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\title{CAMD User Guide}
\author{Patrick R. Amestoy\thanks{ENSEEIHT-IRIT,
2 rue Camichel 31017 Toulouse, France.
email: amestoy@enseeiht.fr.  http://www.enseeiht.fr/$\sim$amestoy.}
\and Yanqing (Morris) Chen
\and Timothy A. Davis\thanks{
email: DrTimothyAldenDavis@gmail.com,
http://www.suitesparse.com.
This work was supported by the National
Science Foundation, under grants ASC-9111263, DMS-9223088, and CCR-0203270.
Portions of the work were done while on sabbatical at Stanford University
and Lawrence Berkeley National Laboratory (with funding from Stanford
University and the SciDAC program).  Ordering constraints added with
support from Sandia National Laboratory (Dept. of Energy).
}
\and Iain S. Duff\thanks{Rutherford Appleton Laboratory, Chilton, Didcot, 
Oxon OX11 0QX, England. email: i.s.duff@rl.ac.uk.  
http://www.numerical.rl.ac.uk/people/isd/isd.html.
This work was supported by the EPSRC under grant GR/R46441.
}}

\input{camd_version.tex}
\maketitle

%------------------------------------------------------------------------------
\begin{abstract}
CAMD is a set of ANSI C routines that implements the approximate minimum degree
ordering algorithm to permute sparse matrices prior to
numerical factorization.  Ordering constraints can be optionally provided.
A MATLAB interface is included.
\end{abstract}
%------------------------------------------------------------------------------

CAMD, Copyright\copyright 2007-2023, Timothy A. Davis, Yanqing Chen, Patrick R.
Amestoy, and Iain S. Duff.  All Rights Reserved.

SPDX-License-Identifier: BSD-3-clause

{\bf Availability:}
    http://www.suitesparse.com.

{\bf Acknowledgments:}

    This work was supported by the National Science Foundation, under
    grants ASC-9111263 and DMS-9223088 and CCR-0203270, and by Sandia
    National Labs (a grant from DOE).
    The conversion to C, the addition of the elimination tree
    post-ordering, and the handling of dense rows and columns
    were done while Davis was on sabbatical at
    Stanford University and Lawrence Berkeley National Laboratory.
    The ordering constraints were added by Chen and Davis.

%------------------------------------------------------------------------------
\newpage
\section{Overview}
%------------------------------------------------------------------------------

CAMD is a set of routines for preordering a sparse matrix prior to
numerical factorization.  It uses an approximate minimum degree ordering
algorithm \cite{AmestoyDavisDuff96,AmestoyDavisDuff04}
to find a permutation matrix $\m{P}$
so that the Cholesky factorization $\m{PAP}\tr=\m{LL}\tr$ has fewer
(often much fewer) nonzero entries than the Cholesky factorization of $\m{A}$.
The algorithm is typically much faster than other ordering methods
and  minimum degree ordering
algorithms that compute an exact degree \cite{GeorgeLiu89}.
Some methods, such as approximate deficiency
\cite{RothbergEisenstat98} and graph-partitioning based methods
\cite{Chaco,KarypisKumar98e,PellegriniRomanAmestoy00,schu:01}
can produce better orderings, depending on the matrix.

The algorithm starts with an undirected graph representation of a
symmetric sparse matrix $\m{A}$.  Node $i$ in the graph corresponds to row
and column $i$ of the matrix, and there is an edge $(i,j)$ in the graph if
$a_{ij}$ is nonzero.
The degree of a node is initialized to the number of off-diagonal nonzeros
in row $i$, which is the size of the set of nodes
adjacent to $i$ in the graph.

The selection of a pivot $a_{ii}$ from the diagonal of $\m{A}$ and the first
step of Gaussian elimination corresponds to one step of graph elimination.
Numerical fill-in causes new nonzero entries in the matrix
(fill-in refers to
nonzeros in $\m{L}$ that are not in $\m{A}$).
Node $i$ is eliminated and edges are added to its neighbors
so that they form a clique (or {\em element}).  To reduce fill-in,
node $i$ is selected as the node of least degree in the graph.
This process repeats until the graph is eliminated.

