A>
<DD>
S.&nbsp;Filippone, M.&nbsp;Colajanni,
<EM>PSBLAS: A Library for Parallel Linear Algebra
Computation on Sparse Matrices</EM>,
ACM Transactions on Mathematical Software, 26 (4), 2000, 527-550. <P></P><DT><A NAME="MPI2">16</A>
<DD>
W.&nbsp;Gropp, S.&nbsp;Huss-Lederman, A.&nbsp;Lumsdaine, E.&nbsp;Lusk, B.&nbsp;Nitzberg, W.&nbsp;Saphir, M.&nbsp;Snir,
<EM>MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions</EM>,
MIT Press, 1998. <P></P><DT><A NAME="blas1">17</A>
<DD>
C.&nbsp;L.&nbsp;Lawson, R.&nbsp;J.&nbsp;Hanson, D.&nbsp;Kincaid, F.&nbsp;T.&nbsp;Krogh,
<I>Basic Linear Algebra Subprograms for FORTRAN usage</I>,
ACM Transactions on Mathematical Software, 5 (3), 1979, 308-323. <P></P><DT><A NAME="SUPERLUDIST">18</A>
<DD>
X.&nbsp;S.&nbsp;Li, J.&nbsp;W.&nbsp;Demmel,
<EM>SuperLU_DIST: A Scalable Distributed-memory
Sparse Direct Solver for Unsymmetric Linear Systems</EM>,
ACM Transactions on Mathematical Software, 29 (2), 2003, 110-140. <P></P><DT><A NAME="Notay2008">19</A>
<DD>
Y.&nbsp;Notay, P.&nbsp;S.&nbsp;Vassilevski,
<EM>Recursive Krylov-based multigrid cycles</EM>,
Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487. <P></P><DT><A NAME="Saad_book">20</A>
<DD>
Y.&nbsp;Saad,
<EM>Iterative methods for sparse linear systems</EM>, 2nd edition, SIAM, 2003. <P></P><DT><A NAME="dd2_96">21</A>
<DD>
B.&nbsp;Smith, P.&nbsp;Bjorstad, W.&nbsp;Gropp,
<EM>Domain Decomposition: Parallel Multilevel Methods for Elliptic
Partial Differential Equations</EM>,
Cambridge University Press, 1996. <P></P><DT><A NAME="MPI1">22</A>
<DD>
M.&nbsp;Snir, S.&nbsp;Otto, S.&nbsp;Huss-Lederman, D.&nbsp;Walker, J.&nbsp;Dongarra,
<EM>MPI: The Complete Reference. Volume 1 - The MPI Core</EM>, second edition,
MIT Press, 1998. <P></P><DT><A NAME="Stuben_01">23</A>
<DD>
K.&nbsp;St&#252;ben,
<EM>An Introduction to Algebraic Multigrid</EM>,
in A.&nbsp;Sch&#252;ller, U.&nbsp;Trottenberg, C.&nbsp;Oosterlee, Multigrid,
Academic Press, 2001. <P></P><DT><A NAME="TUMINARO_TONG">24</A>
<DD>
R.&nbsp;S.&nbsp;Tuminaro, C.&nbsp;Tong,
<EM>Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines</EM>, in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000. <P></P><DT><A NAME="VANEK_MANDEL_BREZINA">25</A>
<DD>
P.&nbsp;Vanek, J.&nbsp;Mandel, M.&nbsp;Brezina,
<EM>Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems</EM>,
Computing, 56 (3) 1996, 179-196.
<P>
The command line arguments were: <BR>
<STRONG>latex2html</STRONG> <TT>-local_icons -noaddress -dir ../../html userhtml.tex</TT>
</DL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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The translation was initiated by Salvatore Filippone on 2017-07-25<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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original version by: Nikos Drakos, CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
@@ -13,7 +13,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -23,39 +23,34 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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This document was generated using the
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Copyright &#169; 1993, 1994, 1995, 1996,
<A HREF="http://cbl.leeds.ac.uk/nikos/personal.html">Nikos Drakos</A>,
@@ -66,13 +61,10 @@ Copyright &#169; 1997, 1998, 1999,
Mathematics Department, Macquarie University, Sydney.
<P>
The command line arguments were: <BR>
<STRONG>latex2html</STRONG> <TT>-html_version 4.0 -dir ../../html userhtml.tex</TT>
<STRONG>latex2html</STRONG> <TT>-local_icons -noaddress -dir ../../html userhtml.tex</TT>
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The translation was initiated by Salvatore Filippone on 2008-07-23
The translation was initiated by Salvatore Filippone on 2017-08-09<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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Salvatore Filippone
2008-07-23
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@@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -65,7 +65,7 @@ where contact points for further information can be also found.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The software is available under a modified BSD license, as specified
in Appendix&nbsp;<A HREF="node28.html#sec:license">A</A>; please note that some of the optional
in Appendix&nbsp;<A HREF="node29.html#sec:license">A</A>; please note that some of the optional
third party libraries may be licensed under a different and more
stringent license, most notably the GPL, and this should be taken into
account when treating derived works.
@@ -89,8 +89,19 @@ constant
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
whose current value is <code>2.1.0</code>
whose current value is <code>2.1.0</code>.
</FONT></FONT></FONT>
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<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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@@ -51,80 +50,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
<BR>
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<H1><A NAME="SECTION00050000000000000000"></A><A NAME="sec:building"></A>
<BR>
Configuring and Building MLD2P4
</H1><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
In order to build MLD2P4 it is necessary to set up a Makefile with appropriate
system-dependent variables; this is done by means of the <code>configure</code>
script. The distribution also includes the autoconf and automake
sources employed to generate the script, but usually this is not needed
to build the software.
