Fixed docs.

docs/html/node13.html

@@ -80,25 +80,25 @@ where <!-- MATH
  $A=(a_{ij}) \in \mathbb{R}^{n \times n}$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="137" HEIGHT="38" 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
- WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img3.png"
  ALT="$A$"></SPAN> has a symmetric sparsity
 pattern.
 </BIG></BIG></BIG>
 <P>
 <BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">Let us consider as finest index space the set of row (column) indices of <SPAN CLASS="MATH"><IMG
- WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img3.png"
  ALT="$A$"></SPAN>, i.e.,
 <!-- MATH
  $\Omega = \{1, 2, \ldots, n\}$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="131" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
  SRC="img6.png"
  ALT="$\Omega = \{1, 2, \ldots, n\}$"></SPAN>. 
 Any algebraic multilevel preconditioners implemented in MLD2P4 generates
@@ -122,39 +122,39 @@ a hierarchy of index spaces and a corresponding hierarchy of matrices,
 <BR CLEAR="ALL">
 <P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
 by using the information contained in <SPAN CLASS="MATH"><IMG
- WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img3.png"
  ALT="$A$"></SPAN>, without assuming any
 knowledge of the geometry of the problem from which <SPAN CLASS="MATH"><IMG
- WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img3.png"
  ALT="$A$"></SPAN> originates.
 A vector space <!-- MATH
  $\mathbb{R}^{n_{k}}$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="34" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img8.png"
  ALT="$\mathbb{R}^{n_{k}}$"></SPAN> is associated with <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="18" 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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img9.png"
  ALT="$\Omega^k$"></SPAN>.
 For all <SPAN CLASS="MATH"><IMG
- WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="70" HEIGHT="34" 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="20" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img12.png"
  ALT="$k$"></SPAN> and <SPAN CLASS="MATH"><IMG
- WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
  SRC="img13.png"
  ALT="$k+1$"></SPAN>:
 </BIG></BIG></BIG>
@@ -168,7 +168,7 @@ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
  -->
 
 <IMG
- WIDTH="255" HEIGHT="30" BORDER="0"
+ WIDTH="253" HEIGHT="30" BORDER="0"
  SRC="img14.png"
  ALT="\begin{displaymath}
 P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
@@ -178,7 +178,7 @@ R^k \in \mathbb{R}^{n_{k+1}\times n_k};
 <BR CLEAR="ALL">
 <P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
 the matrix <SPAN CLASS="MATH"><IMG
- WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="43" HEIGHT="18" 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.,
@@ -192,7 +192,7 @@ A^{k+1}=R^kA^kP^k.
  -->
 
 <IMG
- WIDTH="131" HEIGHT="28" BORDER="0"
+ WIDTH="129" HEIGHT="27" BORDER="0"
  SRC="img16.png"
  ALT="\begin{displaymath}
 A^{k+1}=R^kA^kP^k.
@@ -205,22 +205,22 @@ 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="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img18.png"
  ALT="$M^k$"></SPAN> is set up at each level <SPAN CLASS="MATH"><IMG
- WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="70" HEIGHT="34" 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 
 (for example, setting up a solver based on the <SPAN CLASS="MATH"><IMG
- WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="30" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img19.png"
  ALT="$LU$"></SPAN> factorization means computing
 and storing the <SPAN CLASS="MATH"><IMG
- WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="17" 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"
+ WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img21.png"
  ALT="$U$"></SPAN> factors). The construction of the hierarchy of AMG components
 described so far corresponds to the so-called build phase of the preconditioner.
@@ -262,8 +262,8 @@ end
 \begin{tabbing}
 \quad \=\quad \=\quad...
 ...[1mm]
-\&gt;endif  [1mm]
-\&gt;return $u^k$  [1mm]
+\&gt;endif \\ [1mm]
+\&gt;return $u^k$\ \\ [1mm]
 end
 \end{tabbing}\end{minipage}}">
 