The clique is represented implicitly.  Rather than listing all the
new edges in the graph, a single list of nodes is kept which represents
the clique.  This list corresponds to the nonzero pattern of the first
column of $\m{L}$.  As the elimination proceeds, some of these cliques
become subsets of subsequent cliques, and are removed.   This graph
can be stored in place, that is
using the same amount of memory as the original graph.

The most costly part of the minimum degree algorithm is the recomputation
of the degrees of nodes adjacent to the current pivot element.
Rather than keep track of the exact degree, the approximate minimum degree
algorithm finds an upper bound on the degree that is easier to compute.
For nodes of least degree, this bound tends to be tight.  Using the
approximate degree instead of the exact degree leads to a substantial savings
in run time, particularly for very irregularly structured matrices.
It has no effect on the quality of the ordering.

The elimination phase is followed by an
elimination tree post-ordering.  This has no effect on fill-in, but
reorganizes the ordering so that the subsequent numerical factorization is
more efficient.  It also includes a pre-processing phase in which nodes of
very high degree are removed (without causing fill-in), and placed last in the
permutation $\m{P}$ (subject to the constraints).
This reduces the run time substantially if the matrix
has a few rows with many nonzero entries, and has little effect on the quality
of the ordering.
CAMD operates on the
symmetric nonzero pattern of $\m{A}+\m{A}\tr$, so it can be given
an unsymmetric matrix, or either the lower or upper triangular part of
a symmetric matrix.

CAMD has the ability to order the matrix with constraints.  Each
node $i$ in the graph (row/column $i$ in the matrix) has a constraint,
{\tt C[i]}, which is in the range {\tt 0} to {\tt n-1}.  All nodes with
{\tt C[i] = 0} are
ordered first, followed by all nodes with constraint {\tt 1}, and so on.
That is, {\tt C[P[k]]} is monotonically non-decreasing as {\tt k} varies from
{\tt 0} to {\tt n-1}.  If {\tt C} is NULL, no
constraints are used (the ordering will be similar to AMD's ordering,
except that the postordering is different).
The optional {\tt C} parameter is also provided in the MATLAB interface,
({\tt p = camd (A,Control,C)}).

For a discussion of the long history of the minimum degree algorithm,
see \cite{GeorgeLiu89}.

%------------------------------------------------------------------------------
\section{Availability}
%------------------------------------------------------------------------------

CAMD is available at http://www.suitesparse.com.
The Fortran version is available as the routine {\tt MC47} in HSL
(formerly the Harwell Subroutine Library) \cite{hsl:2002}. {\tt MC47} does
not include ordering constraints.

%------------------------------------------------------------------------------
\section{Using CAMD in MATLAB}
%------------------------------------------------------------------------------

To use CAMD in MATLAB, you must first compile the CAMD mexFunction.
Just type {\tt make} in the Unix system shell, while in the {\tt CAMD}
directory.  You can also type {\tt camd\_make} in MATLAB, while in the
{\tt CAMD/MATLAB} directory.  Place the {\tt CAMD/MATLAB} directory in your
MATLAB path.  This can be done on any system with MATLAB, including Windows.
See Section~\ref{Install} for more details on how to install CAMD.

The MATLAB statement {\tt p=camd(A)} finds a permutation vector {\tt p} such
that the Cholesky factorization {\tt chol(A(p,p))} is typically sparser than
{\tt chol(A)}.
If {\tt A} is unsymmetric, {\tt camd(A)} is identical to {\tt camd(A+A')}
(ignoring numerical cancellation).
If {\tt A} is not symmetric positive definite,
but has substantial diagonal entries and a mostly symmetric nonzero pattern,
then this ordering is also suitable for LU factorization.  A partial pivoting
threshold may be required to prevent pivots from being selected off the
diagonal, such as the statement {\tt [L,U,P] = lu (A (p,p), 0.1)}.
Type {\tt help lu} for more details.
The statement {\tt [L,U,P,Q] = lu (A (p,p))} in MATLAB 6.5 is
not suitable, however, because it uses UMFPACK Version 4.0 and thus
does not attempt to select pivots from the diagonal.
UMFPACK Version 4.1 in MATLAB 7.0 and later
uses several strategies, including a symmetric pivoting strategy, and
will give you better results if you want to factorize an unsymmetric matrix
of this type.  Refer to the UMFPACK User Guide for more details, at
http://www.suitesparse.com.

