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<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
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<BR>
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Multi-level Schwarz Preconditioners
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</H2>
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<P>
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The Multilevel preconditioners implemented in MLD2P4 are obtained by combining
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AS preconditioners with coarse-space corrections; therefore
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we first provide a sketch of the AS preconditioners.
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<P>
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Given the linear system ,
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where <!-- MATH
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$A=(a_{ij}) \in \Re^{n \times n}$
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-->
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<IMG
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WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img4.png"
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ALT="$A=(a_{ij}) \in \Re^{n \times n}$"> is a
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nonsingular sparse matrix with a symmetric nonzero pattern,
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let <IMG
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WIDTH="92" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$G=(W,E)$"> be the adjacency graph of <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img2.png"
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ALT="$A$">, where <!-- MATH
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$W=\{1, 2, \ldots, n\}$
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-->
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<IMG
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WIDTH="139" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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ALT="$W=\{1, 2, \ldots, n\}$">
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and <!-- MATH
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$E=\{(i,j) : a_{ij} \neq 0\}$
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-->
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<IMG
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WIDTH="162" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img7.png"
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ALT="$E=\{(i,j) : a_{ij} \neq 0\}$"> are the vertex set and the edge set of <IMG
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WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img8.png"
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ALT="$G$">,
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respectively. Two vertices are called adjacent if there is an edge connecting
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them. For any integer <IMG
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WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img9.png"
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ALT="$\delta > 0$">, a <IMG
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WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img10.png"
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ALT="$\delta$">-overlap
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partition of <IMG
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WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img11.png"
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ALT="$W$"> can be defined recursively as follows.
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Given a 0-overlap (or non-overlapping) partition of <IMG
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WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img11.png"
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ALT="$W$">,
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i.e. a set of <IMG
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WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img12.png"
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ALT="$m$"> disjoint nonempty sets <!-- MATH
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$W_i^0 \subset W$
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-->
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<IMG
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WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img13.png"
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ALT="$W_i^0 \subset W$"> such that
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<!-- MATH
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$\cup_{i=1}^m W_i^0 = W$
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-->
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<IMG
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WIDTH="108" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img14.png"
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ALT="$\cup_{i=1}^m W_i^0 = W$">, a <IMG
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WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img10.png"
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ALT="$\delta$">-overlap
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partition of <IMG
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WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img11.png"
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ALT="$W$"> is obtained by considering the sets
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<!-- MATH
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$W_i^\delta \supset W_i^{\delta-1}$
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-->
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<IMG
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WIDTH="97" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
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SRC="img15.png"
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ALT="$W_i^\delta \supset W_i^{\delta-1}$"> obtained by including the vertices that
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are adjacent to any vertex in <!-- MATH
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$W_i^{\delta-1}$
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-->
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<IMG
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WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
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SRC="img16.png"
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ALT="$W_i^{\delta-1}$">.
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<P>
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Let <IMG
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WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.png"
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ALT="$n_i^\delta$"> be the size of <IMG
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WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img18.png"
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ALT="$W_i^\delta$"> and <!-- MATH
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$R_i^{\delta} \in
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\Re^{n_i^\delta \times n}$
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-->
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<IMG
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WIDTH="93" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
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SRC="img19.png"
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ALT="$R_i^{\delta} \in
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\Re^{n_i^\delta \times n}$"> the restriction operator that maps
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a vector <IMG
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WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img20.png"
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ALT="$v \in \Re^n$"> onto the vector <!-- MATH
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$v_i^{\delta} \in \Re^{n_i^\delta}$
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-->
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<IMG
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WIDTH="70" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
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SRC="img21.png"
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ALT="$v_i^{\delta} \in \Re^{n_i^\delta}$">
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containing the components of <IMG
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WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img22.png"
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ALT="$v$"> corresponding to the vertices in
|
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<IMG
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WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img18.png"
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ALT="$W_i^\delta$">. The transpose of <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img23.png"
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ALT="$R_i^{\delta}$"> is a
|
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prolongation operator from <!-- MATH
|
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$\Re^{n_i^\delta}$
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-->
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<IMG
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WIDTH="33" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
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SRC="img24.png"
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ALT="$\Re^{n_i^\delta}$"> to <IMG
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WIDTH="26" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$\Re^n$">.
