AMD Version 1.1 User Guide
Patrick R. Amestoy* Timothy A. Davist lain S. Duff
January 29, 2004
Abstract
AMD is a set of routines that implements the approximate minimum degree ordering algo
rithm to permute sparse matrices prior to numerical factorization. There are versions written
in both C and Fortran 77. A MATLAB interface is included.
Technical report TR04002, CISE Department, University of Florida, Gainesville, FL, 2004.
AMD Version 1.1 (Jan. 21, 2004), Copyright@2004 by Timothy A. Davis, Patrick R. A n. I ,,
and Iain S. Duff. All Rights Reserved.
AMD License: Your use or distribution of AMD or any modified version of AMD implies that
you agree to this License.
THIS MATERIAL IS PROVIDED AS IS, WITH ABSOLUTELY NO WARRANTY EXPRESSED
OR IMPLIED. ANY USE IS AT YOUR OWN RISK.
Permission is hereby granted to use or copy this program, provided that the Colp, iht. this
License, and the Availability of the original version is retained on all copies. User documentation
of any code that uses AMD or any modified version of AMD code must cite the Copl,riit. this
License, the Availability note, and "Used by permission." Permission to modify the code and to
distribute modified code is granted, provided the Copyright, this License, and the Availability note
are retained, and a notice that the code was modified is included. This software was developed
with support from the National Science Foundation, and is provided to you free of charge.
Availability: http://www.cise.ufl.edu/research/sparse/amd
Acknowledgments:
This work was supported by the National Science Foundation, under grants ASC9111263 and
DMS'22i :'i and CCR0203270. 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.
*ENSEEIHTIRIT, 2 rue Camichel 31017 Toulouse, France. email: ...... .. ..... ,,1 II.
http://www.enseeiht.fr/~amestoy.
tDept. of Computer and Information Science and Engineering, Univ. of Florida, Gainesville, FL, USA. email:
davis@cise.ufl.edu. http://www.cise.ufl.edu/~davis. This work was supported by the National Science Foundation,
under grants ASC9111263, DMS9223088, and CCR0203270. 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).
tRutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 OQX, 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.
1 Overview
AMD is a set of routines for preordering a sparse matrix prior to numerical factorization. It uses
an approximate minimum degree ordering algorithm [1] to find a permutation matrix P so that
the C'!i .1. l., factorization PAPT LLT has fewer (often much fewer) nonzero entries than the
C'!i, .1 lI: factorization of A. The algorithm is typically much faster than other ordering methods
and minimum degree ordering algorithms that compute an exact degree [3]. Some methods, such
as approximate deficiency [8] and graphpartitioning based methods [4, 6, 7, 9] can produce better
orderings, depending on the matrix.
The algorithm starts with an undirected graph representation of a symmetric sparse matrix 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 aij is nonzero. The degree of a node is initialized to the number of offdiagonal
nonzeros in row i, which is the size of the set of nodes adjacent to i in the graph.
The selection of a pivot aii from the diagonal of A and the first step of Gaussian elimination
corresponds to one step of graph elimination. Numerical fillin causes new nonzero entries in the
matrix (fillin refers to nonzeros in L that are not in A). Node i is eliminated and edges are added
to its neighbors so that they form a clique (or element). To reduce fillin, 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 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 postordering.
This has no effect on fillin, but reorganizes the ordering so that the subsequent numerical factor
ization is more efficient. It also includes a preprocessing phase in which nodes of very high degree
are removed (without causing fillin), and placed last in the permutation 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 A+AT, 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 [1] (approximate minimum external degree, both with and without
i:_:_. i\. absorption). For a discussion of the long history of the minimum degree algorithm, see
[3].
2 Availability
In addition to appearing as a Collected Algorithm of the ACM, AMD Version 1.1 is available at
http://www.cise.ufl.edu/research/sparse. The Fortran version is available as the routine MC47 in
HSL (formerly the Harwell Subroutine Library) [5].
3 Using AMD in MATLAB
To use AMD in MATLAB, you must first compile the AMD mexFunction. Just type make in the
Unix , I. in shell, while in the AMD directory. You can also type amd_make in MATLAB, while in
the AMD/MATLAB directory. Place the AMD/MATLAB directory in your MATLAB path. This can be
done on any system with MATLAB, including Windows. See Section 8 for more details on how to
install AMD.
The MATLAB statement p=amd(A) finds a permutation vector p such that the C'!, .I lI: factor
ization chol(A(p,p)) is typically sparser than chol(A). If A is unsymmetric, amd(A) is identical
to amd(A+A') (ignoring numerical cancellation). If 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 [L,U,P] = lu (A (p,p), 0.1). Type help
lu for more details. The statement [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 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.cise.ufl.edu/research/sparse/umfpack.
