## TG01HD

### Orthogonal reduction of a descriptor system to the controllability staircase form

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

Purpose

      To compute orthogonal transformation matrices Q and Z which
      reduce the N-th order descriptor system (A-lambda*E,B,C)
      to the form

                 ( Ac  *  )             ( Ec  *  )           ( Bc )
        Q'*A*Z = (        ) ,  Q'*E*Z = (        ) ,  Q'*B = (    ) ,
                 ( 0  Anc )             ( 0  Enc )           ( 0  )

           C*Z = ( Cc Cnc ) ,

      where the NCONT-th order descriptor system (Ac-lambda*Ec,Bc,Cc)
      is a finite and/or infinite controllable. The pencil
      Anc - lambda*Enc is regular of order N-NCONT and contains the
      uncontrollable finite and/or infinite eigenvalues of the pencil
      A-lambda*E.

      For JOBCON = 'C' or 'I', the pencil ( Bc Ec-lambda*Ac ) has full
      row rank NCONT for all finite lambda and is in a staircase form
      with
                      _      _          _        _
                    ( E1,0   E1,1  ...  E1,k-1   E1,k  )
                    (        _          _        _     )
        ( Bc Ec ) = (  0     E2,1  ...  E2,k-1   E2,k  ) ,  (1)
                    (              ...  _        _     )
                    (  0       0   ...  Ek,k-1   Ek,k  )

                      _          _        _
                    ( A1,1  ...  A1,k-1   A1,k  )
                    (            _        _     )
          Ac      = (   0   ...  A2,k-1   A2,k  ) ,         (2)
                    (       ...           _     )
                    (   0   ...    0      Ak,k  )
            _
      where Ei,i-1 is an rtau(i)-by-rtau(i-1) full row rank matrix
                             _
      (with rtau(0) = M) and Ai,i is an rtau(i)-by-rtau(i)
      upper triangular matrix.

      For JOBCON = 'F', the pencil ( Bc Ac-lambda*Ec ) has full
      row rank NCONT for all finite lambda and is in a staircase form
      with
                      _     _          _        _
                    ( A1,0  A1,1  ...  A1,k-1   A1,k  )
                    (       _          _        _     )
        ( Bc Ac ) = (  0    A2,1  ...  A2,k-1   A2,k  ) ,   (3)
                    (             ...  _        _     )
                    (  0      0   ...  Ak,k-1   Ak,k  )

                      _          _        _
                    ( E1,1  ...  E1,k-1   E1,k  )
                    (            _        _     )
          Ec      = (   0   ...  E2,k-1   E2,k  ) ,         (4)
                    (       ...           _     )
                    (   0   ...    0      Ek,k  )
            _
      where Ai,i-1 is an rtau(i)-by-rtau(i-1) full row rank matrix
                             _
      (with rtau(0) = M) and Ei,i is an rtau(i)-by-rtau(i)
      upper triangular matrix.

      For JOBCON = 'C', the (N-NCONT)-by-(N-NCONT) regular pencil
      Anc - lambda*Enc has the form

                          ( Ainc - lambda*Einc         *          )
       Anc - lambda*Enc = (                                       ) ,
                          (        0           Afnc - lambda*Efnc )

      where:
        1) the NIUCON-by-NIUCON regular pencil Ainc - lambda*Einc,
           with Ainc upper triangular and nonsingular, contains the
           uncontrollable infinite eigenvalues of A - lambda*E;
        2) the (N-NCONT-NIUCON)-by-(N-NCONT-NIUCON) regular pencil
           Afnc - lambda*Efnc, with Efnc upper triangular and
           nonsingular, contains the uncontrollable finite
           eigenvalues of A - lambda*E.

      Note: The significance of the two diagonal blocks can be
            interchanged by calling the routine with the
            arguments A and E interchanged. In this case,
            Ainc - lambda*Einc contains the uncontrollable zero
            eigenvalues of A - lambda*E, while Afnc - lambda*Efnc
            contains the uncontrollable nonzero finite and infinite
            eigenvalues of A - lambda*E.

