TG01AD

Balancing the matrices of the system pencil corresponding to a descriptor triple (A-lambda E,B,C)

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

Purpose

  To balance the matrices of the system pencil

          S =  ( A  B ) - lambda ( E  0 ) :=  Q - lambda Z,
               ( C  0 )          ( 0  0 )

  corresponding to the descriptor triple (A-lambda E,B,C),
  by balancing. This involves diagonal similarity transformations
  (Dl*A*Dr - lambda Dl*E*Dr, Dl*B, C*Dr) applied to the system
  (A-lambda E,B,C) to make the rows and columns of system pencil
  matrices

               diag(Dl,I) * S * diag(Dr,I)

  as close in norm as possible. Balancing may reduce the 1-norms
  of the matrices of the system pencil S.

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

           S = A-lambda E,

           S = ( A-lambda E  B ),    or

           S = ( A-lambda E ).
               (     C      )

Specification
      SUBROUTINE TG01AD( JOB, L, N, M, P, THRESH, A, LDA, E, LDE,
     $                   B, LDB, C, LDC, LSCALE, RSCALE, DWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER          JOB
      INTEGER            INFO, L, LDA, LDB, LDC, LDE, M, N, P
      DOUBLE PRECISION   THRESH
C     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   DWORK( * ), E( LDE, * ), LSCALE( * ),
     $                   RSCALE( * )

Arguments

Mode Parameters

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

Input/Output Parameters
  L       (input) INTEGER
          The number of rows of matrices A, B, and E.  L >= 0.

  N       (input) INTEGER
          The number of columns of matrices A, E, and C.  N >= 0.

  M       (input) INTEGER
          The number of columns of matrix B.  M >= 0.

  P       (input) INTEGER
          The number of rows of matrix C.  P >= 0.

  THRESH  (input) DOUBLE PRECISION
          Threshold value for magnitude of elements:
          elements with magnitude less than or equal to
          THRESH are ignored for balancing. THRESH >= 0.

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

  LDA     INTEGER
          The leading dimension of array A.  LDA >= MAX(1,L).

  E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
          On entry, the leading L-by-N part of this array must
          contain the descriptor matrix E.
          On exit, the leading L-by-N part of this array contains
          the balanced matrix Dl*E*Dr.

  LDE     INTEGER
          The leading dimension of array E.  LDE >= MAX(1,L).

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

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

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

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

  LSCALE  (output) DOUBLE PRECISION array, dimension (L)
          The scaling factors applied to S from left.  If Dl(j) is
          the scaling factor applied to row j, then
          SCALE(j) = Dl(j), for j = 1,...,L.

  RSCALE  (output) DOUBLE PRECISION array, dimension (N)
          The scaling factors applied to S from right.  If Dr(j) is
          the scaling factor applied to column j, then
          SCALE(j) = Dr(j), for j = 1,...,N.

Workspace
  DWORK   DOUBLE PRECISION array, dimension (3*(L+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(Dl,I)  * S * diag(Dr,I)

  to make the 1-norms of each row of the first L rows of S and its
  corresponding N columns nearly equal.

  Information about the diagonal matrices Dl and Dr are returned in
  the vectors LSCALE and RSCALE, respectively.

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.

  [2] R.C. Ward, R. C.
      Balancing the generalized eigenvalue problem.
      SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

Numerical Aspects
  None.