The AMD mexFunction is much faster than the builtin MATLAB symmetric minimum degree
ordering methods, SYMAMD and SYMMMD. Its ordering quality is comparable to SYMAMD,
and better than SYMMMD [2].
An optional input argument can be used to modify the control parameters for AMD ( ::_ressive
absorption, dense row/column Ii i _,lli :_. and printing of statistics). An optional output argument
provides statistics on the ordering, including an .111 I1, i, of the fillin and the floatingpoint oper
ation count for a subsequent factorization. For more details (once AMD is installed), type help
amd in the MATLAB command window.
4 Using AMD in a C program
The Ccallable AMD library consists of five usercallable routines and one include file. There are
two versions of each of the routines, with int and long integers. The routines with prefix amd_l_ use
long integer arguments; the others use int integer arguments. If you compile AMD in the standard
ILP32 mode (32bit into's, 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 int remains 32bits, but the size of a long and a pointer both get promoted
to 64bits.
The following routines are fully described in Section 9:
Samd_order (long version: amd_l_order)
#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)
Computes the approximate minimum degree ordering of an nb, ,, matrix A. Returns a
permutation vector P of size n, where P [k] = i if row and column i are the kth 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 A + AT. It computes statistics about the
matrix A, such as the symmetry of its nonzero pattern, the number of nonzeros in L, and
the number of floatingpoint operations required for C'! .I. lI: and LU factorizations (which
are returned in the Info array). The user's input matrix is not modified. It returns AMD_OK
if successful, AMD_INVALID if the matrix is invalid, or AMD_OUT_OF_MEMORY if out of memory.
amd_defaults (long version: amd_l_defaults)
#include "amd.h"
double Control [AMDCONTROL] ;
amddefaults (Control) ;
Sets the default control parameters in the Control array. These can then be modified as
desired before passing the ii i', to the other AMD routines.
amd_control (long version: amd_l_control)
#include "amd.h"
double Control [AMD_CONTROL] ;
amd_control (Control) ;
Prints a description of the control parameters, and their values.
amd_info (long version: amd_l_info)
#include "amd.h"
double Info [AMD_INFO]
amd_info (Info) ;
Prints a description of the statistics computed by AMD, and their values.
amd_preprocess (long version: amd_l_info)
#include "amd.h"
int n, Ap [n+1], Ai [nz], Rp [n+1], Ri [nz] ;
int result = amd_preprocess (n, Ap, Ai, Rp, Ri)
Removes duplicate entries and sorts each column of its input A, and returns the nonzero
pattern of the transpose, R AT. It returns the transpose because this is the simplest way
to sort a matrix and remove duplicate entries. Either A or AT can be passed to amd_order
with little effect on the ordering (except for minor tiebreaking changes).
The nonzero pattern of the matrix A is represented in compressed column form. For an nb, ,
matrix A with nz nonzero entries, the representation consists of two arrays: Ap of size n+1 and
Ai of size nz. The row indices of entries in column j are stored in Ai [Ap [j] ... Ap[j+1]1]. For
amd_order, no duplicate row indices may be present, and the row indices in any given column must
be sorted in ascending order. The matrix is 0based, and thus row indices must be in the range 0
to n1. The first entry Ap [0] must be zero. The total number of entries in the matrix is thus nz
= Ap[n].
The matrix must be square, but it does not need to be symmetric. The amd_order routine
constructs the nonzero pattern of B A + AT (without forming AT explicitly), and then orders
the matrix B. Thus, either the lower triangular part of A, the upper triangular part, or any
combination may be passed. The transpose AT may also be passed to amd_order. The diagonal
entries may be present, but are ignored.
The input to amd_order must have sorted columns because it uses an inplace algorithm to
construct A + AT without first constructing AT. This saves memory, but places an additional
restriction on the input. If the input matrix has columns with unsorted and/or duplicate row
indices, it is not valid as input to amd_order. To handle this case, the amd_preprocess routine is
provided. It sorts, transposes, and removes duplicate entries from its input matrix, returning its
result as another compressedcolumn matrix R which can then be passed to amd_order.
4.1 Control parameters
Control parameters are set an optional Control array. It is optional in the sense that if a NULL
pointer is passed for the Control input argument, then default control parameters are used.
Control [AMD_DENSE] (or Control (1) in MATLAB): controls the threshold for I ii. rows/columns.
