SPRAL_SCALINGv1.0.0
Sparse Matrix Scalings
C User Guide
This package generates various scalings (and
matchings) of real sparse matrices.
Given a symmetric matrix ,
it finds a diagonal matrix
such that the scaled matrix
has specific numerical properties.
Given an unsymmetric or rectangular matrix ,
it finds diagonal matrices
and
such that the scaled matrix
has specific numerical properties.
The specific numerical properties delivered depends on the algorithm used:
-
Matching-based
- algorithms scale
such that the maximum (absolute) value in each row and column of
is exactly 1.0, where the entries of maximum value form a maximum cardinality matching. The
Hungarian algorithm delivers an optimal matching slowly, whereas the auction algorithm delivers
an approximate matching quickly.
-
Norm-equilibration
- algorithms scale
such that the infinity norm of each row and column of
is
(for some user specified tolerance ).
Jonathan Hogg (STFC Rutherford Appleton Laboratory)
Major version history
-
2014-12-17 Version 1.0.0
- Initial public release
4.1 Installation
Please see the SPRAL install documentation.
4.2 Usage overview
4.2.1 Calling sequences
Access to the package requires inclusion of either spral.h (for the entire SPRAL library) or spral_scaling.h (for
just the relevant routines). i.e.
#include "spral.h"
The following functions are available to the user:
- spral_scaling_default_auction_options() initializes the options structure for the auction
algorithm.
- spral_scaling_default_equilib_options() initializes the options structure for the norm
equilibriation algorithm.
- spral_scaling_default_hungarian_options() initializes the options structure for the Hungarian
algorithm.
- spral_scaling_auction_sym() and
spral_scaling_auction_unsym() generate approximate matching-based scalings for symmetric and
unsymmetric/rectangular matrices respectively using an auction algorithm.
- spral_scaling_equilib_sym() and spral_scaling_equilib_unsym() generate norm-equilibration
scalings for symmetric and unsymmetric/rectangular matrices respectively.
- spral_scaling_hungarian_sym() and spral_scaling_hungarian_unsym() generate matching-based
scalings for a symmetric and unsymmetric/rectangular matrices respectively using the Hungarian
algorithm.
4.2.2 Data formats
Compressed Sparse Column (CSC) Format
This standard data format consists of the following data:
int m; /* number of rows (unsymmetric matrix) */
int n; /* number of columns */
int ptr[ n+1 ]; /* column pointers */
int row[ ptr[n]-1 ]; /* row indices */
double val[ ptr[n]-1 ]; /* numerical values */
Non-zero matrix entries are ordered by increasing column index and stored in the arrays row[] and val[] such that
row[k] holds the row number and val[k] holds the value of the k-th entry. The ptr[] array stores column pointers
such that ptr[i] is the position in row[] and val[] of the first entry in the i-th column, and ptr[n] is one more
than the total number of entries. There must be no duplicate or out of range entries. Entries that are zero, including
those on the diagonal, need not be specified.
For symmetric matrices, only the lower triangular entries of
should
be supplied. For unsymmetric matrices, all entries in the matrix should be supplied.
Note that these routines offer no checking of user data, and the behaviour of these routines with misformatted
data is undefined.
To illustrate the CSC format, the following arrays describe the symmetric matrix shown in Figure 4.1.
int n = 5;
int ptr[] = { 0, 3, 4, 6, 8, 9 };
int row[] = { 0, 1, 3, 2, 2, 4, 3, 4, 4 };
double val[] = { 1.1, 2.2, 3.3, 4.4, 5.5, 6.6, 7.7, 8.8, 9.9 };
4.3 Auction Algorithm
4.3.1 spral_scaling_auction_default_options()
To initialize a variable of type struct spral_scaling_auction_options the following routine is
provided.
void spral_scaling_auction_default_options(struct spral_scaling_auction_options *options);
-
*options
- is the instance to be initialized.
4.3.2 spral_scaling_auction_sym()
To generate a scaling for a symmetric matrix using an auction algorithm such that the entry of
maximum absolute value in each row and column is approximately 1.0,
void spral_scaling_auction_sym(int n, const int *ptr, const int *row, const double *val,
double *scaling, int *match, const struct spral_scaling_auction_options *options,
struct spral_scaling_auction_inform *inform);
-
n, ptr[n+1], row[ptr[n]], val[ptr[n]]
- must hold the lower triangular part of
in compressed sparse column format as described in Section 4.2.2.
