AMD_UserGuide.tex

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\title{AMD 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 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).
}
\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.
}}

\date{VERSION 2.4.6, May 4, 2016}
\maketitle

%------------------------------------------------------------------------------
\begin{abstract}
AMD is a set of routines that implements the approximate minimum degree ordering
algorithm to permute sparse matrices prior to
numerical factorization.
There are versions written in both C and Fortran 77.
A MATLAB interface is included.
\end{abstract}
%------------------------------------------------------------------------------

AMD Copyright\copyright 2013 by Timothy A.
Davis, Patrick R. Amestoy, and Iain S. Duff.  All Rights Reserved.
AMD is available under alternate licences; contact T. Davis for details.

{\bf AMD License:} Refer to the AMD/Doc/License.txt file for the license
for your particular copy of AMD.

{\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.
    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.

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

AMD 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.

In the C version of AMD, 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 $\m{P}$.  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.
The C version 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.

The two Fortran versions of AMD are essentially identical to two versions of
the AMD algorithm discussed in an earlier paper \cite{AmestoyDavisDuff96}
(approximate minimum external degree, both with and without aggressive
absorption).
For a discussion of the long history of the minimum degree algorithm,
see \cite{GeorgeLiu89}.

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

In addition to appearing as a Collected Algorithm of the ACM, \newline
AMD 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}.

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

The MATLAB function {\tt amd} is now a built-in function in MATLAB 7.3
(R2006b).  The built-in {\tt amd} and the {\tt amd2} function provided here
differ in how the optional parameters are passed
(the 2nd input parameter).

To use AMD2 in MATLAB, you must first compile the AMD2 mexFunction.
Just type {\tt make} in the Unix system shell, while in the {\tt AMD/MATLAB}
directory.  You can also type {\tt amd\_make} in MATLAB, while in the
{\tt AMD/MATLAB} directory.  Place the {\tt AMD/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 AMD.

The MATLAB statement {\tt p=amd(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 amd(A)} is identical to {\tt amd(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 AMD mexFunction is much faster than the built-in MATLAB symmetric minimum
degree ordering methods, SYMAMD and SYMMMD.  Its ordering quality is
comparable to SYMAMD, and better than SYMMMD
\cite{DavisGilbertLarimoreNg04}.

An optional input argument can be used to modify the control parameters for
AMD (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 AMD is installed),
type {\tt help amd} in the MATLAB command window.

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

The C-callable AMD library consists of seven user-callable routines and one
include file.  There are two versions of each of the routines, with
{\tt int} and {\tt long} integers.
The routines with prefix
{\tt amd\_l\_} use {\tt long} integer arguments; the others use
{\tt int} integer arguments.  If you compile AMD in the standard
ILP32 mode (32-bit {\tt int}'s, {\tt long}'s, and pointers) then the versions
are essentially identical.  You will be able to solve problems using up to 2GB
of memory.  If you compile AMD in the standard LP64 mode, the size of an
{\tt int} remains 32-bits, but the size of a {\tt long} and a pointer both get
promoted to 64-bits.

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

\begin{itemize}
\item {\tt amd\_order}
({\tt long} version: {\tt amd\_l\_order})
    {\footnotesize
    \begin{verbatim}
    #include "amd.h"
    int n, Ap [n+1], Ai [nz], P [n] ;
    double Control [AMD_CONTROL], Info [AMD_INFO] ;
    int result = amd_order (n, Ap, Ai, P, Control, Info) ;
    \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 AMD\_OK} if successful,
    {\tt AMD\_OK\_BUT\_JUMBLED} if successful (but the matrix had unsorted
    and/or duplicate row indices),
    {\tt AMD\_INVALID} if the matrix is invalid,
    {\tt AMD\_OUT\_OF\_MEMORY} if out of memory.

\item {\tt amd\_defaults}
({\tt long} version: {\tt amd\_l\_defaults})
    {\footnotesize
    \begin{verbatim}
    #include "amd.h"
    double Control [AMD_CONTROL] ;
    amd_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 AMD
    routines.

\item {\tt amd\_control}
({\tt long} version: {\tt amd\_l\_control})
    {\footnotesize
    \begin{verbatim}
    #include "amd.h"
    double Control [AMD_CONTROL] ;
    amd_control (Control) ;
    \end{verbatim}
    }
    Prints a description of the control parameters, and their values.