A dense row/column in A + AT can cause AMD to spend significant time in ordering the
matrix. If Control [AMD_DENSE] > 0, rows/columns with more than Control [AMD_DENSE]
'/n entries are ignored during the ordering, and placed last in the output order. The default
value of Control [AMDDENSE] is 10. If negative, no rows/columns are treated as ,I ii,
Rows/columns with 16 or fewer offdiagonal entries are never considered .I. i "
Control [AMDAGGRESSIVE] (or Control(2) in MATLAB): controls whether or not to use
i:_:_ressive 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 adi II. in to the current pivot element (refer to
[1] for more details). The default value is nonzero, which means that :_:ressive absorption
will be performed. This nearly , 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 Control [AMD_AGGRESSIVE] to 0.
Statistics are returned in the Info array (if Info is NULL, then no statistics are returned). Refer
to amd.h file, for more details (14 different statistics are returned, so the list is not included here).
4.2 Sample C program
The following program, amd_demo.c, illustrates the basic use of AMD. See Section 5 for a short
description of each calling sequence.
#include
#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) ;
The Ap and Ai arrays represent the binary matrix
S1 0 0 0
1 1 1 0 1
A 0 1 1 1 0
0 0 1 1 0
0 1 1 0 1
The diagonal entries are ignored. AMD constructs the pattern of A+AT, 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 (Control and Info
are both (double *) NULL). It also ignores the status value returned by amdorder.
More example programs are included with the AMD package. The amd_demo. c program provides
a more detailed demo of AMD. Another example is the AMD mexFunction, amd_mex.c.
4.3 A note about zerosized arrays
AMD uses several userprovided 11i 1, of size n or nz. Either n or nz can be zero. If you attempt
to malloc an array of size zero, however, malloc will return a null pointer which AMD will report
as invalid. If you malloc an array of size n or nz to pass to AMD, make sure that you handle the
n = 0 and nz = 0 cases correctly.
5 Synopsis of Ccallable routines
The matrix A is nbyn with nz entries.
#include "amd.h"
int n, status, Ap [n+1], Ai [nz], P [n], Rp [n+1], Ri [nz]
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) ;
amd_preprocess (n, Ap, Ai, Rp, Ri)
The amd_l_* routines are identical, except that all int arguments become long:
#include "amd.h"
long n, status, Ap [n+1], Ai [nz], P [n], Rp [n+1], Ri [nz] ;
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) ;
amd_l_preprocess (n, Ap, Ai, Rp, Ri)
6 Using AMD in a Fortran program
Two Fortran versions of AMD are provided. The AMD routine computes the approximate minimum
degree oil1, i ir:. using ::zressive absorption. The AMDBAR routine is identical, except that it does
not perform ::zressive absorption. The AMD routine is essentially identical to the HSL routine
MC47B/BD. Note that earlier versions of the Fortran AMD and AMDBAR routines included an 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 MC47A/AD wrapper in HSL allows duplicates, ignores outof
range entries, and only uses entries from the upper triangular part of the matrix. Although we have
an experimental Fortran code for treating 1. ii, rows, the Fortran codes in this release do not
treat I, ii rows and columns of 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 postordering of the elimination
tree, compute statistics on the oicl. iiir:. or check the validity of their input arguments. These
facilities are provided by MC47A/AD and other subroutines from HSL. Only one 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 AMD Fortran routine. The AMDBAR routine is identical except for the routine name.
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)
The input matrix is provided to AMD and AMDBAR in three arrays, PE, of size N, LEN, of size N,
and IW, of size IWLEN. The size of IW must be at least NZ+N. The recommended size is 1.2*NZ +
N. On input, the indices of nonzero entries in row I are stored in IW. PE(I) is the index in IW
of the start of row I. LEN(I) is the number of entries in row I. The matrix is 1based, with row
and column indices in the range 1 to N. Row I is contained in IW (PE(I) ... 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 LEN(I) is zero, then there are no offdiagonal
entries in row I, and PE(I) is ignored. The integer PFREE defines what part of IW contains the
user's input matrix, which is held in IW(1 ... PFREE1). The contents of IW and LEN are undefined
on output, and PE is modified to contain information about the ordering.
As the algorithm proceeds, it modifies the IW 11 1,, placing the pattern of the partially elim
inated matrix in IW(PFREE ... IWLEN). If this space is exhausted, the space is compressed. The
number of compressions performed on the IW array is returned in the scalar NCMPA. The value of
PFREE on output is the length of IW required for no compressions to be needed.