Further Comments
  None
Example

Program Text

*     TG01AD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          LMAX, NMAX, MMAX, PMAX
      PARAMETER        ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
      INTEGER          LDA, LDB, LDC, LDE
      PARAMETER        ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
     $                   LDE = LMAX )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = MAX( 1, 3*(LMAX+NMAX ) ) )
*     .. Local Scalars ..
      CHARACTER*1      JOBS
      INTEGER          I, INFO, J, L, M, N, P
      DOUBLE PRECISION ABCNRM, ENORM, SABCNM, SENORM, THRESH
*     .. Local Arrays ..
      DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
     $                 DWORK(LDWORK), E(LDE,NMAX), LSCALE(LMAX),
     $                 RSCALE(NMAX)
*     .. External Functions ..
      DOUBLE PRECISION DLANGE
      EXTERNAL         DLANGE
*     .. External Subroutines ..
      EXTERNAL         TG01AD
*     .. Intrinsic Functions ..
      INTRINSIC        MAX
*     .. Executable Statements ..
*
      WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
      READ ( NIN, FMT = '()' )
      READ ( NIN, FMT = * ) L, N, M, P, JOBS, THRESH
      IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
         WRITE ( NOUT, FMT = 99989 ) L
      ELSE
         IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
            WRITE ( NOUT, FMT = 99988 ) N
         ELSE
            READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
            READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
            IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
               WRITE ( NOUT, FMT = 99987 ) M
            ELSE
               READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
               IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
                  WRITE ( NOUT, FMT = 99986 ) P
               ELSE
                  READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
*                 Compute norms before scaling
                  ABCNRM = MAX( DLANGE( '1', L, N, A, LDA, DWORK ),
     $                          DLANGE( '1', L, M, B, LDB, DWORK ),
     $                          DLANGE( '1', P, N, C, LDC, DWORK ) )
                  ENORM = DLANGE( '1', L, N, E, LDE, DWORK )
*                 Find the transformed descriptor system
*                 (A-lambda E,B,C).
                  CALL TG01AD( JOBS, L, N, M, P, THRESH, A, LDA, E, LDE,
     $                         B, LDB, C, LDC, LSCALE, RSCALE, DWORK,
     $                         INFO )
*
                  IF ( INFO.NE.0 ) THEN
                     WRITE ( NOUT, FMT = 99998 ) INFO
                  ELSE
                     SABCNM = MAX( DLANGE( '1', L, N, A, LDA, DWORK ),
     $                             DLANGE( '1', L, M, B, LDB, DWORK ),
     $                             DLANGE( '1', P, N, C, LDC, DWORK ) )
                     SENORM = DLANGE( '1', L, N, E, LDE, DWORK )
                     WRITE ( NOUT, FMT = 99997 )
                     DO 10 I = 1, L
                        WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
   10                CONTINUE
                     WRITE ( NOUT, FMT = 99996 )
                     DO 20 I = 1, L
                        WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
   20                CONTINUE
                     WRITE ( NOUT, FMT = 99993 )
                     DO 30 I = 1, L
                        WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
   30                CONTINUE
                     WRITE ( NOUT, FMT = 99992 )
                     DO 40 I = 1, P
                        WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
   40                CONTINUE
                     WRITE ( NOUT, FMT = 99991 )
                     WRITE ( NOUT, FMT = 99995 ) ( LSCALE(I), I = 1,L )
                     WRITE ( NOUT, FMT = 99990 )
                     WRITE ( NOUT, FMT = 99995 ) ( RSCALE(J), J = 1,N )
                     WRITE ( NOUT, FMT = 99994 )
     $                       ABCNRM, SABCNM, ENORM, SENORM
                  END IF
               END IF
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT (' TG01AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01AD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Dl*A*Dr is ')
99996 FORMAT (/' The transformed descriptor matrix Dl*E*Dr is ')
99995 FORMAT (20(1X,F9.4))
99994 FORMAT (/' Norm of [ A B; C 0]         =', 1PD10.3/
     $         ' Norm of scaled [ A B; C 0]  =', 1PD10.3/
     $         ' Norm of E                   =', 1PD10.3/
     $         ' Norm of scaled E            =', 1PD10.3)
99993 FORMAT (/' The transformed input/state matrix Dl*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Dr is ')
99991 FORMAT (/' The diagonal of left scaling matrix Dl is ')
99990 FORMAT (/' The diagonal of right scaling matrix Dr is ')
99989 FORMAT (/' L is out of range.',/' L = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
      END
Program Data
TG01AD EXAMPLE PROGRAM DATA
  4    4     2     2     A   0.0
        -1         0         0    0.003
         0         0    0.1000    0.02
       100        10         0    0.4
         0         0         0    0.0
         1       0.2         0    0.0
         0         1         0    0.01
       300        90         6    0.3
         0         0        20    0.0
        10         0
         0         0
         0      1000
     10000     10000
      -0.1      0.0    0.001    0.0
       0.0      0.01  -0.001    0.0001

Program Results
 TG01AD EXAMPLE PROGRAM RESULTS


 The transformed state dynamics matrix Dl*A*Dr is 
   -1.0000    0.0000    0.0000    0.3000
    0.0000    0.0000    1.0000    2.0000
    1.0000    0.1000    0.0000    0.4000
    0.0000    0.0000    0.0000    0.0000

 The transformed descriptor matrix Dl*E*Dr is 
    1.0000    0.2000    0.0000    0.0000
    0.0000    1.0000    0.0000    1.0000
    3.0000    0.9000    0.6000    0.3000
    0.0000    0.0000    0.2000    0.0000

 The transformed input/state matrix Dl*B is 
  100.0000    0.0000
    0.0000    0.0000
    0.0000  100.0000
  100.0000  100.0000

 The transformed state/output matrix C*Dr is 
   -0.0100    0.0000    0.0010    0.0000
    0.0000    0.0010   -0.0010    0.0010

 The diagonal of left scaling matrix Dl is 
   10.0000   10.0000    0.1000    0.0100

 The diagonal of right scaling matrix Dr is 
    0.1000    0.1000    1.0000   10.0000

 Norm of [ A B; C 0]         = 1.100D+04
 Norm of scaled [ A B; C 0]  = 2.000D+02
 Norm of E                   = 3.010D+02
 Norm of scaled E            = 4.000D+00

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