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The matrix <!-- MATH
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$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
|
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\Re^{n_i^\delta \times n_i^\delta}$
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-->
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<IMG
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WIDTH="201" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
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SRC="img26.png"
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ALT="$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
|
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\Re^{n_i^\delta \times n_i^\delta}$"> can be considered
|
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as a restriction of <IMG
|
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img2.png"
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ALT="$A$"> corresponding to the set <IMG
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WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img27.png"
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ALT="$W_i^{\delta}$">.
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<P>
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The <I>classical one-level AS</I> preconditioner is defined by
|
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<BR><P></P>
|
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<DIV ALIGN="CENTER">
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<!-- MATH
|
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\begin{displaymath}
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M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
|
(A_i^\delta)^{-1} R_i^{\delta},
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|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="207" HEIGHT="58" BORDER="0"
|
|
SRC="img28.png"
|
|
ALT="\begin{displaymath}
|
|
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
|
(A_i^\delta)^{-1} R_i^{\delta},
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where <IMG
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img29.png"
|
|
ALT="$A_i^\delta$"> is assumed to be nonsingular. Its application
|
|
to a vector <IMG
|
|
WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img20.png"
|
|
ALT="$v \in \Re^n$"> within a Krylov solver requires the following
|
|
three steps:
|
|
|
|
<OL>
|
|
<LI>restriction of <IMG
|
|
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img22.png"
|
|
ALT="$v$"> as <!-- MATH
|
|
$v_i = R_i^{\delta} v$
|
|
-->
|
|
<IMG
|
|
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img30.png"
|
|
ALT="$v_i = R_i^{\delta} v$">, <IMG
|
|
WIDTH="96" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img31.png"
|
|
ALT="$i=1,\ldots,m$">;
|
|
</LI>
|
|
<LI>solution of the linear systems <!-- MATH
|
|
$A_i^\delta w_i = v_i$
|
|
-->
|
|
<IMG
|
|
WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img32.png"
|
|
ALT="$A_i^\delta w_i = v_i$">,
|
|
<IMG
|
|
WIDTH="96" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img31.png"
|
|
ALT="$i=1,\ldots,m$">;
|
|
</LI>
|
|
<LI>prolongation and sum of the <IMG
|
|
WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img33.png"
|
|
ALT="$w_i$">'s, i.e. <!-- MATH
|
|
$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$
|
|
-->
|
|
<IMG
|
|
WIDTH="145" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img34.png"
|
|
ALT="$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$">.
|
|
</LI>
|
|
</OL>
|
|
Note that the linear systems at step 2 are usually solved approximately,
|
|
e.g. using incomplete LU factorizations such as ILU(<IMG
|
|
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img35.png"
|
|
ALT="$p$">), MILU(<IMG
|
|
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img35.png"
|
|
ALT="$p$">) and
|
|
ILU(<IMG
|
|
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img36.png"
|
|
ALT="$p,t$">) [<A
|
|
HREF="node25.html#Saad_book">21</A>, Chapter 10].
|
|
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<P>
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|
A variant of the classical AS preconditioner that outperforms it
|
|
in terms of convergence rate and of computation and communication
|
|
time on parallel distributed-memory computers is the so-called <I>Restricted AS
|
|
(RAS)</I> preconditioner [<A
|
|
HREF="node25.html#CAI_SARKIS">5</A>,<A
|
|
HREF="node25.html#EFSTATHIOU">14</A>]. It
|
|
is obtained by zeroing the components of <IMG
|
|
WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img33.png"
|
|
ALT="$w_i$"> corresponding to the
|
|
overlapping vertices when applying the prolongation. Therefore,
|
|
RAS differs from classical AS by the prolongation operators,
|
|
which are substituted by <!-- MATH
|
|
$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$
|
|
-->
|
|
<IMG
|
|
WIDTH="117" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img37.png"
|
|
ALT="$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$">,
|
|
where <IMG
|
|
WIDTH="26" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img38.png"
|
|
ALT="$\tilde{R}_i^0$"> is obtained by zeroing the rows of <IMG
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img39.png"
|
|
ALT="$R_i^\delta$">
|
|
corresponding to the vertices in <!-- MATH
|
|
$W_i^\delta \backslash W_i^0$
|
|
-->
|
|
<IMG
|
|
WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img40.png"
|
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ALT="$W_i^\delta \backslash W_i^0$">:
|
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<BR><P></P>
|
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<DIV ALIGN="CENTER">
|
|
<!-- MATH
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|
\begin{displaymath}
|
|
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
|
(A_i^\delta)^{-1} R_i^{\delta}.