<H2><A NAME="SECTION00041000000000000000">
Contributors</A>
</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
Contributors to version 2:
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">MLD2P4 is implemented almost entirely in Fortran&nbsp;2003, with some
interfaces to external libraries in C; the Fortran compiler
must support the Fortran&nbsp;2003 standard plus the extension <code>MOLD=</code>
feature, which enhances the usability of <code>ALLOCATE</code>.
Many compilers do this; in particular, this is
supported by the GNU Fortran compiler, for which we
recommend to use at least version 4.8.
The software defines data types and interfaces for
real and complex data, in both single and double precision.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Building MLD2P4 requires some base libraries (see Section&nbsp;<A HREF="node6.html#sec:prerequisites">3.1</A>);
interfaces to optional third-party libraries, which extend the functionalities of MLD2P4
(see Section&nbsp;<A HREF="node7.html#sec:third-party">3.2</A>), are also available. Many Linux distributions
(e.g., Ubuntu, Fedora, CentOS) provide precompiled packages for the prerequisite and
optional software. In many cases these packages are split between a runtime part and a
``developer'' part; in order to build MLD2P4 you need both. A description of the base and
optional software used by MLD2P4 is given in the next sections.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><HR>
<!--Table of Child-Links-->
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<UL>
<LI><A NAME="tex2html141"
HREF="node6.html">Prerequisites</A>
<LI><A NAME="tex2html142"
HREF="node7.html">Optional third party libraries</A>
<LI><A NAME="tex2html143"
HREF="node8.html">Configuration options</A>
<LI><A NAME="tex2html144"
HREF="node9.html">Bug reporting</A>
<LI><A NAME="tex2html145"
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<LI>Salvatore Filippone, Cranfield University, UK;
</LI>
<LI>Pasqua D'Ambra, IAC-CNR, Naples, IT;
</LI>
<LI>Daniela di Serafino, University of Campania ``L. Vanvitelli'', Caserta, IT;
</LI>
<LI>Ambra Abdullahi Hassan, University of Rome ``Tor Vergata'', IT.
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
Contributors to version 1:
</FONT></FONT></FONT>
<UL>
<LI>Salvatore Filippone;
</LI>
<LI>Pasqua D'Ambra;
</LI>
<LI>Daniela di Serafino;
</LI>
<LI>Alfredo Buttari, CNRS-IRIT, Toulouse, F.
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT>
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<H2><A NAME="SECTION00051000000000000000"></A><A NAME="sec:prerequisites"></A>
<H1><A NAME="SECTION00050000000000000000"></A><A NAME="sec:building"></A>
<BR>
Prerequisites
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The following base libraries are needed:
</FONT></FONT></FONT><DL>
<DT><STRONG>BLAS</STRONG></DT>
<DD>[<A
HREF="node29.html#blas3">11</A>,<A
HREF="node29.html#blas2">12</A>,<A
HREF="node29.html#blas1">17</A>] Many vendors provide optimized versions
of BLAS; if no vendor version is
available for a given platform, the ATLAS software
(<TT><A NAME="tex2html1"
HREF="math-atlas.sourceforge.net">math-atlas.sourceforge.net</A></TT>)
may be employed. The reference BLAS from Netlib
(<TT><A NAME="tex2html2"
HREF="www.netlib.org/blas">www.netlib.org/blas</A></TT>) are meant to define the standard
behaviour of the BLAS interface, so they are not optimized for any
particular plaftorm, and should only be used as a last
resort. Note that BLAS computations form a relatively small part of
the MLD2P4/PSBLAS computations; they are however critical when using
preconditioners based on MUMPS, UMFPACK or SuperLU third party
libraries. Note that UMFPACK requires a full LAPACK library; our
experience is that configuring ATLAS for building full LAPACK does not
work in the correct way. Our advice is first to download the LAPACK tarfile from
<TT><A NAME="tex2html3"
HREF="www.netlib.org/lapack">www.netlib.org/lapack</A></TT> and install it independently of ATLAS. In this case,
you need to modify the OPTS and NOOPT definitions for including -fPIC compilation option
in the make.inc file of the LAPACK library.
</DD>
<DT><STRONG>MPI</STRONG></DT>
<DD>[<A
HREF="node29.html#MPI2">16</A>,<A
HREF="node29.html#MPI1">22</A>] A version of MPI is available on most
high-performance computing systems.
</DD>
<DT><STRONG>PSBLAS</STRONG></DT>
<DD>[<A
HREF="node29.html#PSBLASGUIDE">13</A>,<A
HREF="node29.html#psblas_00">15</A>] Parallel Sparse BLAS (PSBLAS) is
available from <TT><A NAME="tex2html4"
HREF="www.ce.uniroma2.it/psblas">www.ce.uniroma2.it/psblas</A></TT>; version
3.5.0 (or later) is required. Indeed, all the prerequisites
listed so far are also prerequisites of PSBLAS.
</DD>
</DL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
Please note that the four previous libraries must have Fortran
interfaces compatible with MLD2P4;
usually this means that they should all be built with the same
compiler as MLD2P4.