The output permutation is returned in the array LAST, of size N. If I=LAST(K), then I is the
Kth row in the permuted matrix. The inverse permutation is returned in the ,i , ELEN, where
K=ELEN(I) if I is the Kth row in the permuted matrix. On output, the PE and NV arrays hold the
assembly tree, a supernodal elimination tree that represents the relationship between columns of
the C'! .. lI factor L. If NV(I) > 0, then I is a node in the assembly tree, and the parent of I is
PE(I). If I is a root of the tree, then PE(I) is zero. The value of NV(I) is the number of entries in
the corresponding column of L, including the diagonal. If NV(I) is zero, then I is a nonprincipal
node that is not in the assembly tree. Node PE(I) is the parent of node 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 N arrays (DEGREE, HEAD, NEXT, and W) are used as workspace, and
are not defined on input or output.
If you want to use a simpler userinterface and compute the elimination tree postordering, you
should be able to call the C routines amd_order or amd_l_order from a Fortran program. Just be
sure to take into account the 0based indexing in the P, Ap, and Ai arguments to amd_order
and amd_l_order. A sample interface is provided in the files AMD/Demo/amd_f77cross.f and
AMD/Demo/amd_f77wrapper.c. To compile the amd_f77cross program, type make cross in the
AMD/Demo directory. The FortrantoC calling conventions are highly nonportable, so this example
is not guaranteed to work with your compiler C and Fortran compilers. The output of amd_f77cross
is in amd_f77cross .out.
7 Sample Fortran main program
The following program illustrates the basic usage of the Fortran version of AMD. The AP and AI
arrays represent the binary matrix
S1 0 0 0
1 1 1 0 1
A 0 1 1 1 1
0 0 1 1 0
0 1 1 0 1
in a conventional 1based columnoriented form, except that the diagonal entries are not present.
The matrix has the same as nonzero pattern of A + AT in the C program, in Section 4. The output
permutation is (4, 1, 3, 5, 2). It differs from the permutation returned by the C routine amdorder
because a postorder of the elimination tree has not yet been performed.
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, II/
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 (1,12,1) = 1, 12)
60 CONTINUE
END
00110
01101
in a conventional 1based columnoriented form, except that the diagonal entries are not present.
The matrix has the same as nonzero pattern of A + AT in the C program, in Section 4. The output
permutation is (4, 1, 3, 5, 2). It differs from the permutation returned by the C routine amdorder
because a postorder of the elimination tree has not yet been performed.
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 = I,N
PE (J) = AP (J)
LEN (J) = AP (J+i) AP (J)
10 CONTINUE
DO 20 P = 1,NZ
IW (P) = AI (P)
20 CONTINUE
PFREE= NZ + I
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 (',12,')= ', 12)
60 CONTINUE
END
The Demo directory contains an example of how the C version may be called from a Fortran
program, but this is highly nonportable. For this reason, it is placed in the Demo directory, not in
the primary Source directory.
8 Installation
The following discussion assumes you have the make program, either in Unix, or in Windows with
Cygwin.
Systemdependent configurations are in the AMD/Make directory. You can edit the Make. include
file in that directory to customize the compilation. The default settings will work on most systems.
Sample configuration files are provided for Linux, Sun Solaris, SGI IRIX, IBM AIX, and the
DEC/Compaq Alpha.
To compile and install the Ccallable AMD library, go to the AMD directory and type make. The
library will be placed in AMD/Lib/libamd. a. Two demo programs of the AMD ordering routine will
be compiled and tested in the AMD/Demo directory. The outputs of these demo programs will then
be compared with output files in the distribution. The AMD mexFunction for use in MATLAB
will also be compiled. If you do not have MATLAB type make lib instead.
To compile and install the Fortrancallable AMD library, go to the AMD directory and type make
fortran. The library will be placed in AMD/Lib/libamdf77. a. A demo program will be compiled
and tested in the AMD/Demo directory. The output will be compared with an output file in the
distribution.
Typing make clean will remove all but the final compiled libraries and demo programs. Typing
make purge removes all files not in the original distribution. If you compile AMD and then later
change the Make. include file or your systemspecific configuration file such as Make.linux, then
you should type make purge and then make to recompile.
Here are the various parameters that you can control in your Make. include file:
CC = your C compiler, such as cc.
RANLIB = your ,I. i 's ranlib program, if needed.
CFLAGS = optimization flags, such as 0.
LIB = your libraries, such as lm or lblas.
RM = the command to delete a file.
MV = the command to rename a file.
MEX = the command to compile a MATLAB mexFunction.
F77 = the command to compile a Fortran program (optional).
F77FLAGS = the Fortran compiler flags (optional).
F77LIB = the Fortran libraries (optional).