-
scaling[n]
- holds, on exit, the diagonal of .
scaling[i] holds ,
the scaling corresponding to the i-th row and column.
-
match[m]
- may be NULL. If it is non-NULL, then on exit it specifies the matching of rows to columns. Row i is
matched to column match[i], or is unmatched if match[i]-1.
-
*options
- specifies the algorithmic options used by the subroutine, as explained in Section 4.3.4.
-
*inform
- is used to return information about the execution of the subroutine, as explained in Section 4.3.5.
4.3.3 spral_scaling_auction_unsym()
To generate a scaling for an unsymmetric or rectangular matrix using an auction algorithm such
that the entry of maximum absolute value in each row and column is approximately 1.0,
void spral_scaling_auction_unsym(int m, int n, const int *ptr, const int *row,
const double *val, double *rscaling, double *cscaling, int *match,
const struct spral_scaling_auction_options *options,
struct spral_scaling_auction_inform *inform);
-
m, n, ptr[n+1], row[ptr[n]], val[ptr[n]]
- must hold the lower triangular part of
in compressed sparse column format as described in Section 4.2.2.
-
rscaling[m]
- holds, on exit, the diagonal of .
rscaling[i] holds ,
the scaling corresponding to the i-th row.
-
cscaling[n]
- holds, on exit, the diagonal of .
cscaling[j] holds ,
the scaling corresponding to the j-th column.
-
match[m]
- may be NULL. If it is non-NULL, then on exit it specifies the matching of rows to columns. Row i is
matched to column match[i], or is unmatched if match[i]-1.
-
*options
- specifies the algorithmic options used by the subroutine, as explained in Section 4.3.4.
-
*inform
- is used to return information about the execution of the subroutine, as explained in Section 4.3.5.
4.3.4 struct spral_scaling_auction_options
The structure spral_scaling_auction_options is used to specify the options used by the routines
spral_scaling_auction_sym() and spral_scaling_auction_unsym(). The components, that must be given
default values through a call to spral_scaling_default_auction_options(), are:
-
int array_base
- specifies the array indexing base. It must have the value either 0 (C indexing) or 1 (Fortran
indexing). If array_base1,
the entries of arrays ptr[],row[] and match[] start at 1, not 0. Further, entries of match[] that are
unmatched are indicated by a value of 0, not -1. The default value is array_base0.
-
float eps_initial
- specifies the initial value of the minimum improvement parameter
as described in Section 4.3.7.
-
int max_iterations
- specifies the maximum number of iterations the algorithm may perform. The default is
max_iterations30000.
-
int max_unchanged[3]
- specifies, together with min_proportion[], the termination conditions for the algorithm,
as described in Section 4.3.7. The default is max_unchanged[]
{ 10, 100, 100 }.
-
float min_proportion[3]
- specifies, together with max_unchanged(:), the termination conditions for the
algorithm, as described in Section 4.3.7. The default is min_proportion[]
{ 0.90, 0.0, 0.0 }.
4.3.5 struct spral_scaling_auction_inform
The structure spral_scaling_auction_inform is used to hold parameters that give information about the progress
of the routines spral_scaling_auction_sym() and spral_scaling_auction_unsym(). The components
are:
-
int flag
- gives the exit status of the algorithm (details in Section 4.3.6).
-
int iterations
- holds the number of iterations performed.
-
int matched
- holds the number of rows and columns that have been matched. As the algorithm may terminate
before a full matching is obtained, this only provides a lower bound on the structural rank.
-
int stat
- holds, in the event of an allocation error, the Fortran stat parameter if it is available (and is set
to 0 otherwise).
-
int unmatchable
- holds the number of columns designated as unmatchable. A column is designated as
unmatchable if there is no way to match it that improves the quality of the matching. It provides an
approximate lower bound on the structural rank deficiency.
4.3.6 Error Flags
A successful return from a routine is indicated by inform.flag having the value zero. A negative value is associated
with an error message.
Possible negative (error) values are:
-
-1
- Allocation error. If available, the Fortran stat parameter is returned in inform.stat.
4.3.7 Algorithm description
This algorithm finds a fast approximation to the matching and scaling produced by the HSL package MC64. If an
optimal matching is required, use the Hungarian algorithm instead. The algorithm works by solving the following
maximum product optimization problem using an auction algorithm. The scaling is derived from the dual variables
associated with the solution.