Configuring and Building MLD2P4
</H1><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
In order to build MLD2P4 it is necessary to set up a Makefile with appropriate
system-dependent variables; this is done by means of the <code>configure</code>
script. The distribution also includes the autoconf and automake
sources employed to generate the script, but usually this is not needed
to build the software.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">MLD2P4 is implemented almost entirely in Fortran&nbsp;2003, with some
interfaces to external libraries in C; the Fortran compiler
must support the Fortran&nbsp;2003 standard plus the extension <code>MOLD=</code>
feature, which enhances the usability of <code>ALLOCATE</code>.
Many compilers do this; in particular, this is
supported by the GNU Fortran compiler, for which we
recommend to use at least version 4.8.
The software defines data types and interfaces for
real and complex data, in both single and double precision.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Building MLD2P4 requires some base libraries (see Section&nbsp;<A HREF="node7.html#sec:prerequisites">3.1</A>);
interfaces to optional third-party libraries, which extend the functionalities of MLD2P4
(see Section&nbsp;<A HREF="node8.html#sec:third-party">3.2</A>), are also available. Many Linux distributions
(e.g., Ubuntu, Fedora, CentOS) provide precompiled packages for the prerequisite and
optional software. In many cases these packages are split between a runtime part and a
``developer'' part; in order to build MLD2P4 you need both. A description of the base and
optional software used by MLD2P4 is given in the next sections.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><HR>
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<LI><A NAME="tex2html153"
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<LI><A NAME="tex2html154"
HREF="node8.html">Optional third party libraries</A>
<LI><A NAME="tex2html155"
HREF="node9.html">Configuration options</A>
<LI><A NAME="tex2html156"
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<H2><A NAME="SECTION00051000000000000000"></A><A NAME="sec:prerequisites"></A>
<BR>
Optional third party libraries
Prerequisites
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">We provide interfaces to the following third-party software libraries;
note that these are optional, but if you enable them some defaults
for multi-level preconditioners may change to reflect their presence.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><DL>
<DT><STRONG>UMFPACK</STRONG></DT>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The following base libraries are needed:
</FONT></FONT></FONT><DL>
<DT><STRONG>BLAS</STRONG></DT>
<DD>[<A
HREF="node29.html#UMFPACK">9</A>]
A sparse LU factorization package included in the SuiteSparse library, available from
<TT><A NAME="tex2html5"
HREF="faculty.cse.tamu.edu/davis/suitesparse.html">faculty.cse.tamu.edu/davis/suitesparse.html</A></TT>;
it provides sequential factorization and triangular system solution for double
precision real and complex data. We tested version 4.5.4 of SuiteSparse.
Note that for configuring SuiteSparse you should provide the right path to the BLAS
and LAPACK libraries in the <code>SuiteSparse_config/SuiteSparse_config.mk</code> file.
HREF="node30.html#blas3">11</A>,<A
HREF="node30.html#blas2">12</A>,<A
HREF="node30.html#blas1">17</A>] Many vendors provide optimized versions
of BLAS; if no vendor version is
available for a given platform, the ATLAS software
(<TT><A NAME="tex2html1"
HREF="math-atlas.sourceforge.net">math-atlas.sourceforge.net</A></TT>)
may be employed. The reference BLAS from Netlib
(<TT><A NAME="tex2html2"
HREF="www.netlib.org/blas">www.netlib.org/blas</A></TT>) are meant to define the standard
behaviour of the BLAS interface, so they are not optimized for any
particular plaftorm, and should only be used as a last
resort. Note that BLAS computations form a relatively small part of
the MLD2P4/PSBLAS computations; they are however critical when using
preconditioners based on MUMPS, UMFPACK or SuperLU third party
libraries. Note that UMFPACK requires a full LAPACK library; our
experience is that configuring ATLAS for building full LAPACK does not
work in the correct way. Our advice is first to download the LAPACK tarfile from
<TT><A NAME="tex2html3"
HREF="www.netlib.org/lapack">www.netlib.org/lapack</A></TT> and install it independently of ATLAS. In this case,
you need to modify the OPTS and NOOPT definitions for including -fPIC compilation option
in the make.inc file of the LAPACK library.
</DD>
<DT><STRONG>MUMPS</STRONG></DT>
<DT><STRONG>MPI</STRONG></DT>
<DD>[<A
HREF="node29.html#MUMPS">1</A>]
A sparse LU factorization package available from <TT><A NAME="tex2html6"
HREF="mumps.enseeiht.fr">mumps.enseeiht.fr</A></TT>;
it provides sequential and parallel factorizations and triangular system solution
for single and double precision, real and complex data.
We tested versions 4.10.0 and 5.0.1.
</DD>
<DT><STRONG>SuperLU</STRONG></DT>
<DD>[<A
HREF="node29.html#SUPERLU">10</A>]
A sparse LU factorization package available from
<TT><A NAME="tex2html7"
HREF="crd.lbl.gov/~xiaoye/SuperLU/">crd.lbl.gov/~xiaoye/SuperLU/</A></TT>; it provides sequential
factorization and triangular system solution for single and double precision,
real and complex data. We tested versions 4.3 and 5.0. If you installed BLAS from
ATLAS, remember to define the BLASLIB variable in the make.inc file.
HREF="node30.html#MPI2">16</A>,<A
HREF="node30.html#MPI1">22</A>] A version of MPI is available on most
high-performance computing systems.
</DD>
<DT><STRONG>SuperLU_Dist</STRONG></DT>
<DT><STRONG>PSBLAS</STRONG></DT>
<DD>[<A
HREF="node29.html#SUPERLUDIST">18</A>]
A sparse LU factorization package available
from the same site as SuperLU; it provides parallel factorization and
triangular system solution for double precision real and complex data.