@@ -276,7 +276,7 @@ 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="81" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="82" HEIGHT="21" 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"

docs/html/node14.html

@@ -54,11 +54,11 @@ 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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" 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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img15.png"
  ALT="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
 algorithm described in [<A
@@ -66,15 +66,15 @@ algorithm described in [<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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="43" HEIGHT="18" 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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="18" 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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" 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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img9.png"
  ALT="$\Omega^k$"></SPAN> to obtain <SPAN CLASS="MATH"><IMG
- WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="43" HEIGHT="18" 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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img25.png"
  ALT="$P^k$"></SPAN>;
 </LI>
 <LI>application of <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" 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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img15.png"
  ALT="$A^{k+1}$"></SPAN>.
 </LI>
@@ -116,13 +116,13 @@ each index <!-- MATH
  $j \in \Omega^{k+1}$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="72" 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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="18" 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="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img30.png"
  ALT="$i$"></SPAN>, i.e.,
 </BIG></BIG></BIG>
@@ -149,7 +149,7 @@ 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="49" BORDER="0"
+ WIDTH="387" HEIGHT="48" BORDER="0"
  SRC="img31.png"
  ALT="\begin{displaymath}
 \Omega^k_j \subset \mathcal{N}_i^k(\theta) =
@@ -164,7 +164,7 @@ for a given threshold <!-- MATH
  $\theta \in [0,1]$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
  SRC="img32.png"
  ALT="$\theta \in [0,1]$"></SPAN> (see&nbsp;[<A
  HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>] for the details).
@@ -175,7 +175,7 @@ distribution. This version is embarrassingly parallel, since it does not require
 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 <SPAN CLASS="MATH"><IMG
- WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img3.png"
  ALT="$A$"></SPAN>. Nevertheless, this parallel algorithm has been chosen for
 MLD2P4, since it has been shown to produce good results in practice
@@ -186,7 +186,7 @@ MLD2P4, since it has been shown to produce good results in practice
 </BIG></BIG></BIG>
 <P>
 <BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The prolongator <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img25.png"
  ALT="$P^k$"></SPAN> is built starting from a tentative prolongator
 <!-- MATH
@@ -212,7 +212,7 @@ 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="286" HEIGHT="52" BORDER="0"
+ WIDTH="286" HEIGHT="51" BORDER="0"
  SRC="img34.png"
  ALT="\begin{displaymath}
 \bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
@@ -229,21 +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="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="18" 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="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
  SRC="img27.png"
  ALT="$j \in \Omega^{k+1}$"></SPAN>.
 <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img25.png"
  ALT="$P^k$"></SPAN> is obtained by applying to <SPAN CLASS="MATH"><IMG
- WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img35.png"
  ALT="$\bar{P}^k$"></SPAN> a smoother
 <!-- MATH
@@ -263,7 +263,7 @@ P^k = S^k \bar{P}^k,
  -->
 
 <IMG
- WIDTH="91" HEIGHT="30" BORDER="0"
+ WIDTH="90" HEIGHT="30" BORDER="0"
  SRC="img37.png"
  ALT="\begin{displaymath}
 P^k = S^k \bar{P}^k,
@@ -277,7 +277,7 @@ method&nbsp;[<A
  HREF="node36.html#BREZINA_VANEK">2</A>,<A
  HREF="node36.html#Stuben_01">24</A>].
 A simple choice for <SPAN CLASS="MATH"><IMG
- WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img38.png"
  ALT="$S^k$"></SPAN> is the damped Jacobi smoother:
 </BIG></BIG></BIG>
@@ -290,7 +290,7 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
  -->
 
 <IMG
- WIDTH="176" HEIGHT="32" BORDER="0"
+ WIDTH="175" HEIGHT="31" BORDER="0"
  SRC="img39.png"
  ALT="\begin{displaymath}
 S^k = I - \omega^k (D^k)^{-1} A^k_F ,
@@ -299,17 +299,17 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
 <BR CLEAR="ALL">
 <P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
 where <SPAN CLASS="MATH"><IMG
- WIDTH="28" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="28" HEIGHT="18" 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="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img41.png"
  ALT="$A^k$"></SPAN>,
 <!-- MATH
  $A^k_F = (\bar{a}_{ij}^k)$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="87" 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>
@@ -344,7 +344,7 @@ a_{ij}^k &amp; \m...
 </TABLE>
 <BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
 and <SPAN CLASS="MATH"><IMG
- WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="24" HEIGHT="18" 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"
@@ -373,14 +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="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img41.png"
  ALT="$A^k$"></SPAN> has little or no effect, and
 <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img41.png"
  ALT="$A^k$"></SPAN> can be used instead of <SPAN CLASS="MATH"><IMG
- WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="29" 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>

