SCALING: Sparse Matrix Scalings

Sparse Matrix Scalings

Fortran User Guide
This package generates various scalings (and matchings) of real sparse matrices.

Given a symmetric matrix $A$, it ﬁnds a diagonal matrix $D$ such that the scaled matrix

$Â=DAD$

has speciﬁc numerical properties.

Given a unsymmetric or rectangular matrix $A$, it ﬁnds diagonal matrices ${D}_{r}$ and ${D}_{c}$ such that the scaled matrix

$Â={D}_{r}A{D}_{c}$

has speciﬁc numerical properties.

The speciﬁc numerical properties delivered depends on the algorithm used:

Matching-based
algorithms scale $A$ such that the maximum (absolute) value in each row and column of $Â$ 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 $A$ such that the inﬁnity norm of each row and column of $Â$ is $1.0±\tau$ (for some user speciﬁed tolerance $\tau$).

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

use spral_scaling

The following procedures are available to the user:

• auction_scale_sym() and auction_scale_unsym() generate approximate matching-based scalings for symmetric and unsymmetric/rectangular matrices respectively using an auction algorithm.
• equilib_scale_sym() and equilib_scale_unsym() generate norm-equilibration scalings for symmetric and unsymmetric/rectangular matrices respectively.
• hungarian_scale_sym() and hungarian_scale_unsym() generate matching-based scalings for a symmetric and unsymmetric/rectangular matrices respectively using the Hungarian algorithm.

4.2.2 Optional arguments

We use square brackets [ ] to indicate optional arguments. In each call, optional arguments appear last in argument list. Since we reserve the right to add additional optional arguments in future releases of the code, we strongly recommend that all optional arguments be called by keyword, not by position.

4.2.3 Integer, real and package types

INTEGER denotes default INTEGER and INTEGER(long) denotes INTEGER(kind=selected_int_kind(18)).

REAL denotes double precision real. We also use the term package type to mean the same.

4.2.4 Data formats

Figure 4.1: Data format example matrix (symmetric)
$\left(\begin{array}{ccccc}\hfill 1.1\hfill & \hfill 2.2\hfill & \hfill \hfill & \hfill 3.3\hfill & \hfill \hfill \\ \hfill 2.2\hfill & \hfill \hfill & \hfill 4.4\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill 4.4\hfill & \hfill 5.5\hfill & \hfill \hfill & \hfill 6.6\hfill \\ \hfill 3.3\hfill & \hfill \hfill & \hfill \hfill & \hfill 7.7\hfill & \hfill 8.8\hfill \\ \hfill \hfill & \hfill \hfill & \hfill 6.6\hfill & \hfill 8.8\hfill & \hfill 9.9\hfill \end{array}\right)$

Compressed Sparse Column (CSC) Format

This standard data format consists of the following data:

integer                   :: m      ! number of rows (unsymmetric only)
integer                   :: n      ! number of columns
integer, size(n+1)        :: ptr    ! column pointers
integer, size(ptr(n+1)-1) :: row    ! row indices
real,    size(ptr(n+1)-1) :: val    ! 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+1) 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 $A$ 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.

n = 5
ptr(1:6) = (/ 1,             4,   5,        7,        9,    10 /)
row(1:9) = (/ 1,   2,   4,   3,   3,   5,   4,   5,   5 /)
val(1:9) = (/ 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 auction_scale_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,
call auction_scale_sym(n, ptr, row, val, scaling, options, inform[, match])

n, ptr(:), row(:), val(:)
are INTENT(IN) variables that must hold the lower triangular part of $A$ in compressed sparse column format as described in Section 4.2.4.
scaling(n)
is an INTENT(OUT) array of package type. On exit, scaling(i) holds ${d}_{i}$, the scaling corresponding to the i-th row and column.
options
is an INTENT(IN) scalar of type auction_options. Its components specify the algorithmic options used by the subroutine, as explained in Section 4.3.3.
inform
is an INTENT(OUT) scalar of type auction_inform. On exit, its components provide information about the execution of the subroutine, as explained in Section 4.3.4.
match(n)
is an optional INTENT(OUT) array of type INTEGER. If present, 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)$=$0.