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="217" HEIGHT="58" BORDER="0"
|
|
SRC="img41.png"
|
|
ALT="\begin{displaymath}
|
|
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
|
(A_i^\delta)^{-1} R_i^{\delta}.
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
Analogously, the AS variant called <I>AS with Harmonic extension (ASH)</I>
|
|
is defined by
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
|
(A_i^\delta)^{-1} \tilde{R}_i^0.
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="219" HEIGHT="58" BORDER="0"
|
|
SRC="img42.png"
|
|
ALT="\begin{displaymath}M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
|
(A_i^\delta)^{-1} \tilde{R}_i^0.
|
|
\end{displaymath}">
|
|
</DIV>
|
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<BR CLEAR="ALL">
|
|
<P></P>
|
|
We note that for <IMG
|
|
WIDTH="45" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img43.png"
|
|
ALT="$\delta=0$"> the three variants of the AS preconditioner are
|
|
all equal to the block-Jacobi preconditioner.
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<P>
|
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As already observed, the convergence rate of the one-level Schwarz
|
|
preconditioned iterative solvers deteriorates as the number <IMG
|
|
WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img12.png"
|
|
ALT="$m$"> of partitions
|
|
of <IMG
|
|
WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img11.png"
|
|
ALT="$W$"> increases [<A
|
|
HREF="node25.html#dd1_94">7</A>,<A
|
|
HREF="node25.html#dd2_96">22</A>]. To reduce the dependency
|
|
of the number of iterations on the degree of parallelism we may
|
|
introduce a global coupling among the overlapping partitions by defining
|
|
a coarse-space approximation <IMG
|
|
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img44.png"
|
|
ALT="$A_C$"> of the matrix <IMG
|
|
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$A$">.
|
|
In a pure algebraic setting, <IMG
|
|
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img44.png"
|
|
ALT="$A_C$"> is usually built with
|
|
the Galerkin approach. Given a set <IMG
|
|
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img45.png"
|
|
ALT="$W_C$"> of <I>coarse vertices</I>,
|
|
with size <IMG
|
|
WIDTH="26" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img46.png"
|
|
ALT="$n_C$">, and a suitable restriction operator
|
|
<!-- MATH
|
|
$R_C \in \Re^{n_C \times n}$
|
|
-->
|
|
<IMG
|
|
WIDTH="100" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img47.png"
|
|
ALT="$R_C \in \Re^{n_C \times n}$">, <IMG
|
|
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img44.png"
|
|
ALT="$A_C$"> is defined as
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
A_C=R_C A R_C^T
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="109" HEIGHT="31" BORDER="0"
|
|
SRC="img48.png"
|
|
ALT="\begin{displaymath}
|
|
A_C=R_C A R_C^T
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
and the coarse-level correction matrix to be combined with a generic
|
|
one-level AS preconditioner <IMG
|
|
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img49.png"
|
|
ALT="$M_{1L}$"> is obtained as
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="145" HEIGHT="32" BORDER="0"
|
|
SRC="img50.png"
|
|
ALT="\begin{displaymath}
|
|
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where <IMG
|
|
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img44.png"
|
|
ALT="$A_C$"> is assumed to be nonsingular. The application of <IMG
|
|
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img51.png"
|
|
ALT="$M_{C}^{-1}$">
|
|
to a vector <IMG
|
|
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img22.png"
|
|
ALT="$v$"> corresponds to a restriction, a solution and
|
|
a prolongation step; the solution step, involving the matrix <IMG
|
|
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img44.png"
|
|
ALT="$A_C$">,
|
|
may be carried out also approximately.
