Copyright (c) 2003, The Regents of the University of California, through Lawrence Berkeley National Laboratory (subject to receipt of any required approvals from U.S. Dept. of Energy)
All rights reserved.
The source code is distributed under BSD license, see the file License.txt at the top-level directory.
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=== colamd/symamd - a sparse matrix column ordering algorithm ============
colamd: an approximate minimum degree column ordering algorithm, for LU factorization of symmetric or unsymmetric matrices, QR factorization, least squares, interior point methods for linear programming problems, and other related problems.
symamd: an approximate minimum degree ordering algorithm for Cholesky factorization of symmetric matrices.
Purpose:
Colamd computes a permutation Q such that the Cholesky factorization of
(AQ)'(AQ) has less fill-in and requires fewer floating point operations
than A'A. This also provides a good ordering for sparse partial
pivoting methods, P(AQ) = LU, where Q is computed prior to numerical
factorization, and P is computed during numerical factorization via
conventional partial pivoting with row interchanges. Colamd is the
column ordering method used in SuperLU, part of the ScaLAPACK library.
It is also available as built-in function in MATLAB Version 6,
available from MathWorks, Inc. (http://www.mathworks.com). This
routine can be used in place of colmmd in MATLAB.
Symamd computes a permutation P of a symmetric matrix A such that the
Cholesky factorization of PAP' has less fill-in and requires fewer
floating point operations than A. Symamd constructs a matrix M such
that M'M has the same nonzero pattern of A, and then orders the columns
of M using colmmd. The column ordering of M is then returned as the
row and column ordering P of A.
Authors:
The authors of the code itself are Stefan I. Larimore and Timothy A.
Davis (davis@cise.ufl.edu), University of Florida. The algorithm was
developed in collaboration with John Gilbert, Xerox PARC, and Esmond
Ng, Oak Ridge National Laboratory.
Date:
September 8, 2003. Version 2.3.
Acknowledgements:
This work was supported by the National Science Foundation, under
grants DMS-9504974 and DMS-9803599.
Copyright and License:
Copyright (c) 1998-2003 by the University of Florida.
All Rights Reserved.
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, copy, modify, and/or distribute
this program, provided that the Copyright, this License, and the
Availability of the original version is retained on all copies and made
accessible to the end-user of any code or package that includes COLAMD
or any modified version of COLAMD.
Availability:
The colamd/symamd library is available at
http://www.cise.ufl.edu/research/sparse/colamd/
This is the http://www.cise.ufl.edu/research/sparse/colamd/colamd.c
file. It requires the colamd.h file. It is required by the colamdmex.c
and symamdmex.c files, for the MATLAB interface to colamd and symamd.
See the ChangeLog file for changes since Version 1.0.
=== Description of user-callable routines ================================
colamd_recommended:
C syntax:
#include "colamd.h"
int colamd_recommended (int nnz, int n_row, int n_col) ;
or as a C macro
#include "colamd.h"
Alen = COLAMD_RECOMMENDED (int nnz, int n_row, int n_col) ;
Purpose:
Returns recommended value of Alen for use by colamd. Returns -1
if any input argument is negative. The use of this routine
or macro is optional. Note that the macro uses its arguments
more than once, so be careful for side effects, if you pass
expressions as arguments to COLAMD_RECOMMENDED. Not needed for
symamd, which dynamically allocates its own memory.
Arguments (all input arguments):
int nnz ; Number of nonzeros in the matrix A. This must
be the same value as p [n_col] in the call to
colamd - otherwise you will get a wrong value
of the recommended memory to use.
int n_row ; Number of rows in the matrix A.
int n_col ; Number of columns in the matrix A.
colamd_set_defaults:
C syntax:
#include "colamd.h"
colamd_set_defaults (double knobs [COLAMD_KNOBS]) ;
Purpose:
Sets the default parameters. The use of this routine is optional.
Arguments:
double knobs [COLAMD_KNOBS] ; Output only.
Colamd: rows with more than (knobs [COLAMD_DENSE_ROW] * n_col)
entries are removed prior to ordering. Columns with more than
(knobs [COLAMD_DENSE_COL] * n_row) entries are removed prior to
ordering, and placed last in the output column ordering.