We tested versions 3.3 and 4.2. If you installed BLAS from
ATLAS, remember to define the BLASLIB variable in the make.inc file and
to add the <code>-std=c99</code> option to the C compiler options.
Note that this library requires the ParMETIS
library for parallel graph partitioning and fill-reducing matrix ordering, available from
<TT><A NAME="tex2html8"
HREF="glaros.dtc.umn.edu/gkhome/metis/parmetis/overview">glaros.dtc.umn.edu/gkhome/metis/parmetis/overview</A></TT>.
HREF="node30.html#PSBLASGUIDE">13</A>,<A
HREF="node30.html#psblas_00">15</A>] Parallel Sparse BLAS (PSBLAS) is
available from <TT><A NAME="tex2html4"
HREF="www.ce.uniroma2.it/psblas">www.ce.uniroma2.it/psblas</A></TT>; version
3.5.0 (or later) is required. Indeed, all the prerequisites
listed so far are also prerequisites of PSBLAS.
</DD>
</DL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
</DL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
Please note that the four previous libraries must have Fortran
interfaces compatible with MLD2P4;
usually this means that they should all be built with the same
compiler as MLD2P4.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
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<H2><A NAME="SECTION00053000000000000000">
Configuration options</A>
<H2><A NAME="SECTION00052000000000000000"></A><A NAME="sec:third-party"></A>
<BR>
Optional third party libraries
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to build MLD2P4, the first step is to use the <code>configure</code> script
in the main directory to generate the necessary makefile.
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">We provide interfaces to the following third-party software libraries;
note that these are optional, but if you enable them some defaults
for multi-level preconditioners may change to reflect their presence.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">As a minimal example consider the following:
</FONT></FONT></FONT><PRE>
./configure --with-psblas=PSB-INSTALL-DIR
</PRE><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
which assumes that the various MPI compilers and support libraries are
available in the standard directories on the system, and specifies
only the PSBLAS install directory (note that the latter directory must
be specified with an <EM>absolute</EM> path).
The full set of options may be looked at by issuing the command
<code>./configure --help</code>, which produces:
</FONT></FONT></FONT><PRE>
`configure' configures MLD2P4 2.1 to adapt to many kinds of systems.
Usage: ./configure [OPTION]... [VAR=VALUE]...
To assign environment variables (e.g., CC, CFLAGS...), specify them as
VAR=VALUE. See below for descriptions of some of the useful variables.
Defaults for the options are specified in brackets.
Configuration:
-h, --help display this help and exit
--help=short display options specific to this package
--help=recursive display the short help of all the included packages
-V, --version display version information and exit
-q, --quiet, --silent do not print `checking...' messages
--cache-file=FILE cache test results in FILE [disabled]
-C, --config-cache alias for `--cache-file=config.cache'
-n, --no-create do not create output files
--srcdir=DIR find the sources in DIR [configure dir or `..']
Installation directories:
--prefix=PREFIX install architecture-independent files in PREFIX
[/usr/local]
--exec-prefix=EPREFIX install architecture-dependent files in EPREFIX
[PREFIX]
By default, `make install' will install all the files in
`/usr/local/bin', `/usr/local/lib' etc. You can specify
an installation prefix other than `/usr/local' using `--prefix',
for instance `--prefix=$HOME'.
For better control, use the options below.
Fine tuning of the installation directories:
--bindir=DIR user executables [EPREFIX/bin]
--sbindir=DIR system admin executables [EPREFIX/sbin]
--libexecdir=DIR program executables [EPREFIX/libexec]
--sysconfdir=DIR read-only single-machine data [PREFIX/etc]
--sharedstatedir=DIR modifiable architecture-independent data [PREFIX/com]
--localstatedir=DIR modifiable single-machine data [PREFIX/var]
--libdir=DIR object code libraries [EPREFIX/lib]
--includedir=DIR C header files [PREFIX/include]
--oldincludedir=DIR C header files for non-gcc [/usr/include]
--datarootdir=DIR read-only arch.-independent data root [PREFIX/share]
--datadir=DIR read-only architecture-independent data [DATAROOTDIR]
--infodir=DIR info documentation [DATAROOTDIR/info]
--localedir=DIR locale-dependent data [DATAROOTDIR/locale]
--mandir=DIR man documentation [DATAROOTDIR/man]
--docdir=DIR documentation root [DATAROOTDIR/doc/mld2p4]
--htmldir=DIR html documentation [DOCDIR]
--dvidir=DIR dvi documentation [DOCDIR]
--pdfdir=DIR pdf documentation [DOCDIR]
--psdir=DIR ps documentation [DOCDIR]
Program names:
--program-prefix=PREFIX prepend PREFIX to installed program names
--program-suffix=SUFFIX append SUFFIX to installed program names
--program-transform-name=PROGRAM run sed PROGRAM on installed program names
Optional Features:
--disable-option-checking ignore unrecognized --enable/--with options
--disable-FEATURE do not include FEATURE (same as --enable-FEATURE=no)
--enable-FEATURE[=ARG] include FEATURE [ARG=yes]
--disable-dependency-tracking speeds up one-time build
--enable-dependency-tracking do not reject slow dependency extractors
--enable-serial Specify whether to enable a fake mpi library to run
in serial mode.
--enable-long-integers Specify usage of 64 bits integers.