\item {\tt amd\_info}
({\tt long} version: {\tt amd\_l\_info})
    {\footnotesize
    \begin{verbatim}
    #include "amd.h"
    double Info [AMD_INFO] ;
    amd_info (Info) ;
    \end{verbatim}
    }
    Prints a description of the statistics computed by AMD, and their values.

\item {\tt amd\_valid}
({\tt long} version: {\tt amd\_valid})
    {\footnotesize
    \begin{verbatim}
    #include "amd.h"
    int n, Ap [n+1], Ai [nz] ;
    int result = amd_valid (n, n, Ap, Ai) ;
    \end{verbatim}
    }
    Returns {\tt AMD\_OK} or {\tt AMD\_OK\_BUT\_JUMBLED}
    if the matrix is valid as input to {\tt amd\_order};
    the latter is returned if the matrix has unsorted and/or duplicate
    row indices in one or more columns. 
    Returns {\tt AMD\_INVALID} if the matrix cannot be passed to
    {\tt amd\_order}.
    For {\tt amd\_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 AMD, these must both
    equal {\tt n}.

\item {\tt amd\_2}
({\tt long} version: {\tt amd\_l2})
    AMD ordering kernel.  It is faster than {\tt amd\_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 AMD/Source/amd\_2.c} for a description.

\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 amd\_order},
if duplicate row indices are present, or if the row indices in any given
column are not sorted in ascending order, then {\tt amd\_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 amd\_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 amd\_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 amd\_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[AMD\_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 AMD to spend significant time
in ordering the matrix.  If {\tt Control[AMD\_DENSE]} $\ge 0$,
rows/columns with
more than {\tt Control[AMD\_DENSE]} $\sqrt{n}$ entries are ignored during
the ordering, and placed last in the output order.  The default
value of {\tt Control[AMD\_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[AMD\_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[AMD\_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 amd.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 amd\_demo.c}, illustrates the basic use of AMD.
See Section~\ref{Synopsis} for a short description
of each calling sequence.

{\footnotesize
\begin{verbatim}
#include <stdio.h>
#include "amd.h"

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

int main (void)
{
    int k ;
    (void) amd_order (n, Ap, Ai, P, (double *) NULL, (double *) NULL) ;
    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.
%
AMD constructs the pattern of $\m{A}+\m{A}\tr$,
and returns a permutation vector of $(0, 3, 1, 4, 2)$.
%
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 amd\_order}.

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

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

AMD 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 AMD will report
as invalid.  If you {\tt malloc} an array of
size {\tt n} or {\tt nz} to pass to AMD, 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 "amd.h"
int n, status, Ap [n+1], Ai [nz], P [n] ;
double Control [AMD_CONTROL], Info [AMD_INFO] ;
amd_defaults (Control) ;
status = amd_order (n, Ap, Ai, P, Control, Info) ;
amd_control (Control) ;
amd_info (Info) ;
status = amd_valid (n, n, Ap, Ai) ;
\end{verbatim}
}

The {\tt amd\_l\_*} routines are identical, except that all {\tt int}
arguments become {\tt long}:

{\footnotesize
\begin{verbatim}
#include "amd.h"
long n, status, Ap [n+1], Ai [nz], P [n] ;
double Control [AMD_CONTROL], Info [AMD_INFO] ;
amd_l_defaults (Control) ;
status = amd_l_order (n, Ap, Ai, P, Control, Info) ;
amd_l_control (Control) ;
amd_l_info (Info) ;
status = amd_l_valid (n, n, Ap, Ai) ;
\end{verbatim}
}

%------------------------------------------------------------------------------
\section{Using AMD in a Fortran program}
%------------------------------------------------------------------------------

Two Fortran versions of AMD are provided.  The {\tt AMD} routine computes the
approximate minimum degree ordering, using aggressive absorption.  The
{\tt AMDBAR} routine is identical, except that it does not perform aggressive
absorption.  The {\tt AMD} routine is essentially identical to the HSL
routine {\tt MC47B/BD}.
Note that earlier versions of the Fortran
{\tt AMD} and {\tt AMDBAR} routines included an {\tt IOVFLO} argument,
which is no longer present.