|
|
|
|
<P>
|
|
The combination of <IMG
|
|
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img52.png"
|
|
ALT="$M_{C}$"> and <IMG
|
|
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img49.png"
|
|
ALT="$M_{1L}$"> may be
|
|
performed in either an additive or a multiplicative framework.
|
|
In the former case, the <I>two-level additive</I> Schwarz preconditioner
|
|
is obtained:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="166" HEIGHT="32" BORDER="0"
|
|
SRC="img53.png"
|
|
ALT="\begin{displaymath}
|
|
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
Applying <IMG
|
|
WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img54.png"
|
|
ALT="$M_{2L-A}^{-1}$"> to a vector <IMG
|
|
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img22.png"
|
|
ALT="$v$"> within a Krylov solver
|
|
corresponds to applying <IMG
|
|
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img51.png"
|
|
ALT="$M_{C}^{-1}$">
|
|
and <IMG
|
|
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img55.png"
|
|
ALT="$M_{1L}^{-1}$"> to <IMG
|
|
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img22.png"
|
|
ALT="$v$"> independently and then summing up
|
|
the results.
|
|
|
|
<P>
|
|
In the multiplicative case, the combination can be
|
|
performed by first applying the smoother <IMG
|
|
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img55.png"
|
|
ALT="$M_{1L}^{-1}$"> and then
|
|
the coarse-level correction operator <IMG
|
|
WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img51.png"
|
|
ALT="$M_{C}^{-1}$">:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
\begin{array}{l}
|
|
w = M_{1L}^{-1} v, \\
|
|
z = w + M_{C}^{-1} (v-Aw);
|
|
\end{array}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="177" HEIGHT="51" BORDER="0"
|
|
SRC="img56.png"
|
|
ALT="\begin{displaymath}
|
|
\begin{array}{l}
|
|
w = M_{1L}^{-1} v, \\
|
|
z = w + M_{C}^{-1} (v-Aw);
|
|
\end{array}\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
this corresponds to the following <I>two-level hybrid pre-smoothed</I>
|
|
Schwarz preconditioner:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="309" HEIGHT="34" BORDER="0"
|
|
SRC="img57.png"
|
|
ALT="\begin{displaymath}
|
|
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
On the other hand, by applying the smoother after the coarse-level correction,
|
|
i.e. by computing
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
\begin{array}{l}
|
|
w = M_{C}^{-1} v , \\
|
|
z = w + M_{1L}^{-1} (v-Aw) ,
|
|
\end{array}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="177" HEIGHT="51" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="\begin{displaymath}
|
|
\begin{array}{l}
|
|
w = M_{C}^{-1} v , \\
|
|
z = w + M_{1L}^{-1} (v-Aw) ,
|
|
\end{array}\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
the <I>two-level hybrid post-smoothed</I>
|
|
Schwarz preconditioner is obtained:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="318" HEIGHT="34" BORDER="0"
|
|
SRC="img59.png"
|
|
ALT="\begin{displaymath}
|
|
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
One more variant of two-level hybrid preconditioner is obtained by applying
|
|
the smoother before and after the coarse-level correction. In this case, the
|
|
preconditioner is symmetric if <IMG
|
|
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$A$">, <IMG
|
|
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img49.png"
|
|
ALT="$M_{1L}$"> and <IMG
|
|
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img52.png"
|
|
ALT="$M_{C}$"> are symmetric.
|
|
|
|
<P>
|
|
As previously noted, on parallel computers the number of submatrices usually matches
|
|
the number of available processors. When the size of the system to be preconditioned
|
|
is very large, the use of many processors, i.e. of many small submatrices, often
|
|
leads to a large coarse-level system, whose solution may be computationally expensive.