docs/html/node15.html

@@ -68,7 +68,7 @@ the beginning of the current iteration.
 </BIG></BIG></BIG>
 <P>
 <BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In the AS methods, the index space <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img9.png"
  ALT="$\Omega^k$"></SPAN> is divided into <SPAN CLASS="MATH"><IMG
  WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
@@ -78,11 +78,11 @@ subsets <SPAN CLASS="MATH"><IMG
  WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
  SRC="img51.png"
  ALT="$\Omega^k_i$"></SPAN> of size <SPAN CLASS="MATH"><IMG
- WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
  SRC="img52.png"
  ALT="$n_{k,i}$"></SPAN>,  possibly
 overlapping. For each <SPAN CLASS="MATH"><IMG
- WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img30.png"
  ALT="$i$"></SPAN> we consider the restriction
 operator <!-- MATH
@@ -93,13 +93,13 @@ operator <!-- MATH
  SRC="img53.png"
  ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$"></SPAN>
 that maps a vector <SPAN CLASS="MATH"><IMG
- WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img54.png"
  ALT="$x^k$"></SPAN> to the vector <SPAN CLASS="MATH"><IMG
  WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
  SRC="img55.png"
  ALT="$x_i^k$"></SPAN> made of the components of <SPAN CLASS="MATH"><IMG
- WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img54.png"
  ALT="$x^k$"></SPAN>
 with indices in <SPAN CLASS="MATH"><IMG
@@ -120,7 +120,7 @@ with indices in <SPAN CLASS="MATH"><IMG
  WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
  SRC="img57.png"
  ALT="$A_i^k=R_i^kA^kP_i^k$"></SPAN>, which is the restriction of <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img41.png"
  ALT="$A^k$"></SPAN> to the index
 space <SPAN CLASS="MATH"><IMG
@@ -172,7 +172,7 @@ involves
  SRC="img62.png"
  ALT="$\Omega_i^k$"></SPAN> and of the corresponding 
   operators <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
  SRC="img63.png"
  ALT="$R_i^k$"></SPAN> (and <SPAN CLASS="MATH"><IMG
  WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
@@ -205,13 +205,13 @@ multilevel application phase, requires
 </BIG></BIG></BIG>
 <UL>
 <LI>the restriction of <SPAN CLASS="MATH"><IMG
- WIDTH="25" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img67.png"
  ALT="$w^k$"></SPAN> to the subspaces <!-- MATH
  $\mathbb{R}^{n_{k,i}}$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img68.png"
  ALT="$\mathbb{R}^{n_{k,i}}$"></SPAN>,
 	  i.e. <!-- MATH

docs/html/node18.html

@@ -83,14 +83,14 @@ i.e.,
   matrix data structure;
 </LI>
 <LI>the arrays containing the vectors <SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img72.png"
  ALT="$v$"></SPAN> and <SPAN CLASS="MATH"><IMG
- WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img73.png"
  ALT="$w$"></SPAN> involved in
   the preconditioner application <SPAN CLASS="MATH"><IMG
- WIDTH="81" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
  SRC="img23.png"
  ALT="$w=B^{-1}v$"></SPAN> must be of type   
   <code>psb_</code><SPAN  CLASS="textit">x</SPAN><code>vect_type</code> with <SPAN  CLASS="textit">x</SPAN> =    