The Make.include includes some definitions regarding the BLAS. This is so that AMD and
UMFPACK (which requires AMD) can share the same configuration files. If you wish to use AMD
only, then you can ignore any references to the BLAS (the DNBLAS compile flag).
When you compile your program that uses the Ccallable AMD library, you need to add the
AMD/Lib/libamd. a library and you need to tell your compiler to look in the AMD/Include directory
for include files. To compile a Fortran program that calls the Fortran AMD library, you need to
add the AMD/Lib/libamdf77. a library. See AMD/Demo/Makefile for an example.
If all you want to use is the AMD mexFunction in MATLAB, you can skip the use of the make
command entirely. Simply type amd_make in MATLAB while in the AMD/MATLAB directory. This
works on any , I. ii, with MATLAB, including Windows.
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 amd_control. c
and amd_info.c files. Similarly, if you use default parameters (or define your own Control array),
then you can exclude the amd_defaults.c file. The amd_preprocess.c file is optional as well, if
you can ensure that the input matrix to amd_order is '. , sorted and has no duplicate entries.
Each of these files contains the usercallable routines of the same name. None of these auxiliary
routines are directly called by amd_order. The amd_dump. c file contains debugging routines that
are neither used nor compiled unless debugging is enabled. The amd_internal.h file must be
edited to enable debiu:_:_:_. refer to the instructions in that file. Thus, it too can be excluded if
compiled into a larger production program or library. The bare minimum files required to use just
amd_order are amd.h in the Include directory, and amd_l.c, amd_2.c, amd_aat.c, and_order.c,
amd_postorder.c, amd_post_tree.c, amdvalid.c, and amd_internal.h, in the Source directory.
9 The AMD routines
The file AMD/Include/amd.h listed below describes each usercallable routine in the C version of
AMD, and gives details on their use.
/* ======================================== */
/* === AMD: approximate minimum degree ordering =========================== */
/* ======================================== */
/*  */
/* AMD Version 1.1 (Jan. 21, 2004), Copyright (c) 2004 by Timothy A. Davis, */
/* Patrick R. Amestoy, and Iain S. Duff. See ../README for License. */
/* email: davis@cise.ufl.edu CISE Department, Univ. of Florida. */
/* web: http://www.cise.ufl.edu/research/sparse/amd */
/*  */
/* AMD finds a symmetric ordering P of a matrix A so that the Cholesky
* factorization of P*A*P' has fewer nonzeros and takes less work than the
* Cholesky factorization of A. If A is not symmetric, then it performs its
* ordering on the matrix A+A'. Two sets of usercallable routines are
* provided, one for "int" integers and the other for "long" integers.
* The method is based on the approximate minimum degree algorithm, discussed
* in Amestoy, Davis, and Duff, "An approximate degree ordering algorithm",
* SIAM Journal of Matrix Analysis and Applications, vol. 17, no. 4, pp.
* 886905, 1996. This package can perform both the AMD ordering (with
* aggressive absorption), and the AMDBAR ordering (without aggressive
* absorption) discussed in the above paper. This package differs from the
* Fortran codes discussed in the paper:
(1) it can ignore "dense" rows and columns, leading to faster run times
(2) it computes the ordering of A+A' if A is not symmetric
(3) it is followed by a depthfirst postordering of the assembly tree
(or supernodal elimination tree)
For historical reasons, the Fortran versions, amd.f and amdbar.f, have
been left (nearly) unchanged. They compute the identical ordering as
described in the above paper.
#ifndef AMD H
#define AMD H
int amd_order (
int n,
const int Ap [ ],
const int Ai [ ],
int P [ ],
double Control [ ],
double Info [ ]
) ;
long amdlorder (
long n,
const long Ap [ ],
const long Ai [ ],
long P [ ],
double Control [ ],
double Info [ ]
/* returns 0 if OK, negative value if error */
/* A is nbyn. n must be >= 0. */
/* column pointers for A, of size n+1 */
/* row indices of A, of size nz = Ap [n] */
/* output permutation, of size n */
/* input Control settings, of size AMD_CONTROL */
/* output Info statistics, of size AMD_INFO */
/* see above for description of arguments */
);
/* Input arguments (not modified):
* n: the matrix A is nbyn.
* Ap: an int/long array of size n+1, containing the column pointers of A.
* Ai: an int/long array of size nz, containing the row indices of A,
* where nz = Ap [n].
* Control: a double array of size AMD_CONTROL, containing control
* parameters. Defaults are used if Control is NULL.
*
* Output arguments (not defined on input):
* P: an int/long array of size n, containing the output permutation. If
* row i is the kth pivot row, then P [k] = i. In MATLAB notation,
* the reordered matrix is A (P,P).