The array
gives a matching of rows to columns.
By using the transformation
where , the maximum product
problem in is replaced by a
minimum sum problem in
where all entries are positive. By standard optimization theory, there exist dual variables
and
corresponding to the constraints that satisfy the first order optimality conditions
To obtain a scaling we define scaling matrices
and
as
If a symmetric scaling is required, we average these as
By the first order optimality conditions, these scaling matrices guarantee that
To solve the minimum sum problem an auction algorithm is used. The algorithm is not guaranteed to find an optimal
matching. However it can find an approximate matching very quickly. A matching is maintained along with the row pricing
vector .
In each major iteration, we loop over each column in turn. If the column
is unmatched, we
calculate the value for
each entry and find the maximum across the column. If this maximum is positive, the current matching can be improved by matching
column with row
. This may mean that the previous
match of row now becomes unmatched.
We update the price of row ,
that is ,
to reflect this new benefit and continue to the next column.
To prevent incremental shuffling, we insist that the value of adding a new column is at least a threshold value
above zero, where
is based on the last iteration
in which row changed its
match. This is done by adding
to the price ,
where ,
where itr is the current iteration number.
The algorithm terminates if any of the following are satisfied:
- All entries are matched.
- The number of major iterations exceeds options.max_iterations.
- At least options.max_unchanged[0] iterations have passed without the cardinality of the matching
increasing, and the proportion of matched columns is options.min_proportion[0].
- At least options.max_unchanged[1] iterations have passed without the cardinality of the matching
increasing, and the proportion of matched columns is options.min_proportion[1].
- At least options.max_unchanged[2] iterations have passed without the cardinality of the matching
increasing, and the proportion of matched columns is options.min_proportion[2].
The different combinations given by options.max_unchanged[0:2] and options.min_proportion[0:2]
allow a wide range of termination heuristics to be specified by the user depending on their
particular needs. Note that the matching and scaling produced will always be approximate as
is
non-zero.
Further details are given in the following paper:
- J.D. Hogg and J.A. Scott. (2014). On the efficient scaling of sparse symmetric matrices using an auction
algorithm. RAL Technical Report RAL-P-2014-002.
4.3.8 Example of spral_scaling_auction_sym()
The following code shows an example usage of spral_scaling_auction_sym().
/* examples/C/scaling/auction_sym.f90 - Example code for SPRAL_SCALING */
#include <stdlib.h>
#include <stdio.h>
#include "spral.h"
void main(void) {
/* Derived types */
struct spral_scaling_auction_options options;
struct spral_scaling_auction_inform inform;
/* Other variables */
int match[5], i, j;
double scaling[5];
/* Data for symmetric matrix:
* ( 2 1 )
* ( 1 4 1 8 )
* ( 1 3 2 )
* ( 2 )
* ( 8 2 ) */
int n = 5;
int ptr[] = { 0, 2, 5, 7,7, 8 };
int row[] = { 0, 1, 1, 2, 4, 2, 3, 4 };
double val[] = { 2.0, 1.0, 4.0, 1.0, 8.0, 3.0, 2.0, 2.0 };
printf("Initial matrix:\n");
spral_print_matrix(-1, SPRAL_MATRIX_REAL_SYM_INDEF, n, n, ptr, row, val, 0);
/* Perform symmetric scaling */
spral_scaling_auction_default_options(&options);
spral_scaling_auction_sym(n, ptr, row, val, scaling, match, &options, &inform);
if(inform.flag<0) {
printf("spral_scaling_auction_sym() returned with error %5d", inform.flag);
exit(1);
}
/* Print scaling and matching */
printf("Matching:");
for(int i=0; i<n; i++) printf(" %10d", match[i]);
printf("\nScaling: ");
for(int i=0; i<n; i++) printf(" %10.2le", scaling[i]);
printf("\n");
/* Calculate scaled matrix and print it */
for(int i=0; i<n; i++) {
for(int j=ptr[i]; j<ptr[i+1]; j++)
val[j] = scaling[row[j]] * val[j] * scaling[i];
}
printf("Scaled matrix:\n");
spral_print_matrix(-1, SPRAL_MATRIX_REAL_SYM_INDEF, n, n, ptr, row, val, 0);
}
The above code produces the following output.