|
|
On the other hand, the use of few processors often leads to local sumatrices that
|
|
are too expensive to be processed on single processors, because of memory and/or
|
|
computing requirements. Therefore, it seems natural to use a recursive approach,
|
|
in which the coarse-level correction is re-applied starting from the current
|
|
coarse-level system. The corresponding preconditioners, called <I>multi-level</I>
|
|
preconditioners, can significantly reduce the computational cost of preconditioning
|
|
with respect to the two-level case (see [<A
|
|
HREF="node25.html#dd2_96">22</A>, Chapter 3]).
|
|
Additive and hybrid multilevel preconditioners
|
|
are obtained as direct extensions of the two-level counterparts.
|
|
For a detailed descrition of them, the reader is
|
|
referred to [<A
|
|
HREF="node25.html#dd2_96">22</A>, Chapter 3].
|
|
The algorithm for the application of a multi-level hybrid
|
|
post-smoothed preconditioner <IMG
|
|
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img60.png"
|
|
ALT="$M$"> to a vector <IMG
|
|
WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img22.png"
|
|
ALT="$v$">, i.e. for the
|
|
computation of <IMG
|
|
WIDTH="86" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img61.png"
|
|
ALT="$w=M^{-1}v$">, is reported, for
|
|
example, in Figure <A HREF="#fig:mlhpost_alg">1</A>. Here the number of levels
|
|
is denoted by <IMG
|
|
WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img62.png"
|
|
ALT="$nlev$"> and the levels are numbered in increasing order starting
|
|
from the finest one, i.e. the finest level is level 1; the coarse matrix
|
|
and the corresponding basic preconditioner at each level <IMG
|
|
WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img63.png"
|
|
ALT="$l$"> are denoted by <IMG
|
|
WIDTH="23" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img64.png"
|
|
ALT="$A_l$"> and
|
|
<IMG
|
|
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img65.png"
|
|
ALT="$M_l$">, respectively, with <IMG
|
|
WIDTH="62" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img66.png"
|
|
ALT="$A_1=A$">, while the related restriction operator is
|
|
denoted by <IMG
|
|
WIDTH="23" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img67.png"
|
|
ALT="$R_l$">.
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="509"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
|
|
Application of the multi-level hybrid post-smoothed preconditioner.</CAPTION>
|
|
<TR><TD>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
$\framebox{
|
|
\begin{minipage}{.85\textwidth} {\small
|
|
\begin{tabbing}
|
|
\quad \=\quad \=\quad \=\quad \\[-1mm]$v_1 = v$; \\[2mm]
|
|
\textbf{for $l=2, nlev$\ do}\\[1mm]
|
|
\> ! transfer $v_{l-1}$\ to the next coarser level\\
|
|
\> $v_l = R_lv_{l-1}$\ \\[1mm]
|
|
\textbf{endfor} \\[2mm]
|
|
! apply the coarsest-level correction\\[1mm]
|
|
$y_{nlev} = A_{nlev}^{-1} v_{nlev}$\\[2mm]
|
|
\textbf{for $l=nlev -1 , 1, -1$\ do}\\[1mm]
|
|
\> ! transfer $y_{l+1}$\ to the next finer level\\
|
|
\> $y_l = R_{l+1}^T y_{l+1}$;\\[1mm]
|
|
\> ! compute the residual at the current level\\
|
|
\> $r_l = v_l-A_l^{-1} y_l$;\\[1mm]
|
|
\> ! apply the basic Schwarz preconditioner to the residual\\
|
|
\> $r_l = M_l^{-1} r_l$\\[1mm]
|
|
\> ! update $y_l$\\
|
|
\> $y_l = y_l+r_l$\\
|
|
\textbf{endfor} \\[1mm]
|
|
$w = y_1$;
|
|
\end{tabbing}
|
|
}
|
|
\end{minipage}
|
|
}$
|
|
-->
|
|
<IMG
|
|
WIDTH="430" HEIGHT="435" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img68.png"
|
|
ALT="\framebox{
|
|
\begin{minipage}{.85\textwidth} {\small
|
|
\begin{tabbing}
|
|
\quad \=\quad...
|
|
...= y_l+r_l$\\
|
|
\textbf{endfor} [1mm]
|
|
$w = y_1$;
|
|
\end{tabbing}}
|
|
\end{minipage}}">
|
|
|
|
</DIV></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
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