## TB01ID

### Balancing a system matrix corresponding to a triplet (A,B,C)

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

Purpose

      To reduce the 1-norm of a system matrix

              S =  ( A  B )
                   ( C  0 )

      corresponding to the triple (A,B,C), by balancing. This involves
      a diagonal similarity transformation inv(D)*A*D applied
      iteratively to A to make the rows and columns of
                            -1
                   diag(D,I)  * S * diag(D,I)

      as close in norm as possible.

      The balancing can be performed optionally on the following
      particular system matrices

               S = A,    S = ( A  B )    or    S = ( A )
                                                   ( C )

Specification

          SUBROUTINE TB01ID( JOB, N, M, P, MAXRED, A, LDA, B, LDB, C, LDC,
         $                   SCALE, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          JOB
          INTEGER            INFO, LDA, LDB, LDC, M, N, P
          DOUBLE PRECISION   MAXRED
    C     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
         $                   SCALE( * )

Arguments

Mode Parameters

      JOB     CHARACTER*1
              Indicates which matrices are involved in balancing, as
              follows:
              = 'A':  All matrices are involved in balancing;
              = 'B':  B and A matrices are involved in balancing;
              = 'C':  C and A matrices are involved in balancing;
              = 'N':  B and C matrices are not involved in balancing.

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrix A, the number of rows of matrix B
              and the number of columns of matrix C.
              N represents the dimension of the state vector.  N >= 0.

      M       (input) INTEGER.
              The number of columns of matrix B.
              M represents the dimension of input vector.  M >= 0.

      P       (input) INTEGER.
              The number of rows of matrix C.
              P represents the dimension of output vector.  P >= 0.

      MAXRED  (input/output) DOUBLE PRECISION
              On entry, the maximum allowed reduction in the 1-norm of
              S (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 S is non-zero,
              the ratio between the 1-norm of the given matrix and the
              1-norm of the balanced matrix.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the system state matrix A.
              On exit, the leading N-by-N part of this array contains
              the balanced matrix inv(D)*A*D.

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
              On entry, if M > 0, the leading N-by-M part of this array
              must contain the system input matrix B.
              On exit, if M > 0, the leading N-by-M part of this array
              contains the balanced matrix inv(D)*B.
              The array B is not referenced if M = 0.

      LDB     INTEGER
              The leading dimension of the array B.
              LDB >= MAX(1,N) if M > 0.
              LDB >= 1        if M = 0.

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry, if P > 0, the leading P-by-N part of this array
              must contain the system output matrix C.
              On exit, if P > 0, the leading P-by-N part of this array
              contains the balanced matrix C*D.
              The array C is not referenced if P = 0.

      LDC     INTEGER
              The leading dimension of the array C.  LDC >= MAX(1,P).

      SCALE   (output) DOUBLE PRECISION array, dimension (N)
              The scaling factors applied to S.  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
                            -1
                   diag(D,I)  * S * diag(D,I)

      to make the 1-norms of each row of the first N rows of S 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

    *     TB01ID EXAMPLE PROGRAM TEXT.
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          LDA, LDB, LDC
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
    *     .. Local Scalars ..
          CHARACTER*1      JOB
          INTEGER          I, INFO, J, M, N, P
          DOUBLE PRECISION MAXRED
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 SCALE(NMAX)
    *     .. External Subroutines ..
          EXTERNAL         TB01ID, UD01MD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, M, P, JOB, MAXRED
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99993 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99992 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
                IF ( P.LT.0 .OR. P.GT.MMAX ) THEN
                   WRITE ( NOUT, FMT = 99991 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
    *              Balance system matrix S.
                   CALL TB01ID( JOB, N, M, P, MAXRED, A, LDA, B, LDB, C,
         $                      LDC, SCALE, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      CALL UD01MD( N, N, 5, NOUT, A, LDA,
         $                        'The balanced matrix A', INFO )
                      IF ( M.GT.0 )
         $               CALL UD01MD( N, M, 5, NOUT, B, LDB,
         $                            'The balanced matrix B', INFO )
                      IF ( P.GT.0 )
         $               CALL UD01MD( P, N, 5, NOUT, C, LDC,
         $                            'The balanced matrix C', INFO )
                      CALL UD01MD( 1, N, 5, NOUT, SCALE, 1,
         $                        'The scaling vector SCALE', INFO )
                      WRITE ( NOUT, FMT = 99994 ) MAXRED
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TB01ID EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TB01ID = ',I2)
    99994 FORMAT (/' MAXRED is ',E13.4)
    99993 FORMAT (/' N is out of range.',/' N = ',I5)
    99992 FORMAT (/' M is out of range.',/' M = ',I5)
    99991 FORMAT (/' P is out of range.',/' P = ',I5)
          END

Program Data

     TB01ID EXAMPLE PROGRAM DATA
       5     2     5       A    0.0
              0.0  1.0000e+000          0.0          0.0          0.0
     -1.5800e+006 -1.2570e+003          0.0          0.0          0.0
      3.5410e+014          0.0 -1.4340e+003          0.0 -5.3300e+011
              0.0          0.0          0.0          0.0  1.0000e+000
              0.0          0.0          0.0 -1.8630e+004 -1.4820e+000
              0.0          0.0
      1.1030e+002          0.0
              0.0          0.0
              0.0          0.0
              0.0  8.3330e-003
      1.0000e+000          0.0          0.0          0.0          0.0
              0.0          0.0  1.0000e+000          0.0          0.0
              0.0          0.0          0.0  1.0000e+000          0.0
      6.6640e-001          0.0 -6.2000e-013          0.0          0.0
              0.0          0.0 -1.0000e-003  1.8960e+006  1.5080e+002

Program Results

     TB01ID EXAMPLE PROGRAM RESULTS

     The balanced matrix A ( 5X 5)

                1              2              3              4              5
      1    0.0000000D+00  0.1000000D+05  0.0000000D+00  0.0000000D+00  0.0000000D+00
      2   -0.1580000D+03 -0.1257000D+04  0.0000000D+00  0.0000000D+00  0.0000000D+00
      3    0.3541000D+05  0.0000000D+00 -0.1434000D+04  0.0000000D+00 -0.5330000D+03
      4    0.0000000D+00  0.0000000D+00  0.0000000D+00  0.0000000D+00  0.1000000D+03
      5    0.0000000D+00  0.0000000D+00  0.0000000D+00 -0.1863000D+03 -0.1482000D+01

     The balanced matrix B ( 5X 2)

                1              2
      1    0.0000000D+00  0.0000000D+00
      2    0.1103000D+04  0.0000000D+00
      3    0.0000000D+00  0.0000000D+00
      4    0.0000000D+00  0.0000000D+00
      5    0.0000000D+00  0.8333000D+02

     The balanced matrix C ( 5X 5)

                1              2              3              4              5
      1    0.1000000D-04  0.0000000D+00  0.0000000D+00  0.0000000D+00  0.0000000D+00
      2    0.0000000D+00  0.0000000D+00  0.1000000D+06  0.0000000D+00  0.0000000D+00
      3    0.0000000D+00  0.0000000D+00  0.0000000D+00  0.1000000D-05  0.0000000D+00
      4    0.6664000D-05  0.0000000D+00 -0.6200000D-07  0.0000000D+00  0.0000000D+00
      5    0.0000000D+00  0.0000000D+00 -0.1000000D+03  0.1896000D+01  0.1508000D-01

     The scaling vector SCALE ( 1X 5)

                1              2              3              4              5
      1    0.1000000D-04  0.1000000D+00  0.1000000D+06  0.1000000D-05  0.1000000D-03

     MAXRED is    0.3488E+10

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