\par
\section{Prototypes and descriptions of methods in the {\tt
Misc} directory}
\label{section:Misc:proto}
\par
This section contains brief descriptions including prototypes
of all methods in the {\tt Misc} directory.
\par
%=======================================================================
\subsection{Theoretical nested dissection methods}
\begin{enumerate}
%-----------------------------------------------------------------------
\item
\begin{verbatim}
void mkNDperm ( int n1, int n2, int n3, int newToOld[], int west, 
                int east, int south, int north, int bottom, int top ) ;
\end{verbatim}
\index{mkNDperm@{\tt mkNDperm()}}
This method this vector fills a permutation vector with the
nested dissection
new-to-old ordering of the vertices for the subgrid defined by
nodes whose coordinates lie in
\begin{verbatim}
[west, east] x [south, north] x [bottom, top].
\end{verbatim}
The method calls itself recursively.
To find the permutation for an {\tt n1 x n2 x n3} grid, call
\begin{verbatim}
mkNDperm(n1, n2, n3, newToOld, 0, n1-1, 0, n2-1, 0, n3-1) ;
\end{verbatim}
from a driver program.
\par \noindent {\it Error checking:}
If {\tt n1}, {\tt n2} or {\tt n3} are less than or equal to zero,
or if {\tt newToOld} is {\tt NULL},
or if {\tt west}, {\tt south} or {\tt bottom} 
are less than or equal to zero,
of if ${\tt east} \ge {\tt n1}$,
of if ${\tt north} \ge {\tt n2}$,
of if ${\tt top} \ge {\tt n3}$,
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\item
\begin{verbatim}
void mkNDperm2 ( int n1, int n2, int n3, int newToOld[], int west, 
                 int east, int south, int north, int bottom, int top ) ;
\end{verbatim}
\index{mkNDperm2@{\tt mkNDperm2()}}
This method this vector fills a permutation vector with the
nested dissection
new-to-old ordering of the vertices for the subgrid defined by
nodes whose coordinates lie in
\begin{verbatim}
[west, east] x [south, north] x [bottom, top].
\end{verbatim}
There is one important difference between this method and {\tt
mkNDperm()} above; this method finds {\it double-wide} separators,
necessary for an operator with more than nearest neighbor grid
point coupling.
The method calls itself recursively.
To find the permutation for an {\tt n1 x n2 x n3} grid, call
\begin{verbatim}
mkNDperm(n1, n2, n3, newToOld, 0, n1-1, 0, n2-1, 0, n3-1) ;
\end{verbatim}
from a driver program.
\par \noindent {\it Error checking:}
If {\tt n1}, {\tt n2} or {\tt n3} are less than or equal to zero,
or if {\tt newToOld} is {\tt NULL},
or if {\tt west}, {\tt south} or {\tt bottom} 
are less than or equal to zero,
of if ${\tt east} \ge {\tt n1}$,
of if ${\tt north} \ge {\tt n2}$,
of if ${\tt top} \ge {\tt n3}$,
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\item
\begin{verbatim}
void localND2D ( int n1, int n2, int p1, int p2, 
                 int dsizes1[], int dsizes2[], int oldToNew[] ) ;
\end{verbatim}
\index{localND2D@{\tt localND2D()}}
This method finds a local nested dissection ordering 
\cite{bha93-localND} for an {\tt n1 x n2} 2-D grid.
There are {\tt p1 x p2} domains in the grid.
The {\tt dsizes1[]} and {\tt dsizes2[]} vectors are optional;
they allow the user to explicitly input domain sizes.
If {\tt dsizes1[]} and {\tt dsizes2[]} are not {\tt NULL},
the {\tt q = q1 + q2*p1}'th domain contains a
{\tt dsizes1[q1] x dsizes2[q2]} subgrid of points.
\par \noindent {\it Error checking:}
If {\tt n1} or {\tt n2} are less than or equal to zero,
or if {\tt p1} or {\tt p2} are less than or equal to zero,
or if $2{\tt p1} - 1 > {\tt n1}$,
or if $2{\tt p2} - 1 > {\tt n2}$,
or if {\tt oldToNew} is {\tt NULL},
or if {\tt dsizes1[]} and {\tt dsizes2[]} are not {\tt NULL} 
but have invalid entries (all entries must be positive, 
entries in {\tt dsizes1[]} must sum to {\tt n1 - p1 + 1},
and
entries in {\tt dsizes2[]} must sum to {\tt n2 - p2 + 1},
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\item
\begin{verbatim}
void localND3D ( int n1, int n2, int n3, int p1, int p2, int p3,
                 int dsizes1[], int dsizes2[], int dsizes3[],
                 int oldToNew[] ) ;
\end{verbatim}
\index{localND3D@{\tt localND3D()}}
This method finds a local nested dissection ordering 
\cite{bha93-localND} for an {\tt n1 x n2 x n3} 3-D grid.