The CAMD mexFunction is much faster than the built-in MATLAB symmetric minimum
degree ordering methods, SYMAMD and SYMMMD.  Its ordering quality is
essentially identical to AMD, comparable to SYMAMD, and better than SYMMMD
\cite{DavisGilbertLarimoreNg04}.

An optional input argument can be used to modify the control parameters for
CAMD (aggressive absorption, dense row/column handling, and printing of
statistics).  An optional output
argument provides statistics on the ordering, including an analysis of the
fill-in and the floating-point operation count for a subsequent factorization.
For more details (once CAMD is installed),
type {\tt help camd} in the MATLAB command window.

%------------------------------------------------------------------------------
\section{Using CAMD in a C program}
\label{Cversion}
%------------------------------------------------------------------------------

The C-callable CAMD library consists of eight user-callable routines and one
include file.  There are two versions of seven of the routines, with
\verb'int32_t' and \verb'int64_t' integers.
The routines with prefix
{\tt camd\_l\_} use \verb'int64_t' integer arguments; the others use
\verb'int32_t' integer arguments.

The following routines are fully described in Section~\ref{Primary}:

\begin{itemize}
\item {\tt camd\_order}
(\verb'int64_t' version: {\tt camd\_l\_order})
    {\footnotesize
    \begin{verbatim}
    #include "camd.h"
    int32_t n, Ap [n+1], Ai [nz], P [n], C [n] ;
    double Control [CAMD_CONTROL], Info [CAMD_INFO] ;
    int result = camd_order (n, Ap, Ai, P, Control, Info, C) ;
    \end{verbatim}
    }
    Computes the approximate minimum degree ordering of an $n$-by-$n$ matrix
    $\m{A}$.  Returns a permutation vector {\tt P} of size {\tt n}, where
    {\tt P[k] = i} if row and column {\tt i} are the {\tt k}th row and
    column in the permuted matrix.
    This routine allocates its own memory of size $1.2e+9n$ integers,
    where $e$ is the number of nonzeros in $\m{A}+\m{A}\tr$.
    It computes statistics about the matrix $\m{A}$, such as the symmetry of
    its nonzero pattern, the number of nonzeros in $\m{L}$,
    and the number of floating-point operations required for Cholesky and LU
    factorizations (which are returned in the {\tt Info} array).
    The user's input matrix is not modified.
    It returns {\tt CAMD\_OK} if successful,
    {\tt CAMD\_OK\_BUT\_JUMBLED} if successful (but the matrix had unsorted
    and/or duplicate row indices),
    {\tt CAMD\_INVALID} if the matrix is invalid,
    {\tt CAMD\_OUT\_OF\_MEMORY} if out of memory.

    The array {\tt C} provides the ordering constraints.
    On input, {\tt C} may be null (to denote no constraints);
    otherwise, it must be an array size {\tt n}, with entries in the range
    {\tt 0} to {\tt n-1}.
    On output, {\tt C[P[0..n-1]]} is monotonically non-descreasing.  

\item {\tt camd\_defaults}
(\verb'int64_t' version: {\tt camd\_l\_defaults})
    {\footnotesize
    \begin{verbatim}
    #include "camd.h"
    double Control [CAMD_CONTROL] ;
    camd_defaults (Control) ;
    \end{verbatim}
    }
    Sets the default control parameters in the {\tt Control} array.  These can
    then be modified as desired before passing the array to the other CAMD
    routines.

