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Fixed dates in docs.

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Salvatore Filippone
2017-07-24 11:11:25 +01:00
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@@ -53,7 +53,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
<BR>
AMG preconditioners
</H2>␍␍In order to describe the AMG preconditioners available in MLD2P4, we consider a␍linear system␍<BR>
</H2><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">
<!-- MATH
@@ -64,33 +65,33 @@ Ax=b,
<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="img1.png"
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>
where <!-- MATH
<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="img4.png"
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="img2.png"
SRC="img3.png"
ALT="$A$"> is real, but the␍results are valid for the complex case as well. ␍␍Let us assume as finest index space the set of row (column) indices of <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img2.png"
SRC="img3.png"
ALT="$A$">, i.e.,␍<!-- MATH
$\Omega = \{1, 2, \ldots, n\}$
-->
<IMG
WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img5.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,␍<BR><P></P>
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}
@@ -100,42 +101,42 @@ where <!-- MATH
<IMG
WIDTH="398" HEIGHT="30" BORDER="0"
SRC="img6.png"
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>
by using the information contained in <IMG
<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="img2.png"
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="img2.png"
SRC="img3.png"
ALT="$A$"> originates.␍A vector space <!-- MATH
$\mathbb{R}^{n_{k}}$
-->
<IMG
WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img7.png"
SRC="img8.png"
ALT="$\mathbb{R}^{n_{k}}$"> is associated with <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img8.png"
SRC="img9.png"
ALT="$\Omega^k$">,␍where <IMG
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img9.png"
SRC="img10.png"
ALT="$n_k$"> is the size of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img8.png"
SRC="img9.png"
ALT="$\Omega^k$">.␍For all <IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img10.png"
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="img11.png"
SRC="img12.png"
ALT="$k$"> and <IMG
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img12.png"
ALT="$k+1$">:␍<BR><P></P>
SRC="img13.png"
ALT="$k+1$">:␍</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
@@ -145,12 +146,35 @@ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad R^k \in \mathbb{R}^{n_{k+1}\
<IMG
WIDTH="254" HEIGHT="30" BORDER="0"
SRC="img13.png"
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>
<BR><HR>
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
</FONT></FONT></FONT><HR>
<!--Navigation Panel-->
<A NAME="tex2html220"
HREF="node13.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
<A NAME="tex2html216"
HREF="node11.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
<A NAME="tex2html212"
HREF="node11.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
<A NAME="tex2html218"
HREF="node2.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
<BR>
<B> Next:</B> <A NAME="tex2html221"
HREF="node13.html">Getting Started</A>
<B> Up:</B> <A NAME="tex2html217"
HREF="node11.html">Multigrid Background</A>
<B> Previous:</B> <A NAME="tex2html213"
HREF="node11.html">Multigrid Background</A>
&nbsp; <B> <A NAME="tex2html219"
HREF="node2.html">Contents</A></B>
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