4.3.2 auction_scale_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,
call auction_scale_unsym(m, n, ptr, row, val, rscaling, cscaling, options, inform[, match])

m, n, ptr(:), row(:), val(:)
are INTENT(IN) variables that must hold $A$ in compressed sparse column format as described in Section 4.2.4.
rscaling(m)
is an INTENT(OUT) array of package type. On exit, rscaling(i) holds ${d}_{i}^{r}$, the scaling corresponding to the i-th row.
cscaling(n)
is an INTENT(OUT) array of package type. On exit, scaling(j) holds ${d}_{j}^{c}$, the scaling corresponding to the j-th column.
options
is an INTENT(IN) scalar of type auction_options. Its components specify the algorithmic options used by the subroutine, as explained in Section 4.3.3.
inform
is an INTENT(OUT) scalar of type auction_inform. On exit, its components provide information about the execution of the subroutine, as explained in Section 4.3.4.
match(m)
is an optional INTENT(OUT) array of type INTEGER. If present, 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)$=$0.

4.3.3 type(auction_options)

The derived data type auction_options is used to specify the options used by the routines auction_scale_sym() and auction_scale_unsym(). The components, that are automatically given default values in the deﬁnition of the type, are:

eps_initial
is a scalar of type default REAL that speciﬁes the initial value of the minimum improvement parameter $𝜖$ as described in Section 4.3.6.
max_iterations
is a scalar of type INTEGER that speciﬁes the maximum number of iterations the algorithm may perform. The default is max_iterations=30000.
max_unchanged(3)
is an array of type INTEGER that, together with min_proportion(:) speciﬁes termination conditions for the algorithm, as described in Section 4.3.6. The default is max_unchanged(:) = (/ 10, 100, 100 /).
min_proportion(3)
is an array of type default REAL that, together with max_unchanged(:) speciﬁes termination conditions for the algorithm, as described in Section 4.3.6. The default is min_proportion(:) = (/ 0.90, 0.0, 0.0 /).

4.3.4 type(auction_inform)

The derived data type auction_inform is used to hold parameters that give information about the progress of the routines auction_scale_sym() and auction_scale_unsym(). The components are:

flag
gives the exit status of the algorithm (details in Section 4.3.5).
iterations
is a scalar of type INTEGER that holds the number of iterations performed.
matched
is a scalar of type integer that 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.
stat
is a scalar of type INTEGER. In the event of an allocation error, it holds the Fortran stat parameter if it is available (and is set to 0 otherwise).
unmatchable
is a scalar of type integer that 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.

4.3.5 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.

4.3.6 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.

$\begin{array}{rcll}\underset{\sigma }{max}& \prod _{i=1}^{m}\prod _{j=1}^{n}|{a}_{ij}|{\sigma }_{ij}& & \text{}\\ s.t.& \sum _{i=1}^{m}{\sigma }_{ij}=1,& \forall j=1,n& \text{}\\ & \sum _{j=1}^{n}{\sigma }_{ij}=1,& \forall i=1,m& \text{}\\ & {\sigma }_{ij}\in \left\{0,1\right\}.& & \text{}\end{array}$

The array $\sigma$ gives a matching of rows to columns.

