Fixup the documentation.

7 years ago · 4b43164668
parent 8ee76a1a82
commit 4b43164668
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--- a/docs/html/index.html
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@ -1,6 +1,6 @@
 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">

-<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
+<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
 <HTML>
 <HEAD>
 <TITLE>userhtml</TITLE>
@ -9,7 +9,7 @@
 <META NAME="resource-type" CONTENT="document">
 <META NAME="distribution" CONTENT="global">

-<META NAME="Generator" CONTENT="LaTeX2HTML v2017.2">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
 <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">

 <LINK REL="STYLESHEET" HREF="userhtml.css">
@ -69,17 +69,15 @@ based on PSBLAS</BIG></SPAN>
 <BR>
 <BR>
 <BR>
-Software version: 2.1
+Software version: 2.2
 <BR>
-July 31, 2017
+July 31, 2018

 <BR>

 </BIG></BIG></BIG>
 <P>
-<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">   
-    
-</BIG></BIG></BIG>
+<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">       </BIG></BIG></BIG>
 <P>
 <BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG></BIG>
 <BR><HR>
--- a/docs/html/node1.html
+++ b/docs/html/node1.html
@ -1,6 +1,6 @@
 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">

-<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
+<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
 <HTML>
 <HEAD>
 <TITLE>Abstract</TITLE>
@ -9,7 +9,7 @@
 <META NAME="resource-type" CONTENT="document">
 <META NAME="distribution" CONTENT="global">

-<META NAME="Generator" CONTENT="LaTeX2HTML v2017.2">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
 <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">

 <LINK REL="STYLESHEET" HREF="userhtml.css">
--- a/docs/html/node10.html
+++ b/docs/html/node10.html
@ -1,6 +1,6 @@
 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">

-<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
+<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
 <HTML>
 <HEAD>
 <TITLE>Bug reporting</TITLE>
@ -9,7 +9,7 @@
 <META NAME="resource-type" CONTENT="document">
 <META NAME="distribution" CONTENT="global">

-<META NAME="Generator" CONTENT="LaTeX2HTML v2017.2">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
 <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">

 <LINK REL="STYLESHEET" HREF="userhtml.css">
--- a/docs/html/node11.html
+++ b/docs/html/node11.html
@ -1,6 +1,6 @@
 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">

-<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
+<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
 <HTML>
 <HEAD>
 <TITLE>Example and test programs</TITLE>
@ -9,7 +9,7 @@
 <META NAME="resource-type" CONTENT="document">
 <META NAME="distribution" CONTENT="global">

-<META NAME="Generator" CONTENT="LaTeX2HTML v2017.2">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
 <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">

 <LINK REL="STYLESHEET" HREF="userhtml.css">
--- a/docs/html/node12.html
+++ b/docs/html/node12.html
@ -1,6 +1,6 @@
 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">

-<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
+<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
 <HTML>
 <HEAD>
 <TITLE>Multigrid Background</TITLE>
@ -9,7 +9,7 @@
 <META NAME="resource-type" CONTENT="document">
 <META NAME="distribution" CONTENT="global">

-<META NAME="Generator" CONTENT="LaTeX2HTML v2017.2">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
 <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">

 <LINK REL="STYLESHEET" HREF="userhtml.css">
@ -88,19 +88,19 @@ are considered. The second approach performs a fully automatic coarsening and en
 interplay between fine and coarse level by suitably choosing the coarse space and
 the coarse-to-fine interpolation (see, e.g., [<A
 HREF="node36.html#Briggs2000">3</A>,<A
- HREF="node36.html#Stuben_01">23</A>,<A
- HREF="node36.html#dd2_96">21</A>] for details.)
+ HREF="node36.html#Stuben_01">24</A>,<A
+ HREF="node36.html#dd2_96">22</A>] for details.)
 </BIG></BIG></BIG>
 <P>
 <BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">MLD2P4 uses a pure algebraic approach, based on the smoothed 
 aggregation algorithm [<A
 HREF="node36.html#BREZINA_VANEK">2</A>,<A
- HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>],
+ HREF="node36.html#VANEK_MANDEL_BREZINA">26</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="node36.html#TUMINARO_TONG">24</A>].
+ HREF="node36.html#TUMINARO_TONG">25</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
--- a/docs/html/node13.html
+++ b/docs/html/node13.html
@ -1,6 +1,6 @@
 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">

-<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
+<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
 <HTML>
 <HEAD>
 <TITLE>AMG preconditioners</TITLE>
@ -9,7 +9,7 @@
 <META NAME="resource-type" CONTENT="document">
 <META NAME="distribution" CONTENT="global">

-<META NAME="Generator" CONTENT="LaTeX2HTML v2017.2">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
 <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">