Optional Packages:
--with-PACKAGE[=ARG] use PACKAGE [ARG=yes]
--without-PACKAGE do not use PACKAGE (same as --with-PACKAGE=no)
--with-psblas=DIR The install directory for PSBLAS, for example,
--with-psblas=/opt/packages/psblas-3.5
--with-psblas-incdir=DIR
Specify the directory for PSBLAS includes.
--with-psblas-libdir=DIR
Specify the directory for PSBLAS library.
--with-ccopt additional CCOPT flags to be added: will prepend
to CCOPT
--with-fcopt additional FCOPT flags to be added: will prepend
to FCOPT
--with-libs List additional link flags here. For example,
--with-libs=-lspecial_system_lib or
--with-libs=-L/path/to/libs
--with-clibs additional CLIBS flags to be added: will prepend
to CLIBS
--with-flibs additional FLIBS flags to be added: will prepend
to FLIBS
--with-library-path additional LIBRARYPATH flags to be added: will
prepend to LIBRARYPATH
--with-include-path additional INCLUDEPATH flags to be added: will
prepend to INCLUDEPATH
--with-module-path additional MODULE_PATH flags to be added: will
prepend to MODULE_PATH
--with-extra-libs List additional link flags here. For example,
--with-extra-libs=-lspecial_system_lib or
--with-extra-libs=-L/path/to/libs
--with-blas=&lt;lib&gt; use BLAS library &lt;lib&gt;
--with-blasdir=&lt;dir&gt; search for BLAS library in &lt;dir&gt;
--with-lapack=&lt;lib&gt; use LAPACK library &lt;lib&gt;
--with-mumps=LIBNAME Specify the libname for MUMPS. Default: autodetect
with minimum "-lmumps_common -lpord"
--with-mumpsdir=DIR Specify the directory for MUMPS library and
includes. Note: you will need to add auxiliary
libraries with --extra-libs; this depends on how
MUMPS was configured and installed, at a minimum you
will need SCALAPACK and BLAS
--with-mumpsincdir=DIR Specify the directory for MUMPS includes.
--with-mumpslibdir=DIR Specify the directory for MUMPS library.
--with-umfpack=LIBNAME Specify the library name for UMFPACK and its support
libraries. Default: "-lumfpack -lamd"
--with-umfpackdir=DIR Specify the directory for UMFPACK library and
includes.
--with-umfpackincdir=DIR
Specify the directory for UMFPACK includes.
--with-umfpacklibdir=DIR
Specify the directory for UMFPACK library.
--with-superlu=LIBNAME Specify the library name for SUPERLU library.
Default: "-lsuperlu"
--with-superludir=DIR Specify the directory for SUPERLU library and
includes.
--with-superluincdir=DIR
Specify the directory for SUPERLU includes.
--with-superlulibdir=DIR
Specify the directory for SUPERLU library.
--with-superludist=LIBNAME
Specify the libname for SUPERLUDIST library.
Requires you also specify SuperLU. Default:
"-lsuperlu_dist"
--with-superludistdir=DIR
Specify the directory for SUPERLUDIST library and
includes.
--with-superludistincdir=DIR
Specify the directory for SUPERLUDIST includes.
--with-superludistlibdir=DIR
Specify the directory for SUPERLUDIST library.
Some influential environment variables:
FC Fortran compiler command
FCFLAGS Fortran compiler flags
LDFLAGS linker flags, e.g. -L&lt;lib dir&gt; if you have libraries in a
nonstandard directory &lt;lib dir&gt;
LIBS libraries to pass to the linker, e.g. -l&lt;library&gt;
CC C compiler command
CFLAGS C compiler flags
CPPFLAGS C/C++/Objective C preprocessor flags, e.g. -I&lt;include dir&gt; if
you have headers in a nonstandard directory &lt;include dir&gt;
MPICC MPI C compiler command
MPIFC MPI Fortran compiler command
CPP C preprocessor
Use these variables to override the choices made by `configure' or to help
it to find libraries and programs with nonstandard names/locations.
Report bugs to &lt;bugreport@mld2p4.it&gt;.
</PRE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><DL>
<DT><STRONG>UMFPACK</STRONG></DT>
<DD>[<A
HREF="node30.html#UMFPACK">9</A>]
A sparse LU factorization package included in the SuiteSparse library, available from
<TT><A NAME="tex2html5"
HREF="faculty.cse.tamu.edu/davis/suitesparse.html">faculty.cse.tamu.edu/davis/suitesparse.html</A></TT>;
it provides sequential factorization and triangular system solution for double
precision real and complex data. We tested version 4.5.4 of SuiteSparse.
Note that for configuring SuiteSparse you should provide the right path to the BLAS
and LAPACK libraries in the <code>SuiteSparse_config/SuiteSparse_config.mk</code> file.
</DD>
<DT><STRONG>MUMPS</STRONG></DT>
<DD>[<A
HREF="node30.html#MUMPS">1</A>]
A sparse LU factorization package available from <TT><A NAME="tex2html6"
HREF="mumps.enseeiht.fr">mumps.enseeiht.fr</A></TT>;
it provides sequential and parallel factorizations and triangular system solution
for single and double precision, real and complex data.
We tested versions 4.10.0 and 5.0.1.
</DD>
<DT><STRONG>SuperLU</STRONG></DT>
<DD>[<A
HREF="node30.html#SUPERLU">10</A>]
A sparse LU factorization package available from
<TT><A NAME="tex2html7"
HREF="crd.lbl.gov/~xiaoye/SuperLU/">crd.lbl.gov/~xiaoye/SuperLU/</A></TT>; it provides sequential
factorization and triangular system solution for single and double precision,
real and complex data. We tested versions 4.3 and 5.0. If you installed BLAS from
ATLAS, remember to define the BLASLIB variable in the make.inc file.