In contrast to the C version, the Fortran routines require a symmetric
nonzero pattern, with no diagonal entries present although the {\tt MC47A/AD}
wrapper in HSL allows duplicates, ignores out-of-range entries, and only
uses entries from the upper triangular part of the matrix.  Although we
have an experimental Fortran code for treating ``dense'' rows, the Fortran
codes in this release do not treat
``dense'' rows and columns of $\m{A}$ differently, and thus their run time
can be high if there are a few dense rows and columns in the matrix.
They do not perform a post-ordering of the elimination tree,
compute statistics on the ordering, or check the validity of their input
arguments. These facilities are provided by {\tt MC47A/AD} and other
subroutines from HSL.
Only one {\tt integer}
version of each Fortran routine is provided.  
Both Fortran routines overwrite the user's input
matrix, in contrast to the C version.  
%
The C version does not return the elimination or assembly tree.
The Fortran version returns an assembly tree;
refer to the User Guide for details.
The following is the syntax of the {\tt AMD} Fortran routine.
The {\tt AMDBAR} routine is identical except for the routine name.

{\footnotesize
\begin{verbatim}
        INTEGER N, IWLEN, PFREE, NCMPA, IW (IWLEN), PE (N), DEGREE (N), NV (N),
     $          NEXT (N), LAST (N), HEAD (N), ELEN (N), W (N), LEN (N)
        CALL AMD (N, PE, IW, LEN, IWLEN, PFREE, NV, NEXT,
     $          LAST, HEAD, ELEN, DEGREE, NCMPA, W)
        CALL AMDBAR (N, PE, IW, LEN, IWLEN, PFREE, NV, NEXT,
     $          LAST, HEAD, ELEN, DEGREE, NCMPA, W)
\end{verbatim}
}

The input matrix is provided to {\tt AMD} and {\tt AMDBAR}
in three arrays, {\tt PE}, of size {\tt N},
{\tt LEN}, of size {\tt N}, and {\tt IW}, of size {\tt IWLEN}.  The size of
{\tt IW} must be at least {\tt NZ+N}.  The recommended size is
{\tt 1.2*NZ + N}.
On input, the indices of nonzero entries in row {\tt I} are stored in {\tt IW}.
{\tt PE(I)} is the index in {\tt IW} of the start of row {\tt I}.
{\tt LEN(I)} is the number of entries in row {\tt I}.
The matrix is 1-based, with row and column indices in the range 1 to {\tt N}.
Row {\tt I} is contained in
{\tt IW (PE(I)} $\ldots \:$ {\tt PE(I) + LEN(I) - 1)}.
The diagonal entries must not be present.  The indices within each row must
not contain any duplicates, but they need not be sorted.  The rows
themselves need not be in any particular order, and there may be empty space
between the rows.  If {\tt LEN(I)} is zero, then there are no off-diagonal
entries in row {\tt I}, and {\tt PE(I)} is ignored.  The integer
{\tt PFREE} defines what part of {\tt IW} contains the user's input matrix,
which is held in {\tt IW(1}~$\ldots~\:${\tt PFREE-1)}.
The contents of {\tt IW} and {\tt LEN} are undefined on output,
and {\tt PE} is modified to contain information about the ordering.

As the algorithm proceeds, it modifies the {\tt IW} array, placing the
pattern of the partially eliminated matrix in
{\tt IW(PFREE} $\ldots \:${\tt IWLEN)}.
If this space is exhausted, the space is compressed.
The number of compressions performed on the {\tt IW} array is
returned in the scalar {\tt NCMPA}.  The value of {\tt PFREE} on output is the
length of {\tt IW} required for no compressions to be needed.