There are {\tt p1 x p2 x p3} domains in the grid.
The {\tt q}'th domain contains a
{\tt dsizes1[q] x dsizes2[q] x dsizes3[q]} 
subgrid of points.
The {\tt dsizes1[]}, {\tt dsizes2[]} and {\tt dsizes3[]} vectors 
are optional;
they allow the user to explicitly input domain sizes.
If {\tt dsizes1[]}, {\tt dsizes2[]} and {\tt dsizes3[]} 
are not {\tt NULL},
the {\tt q = q1 + q2*p1+ q3*p1*p2}'th domain contains a
{\tt dsizes1[q1] x dsizes2[q2] x disizes3[q3]} subgrid of points.
\par \noindent {\it Error checking:}
If {\tt n1}, {\tt n2} or {\tt n3} are less than or equal to zero,
or if {\tt p1}, {\tt p2} or {\tt p3} are less than or equal to zero,
or if $2{\tt p1} - 1 > {\tt n1}$,
or if $2{\tt p2} - 1 > {\tt n2}$,
or if $2{\tt p3} - 1 > {\tt n3}$,
or if {\tt oldToNew} is {\tt NULL},
or if {\tt dsizes1[]}, {\tt disizes2[]} and {\tt dsizes3[]} 
are not {\tt NULL} 
but have invalid entries (all entries must be positive, 
entries in {\tt dsizes1[]} must sum to {\tt n1 - p1 + 1},
entries in {\tt dsizes2[]} must sum to {\tt n2 - p2 + 1},
and
entries in {\tt dsizes3[]} must sum to {\tt n3 - p3 + 1},
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\item
\begin{verbatim}
void fp2DGrid ( int n1, int n2, int ivec[], FILE *fp ) ;
\end{verbatim}
\index{fp2DGrid@{\tt fp2DGrid()}}
This method writes the {\tt ivec[]} vector onto an {\tt n1 x n2}
grid to file {\tt fp}.
This is useful to visualize an ordering or a metric on a grid.
\par \noindent {\it Error checking:}
If {\tt n1} or {\tt n2} are less than or equal to zero,
or if {\tt ivec} or {\tt fp} are {\tt NULL},
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\item
\begin{verbatim}
void fp3DGrid ( int n1, int n2, int n3, int ivec[], FILE *fp ) ;
\end{verbatim}
\index{fp3DGrid@{\tt fp3DGrid()}}
This method writes the {\tt ivec[]} vector onto an {\tt n1 x n2 x n3}
grid to file {\tt fp}.
This is useful to visualize an ordering or a metric on a grid.
\par \noindent {\it Error checking:}
If {\tt n1}, {\tt n2} or {\tt n3} are less than or equal to zero, 
or if {\tt ivec} or {\tt fp} are {\tt NULL},
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\end{enumerate}
\par
%=======================================================================
\subsection{Multiple minimum degree, Nested dissection 
            and multisection wrapper methods}
\par
There are three simple methods to find minimum degree, nested
dissection and multisection orderings.
In addition, there is one method that finds the better of two
methods -- nested dissection and multisection.
(Much of the work to find either nested dissection or multisection
is identical, so this method takes little more time than either of
the two separately.)
\par
To properly specify these methods there are many parameters
--- these three wrapper methods insulate the user from all but one
or two of the parameters.
As a result, the quality of the ordering may not be as good as can
be found by using non-default settings of the parameters.
\par
One wrapper method computes a minimum degree ordering --- the only
input parameter is a random number seed.
Two wrappers methods compute the nested dissection and multisection
orderings --- in addition to a random number seed there is a upper
bound on the subgraph size used during the graph partition.
This is the most sensitive of the parameters.
\par
The user interested in more customized orderings should consult the
chapters on the 
the {\tt GPart}, {\tt DSTree} and {\tt MSMD} objects
that perform the three steps of the ordering process:
perform an incomplete nested dissection of the graph,
construct the map from vertices to stages in which they will be
eliminated, and perform the multi-stage minimum degree ordering.
The driver programs in the {\tt GPart} and {\tt MSMD} directories
fully exercise the graph partition and ordering strategies by
giving the user access to all input parameters.
\par
\begin{enumerate}
%-----------------------------------------------------------------------
\item
\begin{verbatim}
ETree * orderViaMMD ( Graph *graph, int seed, int msglvl, FILE *msgFile ) ;
\end{verbatim}
\index{orderViaMMD@{\tt orderViaMMD()}}
This method returns a front tree {\tt ETree} object for a multiple
minimum degree ordering of the graph {\tt graph}.