Symamd: uses only knobs [COLAMD_DENSE_ROW], which is knobs [0].
Rows and columns with more than (knobs [COLAMD_DENSE_ROW] * n)
entries are removed prior to ordering, and placed last in the
output ordering.
COLAMD_DENSE_ROW and COLAMD_DENSE_COL are defined as 0 and 1,
respectively, in colamd.h. Default values of these two knobs
are both 0.5. Currently, only knobs [0] and knobs [1] are
used, but future versions may use more knobs. If so, they will
be properly set to their defaults by the future version of
colamd_set_defaults, so that the code that calls colamd will
not need to change, assuming that you either use
colamd_set_defaults, or pass a (double *) NULL pointer as the
knobs array to colamd or symamd.
colamd:
C syntax:
#include "colamd.h"
int colamd (int n_row, int n_col, int Alen, int *A, int *p,
double knobs [COLAMD_KNOBS], int stats [COLAMD_STATS]) ;
Purpose:
Computes a column ordering (Q) of A such that P(AQ)=LU or
(AQ)'AQ=LL' have less fill-in and require fewer floating point
operations than factorizing the unpermuted matrix A or A'A,
respectively.
Returns:
TRUE (1) if successful, FALSE (0) otherwise.
Arguments:
int n_row ; Input argument.
Number of rows in the matrix A.
Restriction: n_row >= 0.
Colamd returns FALSE if n_row is negative.
int n_col ; Input argument.
Number of columns in the matrix A.
Restriction: n_col >= 0.
Colamd returns FALSE if n_col is negative.
int Alen ; Input argument.
Restriction (see note):
Alen >= 2*nnz + 6*(n_col+1) + 4*(n_row+1) + n_col
Colamd returns FALSE if these conditions are not met.
Note: this restriction makes an modest assumption regarding
the size of the two typedef's structures in colamd.h.
We do, however, guarantee that
Alen >= colamd_recommended (nnz, n_row, n_col)
or equivalently as a C preprocessor macro:
Alen >= COLAMD_RECOMMENDED (nnz, n_row, n_col)
will be sufficient.
int A [Alen] ; Input argument, undefined on output.
A is an integer array of size Alen. Alen must be at least as
large as the bare minimum value given above, but this is very
low, and can result in excessive run time. For best
performance, we recommend that Alen be greater than or equal to
colamd_recommended (nnz, n_row, n_col), which adds
nnz/5 to the bare minimum value given above.
On input, the row indices of the entries in column c of the
matrix are held in A [(p [c]) ... (p [c+1]-1)]. The row indices
in a given column c need not be in ascending order, and
duplicate row indices may be be present. However, colamd will
work a little faster if both of these conditions are met
(Colamd puts the matrix into this format, if it finds that the
the conditions are not met).
The matrix is 0-based. That is, rows are in the range 0 to
n_row-1, and columns are in the range 0 to n_col-1. Colamd
returns FALSE if any row index is out of range.
The contents of A are modified during ordering, and are
undefined on output.
int p [n_col+1] ; Both input and output argument.
p is an integer array of size n_col+1. On input, it holds the
"pointers" for the column form of the matrix A. Column c of
the matrix A is held in A [(p [c]) ... (p [c+1]-1)]. The first
entry, p [0], must be zero, and p [c] <= p [c+1] must hold
for all c in the range 0 to n_col-1. The value p [n_col] is
thus the total number of entries in the pattern of the matrix A.
Colamd returns FALSE if these conditions are not met.
On output, if colamd returns TRUE, the array p holds the column
permutation (Q, for P(AQ)=LU or (AQ)'(AQ)=LL'), where p [0] is
the first column index in the new ordering, and p [n_col-1] is
the last. That is, p [k] = j means that column j of A is the
kth pivot column, in AQ, where k is in the range 0 to n_col-1
(p [0] = j means that column j of A is the first column in AQ).
If colamd returns FALSE, then no permutation is returned, and
p is undefined on output.
double knobs [COLAMD_KNOBS] ; Input argument.
See colamd_set_defaults for a description.
int stats [COLAMD_STATS] ; Output argument.
Statistics on the ordering, and error status.