* Info: a double array of size AMD_INFO, containing statistical
* information. Ignored if Info is NULL.
* On input, the matrix A is stored in columnoriented form. The row indices
* of nonzero entries in column j are stored in Ai [Ap [j] ... Ap [j+1]1].
* The row indices must appear in ascending order in each column, and there
* must not be any duplicate entries. Row indices must be in the range 0 to
* n1. Ap [0] must be zero, and thus nz = Ap [n] is the number of nonzeros
* in A. The array Ap is of size n+1, and the array Ai is of size nz = Ap [n].
* The matrix does not need to be symmetric, and the diagonal does not need to
* be present (if diagonal entries are present, they are ignored except for
* the output statistic Info [AMD_NZDIAG]). The arrays Ai and Ap are not
* modified. This form of the Ap and Ai arrays to represent the nonzero
* pattern of the matrix A is the same as that used internally by MATLAB.
* If you wish to use a more flexible input structure, please see the
* umfpack_*_triplet_to_col routines in the UMFPACK package, at
* http://www.cise.ufl.edu/research/sparse/umfpack, or use the amd_preprocess
* routine discussed below.
* Restrictions: n >= 0. Ap [0] = 0. Ap [j] <= Ap [j+1] for all j in the
* range 0 to n1. nz = Ap [n] >= 0. For all j in the range 0 to n1,
* and for all p in the range Ap [j] to Ap [j+1]2, Ai [p] < Ai [p+1] must
* hold. Ai [O..nz1] must be in the range 0 to n1. To avoid integer
* overflow, (2.4*nz + 8*n) < INT_MAX / sizeof (int) for must hold for the
* "int" version. (2.4*nz + 8*n) < LONG_MAX / sizeof (long) must hold
* for the "long" version. Finally, Ai, Ap, and P must not be NULL. If
* any of these restrictions are not met, AMD returns AMD_INVALID.
*
* AMD returns:
* AMD_OK if the matrix is valid and sufficient memory can be allocated to
* perform the ordering.
* AMDOUTOFMEMORY if not enough memory can be allocated.
* AMD_INVALID if the input arguments n, Ap, Ai are invalid, or if P is
* NULL.
* The AMD routine first forms the pattern of the matrix A+A', and then
* computes a fillreducing ordering, P. If P [k] = i, then row/column i of
* the original is the kth pivotal row. In MATLAB notation, the permuted
* matrix is A (P,P), except that Obased indexing is used instead of the
* 1based indexing in MATLAB.
* The Control array is used to set various parameters for AMD. If a NULL
* pointer is passed, default values are used. The Control array is not
* modified.
* Control [AMD_DENSE]: controls the threshold for "dense" rows/columns.
* A dense row/column in A+A' can cause AMD to spend a lot of time in
* ordering the matrix. If Control [AMD_DENSE] >= 0, rows/columns
* with more than Control [AMD_DENSE] sqrt (n) entries are ignored
* during the ordering, and placed last in the output order. The
* default value of Control [AMD_DENSE] is 10. If negative, no
* rows/columns are treated as "dense". Rows/columns with 16 or
* fewer offdiagonal entries are never considered "dense".
* Control [AMD_AGGRESSIVE]: controls whether or not to use aggressive
* absorption, in which a prior element is absorbed into the current
* element if is a subset of the current element, even if it is not
* adjacent to the current pivot element (refer to Amestoy, Davis,
* & Duff, 1996, for more details). The default value is nonzero,
* which means to perform aggressive absorption. 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 Control [AMD_AGGRESSIVE] to 0.
* Control [2..4] are not used in the current version, but may be used in
* future versions.
* The Info array provides statistics about the ordering on output. If it is
* not present, the statistics are not returned. This is not an error
* condition.
* Info [AMD_STATUS]: the return value of AMD, either AMD_OK,
* AMDOUTOFMEMORY, or AMD_INVALID.
* Info [AMD_N]: n, the size of the input matrix
* Info [AMD_NZ]: the number of nonzeros in A, nz = Ap [n]
* Info [AMD_SYMMETRY]: the symmetry of the matrix A. It is the number
* of "matched" offdiagonal entries divided by the total number of
* offdiagonal entries. An entry A(i,j) is matched if A(j,i) is also
* an entry, for any pair (i,j) for which i != j. In MATLAB notation,
* S = spones (A) ;
* B = tril (S, 1) + triu (S, 1)
* symmetry = nnz (B & B') / nnz (B) ;
* Info [AMD_NZDIAG]: the number of entries on the diagonal of A.