      For JOBCON = 'F', the pencil Anc - lambda*Enc has the form

         Anc - lambda*Enc = Afnc - lambda*Efnc ,

      where the regular pencil Afnc - lambda*Efnc, with Efnc
      upper triangular and nonsingular, contains the uncontrollable
      finite eigenvalues of A - lambda*E.

      For JOBCON = 'I', the pencil Anc - lambda*Enc has the form

         Anc - lambda*Enc = Ainc - lambda*Einc ,

      where the regular pencil Ainc - lambda*Einc, with Ainc
      upper triangular and nonsingular, contains the uncontrollable
      nonzero finite and infinite eigenvalues of A - lambda*E.

      The left and/or right orthogonal transformations Q and Z
      performed to reduce the system matrices can be optionally
      accumulated.

      The reduced order descriptor system (Ac-lambda*Ec,Bc,Cc) has
      the same transfer-function matrix as the original system
      (A-lambda*E,B,C).

Specification

          SUBROUTINE TG01HD( JOBCON, COMPQ, COMPZ, N, M, P, A, LDA, E, LDE,
         $                   B, LDB, C, LDC, Q, LDQ, Z, LDZ, NCONT, NIUCON,
         $                   NRBLCK, RTAU, TOL, IWORK, DWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ, COMPZ, JOBCON
          INTEGER            INFO, LDA, LDB, LDC, LDE, LDQ, LDZ,
         $                   M, N, NCONT, NIUCON, NRBLCK, P
          DOUBLE PRECISION   TOL
    C     .. Array Arguments ..
          INTEGER            IWORK( * ), RTAU( * )
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, *  ),
         $                   DWORK( * ),  E( LDE, * ), Q( LDQ, * ),
         $                   Z( LDZ, * )

Arguments

Mode Parameters

      JOBCON  CHARACTER*1
              = 'C':  separate both finite and infinite uncontrollable
                      eigenvalues;
              = 'F':  separate only finite uncontrollable eigenvalues:
              = 'I':  separate only nonzero finite and infinite
                      uncontrollable eigenvalues.

      COMPQ   CHARACTER*1
              = 'N':  do not compute Q;
              = 'I':  Q is initialized to the unit matrix, and the
                      orthogonal matrix Q is returned;
              = 'U':  Q must contain an orthogonal matrix Q1 on entry,
                      and the product Q1*Q is returned.

      COMPZ   CHARACTER*1
              = 'N':  do not compute Z;
              = 'I':  Z is initialized to the unit matrix, and the
                      orthogonal matrix Z is returned;
              = 'U':  Z must contain an orthogonal matrix Z1 on entry,
                      and the product Z1*Z is returned.

Input/Output Parameters

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

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

      P       (input) INTEGER
              The dimension of descriptor system output vector; also the
              number of rows of matrix C.  P >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the N-by-N state matrix A.
              On exit, the leading N-by-N part of this array contains
              the transformed state matrix Q'*A*Z,

                                 ( Ac   *  )
                        Q'*A*Z = (         ) ,
                                 ( 0   Anc )

              where Ac is NCONT-by-NCONT and Anc is
              (N-NCONT)-by-(N-NCONT).
              If JOBCON = 'F', the matrix ( Bc Ac ) is in the
              controllability staircase form (3).
              If JOBCON = 'C' or 'I', the submatrix Ac is upper
              triangular.
              If JOBCON = 'C', the Anc matrix has the form

                              ( Ainc   *  )
                        Anc = (           ) ,
                              (  0   Afnc )

              where the NIUCON-by-NIUCON matrix Ainc is nonsingular and
              upper triangular.
              If JOBCON = 'I', Anc is nonsingular and upper triangular.