Initial matrix:
Real symmetric indefinite matrix, dimension 5x5 with 8 entries.
0: 2.0000E+00 1.0000E+00
1: 1.0000E+00 4.0000E+00 1.0000E+00 8.0000E+00
2: 1.0000E+00 3.0000E+00 2.0000E+00
3: 2.0000E+00
4: 8.0000E+00 2.0000E+00
Matching: 0 4 3 2 1
Scaling: 7.07e-01 1.62e-01 2.78e-01 1.80e+00 7.72e-01
Scaled matrix:
Real symmetric indefinite matrix, dimension 5x5 with 8 entries.
0: 1.0000E+00 1.1443E-01
1: 1.1443E-01 1.0476E-01 4.5008E-02 1.0000E+00
2: 4.5008E-02 2.3204E-01 1.0000E+00
3: 1.0000E+00
4: 1.0000E+00 1.1932E+00
4.4 Norm-equilibration algorithm
4.4.1 spral_scaling_equilib_default_options()
To initialize a variable of type struct spral_scaling_equilib_options the following routine is
provided.
void spral_scaling_equilib_default_options(struct spral_scaling_equilib_options *options);
-
*options
- is the instance to be initialized.
4.4.2 spral_scaling_equilib_sym()
To generate a scaling for a symmetric matrix using a norm equilibration
algorithm such that the infinity norm of each row and column is equal to
,
void spral_scaling_equilib_sym(int n, const int *ptr, const int *row, const double *val,
double *scaling, const struct spral_scaling_equilib_options *options,
struct spral_scaling_equilib_inform *inform);
-
n, ptr[n+1], row[ptr[n]], val[ptr[n]]
- must hold the lower triangular part of
in compressed sparse column format as described in Section 4.2.2.
-
scaling[n]
- holds, on exit, the diagonal of .
scaling[i] holds ,
the scaling corresponding to the i-th row and column.
-
*options
- specifies the algorithmic options used by the subroutine, as explained in Section 4.4.4.
-
*inform
- is used to return information about the execution of the subroutine, as explained in Section 4.4.5.
4.4.3 spral_scaling_equilib_unsym()
To generate a scaling for an unsymmetric or rectangular matrix using a norm
equilibration algorithm such that the infinity norm of each row and column is equal to
,
void spral_scaling_equilib_unsym(int m, int n, const int *ptr, const int *row,
const double *val, double *rscaling, double *cscaling,
const struct spral_scaling_equilib_options *options,
struct spral_scaling_equilib_inform *inform);
-
m, n, ptr[n+1], row[ptr[n]], val[ptr[n]]
- must hold the lower triangular part of
in compressed sparse column format as described in Section 4.2.2.
-
rscaling[m]
- holds, on exit, the diagonal of .
scaling[i] holds ,
the scaling corresponding to the i-th row.
-
cscaling[n]
- holds, on exit, the diagonal of .
scaling[j] holds ,
the scaling corresponding to the j-th column.
-
*options
- specifies the algorithmic options used by the subroutine, as explained in Section 4.4.4.
-
*inform
- is used to return information about the execution of the subroutine, as explained in Section 4.4.5.
4.4.4 struct spral_scaling_equilib_options
The structure spral_scaling_equilib_options is used to specify the options used by the routines
spral_scaling_equilib_sym() and spral_scaling_equilib_unsym(). The components, that must be given
default values through a call to spral_scaling_default_equilib_options(), are:
-
int array_base
- specifies the array indexing base. It must have the value either 0 (C indexing) or 1 (Fortran
indexing). If array_base1,
the entries of arrays ptr[] and row[] start at 1, not 0. The default value is array_base0.
-
int max_iterations
- specifies the maximum number of iterations the algorithm may perform. The default
is max_iterations=10.
-
float tol
- specifies the convergence tolerance for the algorithm (though often termination is based on
max_iterations). The default is tol = 1e-8.
4.4.5 struct spral_scaling_equilib_inform
The structure spral_scaling_equilib_inform is used to hold parameters that give information about the progress
of the routines spral_scaling_equilib_sym() and spral_scaling_equilib_unsym(). The components
are:
-
int flag
- gives the exit status of the algorithm (details in Section 4.4.6).
-
int iterations
- holds, on exit, the number of iterations performed.
-
int stat
- holds, in the event of an allocation error or deallocation error, the Fortran stat parameter if it is
available (and is set to 0 otherwise).
