Contents
dgebal - balance a general real matrix A
SUBROUTINE DGEBAL(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
CHARACTER * 1 JOB
INTEGER N, LDA, ILO, IHI, INFO
DOUBLE PRECISION A(LDA,*), SCALE(*)
SUBROUTINE DGEBAL_64(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
CHARACTER * 1 JOB
INTEGER*8 N, LDA, ILO, IHI, INFO
DOUBLE PRECISION A(LDA,*), SCALE(*)
F95 INTERFACE
SUBROUTINE GEBAL(JOB, [N], A, [LDA], ILO, IHI, SCALE, [INFO])
CHARACTER(LEN=1) :: JOB
INTEGER :: N, LDA, ILO, IHI, INFO
REAL(8), DIMENSION(:) :: SCALE
REAL(8), DIMENSION(:,:) :: A
SUBROUTINE GEBAL_64(JOB, [N], A, [LDA], ILO, IHI, SCALE, [INFO])
CHARACTER(LEN=1) :: JOB
INTEGER(8) :: N, LDA, ILO, IHI, INFO
REAL(8), DIMENSION(:) :: SCALE
REAL(8), DIMENSION(:,:) :: A
C INTERFACE
#include <sunperf.h>
void dgebal(char job, int n, double *a, int lda, int *ilo,
int *ihi, double *scale, int *info);
void dgebal_64(char job, long n, double *a, long lda, long
*ilo, long *ihi, double *scale, long *info);
dgebal balances a general real matrix A. This involves,
first, permuting A by a similarity transformation to isolate
eigenvalues in the first 1 to ILO-1 and last IHI+1 to N ele-
ments on the diagonal; and second, applying a diagonal simi-
larity transformation to rows and columns ILO to IHI to make
the rows and columns as close in norm as possible. Both
steps are optional.
Balancing may reduce the 1-norm of the matrix, and improve
the accuracy of the computed eigenvalues and/or eigenvec-
tors.
JOB (input)
Specifies the operations to be performed on A:
= 'N': none: simply set ILO = 1, IHI = N,
SCALE(I) = 1.0 for i = 1,...,N; = 'P': permute
only;
= 'S': scale only;
= 'B': both permute and scale.
N (input) The order of the matrix A. N >= 0.
A (input/output)
On entry, the input matrix A. On exit, A is
overwritten by the balanced matrix. If JOB = 'N',
A is not referenced. See Further Details.
LDA (input)
The leading dimension of the array A. LDA >=
max(1,N).
ILO (output)
ILO and IHI are set to integers such that on exit
A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I =
IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI
= N.
IHI (output)
See the description for ILO.
SCALE (output)
Details of the permutations and scaling factors
applied to A. If P(j) is the index of the row and
column interchanged with row and column j and D(j)
is the scaling factor applied to row and column j,
then SCALE(j) = P(j) for j = 1,...,ILO-1 = D(j)
for j = ILO,...,IHI = P(j) for j = IHI+1,...,N.
The order in which the interchanges are made is N
to IHI+1, then 1 to ILO-1.
INFO (output)
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille-
gal value.
The permutations consist of row and column interchanges
which put the matrix in the form
( T1 X Y )
P A P = ( 0 B Z )
( 0 0 T2 )
where T1 and T2 are upper triangular matrices whose eigen-
values lie along the diagonal. The column indices ILO and
IHI mark the starting and ending columns of the submatrix B.
Balancing consists of applying a diagonal similarity
transformation inv(D) * B * D to make the 1-norms of each
row of B and its corresponding column nearly equal. The
output matrix is
( T1 X*D Y )
( 0 inv(D)*B*D inv(D)*Z ).
( 0 0 T2 )
Information about the permutations P and the diagonal matrix
D is returned in the vector SCALE.
This subroutine is based on the EISPACK routine BALANC.
Modified by Tzu-Yi Chen, Computer Science Division, Univer-
sity of
California at Berkeley, USA