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<H2><A NAME="SECTION00082000000000000000"></A><A NAME="sec:precset"></A>
<BR>
Subroutine mld_precset
</H2>
<P>
<DIV ALIGN="CENTER">
<code>mld_precset(p,what,val,info)</code>
<BR>
</DIV>
<P>
This routine sets the parameters defining the preconditioner. More
precisely, the parameter identified by <code>what</code> is assigned the value
contained in <code>val</code>.
<P>
<FONT SIZE="+1"><B>Arguments</B></FONT>
<P>
<TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>p</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>type(mld_</code><I>x</I><code>prec_type), intent(inout)</code>.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The preconditioner data structure. Note that <I>x</I> must
be chosen according to the real/complex, single/double precision
version of MLD2P4 under use.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>what</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer, intent(in)</code>.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The number identifying the parameter to be set.
A mnemonic constant has been associated to each of these
numbers, as reported in Tables&nbsp;<A HREF="#tab:p_type">2</A>-<A HREF="#tab:p_coarse">5</A>.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>val </code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><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>.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>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_type">2</A>-<A HREF="#tab:p_coarse">5</A>.
When the value is of type <code>character(len=*)</code>,
it is also treated as case insensitive.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>info</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer, intent(out)</code>.</TD>
</TR>
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>Error code. If no error, 0 is returned. See Section&nbsp;<A HREF="node22.html#sec:errors">7</A>
for details.</TD>
</TR>
</TABLE>
<P>
<BR>
A variety of (one-level and multi-level) 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
<OL>
<LI>the type of multi-level preconditioner;
</LI>
<LI>the one-level preconditioner used as smoother;
</LI>
<LI>the aggregation algorithm;
</LI>
<LI>the coarse-space correction at the coarsest level.
</LI>
</OL>
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_type">2</A>-<A HREF="#tab:p_coarse">5</A>.
For a detailed description of the meaning of the parameters, please
refer to Section&nbsp;<A HREF="node10.html#sec:background">4</A>.
<P>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1246"></A>
<TABLE>
<CAPTION><STRONG>Table 2:</STRONG>
Parameters defining the type of multi-level preconditioner.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT"><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"><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_ml_type_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57><TT>'ADD'</TT> <TT>'MULT'</TT></TD>
<TD ALIGN="LEFT"><TT>'MULT'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Basic multi-level framework: additive or multiplicative
among the levels (always additive inside a level).</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_smoother_type_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57><TT>'DIAG'</TT> <TT>'BJAC'</TT> <TT>'AS'</TT></TD>
<TD ALIGN="LEFT"><TT>'AS'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Basic one-level preconditioner (i.e. smoother): diagonal,
block Jacobi, AS.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_smoother_pos_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57><TT>'PRE'</TT> <TT>'POST'</TT> <TT>'TWOSIDE'</TT></TD>
<TD ALIGN="LEFT"><TT>'POST'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>``Position'' of the smoother: pre-smoother, post-smoother,
pre- and post-smoother.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR>
<P>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1248"></A>
<TABLE>
<CAPTION><STRONG>Table 3:</STRONG>
Parameters defining the one-level preconditioner used as smoother.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT"><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><code>val</code></TD>
<TD ALIGN="LEFT"><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_sub_ovr_</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>any&nbsp;int.&nbsp;num.&nbsp;<IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img88.png"
ALT="$\ge 0$"></TD>
<TD ALIGN="LEFT">1</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Number of overlap layers.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_sub_restr_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'HALO'</TT> <TT>'NONE'</TT></TD>
<TD ALIGN="LEFT"><TT>'HALO'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Type of restriction operator:
<TT>'HALO'</TT> for taking into account the overlap, <TT>'NONE'</TT>
for neglecting it.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_sub_prol_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'SUM'</TT> <TT>'NONE'</TT></TD>
<TD ALIGN="LEFT"><TT>'NONE'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Type of prolongation operator:
<TT>'SUM'</TT> for adding the contributions from the overlap, <TT>'NONE'</TT>
for neglecting them.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_sub_solve_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'ILU'</TT> <TT>'MILU'</TT> <TT>'ILUT'</TT>
<TT>'UMF'</TT> <TT>'SLU'</TT></TD>
<TD ALIGN="LEFT"><TT>'UMF'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Local solver: ILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img34.png"
ALT="$p$">), MILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img34.png"
ALT="$p$">), ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img35.png"
ALT="$p,t$">), LU from UMFPACK, LU from SuperLU
(plus triangular solve).</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_sub_fillin_</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any&nbsp;int.&nbsp;num.&nbsp;<IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img88.png"
ALT="$\ge 0$"></TD>
<TD ALIGN="LEFT">0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Fill-in level <IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img34.png"
ALT="$p$"> of the incomplete LU factorizations.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_sub_iluthrs_</code></TD>
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any&nbsp;real&nbsp;num.&nbsp;<IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img88.png"
ALT="$\ge 0$"></TD>
<TD ALIGN="LEFT">0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Drop tolerance <IMG
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img89.png"
ALT="$t$"> in the ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img35.png"
ALT="$p,t$">) factorization.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_sub_ren_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'RENUM_NONE'</TT> <TT>'RENUM_GLOBAL'</TT> </TD>
<TD ALIGN="LEFT"><TT>'RENUM_NONE'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Row and column reordering of the local submatrices: no reordering,
reordering according to the global numbering of the rows and columns of
the whole matrix.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR>
<P>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1250"></A>
<TABLE>
<CAPTION><STRONG>Table 4:</STRONG>
Parameters defining the aggregation algorithm.
