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\title{Cuthill McKee}
\author{Andreas Stahel}



\begin{document}
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% \noindent
% \copyright Andreas Stahel, 1999\\
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\section{Some results}

Below find a sample input file for \texttt{triangle}.  The coordinates of
10 nodes are given first. The nodes 1 through 6 describe the outer
boundary, counterclockwise, starting at the lower left corner. Nodes 7
through 10 describe the quadratic hole. Then the segments forming the
boundary of the domain are given, including the hole. At last one point inside
the hole has to be given. The data is stored in a file \com{mytest.poly},
in one column of data.
\begin{verbatim}
# nodes                  # segments              # hole
10 2 0 0                 10 1                    7 7 8 3
1 0 0                    1 1 2 1                 8 8 9 3
2 4 0                    2 2 3 1                 9 9 10 3
3 4 2                    3 3 4 1                 10 10 7 3
4 1 2                    4 4 5 1                                        
5 1 1                    5 5 6 2                 1
6 0 1                    6 6 1 2                 1 2.5 1
7 2 0.5                    
8 3 0.5                    
9 3 1.5                    
10 2 1.5                   
\end{verbatim}
Then run the command \com{triangle -pqa0.01 mytest.poly} to generate the
mesh in figure~\ref{fig:CuthillMesh}. The information is stored in a few files
with names \com{mytest.1.*}~. Some statistics are shown by
\textit{triangle} on the screen.
\begin{verbatim}
Statistics:
  Input points: 10
  Input segments: 10
  Input holes: 1
  Mesh points: 518
  Mesh triangles: 907
  Mesh edges: 1425
  Mesh boundary edges: 1
\end{verbatim}
\textit{triangle} is generating meshes suitable for finite element
computations, e.g. there are no triangles with small angles. But
\textit{triangle} is not concerned with the bandwidth of the resulting
matrix for the FEM problem, i.e. neighboring nodes can have a vastly
differing numbers.
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \epsfig{file=\currdir mytestAmesh.eps,height=5cm}
    \caption{Mesh}
    \label{fig:CuthillMesh}
  \end{center}
\end{figure}

The algorithm of \textbf{Cuthill--McKee} will renumber the nodes to reduce
the bandwidth.  The command \com{CuthillMcKee -v -g mytest.1} reduces the
semibandwidth from 515 to 28. In figure~\ref{fig:CuthillSparse} the nonzero
entries of the matrix are shown. The band structure of the renumbered
problem is evident.


\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \epsfig{file=\currdir mytestAorig.eps,height=6.5cm,width=8cm}\hfil
    \epsfig{file=\currdir mytestACuthill.eps,height=6.5cm,width=8cm}
    \caption{Original numbering  and after renumbering by Cuthill--McKee}
    \label{fig:CuthillSparse}
  \end{center}
\end{figure}

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