The {\tt seed} parameter is a random number seed.
The {\tt msglvl} and {\tt msgFile} parameters govern the
diagnostics output.
Use {\tt msglvl = 0} for no output, {\tt msglvl = 1} for timings
and scalar statistics, and use {\tt msglvl > 1} with care, for it
can generate huge amounts of output.
\par \noindent {\it Error checking:}
If {\tt graph} is {\tt NULL},
or if {\tt msglvl > 0} and {\tt msgFile} is {\tt NULL}, 
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\item
\begin{verbatim}
ETree * orderViaND ( Graph *graph, int maxdomainsize, int seed, 
                     int msglvl, FILE *msgFile ) ;
\end{verbatim}
\index{orderViaND@{\tt orderViaND()}}
This method returns a front tree {\tt ETree} object for a nested
dissection ordering of the graph {\tt graph}.
If a subgraph has more vertices than the {\tt maxdomainsize} parameter,
it is split.
The {\tt seed} parameter is a random number seed.
The {\tt msglvl} and {\tt msgFile} parameters govern the
diagnostics output.
Use {\tt msglvl = 0} for no output, {\tt msglvl = 1} for timings
and scalar statistics, and use {\tt msglvl > 1} with care, for it
can generate huge amounts of output.
\par \noindent {\it Error checking:}
If {\tt graph} is {\tt NULL},
or if ${\tt maxdomainsize} \le 0$,
or if {\tt msglvl > 0} and {\tt msgFile} is {\tt NULL}, 
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\item
\begin{verbatim}
ETree * orderViaMS ( Graph *graph, int maxdomainsize, int seed, 
                     int msglvl, FILE *msgFile ) ;
\end{verbatim}
\index{orderViaMS@{\tt orderViaMS()}}
This method returns a front tree {\tt ETree} object for a 
multisection ordering of the graph {\tt graph}.
If a subgraph has more vertices than the {\tt maxdomainsize} parameter,
it is split.
The {\tt seed} parameter is a random number seed.
The {\tt msglvl} and {\tt msgFile} parameters govern the
diagnostics output.
Use {\tt msglvl = 0} for no output, {\tt msglvl = 1} for timings
and scalar statistics, and use {\tt msglvl > 1} with care, for it
can generate huge amounts of output.
\par \noindent {\it Error checking:}
If {\tt graph} is {\tt NULL},
or if ${\tt maxdomainsize} \le 0$,
or if {\tt msglvl > 0} and {\tt msgFile} is {\tt NULL}, 
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\item
\begin{verbatim}
ETree * orderViaBestOfNDandMS ( Graph *graph, int maxdomainsize, int maxzeros,
                                int maxsize, int seed, int msglvl, FILE *msgFile ) ;
\end{verbatim}
\index{orderViaBestOfNDandMS@{\tt orderViaBestOfNDandMS()}}
This method returns a front tree {\tt ETree} object for a 
better of two orderings, a nested dissection 
and multisection ordering.
If a subgraph has more vertices than the {\tt maxdomainsize} parameter,
it is split.
The {\tt seed} parameter is a random number seed.
This method also transforms the front tree using the {\tt maxzeros}
and {\tt maxsize} parameters.
See the {\tt ETree\_transform()} method 
in Section~\ref{subsection:ETree:proto:transformation}.
The {\tt msglvl} and {\tt msgFile} parameters govern the
diagnostics output.
Use {\tt msglvl = 0} for no output, {\tt msglvl = 1} for timings
and scalar statistics, and use {\tt msglvl > 1} with care, for it
can generate huge amounts of output.
\par \noindent {\it Error checking:}
If {\tt graph} is {\tt NULL},
or if ${\tt maxdomainsize} \le 0$,
or if {\tt msglvl > 0} and {\tt msgFile} is {\tt NULL}, 
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\end{enumerate}
\par
%=======================================================================
\subsection{Graph drawing method}
\par
\begin{enumerate}
%-----------------------------------------------------------------------
\item
\begin{verbatim}
void drawGraphEPS ( Graph *graph, Coords *coords, IV *tagsIV, 
                    double bbox[4], double rect[4], double linewidth1,
                    double linewidth2, double radius, char *epsFileName,
                    int msglvl, FILE *msgFile ) ;
\end{verbatim}
\index{drawGraphEPS@{\tt drawGraphEPS()}}
This method is used to create an EPS (Encapsulated Postscript) file
that contains a picture of a graph in two dimensions.
We use this to visualize separators and domain decompositions,
mostly of regular grids and triangulations of a planar region.
\par
The {\tt graph} object defines the connectivity of the
vertices.
The {\tt coords} object defines the locations of the vertices.