 <LINK REL="STYLESHEET" HREF="userhtml.css">
@ -67,7 +67,7 @@ Ax=b,
 <A NAME="eq:system"></A>
 <TABLE WIDTH="100%" ALIGN="CENTER">
 <TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:system"></A><IMG
- WIDTH="58" HEIGHT="30" BORDER="0"
+ WIDTH="57" HEIGHT="30" BORDER="0"
 SRC="img2.png"
 ALT="\begin{displaymath}
 Ax=b,
@ -80,7 +80,7 @@ where <!-- MATH
 $A=(a_{ij}) \in \mathbb{R}^{n \times n}$
 -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.png"
 ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"></SPAN> is a nonsingular sparse matrix;
 for ease of presentation we assume <SPAN CLASS="MATH"><IMG
@ -98,7 +98,7 @@ pattern.
 $\Omega = \{1, 2, \ldots, n\}$
 -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="131" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img6.png"
 ALT="$\Omega = \{1, 2, \ldots, n\}$"></SPAN>. 
 Any algebraic multilevel preconditioners implemented in MLD2P4 generates
@ -116,7 +116,8 @@ a hierarchy of index spaces and a corresponding hierarchy of matrices,
 <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}">
+ 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><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
@ -132,28 +133,28 @@ A vector space <!-- MATH
 $\mathbb{R}^{n_{k}}$
 -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img8.png"
 ALT="$\mathbb{R}^{n_{k}}$"></SPAN> is associated with <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img9.png"
 ALT="$\Omega^k$"></SPAN>,
 where <SPAN CLASS="MATH"><IMG
 WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img10.png"
 ALT="$n_k$"></SPAN> is the size of <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img9.png"
 ALT="$\Omega^k$"></SPAN>.
 For all <SPAN CLASS="MATH"><IMG
- WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$k &lt; nlev$"></SPAN>, a restriction operator and a prolongation one are built,
 which connect two levels <SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="14" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img12.png"
 ALT="$k$"></SPAN> and <SPAN CLASS="MATH"><IMG
- WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img13.png"
 ALT="$k+1$"></SPAN>:
 </BIG></BIG></BIG>
@ -167,14 +168,17 @@ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
 -->

 <IMG
- WIDTH="254" HEIGHT="30" BORDER="0"
+ WIDTH="255" 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}">
+ 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><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
 the matrix <SPAN CLASS="MATH"><IMG
- WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img15.png"
 ALT="$A^{k+1}$"></SPAN> is computed by using the previous operators according
 to the Galerkin approach, i.e.,
@ -188,9 +192,11 @@ A^{k+1}=R^kA^kP^k.
 -->

 <IMG
- WIDTH="131" HEIGHT="27" BORDER="0"
+ WIDTH="131" HEIGHT="28" BORDER="0"
 SRC="img16.png"
- ALT="\begin{displaymath}
 A^{k+1}=R^kA^kP^k.
\end{displaymath}">
+ ALT="\begin{displaymath}
+A^{k+1}=R^kA^kP^k.
+\end{displaymath}">
 </DIV>
 <BR CLEAR="ALL">
 <P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
@ -199,10 +205,10 @@ In the current implementation of MLD2P4 we have <SPAN CLASS="MATH"><IMG
 SRC="img17.png"
 ALT="$R^k=(P^k)^T$"></SPAN>
 A smoother with iteration matrix <SPAN CLASS="MATH"><IMG
- WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img18.png"
 ALT="$M^k$"></SPAN> is set up at each level <SPAN CLASS="MATH"><IMG
- WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$k &lt; nlev$"></SPAN>, and a solver
 is set up at the coarsest level, so that they are ready for application 
@ -211,7 +217,7 @@ is set up at the coarsest level, so that they are ready for application
 SRC="img19.png"
 ALT="$LU$"></SPAN> factorization means computing
 and storing the <SPAN CLASS="MATH"><IMG
- WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img20.png"
 ALT="$L$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
@ -251,8 +257,15 @@ end
 <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}
}">
+ ALT="\framebox{
+\begin{minipage}{.85\textwidth}
+\begin{tabbing}
+\quad \=\quad \=\quad...
+...[1mm]
+\&gt;endif  [1mm]
+\&gt;return $u^k$  [1mm]
+end
+\end{tabbing}\end{minipage}}">

 </DIV></TD></TR>
 </TABLE>
@ -263,14 +276,14 @@ end
 to obtain different multilevel preconditioners;
 this is  done in the application phase, i.e., in the computation of a vector
 of type <SPAN CLASS="MATH"><IMG
- WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="81" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img23.png"
 ALT="$w=B^{-1}v$"></SPAN>, where <SPAN CLASS="MATH"><IMG
 WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img24.png"
 ALT="$B$"></SPAN> denotes the preconditioner, usually within an iteration
 of a Krylov solver [<A
- HREF="node36.html#Saad_book">20</A>]. An example of such a combination, known as
+ HREF="node36.html#Saad_book">21</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
@ -278,7 +291,7 @@ performed. Other cycles can be defined; in MLD2P4, we implemented the standard V
 and W-cycle&nbsp;[<A
 HREF="node36.html#Briggs2000">3</A>], and a version of the K-cycle described
 in&nbsp;[<A
- HREF="node36.html#Notay2008">19</A>].  
+ HREF="node36.html#Notay2008">20</A>].  
 </BIG></BIG></BIG>
 <P>
 <BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
--- a/docs/html/node14.html
+++ b/docs/html/node14.html
@ -1,6 +1,6 @@
 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">