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

      E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
              On entry, the leading N-by-N part of this array must
              contain the N-by-N descriptor matrix E.
              On exit, the leading N-by-N part of this array contains
              the transformed descriptor matrix Q'*E*Z,

                                 ( Ec   *  )
                        Q'*E*Z = (         ) ,
                                 ( 0   Enc )

              where Ec is NCONT-by-NCONT and Enc is
              (N-NCONT)-by-(N-NCONT).
              If JOBCON = 'C' or 'I', the matrix ( Bc Ec ) is in the
              controllability staircase form (1).
              If JOBCON = 'F', the submatrix Ec is upper triangular.
              If JOBCON = 'C', the Enc matrix has the form

                              ( Einc   *  )
                        Enc = (           ) ,
                              (  0   Efnc )

              where the NIUCON-by-NIUCON matrix Einc is nilpotent
              and the (N-NCONT-NIUCON)-by-(N-NCONT-NIUCON) matrix Efnc
              is nonsingular and upper triangular.
              If JOBCON = 'F', Enc is nonsingular and upper triangular.

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

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
              On entry, the leading N-by-M part of this array must
              contain the N-by-M input matrix B.
              On exit, the leading N-by-M part of this array contains
              the transformed input matrix

                               ( Bc )
                        Q'*B = (    ) ,
                               ( 0  )

               where Bc is NCONT-by-M.
               For JOBCON = 'C' or 'I', the matrix ( Bc Ec ) is in the
               controllability staircase form (1).
               For JOBCON = 'F', the matrix ( Bc Ac ) is in the
               controllability staircase form (3).

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

      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, the leading P-by-N part of this array contains
              the transformed matrix C*Z.

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

      Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
              If COMPQ = 'N': Q is not referenced.
              If COMPQ = 'I': on entry, Q need not be set;
                              on exit, the leading N-by-N part of this
                              array contains the orthogonal matrix Q,
                              where Q' is the product of transformations
                              which are applied to A, E, and B on
                              the left.
              If COMPQ = 'U': on entry, the leading N-by-N part of this
                              array must contain an orthogonal matrix
                              Qc;
                              on exit, the leading N-by-N part of this
                              array contains the orthogonal matrix
                              Qc*Q.

      LDQ     INTEGER
              The leading dimension of array Q.
              LDQ >= 1,        if COMPQ = 'N';
              LDQ >= MAX(1,N), if COMPQ = 'U' or 'I'.

      Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
              If COMPZ = 'N': Z is not referenced.
              If COMPZ = 'I': on entry, Z need not be set;
                              on exit, the leading N-by-N part of this
                              array contains the orthogonal matrix Z,
                              which is the product of transformations
                              applied to A, E, and C on the right.
              If COMPZ = 'U': on entry, the leading N-by-N part of this
                              array must contain an orthogonal matrix
                              Zc;
                              on exit, the leading N-by-N part of this
                              array contains the orthogonal matrix
                              Zc*Z.

      LDZ     INTEGER
              The leading dimension of array Z.
              LDZ >= 1,        if COMPZ = 'N';
              LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.

      NCONT   (output) INTEGER
              The order of the reduced matrices Ac and Ec, and the
              number of rows of reduced matrix Bc; also the order of
              the controllable part of the pair (A-lambda*E,B).

      NIUCON  (output) INTEGER
              For JOBCON = 'C', the order of the reduced matrices
              Ainc and Einc; also the number of uncontrollable
              infinite eigenvalues of the pencil A - lambda*E.
              For JOBCON = 'F' or 'I', NIUCON has no significance
              and is set to zero.

      NRBLCK  (output) INTEGER
              For JOBCON = 'C' or 'I', the number k, of full row rank
                     _
              blocks Ei,i in the staircase form of the pencil
              (Bc Ec-lambda*Ac) (see (1) and (2)).
              For JOBCON = 'F', the number k, of full row rank blocks
              _
              Ai,i in the staircase form of the pencil (Bc Ac-lambda*Ec)
              (see (3) and (4)).