</DD>
<DT><STRONG>SuperLU_Dist</STRONG></DT>
<DD>[<A
HREF="node30.html#SUPERLUDIST">18</A>]
A sparse LU factorization package available
from the same site as SuperLU; it provides parallel factorization and
triangular system solution for double precision real and complex data.
We tested versions 3.3 and 4.2. If you installed BLAS from
ATLAS, remember to define the BLASLIB variable in the make.inc file and
to add the <code>-std=c99</code> option to the C compiler options.
Note that this library requires the ParMETIS
library for parallel graph partitioning and fill-reducing matrix ordering, available from
<TT><A NAME="tex2html8"
HREF="glaros.dtc.umn.edu/gkhome/metis/parmetis/overview">glaros.dtc.umn.edu/gkhome/metis/parmetis/overview</A></TT>.
</DD>
</DL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">For instance, if a user has built and installed PSBLAS 3.5 under the
<code>/opt</code> directory and is
using the SuiteSparse package (which includes UMFPACK), then MLD2P4
might be configured with:
</FONT></FONT></FONT><PRE>
./configure --with-psblas=/opt/psblas-3.5/ \
--with-umfpackincdir=/usr/include/suitesparse/
</PRE><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
Once the configure script has completed execution, it will have
generated the file <code>Make.inc</code> which will then be used by all
Makefiles in the directory tree; this file will be copied in the
install directory under the name <code>Make.inc.MLD2P4</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">To use the MUMPS solver package,
the user has to add the appropriate options to the configure script;
by default we are looking for the libraries
<code>-ldmumps -lsmumps</code> <code> -lzmumps -lcmumps -mumps_common -lpord</code>.
MUMPS often uses additional packages such as ScaLAPACK, ParMETIS,
SCOTCH, as well as enabling OpenMP; in such cases it is necessary to
add linker options with the <code>--with-extra-libs</code> configure option.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">To build the library the user will now enter
</FONT></FONT></FONT><PRE>
make
</PRE><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
followed (optionally) by
</FONT></FONT></FONT><PRE>
make install
</PRE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
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<H2><A NAME="SECTION00054000000000000000">
Bug reporting</A>
</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
If you find any bugs in our codes, please send an email to <BR><TT>pasqua.dambra@cnr.it</TT>
<BR><TT>daniela.diserafino@unicampania.it</TT>
<BR><TT>salvatore.filippone@cranfield.ac.uk</TT> <BR>
You should be aware that the amount of information needed to reproduce a problem
in a parallel program may vary quite a lot.
<H2><A NAME="SECTION00053000000000000000">
Configuration options</A>
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to build MLD2P4, the first step is to use the <code>configure</code> script
in the main directory to generate the necessary makefile.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<BR><HR>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">As a minimal example consider the following:
</FONT></FONT></FONT><PRE>
./configure --with-psblas=PSB-INSTALL-DIR
</PRE><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
which assumes that the various MPI compilers and support libraries are
available in the standard directories on the system, and specifies
only the PSBLAS install directory (note that the latter directory must
be specified with an <EM>absolute</EM> path).
The full set of options may be looked at by issuing the command
<code>./configure --help</code>, which produces:
</FONT></FONT></FONT><PRE>
`configure' configures MLD2P4 2.1 to adapt to many kinds of systems.
Usage: ./configure [OPTION]... [VAR=VALUE]...
To assign environment variables (e.g., CC, CFLAGS...), specify them as
VAR=VALUE. See below for descriptions of some of the useful variables.
Defaults for the options are specified in brackets.
Configuration:
-h, --help display this help and exit
--help=short display options specific to this package
--help=recursive display the short help of all the included packages
-V, --version display version information and exit
-q, --quiet, --silent do not print `checking...' messages
--cache-file=FILE cache test results in FILE [disabled]
-C, --config-cache alias for `--cache-file=config.cache'
-n, --no-create do not create output files
--srcdir=DIR find the sources in DIR [configure dir or `..']
Installation directories:
--prefix=PREFIX install architecture-independent files in PREFIX
[/usr/local]
--exec-prefix=EPREFIX install architecture-dependent files in EPREFIX
[PREFIX]
By default, `make install' will install all the files in
`/usr/local/bin', `/usr/local/lib' etc. You can specify
an installation prefix other than `/usr/local' using `--prefix',
for instance `--prefix=$HOME'.
For better control, use the options below.
