## MB04MD

### Balancing a general real matrix

[Specification] [Arguments] [Method] [References] [Comments] [Example]

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

      None

Example

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   1.0

Program 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

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