The {\tt tagsIV} object is used to define whether or not an edge
is drawn between two vertices adjacent in the graph.
When {\tt tagsIV} is not {\tt NULL}, 
if there is an edge {\tt (u,v)} in the graph and 
{\tt tags[u] = tags[v]}, then the edge with width {\tt linewidth1}
is drawn.
For edges {\tt (u,v)} in the graph and 
{\tt tags[u] != tags[v]}, then the edge with width {\tt linewidth2}
is drawn, assuming ${\tt linewidth2} > 0$.
If {\tt tagsIV} is {\tt NULL}, than all edges are drawn with 
width {\tt linewidth1}.
Each vertex is draw with a filled circle with radius {\tt radius}.
\par
The graph and its {\tt Coords} object occupy a certain area in 2-D
space.
We try to plot the graph inside the area defined by the {\tt
rect[]} array in such a manner that the relative scales are
preserved (the graph is not stretched in either the $x$ or $y$
direction) and that the larger of the width and height of the graph
fills the area defined by the {\tt rect[]} rectangle.
{\it Note}: hacking postscript is {\it not} an area of expertise of
either author.
Some Postscript viewers give us messages that we are not obeying
the format conventions (this we do not doubt), but we have never
failed to view or print one of these files.
\par \noindent {\it Error checking:}
If the method is unable to open the file, 
an error message is printed and the program exits.
%-----------------------------------------------------------------------
\end{enumerate}
\par
%=======================================================================
\subsection{Linear system construction}
\par
Our driver programs test linear systems where the matrices come
from regular grids using nested dissection orderings.
There are two methods that generate linear systems of this form
along with the front tree and symbolic factorization.
\par
\begin{enumerate}
%-----------------------------------------------------------------------
\item
\begin{verbatim}
void mkNDlinsys ( int n1, int n2, int n3, int maxzeros, int maxsize,
                  int type, int symmetryflag, int nrhs, int seed, int msglvl, 
                  FILE *msgFile, ETree **pfrontETree, IVL **psymbfacIVL, 
                  InpMtx **pmtxA, DenseMtx **pmtxX, DenseMtx **pmtxB ) ;
\end{verbatim}
\index{mkNDlinsys@{\tt mkNDlinsys()}}
This method creates a linear system $AX = B$ for a
${\tt n1} \times {\tt n2} \times {\tt n3}$ grid.
The entries in $A$ and $X$ are random numbers,
$B$ is computed as the product of $A$ with $X$.
$A$ can be real ({\tt type = 1}) or complex ({\tt type = 2}),
and can be symmetric ({\tt symmetryflag = 0}),
Hermitian ({\tt symmetryflag = 1}) or
nonsymmetric ({\tt symmetryflag = 2}).
The number of columns of $X$ is given by {\tt nrhs}.
The linear system is ordered using theoretical nested dissection,
and the front tree is transformed using the {\tt maxzeros} and {\tt
maxsize} parameters.
The addresses of the front tree, symbolic factorization, and three
matrix objects are returned in the last five arguments of the
calling sequence.
\par \noindent {\it Error checking:}
None presently.
%-----------------------------------------------------------------------
\item
\begin{verbatim}
void mkNDlinsysQR ( int n1, int n2, int n3, int type, int nrhs, int seed,
               int msglvl, FILE *msgFile, ETree **pfrontETree, IVL **psymbfacIVL, 
               InpMtx **pmtxA, DenseMtx **pmtxX, DenseMtx **pmtxB ) ;
\end{verbatim}
\index{mkNDlinsysQR@{\tt mkNDlinsysQR()}}
This method creates a linear system $AX = B$ for a
natural factor formulation of a
${\tt n1} \times {\tt n2} \times {\tt n3}$ grid.
If {\tt n1}, {\tt n2} and {\tt n3} are all greater than 1,
the grid is formed of linear hexahedral elements and
the matrix $A$ has {\tt 8*n1*n2*n3} rows.
If one of {\tt n1}, {\tt n2} and {\tt n3} is equal to 1,
the grid is formed of linear quadrilateral elements and
the matrix $A$ has {\tt 4*n1*n2*n3} rows.
The entries in $A$ and $X$ are random numbers,
$B$ is computed as the product of $A$ with $X$.
$A$ can be real ({\tt type = 1}) or complex ({\tt type = 2}).
The number of columns of $X$ is given by {\tt nrhs}.
The linear system is ordered using theoretical nested dissection,
and the front tree is transformed using the {\tt maxzeros} and {\tt
maxsize} parameters.
The addresses of the front tree, symbolic factorization, and three
matrix objects are returned in the last five arguments of the
calling sequence.
\par \noindent {\it Error checking:}
None presently.
%-----------------------------------------------------------------------
\end{enumerate}