</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
<TR><TD ALIGN="LEFT"><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=74><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_aggr_alg_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'DEC'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74><TT>'DEC'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Aggregation algorithm. Currently, only the decoupled aggregation is available.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_aggr_kind_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'SMOOTH'</TT> <TT>'RAW'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74><TT>'SMOOTH'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Type of aggregation: smoothed, raw (i.e. using the tentative prolongator).</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_aggr_thresh_</code></TD>
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any&nbsp;real&nbsp;num. <IMG
WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img90.png"
ALT="$\in [0, 1]$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74>0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Threshold <IMG
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img91.png"
ALT="$\theta$"> in the aggregation algorithm.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_aggr_eig_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'A_NORMI'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74><TT>'A_NORMI'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Estimate of the eigenvalue <IMG
WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img85.png"
ALT="$D^{-1}A$"> with largest modulus,
to build the damping parameter <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img84.png"
ALT="$\omega$"> in the smoothed aggregation.
Currently, only the infinity norm of
the matrix is available.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_aggr_damp_</code></TD>
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any&nbsp;real&nbsp;num.</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74><!-- MATH
$4/(3||D^{-1}A||_\infty)$
-->
<IMG
WIDTH="124" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img92.png"
ALT="$4/(3\vert\vert D^{-1}A\vert\vert _\infty)$"></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Damping parameter <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img84.png"
ALT="$\omega$"> in the smoothed aggregation algorithm.
If the user specifies a negative value, then <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img84.png"
ALT="$\omega$">
is set to its default value;
otherwise, <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img84.png"
ALT="$\omega$"> is set to the value provided by the
user. In the latter case no estimate of the eigenvalue of
<IMG
WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img85.png"
ALT="$D^{-1}A$"> with largest modulus is computed.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
</TABLE>
</DIV><P></P>
<BR>
<P>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="1252"></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"><code>what</code></TD>
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><code>val</code></TD>
<TD ALIGN="LEFT"><SMALL>DEFAULT</SMALL></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198><SMALL>COMMENTS</SMALL></TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_coarse_mat_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'DISTR'</TT> <TT>'REPL'</TT></TD>
<TD ALIGN="LEFT"><TT>'DISTR'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Coarsest matrix: distributed among the processors or
replicated on each of them.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_coarse_solve_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'BJAC'</TT> <TT>'UMF'</TT>
<TT>'SLU'</TT> <TT>'SLUDIST'</TT></TD>
<TD ALIGN="LEFT"><TT>'BJAC'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Solver used at the coarsest level: block Jacobi, sequential
LU from UMFPACK, sequential LU from SuperLU,
distributed LU from SuperLU_Dist.
<TT>'BJAC'</TT> and <TT>'SLUDIST'</TT> require the coarsest
matrix to be distributed, while <TT>'UMF'</TT> and
<TT>'SLU'</TT> require it to be replicated.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_coarse_subsolve_</code></TD>
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'ILU'</TT> <TT>'MILU'</TT>
<TT>'ILUT'</TT>
<TT>'UMF'</TT> <TT>'SLU'</TT></TD>
<TD ALIGN="LEFT"><TT>'UMF'</TT></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>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="img34.png"
ALT="$p$">), MILU(<IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img34.png"
ALT="$p$">),
ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img35.png"
ALT="$p,t$">), LU from UMFPACK,
LU from SuperLU, plus triangular solve.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_coarse_sweeps_</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any&nbsp;int.&nbsp;num.&nbsp;<IMG
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img93.png"
ALT="$&gt; 0$"></TD>
<TD ALIGN="LEFT">4</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Number of Block-Jacobi sweeps when 'BJAC' is used as
coarsest-level solver.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_coarse_fillin_</code></TD>
<TD ALIGN="LEFT"><code>integer</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any&nbsp;int.&nbsp;num.&nbsp;<IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img88.png"
ALT="$\ge 0$"></TD>
<TD ALIGN="LEFT">0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Fill-in level <IMG
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img34.png"
ALT="$p$"> of the incomplete LU factorizations.</TD>
</TR>
<TR><TD ALIGN="LEFT"><code>mld_coarse_iluthrs_</code></TD>
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any&nbsp;real.&nbsp;num.&nbsp;<IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img88.png"
ALT="$\ge 0$"></TD>
<TD ALIGN="LEFT">0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Drop tolerance <IMG
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img89.png"
ALT="$t$"> in the ILU(<IMG
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img35.png"
ALT="$p,t$">) factorization.</TD>
</TR>
</TABLE>
</DIV>
</TD></TR>
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