Fine tuning of the installation directories:
--bindir=DIR user executables [EPREFIX/bin]
--sbindir=DIR system admin executables [EPREFIX/sbin]
--libexecdir=DIR program executables [EPREFIX/libexec]
--sysconfdir=DIR read-only single-machine data [PREFIX/etc]
--sharedstatedir=DIR modifiable architecture-independent data [PREFIX/com]
--localstatedir=DIR modifiable single-machine data [PREFIX/var]
--libdir=DIR object code libraries [EPREFIX/lib]
--includedir=DIR C header files [PREFIX/include]
--oldincludedir=DIR C header files for non-gcc [/usr/include]
--datarootdir=DIR read-only arch.-independent data root [PREFIX/share]
--datadir=DIR read-only architecture-independent data [DATAROOTDIR]
--infodir=DIR info documentation [DATAROOTDIR/info]
--localedir=DIR locale-dependent data [DATAROOTDIR/locale]
--mandir=DIR man documentation [DATAROOTDIR/man]
--docdir=DIR documentation root [DATAROOTDIR/doc/mld2p4]
--htmldir=DIR html documentation [DOCDIR]
--dvidir=DIR dvi documentation [DOCDIR]
--pdfdir=DIR pdf documentation [DOCDIR]
--psdir=DIR ps documentation [DOCDIR]
Program names:
--program-prefix=PREFIX prepend PREFIX to installed program names
--program-suffix=SUFFIX append SUFFIX to installed program names
--program-transform-name=PROGRAM run sed PROGRAM on installed program names
Optional Features:
--disable-option-checking ignore unrecognized --enable/--with options
--disable-FEATURE do not include FEATURE (same as --enable-FEATURE=no)
--enable-FEATURE[=ARG] include FEATURE [ARG=yes]
--disable-dependency-tracking speeds up one-time build
--enable-dependency-tracking do not reject slow dependency extractors
--enable-serial Specify whether to enable a fake mpi library to run
in serial mode.
--enable-long-integers Specify usage of 64 bits integers.
Optional Packages:
--with-PACKAGE[=ARG] use PACKAGE [ARG=yes]
--without-PACKAGE do not use PACKAGE (same as --with-PACKAGE=no)
--with-psblas=DIR The install directory for PSBLAS, for example,
--with-psblas=/opt/packages/psblas-3.5
--with-psblas-incdir=DIR
Specify the directory for PSBLAS includes.
--with-psblas-libdir=DIR
Specify the directory for PSBLAS library.
--with-ccopt additional CCOPT flags to be added: will prepend
to CCOPT
--with-fcopt additional FCOPT flags to be added: will prepend
to FCOPT
--with-libs List additional link flags here. For example,
--with-libs=-lspecial_system_lib or
--with-libs=-L/path/to/libs
--with-clibs additional CLIBS flags to be added: will prepend
to CLIBS
--with-flibs additional FLIBS flags to be added: will prepend
to FLIBS
--with-library-path additional LIBRARYPATH flags to be added: will
prepend to LIBRARYPATH
--with-include-path additional INCLUDEPATH flags to be added: will
prepend to INCLUDEPATH
--with-module-path additional MODULE_PATH flags to be added: will
prepend to MODULE_PATH
--with-extra-libs List additional link flags here. For example,
--with-extra-libs=-lspecial_system_lib or
--with-extra-libs=-L/path/to/libs
--with-blas=&lt;lib&gt; use BLAS library &lt;lib&gt;
--with-blasdir=&lt;dir&gt; search for BLAS library in &lt;dir&gt;
--with-lapack=&lt;lib&gt; use LAPACK library &lt;lib&gt;
--with-mumps=LIBNAME Specify the libname for MUMPS. Default: autodetect
with minimum "-lmumps_common -lpord"
--with-mumpsdir=DIR Specify the directory for MUMPS library and
includes. Note: you will need to add auxiliary
libraries with --extra-libs; this depends on how
MUMPS was configured and installed, at a minimum you
will need SCALAPACK and BLAS
--with-mumpsincdir=DIR Specify the directory for MUMPS includes.
--with-mumpslibdir=DIR Specify the directory for MUMPS library.
--with-umfpack=LIBNAME Specify the library name for UMFPACK and its support
libraries. Default: "-lumfpack -lamd"
--with-umfpackdir=DIR Specify the directory for UMFPACK library and
includes.
--with-umfpackincdir=DIR
Specify the directory for UMFPACK includes.
--with-umfpacklibdir=DIR
Specify the directory for UMFPACK library.
--with-superlu=LIBNAME Specify the library name for SUPERLU library.
Default: "-lsuperlu"
--with-superludir=DIR Specify the directory for SUPERLU library and
includes.
--with-superluincdir=DIR
Specify the directory for SUPERLU includes.
--with-superlulibdir=DIR
Specify the directory for SUPERLU library.
--with-superludist=LIBNAME
Specify the libname for SUPERLUDIST library.
Requires you also specify SuperLU. Default:
"-lsuperlu_dist"
--with-superludistdir=DIR
Specify the directory for SUPERLUDIST library and
includes.
--with-superludistincdir=DIR
Specify the directory for SUPERLUDIST includes.
--with-superludistlibdir=DIR
Specify the directory for SUPERLUDIST library.
Some influential environment variables:
FC Fortran compiler command
FCFLAGS Fortran compiler flags
LDFLAGS linker flags, e.g. -L&lt;lib dir&gt; if you have libraries in a
nonstandard directory &lt;lib dir&gt;
LIBS libraries to pass to the linker, e.g. -l&lt;library&gt;
CC C compiler command
CFLAGS C compiler flags
CPPFLAGS C/C++/Objective C preprocessor flags, e.g. -I&lt;include dir&gt; if
you have headers in a nonstandard directory &lt;include dir&gt;
MPICC MPI C compiler command
MPIFC MPI Fortran compiler command
CPP C preprocessor
Use these variables to override the choices made by `configure' or to help
it to find libraries and programs with nonstandard names/locations.
Report bugs to &lt;bugreport@mld2p4.it&gt;.