See colamd.h for related definitions.
Colamd returns FALSE if stats is not present.
stats [0]: number of dense or empty rows ignored.
stats [1]: number of dense or empty columns ignored (and
ordered last in the output permutation p)
Note that a row can become "empty" if it
contains only "dense" and/or "empty" columns,
and similarly a column can become "empty" if it
only contains "dense" and/or "empty" rows.
stats [2]: number of garbage collections performed.
This can be excessively high if Alen is close
to the minimum required value.
stats [3]: status code. < 0 is an error code.
> 1 is a warning or notice.
0 OK. Each column of the input matrix contained
row indices in increasing order, with no
duplicates.
1 OK, but columns of input matrix were jumbled
(unsorted columns or duplicate entries). Colamd
had to do some extra work to sort the matrix
first and remove duplicate entries, but it
still was able to return a valid permutation
(return value of colamd was TRUE).
stats [4]: highest numbered column that
is unsorted or has duplicate
entries.
stats [5]: last seen duplicate or
unsorted row index.
stats [6]: number of duplicate or
unsorted row indices.
-1 A is a null pointer
-2 p is a null pointer
-3 n_row is negative
stats [4]: n_row
-4 n_col is negative
stats [4]: n_col
-5 number of nonzeros in matrix is negative
stats [4]: number of nonzeros, p [n_col]
-6 p [0] is nonzero
stats [4]: p [0]
-7 A is too small
stats [4]: required size
stats [5]: actual size (Alen)
-8 a column has a negative number of entries
stats [4]: column with < 0 entries
stats [5]: number of entries in col
-9 a row index is out of bounds
stats [4]: column with bad row index
stats [5]: bad row index
stats [6]: n_row, # of rows of matrx
-10 (unused; see symamd.c)
-999 (unused; see symamd.c)
Future versions may return more statistics in the stats array.
Example:
See http://www.cise.ufl.edu/research/sparse/colamd/example.c
for a complete example.
To order the columns of a 5-by-4 matrix with 11 nonzero entries in
the following nonzero pattern
x 0 x 0
x 0 x x
0 x x 0
0 0 x x
x x 0 0
with default knobs and no output statistics, do the following:
#include "colamd.h"
#define ALEN COLAMD_RECOMMENDED (11, 5, 4)
int A [ALEN] = {1, 2, 5, 3, 5, 1, 2, 3, 4, 2, 4} ;
int p [ ] = {0, 3, 5, 9, 11} ;
int stats [COLAMD_STATS] ;
colamd (5, 4, ALEN, A, p, (double *) NULL, stats) ;
The permutation is returned in the array p, and A is destroyed.
symamd:
C syntax:
#include "colamd.h"
int symamd (int n, int *A, int *p, int *perm,
double knobs [COLAMD_KNOBS], int stats [COLAMD_STATS],
void (*allocate) (size_t, size_t), void (*release) (void *)) ;
Purpose:
The symamd routine computes an ordering P of a symmetric sparse
matrix A such that the Cholesky factorization PAP' = LL' remains
sparse. It is based on a column ordering of a matrix M constructed
so that the nonzero pattern of M'M is the same as A. The matrix A
is assumed to be symmetric; only the strictly lower triangular part
is accessed. You must pass your selected memory allocator (usually
calloc/free or mxCalloc/mxFree) to symamd, for it to allocate
memory for the temporary matrix M.
Returns:
TRUE (1) if successful, FALSE (0) otherwise.
Arguments:
int n ; Input argument.
Number of rows and columns in the symmetrix matrix A.
Restriction: n >= 0.
Symamd returns FALSE if n is negative.
int A [nnz] ; Input argument.
A is an integer array of size nnz, where nnz = p [n].
The row indices of the entries in column c of the matrix are
held in A [(p [c]) ... (p [c+1]-1)]. The row indices in a
given column c need not be in ascending order, and duplicate
row indices may be present. However, symamd will run faster
if the columns are in sorted order with no duplicate entries.
The matrix is 0-based. That is, rows are in the range 0 to
n-1, and columns are in the range 0 to n-1. Symamd
returns FALSE if any row index is out of range.
