MB04MD

Balancing a general real matrix

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.

      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( * )

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.

  INFO    INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal
                value.

  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.

  [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.

  None.

  None

Program Text

*     MB04MD EXAMPLE PROGRAM TEXT.
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. 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

 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

 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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