Purpose
To reduce the 1-norm of a general real matrix A by balancing. This involves diagonal similarity transformations applied iteratively to A to make the rows and columns as close in norm as possible. This routine can be used instead LAPACK Library routine DGEBAL, when no reduction of the 1-norm of the matrix is possible with DGEBAL, as for upper triangular matrices. LAPACK Library routine DGEBAK, with parameters ILO = 1, IHI = N, and JOB = 'S', should be used to apply the backward transformation.Specification
SUBROUTINE MB04MD( N, MAXRED, A, LDA, SCALE, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, N
DOUBLE PRECISION MAXRED
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), SCALE( * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
MAXRED (input/output) DOUBLE PRECISION
On entry, the maximum allowed reduction in the 1-norm of
A (in an iteration) if zero rows or columns are
encountered.
If MAXRED > 0.0, MAXRED must be larger than one (to enable
the norm reduction).
If MAXRED <= 0.0, then the value 10.0 for MAXRED is
used.
On exit, if the 1-norm of the given matrix A is non-zero,
the ratio between the 1-norm of the given matrix and the
1-norm of the balanced matrix. Usually, this ratio will be
larger than one, but it can sometimes be one, or even less
than one (for instance, for some companion matrices).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the input matrix A.
On exit, the leading N-by-N part of this array contains
the balanced matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
SCALE (output) DOUBLE PRECISION array, dimension (N)
The scaling factors applied to A. If D(j) is the scaling
factor applied to row and column j, then SCALE(j) = D(j),
for j = 1,...,N.
Error Indicator
INFO INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Balancing consists of applying a diagonal similarity transformation inv(D) * A * D to make the 1-norms of each row of A and its corresponding column nearly equal. Information about the diagonal matrix D is returned in the vector SCALE.References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
None.Further Comments
NoneExample
Program Text
* MB04MD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA
PARAMETER ( LDA = NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, N
DOUBLE PRECISION MAXRED
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), SCALE(NMAX)
* .. External Subroutines ..
EXTERNAL MB04MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, MAXRED
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
* Balance matrix A.
CALL MB04MD( N, MAXRED, A, LDA, SCALE, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 ) ( SCALE(I), I = 1,N )
END IF
END IF
STOP
*
99999 FORMAT (' MB04MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04MD = ',I2)
99997 FORMAT (' The balanced matrix is ')
99996 FORMAT (20(1X,F10.4))
99994 FORMAT (/' SCALE is ',/20(1X,F10.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB04MD EXAMPLE PROGRAM DATA 4 0.0 1.0 0.0 0.0 0.0 300.0 400.0 500.0 600.0 1.0 2.0 0.0 0.0 1.0 1.0 1.0 1.0Program Results
MB04MD EXAMPLE PROGRAM RESULTS
The balanced matrix is
1.0000 0.0000 0.0000 0.0000
30.0000 400.0000 50.0000 60.0000
1.0000 20.0000 0.0000 0.0000
1.0000 10.0000 1.0000 1.0000
SCALE is
1.0000 10.0000 1.0000 1.0000