4.4.6 Error Flags
A successful return from a routine is indicated by inform.flag having the value zero. A negative value is associated
with an error message.
Possible negative (error) values are:
-
-1
- Allocation error. If available, the Fortran stat parameter is returned in inform.stat.
4.4.7 Algorithm description
This algorithm is very similar to that used by the HSL routine MC77. An iterative method is used to
scale the infinity norm of both rows and columns to 1 with an asymptotic linear rate of convergence of
,
preserving symmetry if the matrix is symmetric.
For unsymmetric matrices, the algorithm outline is as follows:
for do
if()
exit
end for
For symmetric matrices,
is symmetric, so
,
and some operations can be skipped.
Further details are given in the following paper:
- P. Knight, D. Ruiz and B. Ucar. (2012). A symmetry preserving algorithm for matrix scaling. INRIA
Research Report 7552.
4.4.8 Example of spral_scaling_equilib_sym()
The following code shows an example usage of spral_scaling_equilib_sym().
/* examples/C/scaling/equilib_sym.f90 - Example code for SPRAL_SCALING */
#include <stdlib.h>
#include <stdio.h>
#include "spral.h"
void main(void) {
/* Derived types */
struct spral_scaling_equilib_options options;
struct spral_scaling_equilib_inform inform;
/* Other variables */
int i, j;
double scaling[5];
/* Data for symmetric matrix:
* ( 2 1 )
* ( 1 4 1 8 )
* ( 1 3 2 )
* ( 2 )
* ( 8 2 ) */
int n = 5;
int ptr[] = { 0, 2, 5, 7,7, 8 };
int row[] = { 0, 1, 1, 2, 4, 2, 3, 4 };
double val[] = { 2.0, 1.0, 4.0, 1.0, 8.0, 3.0, 2.0, 2.0 };
printf("Initial matrix:\n");
spral_print_matrix(-1, SPRAL_MATRIX_REAL_SYM_INDEF, n, n, ptr, row, val, 0);
/* Perform symmetric scaling */
spral_scaling_equilib_default_options(&options);
spral_scaling_equilib_sym(n, ptr, row, val, scaling, &options, &inform);
if(inform.flag<0) {
printf("spral_scaling_equilib_sym() returned with error %5d", inform.flag);
exit(1);
}
/* Print scaling */
printf("Scaling: ");
for(int i=0; i<n; i++) printf(" %10.2le", scaling[i]);
printf("\n");
/* Calculate scaled matrix and print it */
for(int i=0; i<n; i++) {
for(int j=ptr[i]; j<ptr[i+1]; j++)
val[j] = scaling[row[j]] * val[j] * scaling[i];
}
printf("Scaled matrix:\n");
spral_print_matrix(-1, SPRAL_MATRIX_REAL_SYM_INDEF, n, n, ptr, row, val, 0);
}
The above code produces the following output.
Initial matrix:
Real symmetric indefinite matrix, dimension 5x5 with 8 entries.
0: 2.0000E+00 1.0000E+00
1: 1.0000E+00 4.0000E+00 1.0000E+00 8.0000E+00
2: 1.0000E+00 3.0000E+00 2.0000E+00
3: 2.0000E+00
4: 8.0000E+00 2.0000E+00
Scaling: 7.07e-01 3.54e-01 5.77e-01 8.66e-01 3.54e-01
Scaled matrix:
Real symmetric indefinite matrix, dimension 5x5 with 8 entries.
0: 1.0000E+00 2.5000E-01
1: 2.5000E-01 5.0000E-01 2.0412E-01 1.0000E+00
2: 2.0412E-01 1.0000E+00 9.9960E-01
3: 9.9960E-01
4: 1.0000E+00 2.5000E-01
4.5 Hungarian algorithm
4.5.1 spral_scaling_hungarian_default_options()
To initialize a variable of type struct spral_scaling_hungarian_options the following routine is
provided.
void spral_scaling_hungarian_default_options(struct spral_scaling_hungarian_options *options);
-
*options
- is the instance to be initialized.
4.5.2 spral_scaling_hungarian_sym()
To generate a scaling for a symmetric matrix using the Hungarian algorithm such that the entry of
maximum absolute value in each row and column is 1.0,
void spral_scaling_hungarian_sym(int n, const int *ptr, const int *row, const double *val,
double *scaling, int *match, const struct spral_scaling_hungarian_options *options,
struct spral_scaling_hungarian_inform *inform);
-
n, ptr[n+1], row[ptr[n]], val[ptr[n]]
- must hold the lower triangular part of
in compressed sparse column format as described in Section 4.2.2.