-<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
+<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
 <HTML>
 <HEAD>
 <TITLE>Smoothed Aggregation</TITLE>
@ -9,7 +9,7 @@
 <META NAME="resource-type" CONTENT="document">
 <META NAME="distribution" CONTENT="global">

-<META NAME="Generator" CONTENT="LaTeX2HTML v2017.2">
+<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
 <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">

 <LINK REL="STYLESHEET" HREF="userhtml.css">
@ -54,27 +54,27 @@ Smoothed Aggregation
 </H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
 <P>
 <BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to define the prolongator <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$P^k$"></SPAN>, used to compute
 the coarse-level matrix <SPAN CLASS="MATH"><IMG
- WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img15.png"
 ALT="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
 algorithm described in [<A
 HREF="node36.html#BREZINA_VANEK">2</A>,<A
- HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>].
+ HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>].
 The basic idea of this algorithm is to build a coarse set of indices
 <SPAN CLASS="MATH"><IMG
- WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img26.png"
 ALT="$\Omega^{k+1}$"></SPAN> by suitably grouping the indices of <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img9.png"
 ALT="$\Omega^k$"></SPAN> into disjoint
 subsets (aggregates), and to define the coarse-to-fine space transfer operator
 <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$P^k$"></SPAN> by applying a suitable smoother to a simple piecewise constant
 prolongation operator, with the aim of improving the quality of the coarse-space correction.
@ -84,26 +84,26 @@ prolongation operator, with the aim of improving the quality of the coarse-space
 </BIG></BIG></BIG>
 <OL>
 <LI>aggregation of the indices of <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img9.png"
 ALT="$\Omega^k$"></SPAN> to obtain <SPAN CLASS="MATH"><IMG
- WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img26.png"
 ALT="$\Omega^{k+1}$"></SPAN>;
 </LI>
 <LI>construction of the prolongator <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$P^k$"></SPAN>;
 </LI>
 <LI>application of <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$P^k$"></SPAN> and <SPAN CLASS="MATH"><IMG
 WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img17.png"
 ALT="$R^k=(P^k)^T$"></SPAN> to build <SPAN CLASS="MATH"><IMG
- WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img15.png"
 ALT="$A^{k+1}$"></SPAN>.
 </LI>
@ -111,18 +111,18 @@ prolongation operator, with the aim of improving the quality of the coarse-space
 <P>
 <BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to perform the coarsening step, the smoothed aggregation algorithm
 described in&nbsp;[<A
- HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
+ HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>] is used. In this algorithm,
 each index <!-- MATH
 $j \in \Omega^{k+1}$
 -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img27.png"
 ALT="$j \in \Omega^{k+1}$"></SPAN> corresponds to an aggregate <SPAN CLASS="MATH"><IMG
 WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img28.png"
 ALT="$\Omega^k_j$"></SPAN> of <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img9.png"
 ALT="$\Omega^k$"></SPAN>,
 consisting of a suitably chosen index <!-- MATH
@ -133,7 +133,7 @@ consisting of a suitably chosen index <!-- MATH
 SRC="img29.png"
 ALT="$i \in \Omega^k$"></SPAN> and indices that are (usually) contained in a
 strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
- WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img30.png"
 ALT="$i$"></SPAN>, i.e.,
 </BIG></BIG></BIG>
@ -149,11 +149,13 @@ strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
 <A NAME="eq:strongly_coup"></A>
 <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"
+ WIDTH="387" HEIGHT="49" 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>
+\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
+\left\{ r \i...
+...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},
+\end{displaymath}"></TD>
 <TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
 (<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
 </TABLE>
@ -162,10 +164,10 @@ for a given threshold <!-- MATH
 $\theta \in [0,1]$
 -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img32.png"
 ALT="$\theta \in [0,1]$"></SPAN> (see&nbsp;[<A
- HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
+ HREF="node36.html#VANEK_MANDEL_BREZINA">26</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
@ -180,11 +182,11 @@ MLD2P4, since it has been shown to produce good results in practice
 [<A
 HREF="node36.html#aaecc_07">5</A>,<A
 HREF="node36.html#apnum_07">7</A>,<A
- HREF="node36.html#TUMINARO_TONG">24</A>].
+ HREF="node36.html#TUMINARO_TONG">25</A>].
 </BIG></BIG></BIG>
 <P>
 <BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The prolongator <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$P^k$"></SPAN> is built starting from a tentative prolongator
 <!-- MATH
@ -210,10 +212,14 @@ MLD2P4, since it has been shown to produce good results in practice
 <A NAME="eq:tent_prol"></A>
 <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"
+ WIDTH="286" HEIGHT="52" 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.
+ ALT="\begin{displaymath}
+\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
+\left\{...
+...Omega^k_j, \\
+0 &amp; \quad \mbox{otherwise},
+\end{array} \right.
 \end{displaymath}"></TD>
 <TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
 (<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
@ -223,21 +229,21 @@ where <SPAN CLASS="MATH"><IMG
 WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img28.png"
 ALT="$\Omega^k_j$"></SPAN> is the aggregate of <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img9.png"
 ALT="$\Omega^k$"></SPAN>
 corresponding to the index <!-- MATH
 $j \in \Omega^{k+1}$
 -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img27.png"
 ALT="$j \in \Omega^{k+1}$"></SPAN>.
 <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$P^k$"></SPAN> is obtained by applying to <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img35.png"
 ALT="$\bar{P}^k$"></SPAN> a smoother
 <!-- MATH
@ -257,9 +263,11 @@ P^k = S^k \bar{P}^k,
 -->