By using the transformation

${w}_{ij}=log{c}_{j}-log|{a}_{ij}|,$

where ${c}_{j}=\underset{i}{max}|{a}_{ij}|$, the maximum product problem in ${a}_{ij}$ is replaced by a minimum sum problem in ${w}_{ij}$ where all entries are positive. By standard optimization theory, there exist dual variables $u$ and $v$ corresponding to the constraints that satisfy the ﬁrst order optimality conditions

To obtain a scaling we deﬁne scaling matrices ${D}_{r}$ and ${D}_{c}$ as

$\begin{array}{rcll}& {d}_{i}^{r}={e}^{{u}_{i}},& & \text{}\\ & {d}_{i}^{c}={e}^{{v}_{i}}.& & \text{}\end{array}$

If a symmetric scaling is required, we average these as

${d}_{i}=\sqrt{{d}_{i}^{r}{d}_{i}^{c}}.$

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 $u$. In each major iteration, we loop over each column in turn. If the column $j$ is unmatched, we calculate the value ${p}_{i}={w}_{ij}-{u}_{i}$ for each entry and ﬁnd the maximum across the column. If this maximum is positive, the current matching can be improved by matching column $j$ with row $i$. This may mean that the previous match of row $i$ now becomes unmatched. We update the price of row $i$, that is ${u}_{i}$, 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 $𝜖$ above zero, where $𝜖$ is based on the last iteration in which row $i$ changed its match. This is done by adding $𝜖$ to the price ${u}_{i}$, where $𝜖=\mathtt{\text{options%eps_initial}}+\mathtt{\text{itr}}∕\left(n+1\right)$, where itr is the current iteration number.

The algorithm terminates if any of the following are satsiﬁed:

• All entries are matched.
• The number of major iterations exceeds options%max_iterations.
• 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).
• At least options%max_unchanged(3) iterations have passed without the cardinality of the matching increasing, and the proportion of matched columns is options%min_proportion(3).

The diﬀerent combinations given by options%max_unchanged(1:3) and options%min_proportion(1:3) 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 $𝜖$ 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.

4.3.7 Example of auction_scale_sym()

The following code shows an example usage of auction_scale_sym().

! examples/Fortran/scaling/auction_sym.f90 - Example code for SPRAL_SCALING
program auction_scale_sym_example
use spral_scaling
use spral_matrix_util, only : print_matrix, &
SPRAL_MATRIX_REAL_SYM_INDEF
implicit none

! Derived types
type (auction_options)  :: options
type (auction_inform)   :: inform

! Parameters
integer, parameter :: wp = kind(0.0d0)

! Matrix data
integer :: n, ptr(6), row(8)
real(wp) :: val(8)

! Other variables
integer :: match(5), i, j
real(wp) :: scaling(5)

! Data for symmetric matrix:
! ( 2  1         )
! ( 1  4  1    8 )
! (    1  3  2   )
! (       2      )
! (    8       2 )
n = 5
ptr(1:n+1)        = (/ 1,        3,             6,      8,8,   9 /)
row(1:ptr(n+1)-1) = (/ 1,   2,   2,   3,   5,   3,   4,   5   /)
val(1:ptr(n+1)-1) = (/ 2.0, 1.0, 4.0, 1.0, 8.0, 3.0, 2.0, 2.0 /)
write(*, "(a)") "Initial matrix:"
call print_matrix(6, -1, SPRAL_MATRIX_REAL_SYM_INDEF, n, n, ptr, row, val)

! Perform symmetric scaling
call auction_scale_sym(n, ptr, row, val, scaling, options, inform, &
match=match)
if(inform%flag<0) then
write(*, "(a, i5)") "auction_scale_sym() returned with error ", &
inform%flag
stop
endif

! Print scaling and matching
write(*,"(a,10i10)")    ’Matching:’, match(1:n)
write(*,"(a,10es10.2)") ’Scaling: ’, scaling(1:n)