-
scaling[n]
- holds, on exit, the diagonal of .
scaling[i] holds ,
the scaling corresponding to the i-th row and column.
-
match[m]
- may be NULL. If it is non-NULL, then on exit it specifies the matching of rows to columns. Row i is
matched to column match[i], or is unmatched if match[i]-1.
-
*options
- specifies the algorithmic options used by the subroutine, as explained in Section 4.5.4.
-
*inform
- is used to return information about the execution of the subroutine, as explained in Section 4.5.5.
4.5.3 spral_scaling_hungarian_unsym()
To generate a scaling for an unsymmetric or rectangular matrix using the Hungarian
algorithm such that the entry of maximum absolute value in each row and column is 1.0,
void spral_scaling_hungarian_unsym(int m, int n, const int *ptr, const int *row,
const double *val, double *rscaling, double *cscaling, int *match,
const struct spral_scaling_hungarian_options *options,
struct spral_scaling_hungarian_inform *inform);
-
m, n, ptr[n+1], row[ptr[n]], val[ptr[n]]
- must hold the lower triangular part of
in compressed sparse column format as described in Section 4.2.2.
-
rscaling[m]
- holds, on exit, the diagonal of .
rscaling[i] holds ,
the scaling corresponding to the i-th row.
-
cscaling[n]
- holds, on exit, the diagonal of .
cscaling[j] holds ,
the scaling corresponding to the j-th column.
-
match[m]
- may be NULL. If it is non-NULL, then on exit it specifies the matching of rows to columns. Row i is
matched to column match[i], or is unmatched if match[i]-1.
-
*options
- specifies the algorithmic options used by the subroutine, as explained in Section 4.5.4.
-
*inform
- is used to return information about the execution of the subroutine, as explained in Section 4.5.5.
4.5.4 struct spral_scaling_hungarian_options
The structure spral_scaling_hungarian_options is used to specify the options used by the routines
spral_scaling_hungarian_sym() and spral_scaling_hungarian_unsym(). The components, that must be given
default values through a call to spral_scaling_default_hungarian_options(), are:
-
int array_base
- specifies the array indexing base. It must have the value either 0 (C indexing) or 1 (Fortran
indexing). If array_base1,
the entries of arrays ptr[],row[] and match[] start at 1, not 0. Further, entries of match[] that are
unmatched are indicated by a value of 0, not -1. The default value is array_base0.
-
bool scale_if_singular
- specifies whether scaling shuold continue if the matrix
is found to be structurally singular. If scale_if_singulartrue,
and the
is structurally singular, a partial scaling corresponding to a maximum cardinality matching will be
returned and a warning issued. Otherwise an identity scaling will be returned and an error issued.
4.5.5 struct spral_scaling_hungarian_inform
The structure spral_scaling_hungarian_inform is used to hold parameters that give information about the
progress of the routines spral_scaling_hungarian_sym() and spral_scaling_hungarian_unsym(). The
components are:
-
int flag
- gives the exit status of the algorithm (details in Section 4.5.6).
-
int matched
- holds the number of rows and columns that have been matched (i.e. the structural rank).
-
int stat
- holds, in the event of an allocation error or deallocation error, the Fortran stat parameter if it is
available (and is set to 0 otherwise).
4.5.6 Error Flags
A successful return from a routine is indicated by inform.flag having the value zero. A negative value is associated
with an error message and a positive value with a warning.
Possible negative (error) values are:
-
-1
- Allocation error. If available, the Fortran stat parameter is returned in inform.stat.
-
-2
- Matrix
is structurally rank-deficient. This error is returned only if options.scale_if_singularfalse.
The scaling vector is set to 1.0 and a matching of maximum cardinality returned in the optional
argument match[], if present.
Possible positive (warning) values are:
-
+1
- Matrix
is structurally rank-deficient. This warning is returned only if options.scale_if_singulartrue.
4.5.7 Algorithm description
This algorithm is the same as used by the HSL package MC64. A scaling is derived from dual variables
found during the solution of the below maximum product optimization problem using the Hungarian
algorithm.