      RTAU    (output) INTEGER array, dimension (N)
              RTAU(i), for i = 1, ..., NRBLCK, is the row dimension of
                                      _         _
              the full row rank block Ei,i-1 or Ai,i-1 in the staircase
              form (1) or (3) for JOBCON = 'C' or 'I', or
              for JOBCON = 'F', respectively.

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance to be used in rank determinations when
              transforming (A-lambda*E, B). 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 = N*N*EPS,  is used instead, where EPS
              is the machine precision (see LAPACK Library routine
              DLAMCH).  TOL < 1.

Workspace

      IWORK   INTEGER array, dimension (M)

      DWORK   DOUBLE PRECISION array, dimension (MAX(N,2*M))

Error Indicator

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

Method

      The subroutine is based on the reduction algorithms of [1].

References

      [1] A. Varga
          Computation of Irreducible Generalized State-Space
          Realizations.
          Kybernetika, vol. 26, pp. 89-106, 1990.

Numerical Aspects

      The algorithm is numerically backward stable and requires
      0( N**3 )  floating point operations.

Further Comments

      If the system matrices A, E and B are badly scaled, it is
      generally recommendable to scale them with the SLICOT routine
      TG01AD, before calling TG01HD.

Example

Program Text

    *     TG01HD 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, LDE, LDQ, LDZ
          PARAMETER        ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
         $                   LDE = LMAX, LDQ = LMAX, LDZ = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX( 1, NMAX, 2*MMAX ) )
    *     .. Local Scalars ..
          CHARACTER*1      COMPQ, COMPZ, JOBCO
          INTEGER          I, INFO, J, M, N, NCONT, NIUCON, NRBLCK, P
          DOUBLE PRECISION TOL
    *     .. Local Arrays ..
          INTEGER          IWORK(MMAX), RTAU(NMAX)
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX),
         $                 Z(LDZ,NMAX)
    *     .. External Subroutines ..
          EXTERNAL         TG01HD
    *     .. 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 = * ) N, M, P, TOL, JOBCO
          COMPQ = 'I'
          COMPZ = 'I'
          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,N )
             READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
             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,N )
                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 )
    *              Find the transformed descriptor system (A-lambda E,B,C).
                   CALL TG01HD( JOBCO, COMPQ, COMPZ, N, M, P, A, LDA,
         $                      E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ,
         $                      NCONT, NIUCON, NRBLCK, RTAU, TOL, IWORK,
         $                      DWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99994 ) NCONT, NIUCON
                      WRITE ( NOUT, FMT = 99985 )
                      WRITE ( NOUT, FMT = 99984 ) ( RTAU(I), I = 1,NRBLCK )
                      WRITE ( NOUT, FMT = 99997 )
                      DO 10 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
       10             CONTINUE
                      WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
       20             CONTINUE
                      WRITE ( NOUT, FMT = 99993 )
                      DO 30 I = 1, N
                         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 )
                      DO 50 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
       50             CONTINUE
                      WRITE ( NOUT, FMT = 99990 )
                      DO 60 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
       60             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TG01HD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TG01HD = ',I2)
    99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
    99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT (' Dimension of controllable part   =', I5/
         $        ' Number of uncontrollable infinite eigenvalues =', I5)
    99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
    99992 FORMAT (/' The transformed state/output matrix C*Z is ')
    99991 FORMAT (/' The left transformation matrix Q is ')
    99990 FORMAT (/' The right transformation matrix Z 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)
    99985 FORMAT (/' The staircase form row dimensions are ' )
    99984 FORMAT (10I5)
          END