</PRE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">For instance, if a user has built and installed PSBLAS 3.5 under the
<code>/opt</code> directory and is
using the SuiteSparse package (which includes UMFPACK), then MLD2P4
might be configured with:
</FONT></FONT></FONT><PRE>
./configure --with-psblas=/opt/psblas-3.5/ \
--with-umfpackincdir=/usr/include/suitesparse/
</PRE><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
Once the configure script has completed execution, it will have
generated the file <code>Make.inc</code> which will then be used by all
Makefiles in the directory tree; this file will be copied in the
install directory under the name <code>Make.inc.MLD2P4</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">To use the MUMPS solver package,
the user has to add the appropriate options to the configure script;
by default we are looking for the libraries
<code>-ldmumps -lsmumps</code> <code> -lzmumps -lcmumps -mumps_common -lpord</code>.
MUMPS often uses additional packages such as ScaLAPACK, ParMETIS,
SCOTCH, as well as enabling OpenMP; in such cases it is necessary to
add linker options with the <code>--with-extra-libs</code> configure option.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">To build the library the user will now enter
</FONT></FONT></FONT><PRE>
make
</PRE><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
followed (optionally) by
</FONT></FONT></FONT><PRE>
make install
</PRE><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
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HREF="node6.html">Configuring and Building MLD2P4</A>
<UL>
<LI><A NAME="tex2html32"
HREF="node7.html">Optional third party libraries</A>
HREF="node7.html">Prerequisites</A>
<LI><A NAME="tex2html33"
HREF="node8.html">Configuration options</A>
HREF="node8.html">Optional third party libraries</A>
<LI><A NAME="tex2html34"
HREF="node9.html">Bug reporting</A>
HREF="node9.html">Configuration options</A>
<LI><A NAME="tex2html35"
HREF="node10.html">Example and test programs</A>
</UL>
<BR>
HREF="node10.html">Bug reporting</A>
<LI><A NAME="tex2html36"
HREF="node11.html">Multigrid Background</A>
<UL>
HREF="node11.html">Example and test programs</A>
</UL>
<BR>
<LI><A NAME="tex2html37"
HREF="node12.html">AMG preconditioners</A>
HREF="node12.html">Multigrid Background</A>
<UL>
<LI><A NAME="tex2html38"
HREF="node13.html">Smoothed Aggregation</A>
HREF="node13.html">AMG preconditioners</A>
<LI><A NAME="tex2html39"
HREF="node14.html">Smoothers and coarsest-level solvers</A>
</UL>
<BR>
HREF="node14.html">Smoothed Aggregation</A>
<LI><A NAME="tex2html40"
HREF="node15.html">Getting Started</A>
<UL>
HREF="node15.html">Smoothers and coarsest-level solvers</A>
</UL>
<BR>
<LI><A NAME="tex2html41"
HREF="node16.html">Examples</A>
</UL>
<BR>
<LI><A NAME="tex2html42"
HREF="node17.html">User Interface</A>
HREF="node16.html">Getting Started</A>
<UL>
<LI><A NAME="tex2html43"
HREF="node18.html">Subroutine init</A>
<LI><A NAME="tex2html44"
HREF="node19.html">Subroutine set</A>
<LI><A NAME="tex2html45"
HREF="node20.html">Subroutine build</A>
<LI><A NAME="tex2html46"
HREF="node21.html">Subroutine hierarchy_build</A>
<LI><A NAME="tex2html47"
HREF="node22.html">Subroutine smoothers_build</A>
<LI><A NAME="tex2html48"
HREF="node23.html">Subroutine apply</A>
<LI><A NAME="tex2html49"
HREF="node24.html">Subroutine free</A>
<LI><A NAME="tex2html50"
HREF="node25.html">Subroutine descr</A>
<LI><A NAME="tex2html42"
HREF="node17.html">Examples</A>
</UL>
<BR>
<LI><A NAME="tex2html43"
HREF="node18.html">User Interface</A>
<UL>
<LI><A NAME="tex2html44"
HREF="node19.html">Subroutine init</A>
<LI><A NAME="tex2html45"
HREF="node20.html">Subroutine set</A>
<LI><A NAME="tex2html46"
HREF="node21.html">Subroutine build</A>
<LI><A NAME="tex2html47"
HREF="node22.html">Subroutine hierarchy_build</A>
<LI><A NAME="tex2html48"
HREF="node23.html">Subroutine smoothers_build</A>
<LI><A NAME="tex2html49"
HREF="node24.html">Subroutine apply</A>
<LI><A NAME="tex2html50"
HREF="node25.html">Subroutine free</A>
<LI><A NAME="tex2html51"
HREF="node26.html">Adding new smoother and solver objects to MLD2P4</A>
HREF="node26.html">Subroutine descr</A>
</UL>
<BR>
<LI><A NAME="tex2html52"
HREF="node27.html">Error Handling</A>
HREF="node27.html">Adding new smoother and solver objects to MLD2P4</A>
<LI><A NAME="tex2html53"
HREF="node28.html">License</A>
HREF="node28.html">Error Handling</A>
<LI><A NAME="tex2html54"
HREF="node29.html">Bibliography</A>
HREF="node29.html">License</A>
<LI><A NAME="tex2html55"
HREF="node30.html">About this document ...</A>
HREF="node30.html">Bibliography</A>
<LI><A NAME="tex2html56"
HREF="node31.html">About this document ...</A>
</UL>
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(C) Copyright 2008, 2010, 2012, 2017
Salvatore Filippone Cranfield University, Cranfield, UK
Ambra Abdullahi Hassan University of Rome Tor Vergata, Rome, IT
Alfredo Buttari CNRS-IRIT, Toulouse, FR
Pasqua D'Ambra IAC-CNR, Naples, IT
Daniela di Serafino University of Campania L. Vanvitelli, Caserta, IT