## TG01GD

### Reduced descriptor representation without non-dynamic modes

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

Purpose

      To find a reduced descriptor representation (Ar-lambda*Er,Br,Cr)
      without non-dynamic modes for a descriptor representation
      (A-lambda*E,B,C). Optionally, the reduced descriptor system can
      be put into a standard form with the leading diagonal block
      of Er identity.

Specification

          SUBROUTINE TG01GD( JOBS, L, N, M, P, A, LDA, E, LDE, B, LDB,
         $                   C, LDC, D, LDD, LR, NR, RANKE, INFRED, TOL,
         $                   IWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         JOBS
          INTEGER           INFO, INFRED, L, LDA, LDB, LDC, LDD, LDE,
         $                  LDWORK, LR, M, N, NR, P, RANKE
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
         $                  DWORK(*), E(LDE,*)

Arguments

Mode Parameters

      JOBS    CHARACTER*1
              Indicates whether the user wishes to transform the leading
              diagonal block of Er to an identity matrix, as follows:
              = 'S':  make Er with leading diagonal identity;
              = 'D':  keep Er unreduced or upper triangular.

Input/Output Parameters

      L       (input) INTEGER
              The number of rows of the matrices A, E, and B;
              also the number of differential equations.  L >= 0.

      N       (input) INTEGER
              The number of columns of the matrices A, E, and C;
              also the dimension of descriptor state vector.  N >= 0.

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

      P       (input) INTEGER
              The number of rows of the matrix C.
              also the dimension of the output vector.  P >= 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, if NR < N, the leading LR-by-NR part of this
              array contains the reduced order state matrix Ar of a
              descriptor realization without non-dynamic modes.
              Array A contains the original state dynamics matrix if
              INFRED < 0.

      LDA     INTEGER
              The leading dimension of the 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, if INFRED >= 0, the leading LR-by-NR part of this
              array contains the reduced order descriptor matrix Er of a
              descriptor realization without non-dynamic modes.
              In this case, only the leading RANKE-by-RANKE submatrix
              of Er is nonzero and this submatrix is nonsingular and
              upper triangular. Array E contains the original descriptor
              matrix if INFRED < 0. If JOBS = 'S', then the leading
              RANKE-by-RANKE submatrix results in an identity matrix.

      LDE     INTEGER
              The leading dimension of the 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 matrix B.
              On exit, the leading LR-by-M part of this array contains
              the reduced order input matrix Br of a descriptor
              realization without non-dynamic modes. Array B contains
              the original input matrix if INFRED < 0.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= MAX(1,L).

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the leading P-by-N part of this array must
              contain the output matrix C.
              On exit, the leading P-by-NR part of this array contains
              the reduced order output matrix Cr of a descriptor
              realization without non-dynamic modes. Array C contains
              the original output matrix if INFRED < 0.

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

      D       (input/output) DOUBLE PRECISION array, dimension (LDD,M)
              On entry, the leading P-by-M part of this array must
              contain the original feedthrough matrix D.
              On exit, the leading P-by-M part of this array contains
              the feedthrough matrix Dr of a reduced descriptor
              realization without non-dynamic modes.

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

      LR      (output) INTEGER
              The number of reduced differential equations.

      NR      (output) INTEGER
              The dimension of the reduced descriptor state vector.

      RANKE   (output) INTEGER
              The estimated rank of the matrix E.

      INFRED  (output) INTEGER
              This parameter contains information on performed reduction
              and on structure of resulting system matrices, as follows:
              INFRED >= 0 the reduced system is in an SVD-like
                          coordinate form with Er upper triangular;
                          INFRED is the achieved order reduction.
              INFRED  < 0 no reduction achieved and the original
                          system has been restored.

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance to be used in rank determinations when
              transforming (A-lambda*E). If the user sets TOL > 0,
              then the given value of TOL is used as a lower bound for
              reciprocal condition numbers in rank determinations; a
              (sub)matrix whose estimated condition number is less than
              1/TOL is considered to be of full rank.  If the user sets
              TOL <= 0, then an implicitly computed, default tolerance,
              defined by  TOLDEF = L*N*EPS,  is used instead, where EPS
              is the machine precision (see LAPACK Library routine
              DLAMCH).  TOL < 1.

Workspace

      IWORK   INTEGER array, dimension (N)

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= 1, if MIN(L,N) = 0; otherwise,
              LDWORK >= MAX( N+P, MIN(L,N)+MAX(3*N-1,M,L) ).
              If LDWORK >= 2*L*N+L*M+N*P+
                           MAX( 1, N+P, MIN(L,N)+MAX(3*N-1,M,L) ) then
              the original matrices are restored if no order reduction
              is possible. This is achieved by saving system matrices
              before reduction and restoring them if no order reduction
              took place.

              If LDWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the DWORK
              array, returns this value as the first entry of the DWORK
              array, and no error message related to LDWORK is issued by
              XERBLA. The optimal size does not necessarily include the
              space needed for saving the original system matrices.