The array
gives a matching of rows to columns.
By using the transformation
where , the maximum product
problem in is replaced by a
minimum sum problem in
where all entries are positive. By standard optimization theory, there exist dual variables
and
corresponding to the constraints that satisfy the first order optimality conditions
To obtain a scaling we define scaling matrices
and
as
If a symmetric scaling is required, we average these as
By the first order optimality conditions, these scaling matrices guarantee that
To solve the minimum sum problem, the Hungarian algorithm maintains an optimal matching on a subset of the
rows and columns. It proceeds to grow this set by finding augmenting paths from an unmatched row to an
unmatched column. The algorithm is guaranteed to find the optimal solution in a fixed number of steps, but can be
very slow as it may need to explore the full matrix a number of times equal to the dimension of the matrix. To
minimize the solution time, a warmstarting heuristic is used to construct an initial optimal subset
matching.
Further details are given in the following paper:
- I.S. Duff and J. Koster. (1997). The design and use of algorithms for permuting large entries to the
diagonal of sparse matrices. SIAM J. Matrix Anal. Applics. 20(4), pp 889–901.
4.5.8 Example usage of spral_scaling_hungarian_unsym()
The following code shows an example usage of hungarian_scale_unsym().
/* examples/C/scaling/hungarian_unsym.f90 - Example code for SPRAL_SCALING */
#include <stdlib.h>
#include <stdio.h>
#include "spral.h"
void main(void) {
/* Derived types */
struct spral_scaling_hungarian_options options;
struct spral_scaling_hungarian_inform inform;
/* Other variables */
int match[5], i, j;
double rscaling[5], cscaling[5];
/* Data for unsymmetric matrix:
* ( 2 5 )
* ( 1 4 7 )
* ( 1 2 )
* ( 3 )
* ( 8 2 ) */
int m = 5, n = 5;
int ptr[] = { 0, 2, 6, 7, 8, 10 };
int row[] = { 0, 1, 0, 1, 2, 4, 3, 2, 1, 4 };
double val[] = { 2.0, 1.0, 5.0, 4.0, 1.0, 8.0, 3.0, 2.0, 7.0, 2.0 };
printf("Initial matrix:\n");
spral_print_matrix(-1, SPRAL_MATRIX_REAL_UNSYM, m, n, ptr, row, val, 0);
/* Perform symmetric scaling */
spral_scaling_hungarian_default_options(&options);
spral_scaling_hungarian_unsym(m, n, ptr, row, val, rscaling, cscaling, match,
&options, &inform);
if(inform.flag<0) {
printf("spral_scaling_hungarian_unsym() returned with error %5d", inform.flag);
exit(1);
}
/* Print scaling and matching */
printf("Matching:");
for(int i=0; i<n; i++) printf(" %10d", match[i]);
printf("\nRow Scaling: ");
for(int i=0; i<m; i++) printf(" %10.2le", rscaling[i]);
printf("\nCol Scaling: ");
for(int i=0; i<n; i++) printf(" %10.2le", cscaling[i]);
printf("\n");
/* Calculate scaled matrix and print it */
for(int i=0; i<n; i++) {
for(int j=ptr[i]; j<ptr[i+1]; j++)
val[j] = rscaling[row[j]] * val[j] * cscaling[i];
}
printf("Scaled matrix:\n");
spral_print_matrix(-1, SPRAL_MATRIX_REAL_UNSYM, m, n, ptr, row, val, 0);
}
The above code produces the following output.
Initial matrix:
Real unsymmetric matrix, dimension 5x5 with * entries.
0: 2.0000E+00 5.0000E+00
1: 1.0000E+00 4.0000E+00 7.0000E+00
2: 1.0000E+00 2.0000E+00
3: 3.0000E+00
4: 8.0000E+00 2.0000E+00
Matching: 0 4 3 2 1
Row Scaling: 5.22e-01 5.22e-01 5.22e-01 5.22e-01 5.22e-01
Col Scaling: 9.59e-01 2.40e-01 6.39e-01 9.59e-01 2.74e-01
Scaled matrix:
Real unsymmetric matrix, dimension 5x5 with * entries.
0: 1.0000E+00 6.2500E-01
1: 5.0000E-01 5.0000E-01 1.0000E+00
2: 1.2500E-01 1.0000E+00
3: 1.0000E+00
4: 1.0000E+00 2.8571E-01