* Info [AMDNZ_A_PLUSAT]: the number of nonzeros in A+A', excluding the
* diagonal. If A is perfectly symmetric (Info [AMD_SYMMETRY] = 1)
* with a fully nonzero diagonal, then Info [AMDNZ_A_PLUSAT] = nzn
* (the smallest possible value). If A is perfectly unsymmetric
* (Info [AMD_SYMMETRY] = 0, for an upper triangular matrix, for
* example) with no diagonal, then Info [AMD_NZ_A_PLUS_AT] = 2*nz
* (the largest possible value).
* Info [AMD_NDENSE]: the number of "dense" rows/columns of A+A' that were
* removed from A prior to ordering. These are placed last in the
* output order P.
* Info [AMD_MEMORY]: the amount of memory used by AMD, in bytes. In the
* current version, this is 1.2 Info [AMD_NZ_A_PLUS_AT] + 9*n
* times the size of an integer. This is at most 2.4nz + 9n. This
* excludes the size of the input arguments Ai, Ap, and P, which have
* a total size of nz + 2*n + 1 integers.
* Info [AMDNCMPA]: the number of garbage collections performed.
* Info [AMDLNZ]: the number of nonzeros in L (excluding the diagonal).
* This is a slight upper bound because mass elimination is combined
* with the approximate degree update. It is a rough upper bound if
* there are many "dense" rows/columns. The rest of the statistics,
* below, are also slight or rough upper bounds, for the same reasons.
* The postordering of the assembly tree might also not exactly
* correspond to a true elimination tree postordering.
* Info [AMDNDIV]: the number of divide operations for a subsequent LDL'
* or LU factorization of the permuted matrix A (P,P).
* Info [AMDNMULTSUBSLDL]: the number of multiplysubtract pairs for a
* subsequent LDL' factorization of A (P,P).
* Info [AMDNMULTSUBSLU]: the number of multiplysubtract pairs for a
* subsequent LU factorization of A (P,P), assuming that no numerical
* pivoting is required.
* Info [AMDDMAX]: the maximum number of nonzeros in any column of L,
* including the diagonal.
* Info [14..19] are not used in the current version, but may be used in
* future versions.
*/
/*  */
/* AMD preprocess */
/*  */
/* amdpreprocess: sorts, removes duplicate entries, and transposes the
* nonzero pattern of a columnform matrix A, to obtain the matrix R.
* Alternatively, you can consider this routine as constructing a rowform
* matrix from a columnform matrix. Duplicate entries are allowed in A (and
* removed in R). The columns of R are sorted. Checks its input A for errors.
* On input, A can have unsorted columns, and can have duplicate entries.
* Ap [0] must still be zero, and Ap must be monotonically nondecreasing.
* Row indices must be in the range 0 to n1.
* On output, if this routine returns AMD_OK, then the matrix R is a valid
* input matrix for AMD_order. It has sorted columns, with no duplicate
* entries in each column. Since AMD_order operates on the matrix A+A', it
* can just as easily use A or A', so the transpose has no significant effect
* (except for minor tiebreaking, which can lead to a minor effect in the
* quality of the ordering). As an example, compare the output of amddemo.c
* and amd demo2.c.
* This routine transposes A to get R because that's the simplest way to
* sort and remove duplicate entries from a matrix.
* Allocates 2*n integer work arrays, and free's them when done.
* If you wish to call amd_order, but do not know if your matrix has unsorted
* columns or duplicate entries, then you can use the following code, which is
* fairly efficient. amd_order will not allocate any internal matrix until
* it checks that the input matrix is valid, so the method below is memory
* efficient as well. This code snippet assumes that Rp and Ri are already
* allocated, and are the same size as Ap and Ai respectively.
result = amd_order (n, p, Ap, Ai, Control, Info)
if (result == AMD_INVALID)
{
if (amd_preprocess (n, Ap, Ai, Rp, Ri) == AMD_OK)
{
result = amd_order (n, p, Rp, Ri, Control, Info)
}
}
amd_preprocess will still return AMD_INVALID if any row index in Ai is out
of range or if the Ap array is invalid. These errors are not corrected by
amd_preprocess since they represent a more serious error that should be
flagged with the AMD_INVALID error code.
*/
int amd_preprocess
(
int n,
const int Ap [ ],
const int Ai [ ],
int Rp [ ],
int Ri [ ]
long amd_l_preprocess
(
long n,
const long Ap [ ],
const long Ai [ ],
long Rp [ ],
long Ri [ ]
/* Input arguments (not modified):
* n: the matrix A is nbyn.
* Ap: an int/long array of size n+1, containing the column pointers of A.