Error Indicator

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

Method

      The subroutine elliminates the non-dynamics modes in two steps:

      Step 1: Reduce the system to the SVD-like coordinate form
      (Q'*A*Z-lambda*Q'*E*Z, Q'*B, C*Z) , where

               ( A11 A12 A13 )           ( E11 0 0 )         ( B1 )
      Q'*A*Z = ( A21 A22  0  ), Q'*E*Z = (  0  0 0 ), Q'*B = ( B2 ),
               ( A31  0   0  )           (  0  0 0 )         ( B3 )

         C*Z = ( C1  C2  C3 ),

      where E11 and A22 are upper triangular invertible matrices.

      Step 2: Compute the reduced system as (Ar-lambda*Er,Br,Cr,Dr),
      where
           ( A11 - A12*inv(A22)*A21, A13 )        ( E11 0 )
      Ar = (                             ),  Er = (       ),
           (     A31                  0  )        (  0  0 )

           ( B1 - A12*inv(A22)*B2 )
      Br = (                      ),  Cr = ( C1 - C2*inv(A22)*A21, C3 ),
           (        B3            )

      Dr = D - C2*inv(A22)*B2.

      Step 3: If desired (JOBS = 'S'), reduce the descriptor system to
      the standard form

      Ar <- diag(inv(E11),I)*Ar;  Br <- diag(inv(E11),I)*Br;
      Er  = diag(I,0).

      If L = N and LR = NR = RANKE, then if Step 3 is performed,
      the resulting system is a standard state space system.

Numerical Aspects

      If L = N, the algorithm requires 0( N**3 ) floating point
      operations.

Further Comments

      None

Example

Program Text

    *     TG01GD EXAMPLE PROGRAM TEXT
    *
    *     .. 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, LDD, LDE
          PARAMETER        ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
         $                   LDD = PMAX, LDE = LMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MIN( LMAX, NMAX ) +
         $                            MAX( 3*NMAX - 1, MMAX, LMAX ) +
         $                            2*LMAX*NMAX + LMAX*MMAX + PMAX*NMAX )
    *     .. Local Scalars ..
          CHARACTER*1      JOBS
          INTEGER          I, INFO, INFRED, J, L, LR, M, N, NR, P, RANKE
          DOUBLE PRECISION TOL
    *     .. Local Arrays ..
          INTEGER          IWORK(NMAX)
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX),   C(LDC,NMAX),
         $                 D(LDD,MMAX), DWORK(LDWORK), E(LDE,NMAX)
    *     .. External Subroutines ..
          EXTERNAL         TG01GD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX, MIN
    *     .. 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, TOL
          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 )
                      READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
    *                 Find the reduced descriptor system
    *                 (A-lambda E,B,C,D).
                      CALL TG01GD( JOBS, L, N, M, P, A, LDA, E, LDE, B, LDB,
         $                         C, LDC, D, LDD, LR, NR, RANKE, INFRED,
         $                         TOL, IWORK, DWORK, LDWORK, INFO )
    *
                      IF ( INFO.NE.0 ) THEN
                         WRITE ( NOUT, FMT = 99998 ) INFO
                      ELSE
                         WRITE ( NOUT, FMT = 99994 ) RANKE
                         WRITE ( NOUT, FMT = 99997 )
                         DO 10 I = 1, LR
                            WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
       10                CONTINUE
                         WRITE ( NOUT, FMT = 99996 )
                         DO 20 I = 1, LR
                            WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,NR )
       20                CONTINUE
                         WRITE ( NOUT, FMT = 99993 )
                         DO 30 I = 1, LR
                            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,NR )
       40                CONTINUE
                         WRITE ( NOUT, FMT = 99991 )
                         DO 50 I = 1, P
                            WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
       50                CONTINUE
                      END IF
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TG01GD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TG01GD = ',I2)
    99997 FORMAT (/' The reduced state dynamics matrix is ')
    99996 FORMAT (/' The reduced descriptor matrix is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT (' Rank of matrix E   =', I5)
    99993 FORMAT (/' The reduced input/state matrix is ')
    99992 FORMAT (/' The reduced state/output matrix is ')
    99991 FORMAT (/' The transformed feedthrough matrix 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

    TG01GD EXAMPLE PROGRAM DATA
      4    4     2     2     D     0.0
        -1     0     0     3
         0     0     1     2
         1     1     0     4
         0     0     0     0
         1     2     0     0
         0     1     0     1
         3     9     6     3
         0     0     2     0
         1     0
         0     0
         0     1
         1     1
        -1     0     1     0
         0     1    -1     1
         1     0
         1     1

Program Results

     TG01GD EXAMPLE PROGRAM RESULTS

     Rank of matrix E   =    3

     The reduced state dynamics matrix is
       2.5102  -3.8550 -11.4533
      -0.0697   0.0212   0.7015
       0.3798  -0.1156  -3.8250

     The reduced descriptor matrix is
      10.1587   5.8230   1.3021
       0.0000  -2.4684  -0.1896
       0.0000   0.0000   1.0338

     The reduced input/state matrix is
       7.7100   1.6714
       0.7678   1.1070
       2.5428   0.6935

     The reduced state/output matrix is
       0.5477  -2.5000  -6.2610
      -1.0954   1.0000  -0.8944

     The transformed feedthrough matrix is
       4.0000   1.0000
       1.0000   1.0000

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