\item {\tt camd\_control}
(\verb'int64_t' version: {\tt camd\_l\_control})
    {\footnotesize
    \begin{verbatim}
    #include "camd.h"
    double Control [CAMD_CONTROL] ;
    camd_control (Control) ;
    \end{verbatim}
    }
    Prints a description of the control parameters, and their values.

\item {\tt camd\_info}
(\verb'int64_t' version: {\tt camd\_l\_info})
    {\footnotesize
    \begin{verbatim}
    #include "camd.h"
    double Info [CAMD_INFO] ;
    camd_info (Info) ;
    \end{verbatim}
    }
    Prints a description of the statistics computed by CAMD, and their values.

\item {\tt camd\_valid}
(\verb'int64_t' version: {\tt camd\_valid})
    {\footnotesize
    \begin{verbatim}
    #include "camd.h"
    int32_t n, Ap [n+1], Ai [nz] ;
    int result = camd_valid (n, n, Ap, Ai) ;
    \end{verbatim}
    }
    Returns {\tt CAMD\_OK} or {\tt CAMD\_OK\_BUT\_JUMBLED}
    if the matrix is valid as input to {\tt camd\_order};
    the latter is returned if the matrix has unsorted and/or duplicate
    row indices in one or more columns. 
    Returns {\tt CAMD\_INVALID} if the matrix cannot be passed to
    {\tt camd\_order}.
    For {\tt camd\_order}, the matrix must
    also be square.  The first two arguments are the number of rows and the
    number of columns of the matrix.  For its use in CAMD, these must both
    equal {\tt n}.

\item {\tt camd\_2}
(\verb'int64_t' version: {\tt camd\_l2})
    CAMD ordering kernel.  It is faster than {\tt camd\_order}, and
    can be called by the user, but it is difficult to use.
    It does not check its inputs for errors.
    It does not require the columns of its input matrix to be sorted,
    but it destroys the matrix on output.  Additional workspace must be passed.
    Refer to the source file {\tt CAMD/Source/camd\_2.c} for a description.

\item \verb'camd_version': returns the CAMD version

\end{itemize}

The nonzero pattern of the matrix $\m{A}$ is represented in compressed column
form.
For an $n$-by-$n$ matrix $\m{A}$ with {\tt nz} nonzero entries, the
representation consists of two arrays: {\tt Ap} of size {\tt n+1} and {\tt Ai}
of size {\tt nz}.  The row indices of entries in column {\tt j} are stored in
    {\tt Ai[Ap[j]} $\ldots$ {\tt Ap[j+1]-1]}.
For {\tt camd\_order},
if duplicate row indices are present, or if the row indices in any given
column are not sorted in ascending order, then {\tt camd\_order} creates
an internal copy of the matrix with sorted rows and no duplicate entries,
and orders the copy.  This adds slightly to the time and memory usage of
{\tt camd\_order}, but is not an error condition.

The matrix is 0-based, and thus
row indices must be in the range {\tt 0} to {\tt n-1}.
The first entry {\tt Ap[0]} must be zero.
The total number of entries in the matrix is thus {\tt nz = Ap[n]}.

The matrix must be square, but it does not need to be symmetric.
The {\tt camd\_order} routine constructs the nonzero pattern of
$\m{B} = \m{A}+\m{A}\tr$ (without forming $\m{A}\tr$ explicitly if
$\m{A}$ has sorted columns and no duplicate entries),
and then orders the matrix $\m{B}$.  Thus, either the
lower triangular part of $\m{A}$, the upper triangular part,
or any combination may be passed.  The transpose $\m{A}\tr$ may also be
passed to {\tt camd\_order}.
The diagonal entries may be present, but are ignored.