* Ai: an int/long array of size nz, containing the row indices of A,
* where nz = Ap [n].
* The nonzero pattern of column j of A is in Ai [Ap [j] ... Ap [j+l]l].
* Ap [0] must be zero, and Ap [j] <= Ap [j+1] must hold for all j in the
* range 0 to n1. Row indices in Ai must be in the range 0 to n1.
* The row indices in any one column need not be sorted, and duplicates
* may exist.
* Output arguments (not defined on input):
* Rp: an int/long array of size n+1, containing the column pointers of R.
Ri: an int/long array of size rnz, containing the row indices of R,
where rnz = Rp [n]. Note that Rp [n] will be less than Ap [n] if
duplicates appear in A. In general, Rp [n] <= Ap [n].
The data structure for R is the same as A, except that each column of
R contains sorted row indices, and no duplicates appear in any column.
* amd_preprocess returns:
* AMD_OK if the matrix A is valid and sufficient memory can
* to perform the preprocessing.
* AMDOUTOFMEMORY if not enough memory can be allocated.
*
be allocated
AMD_INVALID if the input arguments n, Ap, Ai are invalid, or if Rp or
Ri are NULL.
/                          
/* AMD Control and Info arrays */
/                          
/* amd_defaults: sets the default control settings */
void amd_defaults (double Control [ ]) ;
void amdldefaults (double Control [ ]) ;
/* amd_control: prints the control settings */
void amd_control (double Control [ ]) ;
void amdlcontrol (double Control [ ]) ;
/* amd_info: prints the statistics */
void amd_info (double Info [ ]) ;
void amd 1 info (double Info [ ]) ;
#define AMDCONTROL 5
#define AMD INFO 20
/* contents
#define AMD_
#define AMD_
of Control */
DENSE 0
AGGRESSIVE 1
/* default Control settings */
#define AMDDEFAULTDENSE 10.0
#define AMDDEFAULTAGGRESSIVE 1
/* contents
#define AMD_
#define AMD_
#define AMD_
#define AMD_
#define AMD_
#define AMD_
#define AMD_
#define AMD_
#define AMD_
#define AMD_
#define AMD_
#define AMD
of Info */
STATUS 0
N 1
NZ 2
SYMMETRY 3
NZDIAG 4
NZ_A_PLUS_
NDENSE 6
MEMORY 7
NCMPA 8
LNZ 9
NDIV 10
NMULTSUBS
#define AMDNMULTSUBS_
#define AMD DMAX 13
/* size of Control array */
/* size of Info array */
/* "dense" if degree > Control [0] sqrt (n) */
/* do aggressive absorption if Control [1] != 0 */
/* default "dense" degree 10*sqrt(n) */
/* do aggressive absorption by default */
/* return value of amd_order and amdlorder */
/* A is nbyn */
/* number of nonzeros in A */
/* symmetry of pattern (1 is sym., 0 is unsym.) */
/* # of entries on diagonal */
AT 5 /* nz in A+A' */
/* number of "dense" rows/columns in A */
/* amount of memory used by AMD */
/* number of garbage collections in AMD */
/* approx. nz in L, excluding the diagonal */
/* number of fl. point divides for LU and LDL' */
LDL 11 /* number of fl. point (*,) pairs for LDL' */
LU 12 /* number of fl. point (*,) pairs for LU */
/* max nz. in any column of L, incl. diagonal */
/             
/* return values of AMD */
/      
#define AMDOK 0
#define AMD OUT OF MEMORY 1
#define AMD INVALID 2
/* success */
/* malloc failed */
/* input arguments are not valid */
#endif
References
[1] P. R. Amestoy, T. A. Davis, and I. S. Duff. An approximate minimum degree ordering algorithm.
SIAM J. Matrix Anal. Applic., 17(4):886905, 1996.
[2] T. A. Davis, J. R. Gilbert, S. I. Larimore, and E. G. Ng. A column approximate minimum
degree ordering algorithm. ACM Trans. Math. S..fli, (to appear).
[3] A. George and J. W. H. Liu. The evolution of the minimum degree ordering algorithm. SIAM
Review, 31(1):119, 1989.
[4] B. Hendrickson and E. Rothberg. Improving the runtime and quality of nested dissection
ordering. SIAM J. Sci. Comput., 20:468489, 1999.
[5] HSL. HSL 2002: A collection of Fortran codes for large scale scientific computation, 2002.
www.cse.clrc.ac.uk/nag/hsl.
[6] G. Karypis and V. Kumar. A fast and high quality multilevel scheme for partitioning irregular
graphs. SIAM J. Sci. Comput., 20:359392, 1998.
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