! Calculate scaled matrix and print it
do i = 1, n
do j = ptr(i), ptr(i+1)-1
val(j) = scaling(i) * val(j) * scaling(row(j))
end do
end do
write(*, "(a)") "Scaled matrix:"
call print_matrix(6, -1, SPRAL_MATRIX_REAL_SYM_INDEF, n, n, ptr, row, val)

end program auction_scale_sym_example
The above code produces the following output.
Initial matrix:
Real symmetric indefinite matrix, dimension 5x5 with 8 entries.
1:   2.0000E+00   1.0000E+00
2:   1.0000E+00   4.0000E+00   1.0000E+00                8.0000E+00
3:                1.0000E+00   3.0000E+00   2.0000E+00
4:                             2.0000E+00
5:                8.0000E+00                             2.0000E+00
Matching:         1         5         4         3         2
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.
1:   1.0000E+00   1.1443E-01
2:   1.1443E-01   1.0476E-01   4.5008E-02                1.0000E+00
3:                4.5008E-02   2.3204E-01   1.0000E+00
4:                             1.0000E+00
5:                1.0000E+00                             1.1932E+00

4.4 Norm-equilibration algorithm

4.4.1 equilib_scale_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 $1.0±\tau$,
call equilib_scale_sym(n, ptr, row, val, scaling, options, inform)

n, ptr(:), row(:), val(:)
are INTENT(IN) variables that must hold the lower triangular part of $A$ in compressed sparse column format as described in Section 4.2.4.
scaling(n)
is an INTENT(OUT) array of package type. On exit, scaling(i) holds ${d}_{i}$, the scaling corresponding to the i-th row and column.
options
is an INTENT(IN) scalar of type equilib_options. Its components specify the algorithmic options used by the subroutine, as explained in Section 4.4.3.
inform
is an INTENT(OUT) scalar of type equilib_inform. On exit, its components provide information about the execution of the subroutine, as explained in Section 4.4.4.

4.4.2 equilib_scale_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 $1.0±\tau$,
call equilib_scale_unsym(m, n, ptr, row, val, rscaling, cscaling, options, inform)

m, n, ptr(:), row(:), val(:)
are INTENT(IN) variables that must hold $A$ in compressed sparse column format as described in Section 4.2.4.
rscaling(m)
is an INTENT(OUT) array of package type. On exit, rscaling(i) holds ${d}_{i}^{r}$, the scaling corresponding to the i-th row.
cscaling(n)
is an INTENT(OUT) array of package type. On exit, cscaling(j) holds ${d}_{j}^{c}$, the scaling corresponding to the j-th column.
options
is an INTENT(IN) scalar of type equilib_options. Its components specify the algorithmic options used by the subroutine, as explained in Section 4.4.3.
inform
is an INTENT(OUT) scalar of type equilib_inform. On exit, its components provide information about the execution of the subroutine, as explained in Section 4.4.4.

4.4.3 type(equilib_options)

The derived data type equilib_options is used to specify the options used by the routine equilib_scale_sym(). The components, that are automatically given default values in the deﬁnition of the type, are:

max_iterations
is a scalar of type INTEGER that speciﬁes the maximum number of iterations the algorithm may perform. The default is max_iterations=10.
tol
is a scalar of type default REAL that speciﬁes the convergence tolerance $\tau$ for the algorithm (though often termination is based on max_iterations). The default is tol = 1e-8.

4.4.4 type(equilib_inform)

The derived data type equilib_inform is used to hold parameters that give information about the progress of the routines equilib_scale_sym() and equilib_scale_unsym(). The components are:

flag
gives the exit status of the algorithm (details in Section 4.4.5).
iterations
is a scalar of type INTEGER. On exit, it holds the number of iterations performed.
stat
is a scalar of type INTEGER. In the event of an allocation error, it holds the Fortran stat parameter if it is available (and is set to 0 otherwise).

4.4.5 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.

4.4.6 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 $\frac{1}{2}$, preserving symmetry if the matrix is symmetric.