%------------------------------------------------------------------------------
\subsection{Control parameters}
\label{control_param}
%------------------------------------------------------------------------------

Control parameters are set in an optional {\tt Control} array.
It is optional in the sense that if
a {\tt NULL} pointer is passed for the {\tt Control} input argument,
then default control parameters are used.
%
\begin{itemize}
\item {\tt Control[CAMD\_DENSE]} (or {\tt Control(1)} in MATLAB):
controls the threshold for ``dense''
rows/columns.  A dense row/column in $\m{A}+\m{A}\tr$
can cause CAMD to spend significant time
in ordering the matrix.  If {\tt Control[CAMD\_DENSE]} $\ge 0$,
rows/columns with
more than {\tt Control[CAMD\_DENSE]} $\sqrt{n}$ entries are ignored during
the ordering, and placed last in the output order.  The default
value of {\tt Control[CAMD\_DENSE]} is 10.  If negative, no rows/columns
are treated as ``dense.''  Rows/columns with 16 or fewer off-diagonal
entries are never considered ``dense.''
%
\item {\tt Control[CAMD\_AGGRESSIVE]} (or {\tt Control(2)} in MATLAB):
controls whether or not to use
aggressive absorption, in which a prior element is absorbed into the current
element if it is a subset of the current element, even if it is not
adjacent to the current pivot element (refer
to \cite{AmestoyDavisDuff96,AmestoyDavisDuff04}
for more details).  The default value is nonzero,
which means that aggressive absorption will be performed.  This nearly always
leads to a better ordering (because the approximate degrees are more
accurate) and a lower execution time.  There are cases where it can
lead to a slightly worse ordering, however.  To turn it off, set
{\tt Control[CAMD\_AGGRESSIVE]} to 0.
%
\end{itemize}

Statistics are returned in the {\tt Info} array
(if {\tt Info} is {\tt NULL}, then no statistics are returned).
Refer to {\tt camd.h} file, for more details
(14 different statistics are returned, so the list is not included here).

%------------------------------------------------------------------------------
\subsection{Sample C program}
%------------------------------------------------------------------------------

The following program, {\tt camd\_demo.c}, illustrates the basic use of CAMD.
See Section~\ref{Synopsis} for a short description
of each calling sequence.

{\footnotesize
\begin{verbatim}
#include "camd.h"

int32_t n = 5 ;
int32_t Ap [ ] = { 0,   2,       6,       10,  12, 14} ;
int32_t Ai [ ] = { 0,1, 0,1,2,4, 1,2,3,4, 2,3, 1,4   } ;
int32_t C [ ] = { 2, 0, 0, 0, 1 } ;
int32_t P [5] ;

int main (void)
{
    int32_t k ;
    (void) camd_order (n, Ap, Ai, P, (double *) NULL, (double *) NULL, C) ;
    for (k = 0 ; k < n ; k++) printf ("P [%d] = %d\n", k, P [k]) ;
    return (0) ;
}

\end{verbatim}
}

The {\tt Ap} and {\tt Ai} arrays represent the binary matrix
\[
\m{A} = \left[
\begin{array}{rrrrr}
 1 &  1 &  0 &  0 &  0 \\
 1 &  1 &  1 &  0 &  1 \\
 0 &  1 &  1 &  1 &  0 \\
 0 &  0 &  1 &  1 &  0 \\
 0 &  1 &  1 &  0 &  1 \\
\end{array}
\right].
\]
The diagonal entries are ignored.
%
CAMD constructs the pattern of $\m{A}+\m{A}\tr$,
and returns a permutation vector of $(3, 2, 1, 4, 0)$.
Note that nodes 1, 2, and 3 appear first (they are in the constraint set 0),
node 4 appears next (since {\tt C[4] = 1}), and node 0 appears last.
%
Since the matrix is unsymmetric but with a mostly symmetric nonzero
pattern, this would be a suitable permutation for an LU factorization of a
matrix with this nonzero pattern and whose diagonal entries are not too small.
The program uses default control settings and does not return any statistics
about the ordering, factorization, or solution ({\tt Control} and {\tt Info}
are both {\tt (double *) NULL}).  It also ignores the status value returned by
{\tt camd\_order}.