 <IMG
- WIDTH="90" HEIGHT="30" BORDER="0"
+ WIDTH="91" HEIGHT="30" BORDER="0"
 SRC="img37.png"
- ALT="\begin{displaymath}
P^k = S^k \bar{P}^k,
\end{displaymath}">
+ ALT="\begin{displaymath}
+P^k = S^k \bar{P}^k,
+\end{displaymath}">
 </DIV>
 <BR CLEAR="ALL">
 <P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
@ -267,9 +275,9 @@ in order to remove nonsmooth components from the range of the prolongator,
 and hence to improve the convergence properties of the multilevel
 method&nbsp;[<A
 HREF="node36.html#BREZINA_VANEK">2</A>,<A
- HREF="node36.html#Stuben_01">23</A>].
+ HREF="node36.html#Stuben_01">24</A>].
 A simple choice for <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img38.png"
 ALT="$S^k$"></SPAN> is the damped Jacobi smoother:
 </BIG></BIG></BIG>
@ -282,24 +290,26 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
 -->

 <IMG
- WIDTH="175" HEIGHT="31" BORDER="0"
+ WIDTH="176" HEIGHT="32" BORDER="0"
 SRC="img39.png"
- ALT="\begin{displaymath}
S^k = I - \omega^k (D^k)^{-1} A^k_F , 
\end{displaymath}">
+ ALT="\begin{displaymath}
+S^k = I - \omega^k (D^k)^{-1} A^k_F ,
+\end{displaymath}">
 </DIV>
 <BR CLEAR="ALL">
 <P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
 where <SPAN CLASS="MATH"><IMG
- WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="28" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img40.png"
 ALT="$D^k$"></SPAN> is the diagonal matrix with the same diagonal entries as <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img41.png"
 ALT="$A^k$"></SPAN>,
 <!-- MATH
 $A^k_F = (\bar{a}_{ij}^k)$
 -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img42.png"
 ALT="$A^k_F = (\bar{a}_{ij}^k)$"></SPAN> is the filtered matrix defined as
 </BIG></BIG></BIG>
@ -321,17 +331,20 @@ where <SPAN CLASS="MATH"><IMG
 <A NAME="eq:filtered"></A>
 <TABLE WIDTH="100%" ALIGN="CENTER">
 <TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
- WIDTH="514" HEIGHT="74" BORDER="0"
+ WIDTH="499" HEIGHT="59" 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>
+\bar{a}_{ij}^k =
+\left \{ \begin{array}{ll}
+a_{ij}^k &amp; \m...
+...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
+\end{displaymath}"></TD>
 <TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
 (<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
 </TABLE>
 <BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
 and <SPAN CLASS="MATH"><IMG
- WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img44.png"
 ALT="$\omega^k$"></SPAN> is an approximation of <SPAN CLASS="MATH"><IMG
 WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
@ -360,14 +373,14 @@ of <SPAN CLASS="MATH"><IMG
 SRC="img46.png"
 ALT="$\rho^k$"></SPAN>. Note that for systems coming from uniformly elliptic
 problems, filtering the matrix <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img41.png"
 ALT="$A^k$"></SPAN> has little or no effect, and
 <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img41.png"
 ALT="$A^k$"></SPAN> can be used instead of <SPAN CLASS="MATH"><IMG
- WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img49.png"
 ALT="$A^k_F$"></SPAN>. The latter choice is the default in MLD2P4.
 </BIG></BIG></BIG>
--- a/Show More
+++ b/Show More