The output permutation is returned in the array {\tt LAST}, of size {\tt N}.
If {\tt I=LAST(K)}, then {\tt I} is the {\tt K}th row in the permuted
matrix.  The inverse permutation is returned in the array {\tt ELEN}, where
{\tt K=ELEN(I)} if {\tt I} is the {\tt K}th row in the permuted matrix.
On output, the {\tt PE} and {\tt NV} arrays hold the assembly tree,
a supernodal elimination tree that represents the relationship between
columns of the Cholesky factor $\m{L}$.
If {\tt NV(I)} $> 0$, then {\tt I} is a node in the assembly
tree, and the parent of {\tt I} is {\tt -PE(I)}.  If {\tt I} is a root of
the tree, then {\tt PE(I)} is zero.  The value of {\tt NV(I)} is the
number of entries in the corresponding column of $\m{L}$, including the
diagonal.
If {\tt NV(I)} is zero, then {\tt I} is a non-principal node that is
not in the assembly tree.  Node {\tt -PE(I)} is the parent of node {\tt I}
in a subtree, the root of which is a node in the assembly tree.  All nodes
in one subtree belong to the same supernode in the assembly tree.
The other size {\tt N} arrays
({\tt DEGREE}, {\tt HEAD}, {\tt NEXT}, and {\tt W}) are used as workspace,
and are not defined on input or output.

If you want to use a simpler user-interface and compute the elimination
tree post-ordering, you should be able to call the C routines {\tt amd\_order}
or {\tt amd\_l\_order} from a Fortran program.   Just be sure to take into
account the 0-based indexing in the {\tt P}, {\tt Ap}, and {\tt Ai} arguments
to {\tt amd\_order} and {\tt amd\_l\_order}.  A sample interface is provided
in the files {\tt AMD/Demo/amd\_f77cross.f} and
{\tt AMD/Demo/amd\_f77wrapper.c}.  To compile the {\tt amd\_f77cross} program,
type {\tt make cross} in the {\tt AMD/Demo} directory.  The
Fortran-to-C calling conventions are highly non-portable, so this example
is not guaranteed to work with your compiler C and Fortran compilers.
The output of {\tt amd\_f77cross} is in {\tt amd\_f77cross.out}.

%------------------------------------------------------------------------------
\section{Sample Fortran main program}
%------------------------------------------------------------------------------

The following program illustrates the basic usage of the Fortran version of AMD.
The {\tt AP} and {\tt AI} arrays represent the binary matrix
\[
\m{A} = \left[
\begin{array}{rrrrr}
 1 &  1 &  0 &  0 &  0 \\
 1 &  1 &  1 &  0 &  1 \\
 0 &  1 &  1 &  1 &  1 \\
 0 &  0 &  1 &  1 &  0 \\
 0 &  1 &  1 &  0 &  1 \\
\end{array}
\right]
\]
in a conventional 1-based column-oriented form,
except that the diagonal entries are not present.
The matrix has the same as nonzero pattern of $\m{A}+\m{A}\tr$ in the C
program, in Section~\ref{Cversion}.
The output permutation is $(4, 1, 3, 5, 2)$.
It differs from the permutation returned by the C routine {\tt amd\_order}
because a post-order of the elimination tree has not yet been performed.

{\footnotesize
\begin{verbatim}
        INTEGER N, NZ, J, K, P, IWLEN, PFREE, NCMPA
        PARAMETER (N = 5, NZ = 10, IWLEN = 17)
        INTEGER AP (N+1), AI (NZ), LAST (N), PE (N), LEN (N), ELEN (N),
     $      IW (IWLEN), DEGREE (N), NV (N), NEXT (N), HEAD (N), W (N)
        DATA AP / 1, 2,     5,     8,  9,  11/
        DATA AI / 2, 1,3,5, 2,4,5, 3,  2,3   /
C       load the matrix into the AMD workspace
        DO 10 J = 1,N
            PE (J) = AP (J)
            LEN (J) = AP (J+1) - AP (J)
10      CONTINUE
        DO 20 P = 1,NZ
            IW (P) = AI (P)
20      CONTINUE
        PFREE = NZ + 1
C       order the matrix (destroys the copy of A in IW, PE, and LEN)
        CALL AMD (N, PE, IW, LEN, IWLEN, PFREE, NV, NEXT, LAST, HEAD,
     $      ELEN, DEGREE, NCMPA, W)
        DO 60 K = 1, N
            PRINT 50, K, LAST (K)
50          FORMAT ('P (',I2,') = ', I2)
60      CONTINUE
        END
\end{verbatim}
}

The {\tt Demo} directory contains an example of how the C version
may be called from a Fortran program, but this is highly non-portable.
For this reason, it is placed in the {\tt Demo} directory, not in the
primary {\tt Source} directory.