More example programs are included with the CAMD package.
The {\tt camd\_demo.c} program provides a more detailed demo of CAMD.
Another example is the CAMD mexFunction, {\tt camd\_mex.c}.

%------------------------------------------------------------------------------
\subsection{A note about zero-sized arrays}
%------------------------------------------------------------------------------

CAMD uses several user-provided arrays of size {\tt n} or {\tt nz}.
Either {\tt n} or {\tt nz} can be zero.
If you attempt to {\tt malloc} an array of size zero,
however, {\tt malloc} will return a null pointer which CAMD will report
as invalid.  If you {\tt malloc} an array of
size {\tt n} or {\tt nz} to pass to CAMD, make sure that you handle the
{\tt n} = 0 and {\tt nz = 0} cases correctly.

%------------------------------------------------------------------------------
\section{Synopsis of C-callable routines}
\label{Synopsis}
%------------------------------------------------------------------------------

The matrix $\m{A}$ is {\tt n}-by-{\tt n} with {\tt nz} entries.

{\footnotesize
\begin{verbatim}
#include "camd.h"
int32_t n, status, Ap [n+1], Ai [nz], P [n], C [n] ;
double Control [CAMD_CONTROL], Info [CAMD_INFO] ;
camd_defaults (Control) ;
status = camd_order (n, Ap, Ai, P, Control, Info, C) ;
camd_control (Control) ;
camd_info (Info) ;
status = camd_valid (n, n, Ap, Ai) ;
\end{verbatim}
}

The {\tt camd\_l\_*} routines are identical, except that all \verb'int32_t'
arguments become \verb'int64_t':

{\footnotesize
\begin{verbatim}
#include "camd.h"
int64_t n, status, Ap [n+1], Ai [nz], P [n], C [n] ;
double Control [CAMD_CONTROL], Info [CAMD_INFO] ;
camd_l_defaults (Control) ;
status = camd_l_order (n, Ap, Ai, P, Control, Info, C) ;
camd_l_control (Control) ;
camd_l_info (Info) ;
status = camd_l_valid (n, n, Ap, Ai) ;
\end{verbatim}
}

The \verb'camd_version' function uses plain \verb'int':

{\footnotesize
\begin{verbatim}
#include "camd.h"
int version [3] ;
camd_version (version) ;
\end{verbatim}
}

%------------------------------------------------------------------------------
\section{Installation}
\label{Install}
%------------------------------------------------------------------------------

CAMD now relies primarily on CMake to build the library.  It also includes
a simple \verb'CAMD/Makefile' which uses cmake to do the actual build.
The use of this \verb'CAMD/Makefile' is optional; for Windows, just import
the CMakeLists.txt into MS Visual Studio.

To compile and install the library for both system-wide usage and local
usage:

    \begin{verbatim}
        make
        sudo make install
    \end{verbatim}

To compile/install for just local usage (SuiteSparse/lib and
SuiteSparse/include):

    \begin{verbatim}
        make local
        make install
    \end{verbatim}

To run the demos

    \begin{verbatim}
        make demos
    \end{verbatim}

To remove all files in \verb'CAMD/' not in the original distribution (leaves
SuiteSparse/lib and SuiteSparse/include unchanged):

    \begin{verbatim}
        make clean
    \end{verbatim}

To use the CAMD mexFunction in MATLAB, simply type {\tt camd\_make} in MATLAB
while in the {\tt CAMD/MATLAB} directory.  This works on any system with MATLAB,
including Windows.

%------------------------------------------------------------------------------
\newpage
\section{The CAMD routines}
\label{Primary}
%------------------------------------------------------------------------------

The file {\tt CAMD/Include/camd.h} listed below
describes each user-callable routine in the C version of CAMD,
and gives details on their use.

{\footnotesize
\input{camd_h.tex}
}


%------------------------------------------------------------------------------
\newpage
% References
%------------------------------------------------------------------------------

\bibliographystyle{plain}
\bibliography{CAMD_UserGuide}

\end{document}