The contents of A are not modified.
int p [n+1] ; Input argument.
p is an integer array of size n+1. On input, it holds the
"pointers" for the column form of the matrix A. Column c of
the matrix A is held in A [(p [c]) ... (p [c+1]-1)]. The first
entry, p [0], must be zero, and p [c] <= p [c+1] must hold
for all c in the range 0 to n-1. The value p [n] is
thus the total number of entries in the pattern of the matrix A.
Symamd returns FALSE if these conditions are not met.
The contents of p are not modified.
int perm [n+1] ; Output argument.
On output, if symamd returns TRUE, the array perm holds the
permutation P, where perm [0] is the first index in the new
ordering, and perm [n-1] is the last. That is, perm [k] = j
means that row and column j of A is the kth column in PAP',
where k is in the range 0 to n-1 (perm [0] = j means
that row and column j of A are the first row and column in
PAP'). The array is used as a workspace during the ordering,
which is why it must be of length n+1, not just n.
double knobs [COLAMD_KNOBS] ; Input argument.
See colamd_set_defaults for a description.
int stats [COLAMD_STATS] ; Output argument.
Statistics on the ordering, and error status.
See colamd.h for related definitions.
Symamd returns FALSE if stats is not present.
stats [0]: number of dense or empty row and columns ignored
(and ordered last in the output permutation
perm). Note that a row/column can become
"empty" if it contains only "dense" and/or
"empty" columns/rows.
stats [1]: (same as stats [0])
stats [2]: number of garbage collections performed.
stats [3]: status code. < 0 is an error code.
> 1 is a warning or notice.
0 OK. Each column of the input matrix contained
row indices in increasing order, with no
duplicates.
1 OK, but columns of input matrix were jumbled
(unsorted columns or duplicate entries). Symamd
had to do some extra work to sort the matrix
first and remove duplicate entries, but it
still was able to return a valid permutation
(return value of symamd was TRUE).
stats [4]: highest numbered column that
is unsorted or has duplicate
entries.
stats [5]: last seen duplicate or
unsorted row index.
stats [6]: number of duplicate or
unsorted row indices.
-1 A is a null pointer
-2 p is a null pointer
-3 (unused, see colamd.c)
-4 n is negative
stats [4]: n
-5 number of nonzeros in matrix is negative
stats [4]: # of nonzeros (p [n]).
-6 p [0] is nonzero
stats [4]: p [0]
-7 (unused)
-8 a column has a negative number of entries
stats [4]: column with < 0 entries
stats [5]: number of entries in col
-9 a row index is out of bounds
stats [4]: column with bad row index
stats [5]: bad row index
stats [6]: n_row, # of rows of matrx
-10 out of memory (unable to allocate temporary
workspace for M or count arrays using the
"allocate" routine passed into symamd).
-999 internal error. colamd failed to order the
matrix M, when it should have succeeded. This
indicates a bug. If this (and *only* this)
error code occurs, please contact the authors.
Don't contact the authors if you get any other
error code.
Future versions may return more statistics in the stats array.
void * (*allocate) (size_t, size_t)
A pointer to a function providing memory allocation. The
allocated memory must be returned initialized to zero. For a
C application, this argument should normally be a pointer to
calloc. For a MATLAB mexFunction, the routine mxCalloc is
passed instead.
void (*release) (size_t, size_t)
A pointer to a function that frees memory allocated by the
memory allocation routine above. For a C application, this
argument should normally be a pointer to free. For a MATLAB
mexFunction, the routine mxFree is passed instead.
colamd_report:
C syntax:
#include "colamd.h"
colamd_report (int stats [COLAMD_STATS]) ;
Purpose:
Prints the error status and statistics recorded in the stats
array on the standard error output (for a standard C routine)
or on the MATLAB output (for a mexFunction).
Arguments:
int stats [COLAMD_STATS] ; Input only. Statistics from colamd.
symamd_report:
C syntax:
#include "colamd.h"
symamd_report (int stats [COLAMD_STATS]) ;
Purpose:
Prints the error status and statistics recorded in the stats
array on the standard error output (for a standard C routine)
or on the MATLAB output (for a mexFunction).
Arguments:
int stats [COLAMD_STATS] ; Input only. Statistics from symamd.