For unsymmetric matrices, the algorithm outline is as follows:

${D}_{r}^{\left(0\right)}=I,{D}_{c}^{\left(0\right)}=I$
for $k=1,\mathtt{\text{options%max_iterations}}$ do
${A}^{\left(k-1\right)}={D}_{r}^{\left(k-1\right)}A{D}_{c}^{\left(k-1\right)}$
${\left({D}_{r}^{\left(k\right)}\right)}_{ii}={\left({D}_{r}^{\left(k-1\right)}\right)}_{ii}∕\sqrt{\underset{j}{max}{\left({A}^{\left(k-1\right)}\right)}_{ij}}$
${\left({D}_{c}^{\left(k\right)}\right)}_{jj}={\left({D}_{c}^{\left(k-1\right)}\right)}_{jj}∕\sqrt{\underset{i}{max}{\left({A}^{\left(k-1\right)}\right)}_{ij}}$
if($|1-\parallel {A}^{\left(k-1\right)}{\parallel }_{max}|\le \mathtt{\text{options%tol}}$) exit

end for
For symmetric matrices, ${A}^{\left(k-1\right)}$ is symmetric, so ${D}_{r}^{\left(k\right)}={D}_{c}^{\left(k\right)}$, 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.

4.4.7 Example of equilib_scale_sym()

The following code shows an example usage of equilib_scale_sym().

! examples/Fortran/scaling/equilib_sym.f90 - Example code for SPRAL_SCALING
program equilib_scale_sym_example
use spral_scaling
use spral_matrix_util, only : print_matrix, &
SPRAL_MATRIX_REAL_SYM_INDEF
implicit none

! Derived types
type (equilib_options)  :: options
type (equilib_inform)   :: inform

! Parameters
integer, parameter :: wp = kind(0.0d0)

! Matrix data
integer :: n, ptr(6), row(8)
real(wp) :: val(8)

! Other variables
integer :: i, j
real(wp) :: scaling(5)

! Data for symmetric matrix:
! ( 2  1         )
! ( 1  4  1    8 )
! (    1  3  2   )
! (       2      )
! (    8       2 )
n = 5
ptr(1:n+1)        = (/ 1,        3,             6,      8,8,   9 /)
row(1:ptr(n+1)-1) = (/ 1,   2,   2,   3,   5,   3,   4,   5   /)
val(1:ptr(n+1)-1) = (/ 2.0, 1.0, 4.0, 1.0, 8.0, 3.0, 2.0, 2.0 /)
write(*, "(a)") "Initial matrix:"
call print_matrix(6, -1, SPRAL_MATRIX_REAL_SYM_INDEF, n, n, ptr, row, val)

! Perform symmetric scaling
call equilib_scale_sym(n, ptr, row, val, scaling, options, inform)
if(inform%flag<0) then
write(*, "(a, i5)") "equilib_scale_sym() returned with error ", &
inform%flag
stop
endif

! Print scaling and matching
write(*,"(a,10es10.2)") ’Scaling: ’, scaling(1:n)

! Calculate scaled matrix and print it
do i = 1, n
do j = ptr(i), ptr(i+1)-1
val(j) = scaling(i) * val(j) * scaling(row(j))
end do
end do
write(*, "(a)") "Scaled matrix:"
call print_matrix(6, -1, SPRAL_MATRIX_REAL_SYM_INDEF, n, n, ptr, row, val)

end program equilib_scale_sym_example
The above code produces the following output.
Initial matrix:
Real symmetric indefinite matrix, dimension 5x5 with 8 entries.
1:   2.0000E+00   1.0000E+00
2:   1.0000E+00   4.0000E+00   1.0000E+00                8.0000E+00
3:                1.0000E+00   3.0000E+00   2.0000E+00
4:                             2.0000E+00
5:                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.
1:   1.0000E+00   2.5000E-01
2:   2.5000E-01   5.0000E-01   2.0412E-01                1.0000E+00
3:                2.0412E-01   1.0000E+00   9.9960E-01
4:                             9.9960E-01
5:                1.0000E+00                             2.5000E-01