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

The following discussion assumes you have the {\tt make} program, either in
Unix, or in Windows with Cygwin.

System-dependent configurations are in the
{\tt ../SuiteSparse\_config/SuiteSparse\_config.mk}
file.  You can edit that file to customize the compilation.  The default
settings will work on most systems.
Sample configuration files are provided
for Mac, Linux, Sun Solaris, and IBM AIX.
The system you are on is detected automatically.

To compile and install the C-callable AMD library,
go to the {\tt AMD} directory and type {\tt make}.
A dynamic library is placed in
in {\tt AMD/Lib/libamd.so.*}, ({\tt *.dylib} for the Mac).
Three demo programs of the AMD ordering routine will be compiled and tested in
the {\tt AMD/Demo} directory.
The outputs of these demo programs will then be compared with output
files in the distribution.

To compile and install the Fortran-callable AMD library,
go to the {\tt AMD} directory and type {\tt make fortran}.
The library will be placed in {\tt AMD/Lib/libamdf77.a}.
A demo program will be compiled and tested in the {\tt AMD/Demo} directory.
The output will be compared with an output file in the distribution.

Typing {\tt make clean} will remove all but the final compiled libraries
and demo programs.  Typing {\tt make purge} or {\tt make distclean}
removes all files not in the original distribution.
If you compile AMD and then later change the
{\tt ../SuiteSparse\_config/SuiteSparse\_config.mk} file
then you should type {\tt make purge} and then {\tt make} to recompile.

When you compile your program that uses the C-callable AMD library,
you need to add the {\tt AMD/Lib/libamd.*} library
and you need to tell your compiler to look in the
{\tt AMD/Include} directory for include
files.   To compile a Fortran program that calls the Fortran AMD library,
you need to add the {\tt AMD/Lib/libamdf77.a} library.
See {\tt AMD/Demo/Makefile} for an example.

By doing {\tt make}, all compiled dynamic libraries are also copied into {\tt
SuiteSparse/lib} the include files are coped into {\tt SuiteSparse/include},
and documentation is copied into {\tt SuiteSparse/doc}.

Alternatively, you can install the shared library
and include files in /usr/local/lib and /usr/local/include, and
the documenation in /usr/local/doc.
Just do {\tt make} then {\tt make install}.
Now you can simply use {\tt -lamd} when you compile your own program.

If all you want to use is the AMD2 mexFunction in MATLAB, you can skip
the use of the {\tt make} command entirely.  Simply type
{\tt amd\_make} in MATLAB while in the {\tt AMD/MATLAB} directory.
This works on any system with MATLAB, including Windows.
Alternately, type {\tt make} in the {\tt AMD/MATLAB} directory,
or just use the built-in {\tt amd} in MATLAB 7.3 or later.

If you are including AMD as a subset of a larger library and do not want
to link the C standard I/O library, or if you simply do not need to use
them, you can safely remove the {\tt amd\_control.c} and {\tt amd\_info.c}
files.  Similarly, if you use default parameters (or define your
own {\tt Control} array), then you can exclude the {\tt amd\_defaults.c}
file.
Each of these files contains the user-callable routines of the same
name.  None of these auxiliary routines are directly called by
{\tt amd\_order}.
The {\tt amd\_dump.c} file contains debugging routines
that are neither used nor compiled unless debugging is enabled.
The {\tt amd\_internal.h} file must be edited to enable debugging;
refer to the instructions in that file.
The bare minimum files required to use just {\tt amd\_order} are
{\tt amd.h} and {\tt amd\_internal.h}
in the {\tt Include} directory,
and
{\tt amd\_1.c},
{\tt amd\_2.c},
{\tt amd\_aat.c},
{\tt amd\_global.c},
{\tt and\_order.c},
{\tt amd\_postorder.c},
{\tt amd\_post\_tree.c},
{\tt amd\_preprocess.c},
and
{\tt amd\_valid.c}
in the {\tt Source} directory.

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

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

{\footnotesize
\input{amd_h.tex}
}


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

\bibliographystyle{plain}
\bibliography{AMD_UserGuide}

\end{document}