Program Data

    TG01HD EXAMPLE PROGRAM DATA
      7    3     2     0.0    C
         2     0     2     0    -1     3     1
         0     1     0     0     1     0     0
         0     0     0     1     0     0     1
         0     0     2     0    -1     3     1
         0     0     0     1     0     0     1
         0     1     0     0     1     0     0
         0     0     0     1     0     0     1
         0     0     1     0     0     0     0
         0     0     0     0     0     1     0
         0     0     0     0     0     0     1
         0     0     0     0     0     0     1
         0     0     0     1     0     0     0
         0     0     1     0    -1     0     0
         1     3     0     2     0     0     0
         2     1     0
         0     0     0
         0     0     0
         0     0     0
         0     0     0
         0     0     0
         1     2     3
         1     0     0     1     0     0     1
         0    -1     1     0    -1     1     0

Program Results

     TG01HD EXAMPLE PROGRAM RESULTS

     Dimension of controllable part   =    3
     Number of uncontrollable infinite eigenvalues =    1

     The staircase form row dimensions are
        2    1

     The transformed state dynamics matrix Q'*A*Z is
       0.0000   0.0000   0.0000   0.0000  -1.2627   0.4334   0.4666
       0.0000   2.0000   0.0000  -3.7417  -0.8520   0.2924  -0.4342
       0.0000   0.0000   1.7862   0.3780  -0.2651  -0.7723   0.0000
       0.0000   0.0000   0.0000   3.7417   0.8520  -0.2924   0.4342
       0.0000   0.0000   0.0000   0.0000  -1.5540   0.5334   0.5742
       0.0000   0.0000   0.0000   0.0000  -0.6533   0.2242   0.2414
       0.0000   0.0000   0.0000   0.0000  -0.5892   0.2022   0.2177

     The transformed descriptor matrix Q'*E*Z is
      -1.8325   1.0000   2.3752   0.0000  -0.8214   0.2819   1.8016
       0.4887   0.0000   0.3770  -0.5345   0.1874   0.5461   0.0000
      -0.1728   0.0000  -0.1333  -1.1339   0.1325   0.3861   0.0000
       0.0000   0.0000   0.0000   0.0000   0.8520  -0.2924   0.4342
       0.0000   0.0000   0.0000   0.0000  -1.0260  -0.1496   0.0000
       0.0000   0.0000   0.0000   0.0000   0.0000   1.1937   0.0000
       0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.0000

     The transformed input/state matrix Q'*B is
       1.0000   2.0000   3.0000
       2.0000   1.0000   0.0000
       0.0000   0.0000   0.0000
       0.0000   0.0000   0.0000
       0.0000   0.0000   0.0000
       0.0000   0.0000   0.0000
       0.0000   0.0000   0.0000

     The transformed state/output matrix C*Z is
       0.0000   1.0000   0.0000   0.0000  -1.2627   0.4334   0.4666
       0.3665   0.0000  -0.9803  -1.6036   0.1874   0.5461   0.0000

     The left transformation matrix Q is
       0.0000   1.0000   0.0000   0.0000   0.0000   0.0000   0.0000
       0.0000   0.0000   0.7071   0.0000   0.2740  -0.6519   0.0000
       0.0000   0.0000   0.0000   0.0000   0.8304   0.3491  -0.4342
       0.0000   0.0000   0.0000  -1.0000   0.0000   0.0000   0.0000
       0.0000   0.0000   0.0000   0.0000   0.4003   0.1683   0.9008
       0.0000   0.0000   0.7071   0.0000  -0.2740   0.6519   0.0000
       1.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000

     The right transformation matrix Z is
       0.0000   1.0000   0.0000   0.0000   0.0000   0.0000   0.0000
      -0.6108   0.0000   0.7917   0.0000   0.0000   0.0000   0.0000
       0.4887   0.0000   0.3770  -0.5345   0.1874   0.5461   0.0000
       0.0000   0.0000   0.0000   0.0000  -0.4107   0.1410   0.9008
       0.6108   0.0000   0.4713   0.2673  -0.1874  -0.5461   0.0000
      -0.1222   0.0000  -0.0943  -0.8018  -0.1874  -0.5461   0.0000
       0.0000   0.0000   0.0000   0.0000  -0.8520   0.2924  -0.4342

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