4.5 Hungarian algorithm

4.5.1 hungarian_scale_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,
call hungarian_scale_sym(n, ptr, row, val, scaling, options, inform[, match])

n, ptr(:), row(:), val(:)
are INTENT(IN) variables that must hold the lower triangular part of $A$ in compressed sparse column format as described in Section 4.2.4.
scaling(n)
is an INTENT(OUT) array of package type. On exit, scaling(i) holds ${d}_{i}$, the scaling corresponding to the i-th row and column.
options
is an INTENT(IN) scalar of type hungarian_options. Its components specify the algorithmic options used by the subroutine, as explained in Section 4.5.3.
inform
is an INTENT(OUT) scalar of type hungarian_inform. On exit, its components provide information about the execution of the subroutine, as explained in Section 4.5.4.
match(m)
is an optional INTENT(OUT) array of type INTEGER. If present, 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)$=$0.

4.5.2 hungarian_scale_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,
call hungarian_scale_unsym(m, n, ptr, row, val, rscaling, cscaling, options, inform[, match])

m, n, ptr(:), row(:), val(:)
are INTENT(IN) variables that must hold the lower triangular part of $A$ in compressed sparse column format as described in Section 4.2.4.
rscaling(m)
is an INTENT(OUT) array of package type. On exit, rscaling(i) holds ${d}_{i}^{r}$, the scaling corresponding to the i-th row.
rscaling(n)
is an INTENT(OUT) array of package type. On exit, rscaling(j) holds ${d}_{j}^{c}$, the scaling corresponding to the j-th column.
options
is an INTENT(IN) scalar of type hungarian_options. Its components specify the algorithmic options used by the subroutine, as explained in Section 4.5.3.
inform
is an INTENT(OUT) scalar of type hungarian_inform. On exit, its components provide information about the execution of the subroutine, as explained in Section 4.5.4.
match(m)
is an optional INTENT(OUT) array of type INTEGER. If present, 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)$=$0.

4.5.3 type(hungarian_options)

The derived data type hungarian_options is used to specify the options used by the routines
hungarian_scale_sym() and hungarian_scale_unsym(). The components, that are automatically given default values in the deﬁnition of the type, are:

scale_if_singular
is a scalar of type default LOGICAL that speciﬁes whether scaling should continue if the matrix $A$ is found to be structurally singular. If scale_if_singular$=$.true., and the $A$ 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.

4.5.4 type(hungarian_inform)

The derived data type hungarian_inform is used to hold parameters that give information about the progress of the routine hungarian_scale_sym() and hungarian_scale_unsym(). The components are:

flag
gives the exit status of the algorithm (details in Section 4.5.5).
matched
is a scalar of type INTEGER that holds the number of rows and columns that have been matched (i.e. the structural rank).
stat
is a scalar of type INTEGER. In the event of an allocation error, it holds the Fortran stat parameter if it is available (and is set to 0 otherwise).

4.5.5 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 $A$ is structurally rank-deﬁcient. This error is returned only if options%scale_if_singular$=$.false.. 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 $A$ is structurally rank-deﬁcient. This warning is returned only if options%scale_if_singular$=$.true..

4.5.6 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.

$\begin{array}{rcll}\underset{\sigma }{max}& \prod _{i=1}^{m}\prod _{j=1}^{n}|{a}_{ij}|{\sigma }_{ij}& & \text{}\\ s.t.& \sum _{i=1}^{m}{\sigma }_{ij}=1,& \forall j=1,n& \text{}\\ & \sum _{j=1}^{n}{\sigma }_{ij}=1,& \forall i=1,m& \text{}\\ & {\sigma }_{ij}\in \left\{0,1\right\}.& & \text{}\end{array}$

The array $\sigma$ gives a matching of rows to columns.