docs/html/node20.html

@@ -326,10 +326,10 @@ Parameters defining the aggregation algorithm.
  $\lfloor 40 \sqrt[3]{n} \rfloor$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
  SRC="img76.png"
  ALT="$\lfloor 40 \sqrt[3]{n} \rfloor$"></SPAN>, where <SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img77.png"
  ALT="$n$"></SPAN> is the dimension
                             of the matrix at the finest level</TD>
@@ -376,7 +376,7 @@ Currently, only the
                            <code>SYMDEC</code> option  applies decoupled
                            aggregation to  the sparsity pattern
                            of <SPAN CLASS="MATH"><IMG
- WIDTH="62" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="62" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
  SRC="img79.png"
  ALT="$A+A^T$"></SPAN>.</TD>
 </TR>
@@ -449,12 +449,12 @@ Parameters defining the aggregation algorithm (continued).
 <TD ALIGN="LEFT" VALIGN="TOP" WIDTH=71>Any&nbsp;real 
 <P>
 number&nbsp;<SPAN CLASS="MATH"><IMG
- WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
  SRC="img80.png"
  ALT="$\in [0, 1]$"></SPAN></TD>
 <TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>0.01</TD>
 <TD ALIGN="LEFT" VALIGN="TOP" WIDTH=187>The threshold <SPAN CLASS="MATH"><IMG
- WIDTH="13" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img81.png"
  ALT="$\theta$"></SPAN> 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>.
@@ -643,7 +643,7 @@ number <SPAN CLASS="MATH"><IMG
  ALT="$\ge 0$"></SPAN></TD>
 <TD ALIGN="LEFT" VALIGN="TOP" WIDTH=43>0</TD>
 <TD ALIGN="LEFT" VALIGN="TOP" WIDTH=213>Drop tolerance <SPAN CLASS="MATH"><IMG
- WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img85.png"
  ALT="$t$"></SPAN> in the ILU(<SPAN CLASS="MATH"><IMG
  WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
@@ -866,7 +866,7 @@ Parameters defining the smoother or the details of the one-level preconditioner
 <TD ALIGN="LEFT" VALIGN="TOP" WIDTH=62><SMALL CLASS="SMALL"> 0
                          </SMALL></TD>
 <TD ALIGN="LEFT" VALIGN="TOP" WIDTH=201><SMALL CLASS="SMALL"> Drop tolerance <SPAN CLASS="MATH"><IMG
- WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img85.png"
  ALT="$t$"></SPAN> in the ILU(<SPAN CLASS="MATH"><IMG
  WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"

docs/html/node24.html

@@ -62,7 +62,7 @@ This method computes <!-- MATH
  $y = op(B^{-1})\, x$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
  SRC="img86.png"
  ALT="$y = op(B^{-1})\, x$"></SPAN>, where <SPAN CLASS="MATH"><IMG
  WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
@@ -91,7 +91,7 @@ and hence it is completely transparent to the user.
 <TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> 
               </BIG></BIG></BIG></TD>
 <TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> The local part of the vector <SPAN CLASS="MATH"><IMG
- WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
  SRC="img88.png"
  ALT="$x$"></SPAN>. Note that <SPAN  CLASS="textit">type</SPAN> and   
                 <SPAN  CLASS="textit">kind_parameter</SPAN> must be chosen according
@@ -137,28 +137,28 @@ and hence it is completely transparent to the user.
  $op(B^{-1}) = B^{-1}$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="123" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
  SRC="img90.png"
  ALT="$op(B^{-1}) = B^{-1}$"></SPAN>;
                 if <code>trans</code> = <code>'T','t'</code> then <!-- MATH
  $op(B^{-1}) = B^{-T}$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="125" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
  SRC="img91.png"
  ALT="$op(B^{-1}) = B^{-T}$"></SPAN>
                 (transpose of <SPAN CLASS="MATH"><IMG
- WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
  SRC="img92.png"
  ALT="$B^{-1})$"></SPAN>;  if <code>trans</code> = <code>'C','c'</code> then <!-- MATH
  $op(B^{-1}) = B^{-C}$
  -->
 <SPAN CLASS="MATH"><IMG
- WIDTH="126" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
  SRC="img93.png"
  ALT="$op(B^{-1}) = B^{-C}$"></SPAN>
                 (conjugate transpose of <SPAN CLASS="MATH"><IMG
- WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
+ WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
  SRC="img92.png"
  ALT="$B^{-1})$"></SPAN>.</BIG></BIG></BIG></TD>
 </TR>

docs/html/node3.html

@@ -82,7 +82,7 @@ Ax=b,
 </TABLE>
 <BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"> 
 where <SPAN CLASS="MATH"><IMG
- WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
+ WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
  SRC="img3.png"
  ALT="$A$"></SPAN> is a square, real or complex, sparse matrix. 
 The name of the package comes from its original implementation, containing