By using the transformation

${w}_{ij}=log{c}_{j}-log|{a}_{ij}|,$

where ${c}_{j}=\underset{i}{max}|{a}_{ij}|$, the maximum product problem in ${a}_{ij}$ is replaced by a minimum sum problem in ${w}_{ij}$ where all entries are positive. By standard optimization theory, there exist dual variables $u$ and $v$ corresponding to the constraints that satisfy the ﬁrst order optimality conditions

To obtain a scaling we deﬁne scaling matrices ${D}_{r}$ and ${D}_{c}$ as

$\begin{array}{rcll}& {d}_{i}^{r}={e}^{{u}_{i}},& & \text{}\\ & {d}_{i}^{c}={e}^{{v}_{i}}.& & \text{}\end{array}$

If a symmetric scaling is required, we average these as

${d}_{i}=\sqrt{{d}_{i}^{r}{d}_{i}^{c}}.$

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.

4.5.7 Example usage of hungarian_scale_unsym()

The following code shows an example usage of hungarian_scale_unsym().

! examples/Fortran/scaling/hungarian_unsym.f90 - Example code for SPRAL_SCALING
program hungarian_scale_unsym_example
use spral_scaling
use spral_matrix_util, only : print_matrix, &
SPRAL_MATRIX_REAL_UNSYM
implicit none

! Derived types
type (hungarian_options)  :: options
type (hungarian_inform)   :: inform

! Parameters
integer, parameter :: wp = kind(0.0d0)

! Matrix data
integer :: m, n, ptr(6), row(10)
real(wp) :: val(10)

! Other variables
integer :: match(5), i, j
real(wp) :: rscaling(5), cscaling(5)

! Data for unsymmetric matrix:
! ( 2  5         )
! ( 1  4       7 )
! (    1     2   )
! (       3      )
! (    8       2 )
m = 5; n = 5
ptr(1:n+1)        = (/ 1,        3,                  7,   8,   9,       11 /)
row(1:ptr(n+1)-1) = (/ 1,   2,   1,   2,   3,   5,   4,   3,   2,   5   /)
val(1:ptr(n+1)-1) = (/ 2.0, 1.0, 5.0, 4.0, 1.0, 8.0, 3.0, 2.0, 7.0, 2.0 /)
write(*, "(a)") "Initial matrix:"
call print_matrix(6, -1, SPRAL_MATRIX_REAL_UNSYM, m, n, ptr, row, val)

! Perform unsymmetric scaling
call hungarian_scale_unsym(m, n, ptr, row, val, rscaling, cscaling, options, &
inform, match=match)
if(inform%flag<0) then
write(*, "(a, i5)") "hungarian_scale_unsym() returned with error ", &
inform%flag
stop
endif

! Print scaling and matching
write(*,"(a,10i10)")    ’Matching:’, match(1:m)
write(*,"(a,10es10.2)") ’Row Scaling: ’, rscaling(1:m)
write(*,"(a,10es10.2)") ’Col Scaling: ’, cscaling(1:n)

! Calculate scaled matrix and print it
do i = 1, n
do j = ptr(i), ptr(i+1)-1
val(j) = rscaling(row(j)) * val(j) * cscaling(i)
end do
end do
write(*, "(a)") "Scaled matrix:"
call print_matrix(6, -1, SPRAL_MATRIX_REAL_UNSYM, m, n, ptr, row, val)

end program hungarian_scale_unsym_example
The above code produces the following output.
Initial matrix:
Real unsymmetric matrix, dimension 5x5 with 10 entries.
1:   2.0000E+00   5.0000E+00
2:   1.0000E+00   4.0000E+00                             7.0000E+00
3:                1.0000E+00                2.0000E+00
4:                             3.0000E+00
5:                8.0000E+00                             2.0000E+00
Matching:         1         5         4         3         2
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 10 entries.
1:   1.0000E+00   6.2500E-01
2:   5.0000E-01   5.0000E-01                             1.0000E+00
3:                1.2500E-01                1.0000E+00
4:                             1.0000E+00
5:                1.0000E+00                             2.8571E-01