## TG01HX

### Orthogonal reduction of a descriptor system to a system with the same transfer-function matrix and with no uncontrollable finite eigenvalues

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

Purpose

      Given the descriptor system (A-lambda*E,B,C) with the system
      matrices A, E and B of the form

             ( A1 X1 )        ( E1 Y1 )        ( B1 )
         A = (       ) ,  E = (       ) ,  B = (    ) ,
             ( 0  X2 )        ( 0  Y2 )        ( 0  )

      where
           - B is an L-by-M matrix, with B1 an N1-by-M  submatrix
           - A is an L-by-N matrix, with A1 an N1-by-N1 submatrix
           - E is an L-by-N matrix, with E1 an N1-by-N1 submatrix
               with LBE nonzero sub-diagonals,
      this routine reduces the pair (A1-lambda*E1,B1) to the form

      Qc'*[B1 A1-lambda*E1]*diag(I,Zc) =

                               ( Bc Ac-lambda*Ec      *         )
                               (                                ) ,
                               ( 0     0         Anc-lambda*Enc )

      where:
      1) the pencil ( Bc Ac-lambda*Ec ) has full row rank NR 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  ) ,  (1)
                          (              ...  _        _     )
                          (  0       0   ...  Ak,k-1   Ak,k  )

                            _          _        _
                          ( E1,1  ...  E1,k-1   E1,k  )
                          (            _        _     )
                Ec      = (   0   ...  E2,k-1   E2,k  ) ,         (2)
                          (       ...           _     )
                          (   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.

       2) the pencil Anc-lambda*Enc is regular of order N1-NR with Enc
          upper triangular; this pencil contains the uncontrollable
          finite eigenvalues of the pencil (A1-lambda*E1).

      The transformations are applied to the whole matrices A, E, B
      and C. The left and/or right orthogonal transformations Qc and Zc
      performed to reduce the pencil can be optionally accumulated
      in the matrices Q and Z, respectively.

      The reduced order descriptor system (Ac-lambda*Ec,Bc,Cc) has no
      uncontrollable finite eigenvalues and has the same
      transfer-function matrix as the original system (A-lambda*E,B,C).

Specification

          SUBROUTINE TG01HX( COMPQ, COMPZ, L, N, M, P, N1, LBE, A, LDA,
         $                   E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NR,
         $                   NRBLCK, RTAU, TOL, IWORK, DWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ, COMPZ
          INTEGER            INFO, L, LBE, LDA, LDB, LDC, LDE, LDQ, LDZ, M,
         $                   N, N1, NR, 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

      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

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

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

      N1      (input) INTEGER
              The order of subsystem (A1-lambda*E1,B1,C1) to be reduced.
              MIN(L,N) >= N1 >= 0.

      LBE     (input) INTEGER
              The number of nonzero sub-diagonals of submatrix E1.
              MAX(0,N1-1) >= LBE >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading L-by-N part of this array must
              contain the L-by-N state matrix A in the partitioned
              form
                       ( A1 X1 )
                   A = (       ) ,
                       ( 0  X2 )

              where A1 is N1-by-N1.
              On exit, the leading L-by-N part of this array contains
              the transformed state matrix,

                                   ( Ac  *   * )
                        Qc'*A*Zc = ( 0  Anc  * ) ,
                                   ( 0   0   * )

              where Ac is NR-by-NR and Anc is (N1-NR)-by-(N1-NR).
              The matrix ( Bc Ac ) is in the controllability
              staircase form (1).

      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 L-by-N descriptor matrix E in the partitioned
              form
                       ( E1 Y1 )
                   E = (       ) ,
                       ( 0  Y2 )

              where E1 is N1-by-N1 matrix with LBE nonzero
              sub-diagonals.
              On exit, the leading L-by-N part of this array contains
              the transformed descriptor matrix

                                   ( Ec  *   * )
                        Qc'*E*Zc = ( 0  Enc  * ) ,
                                   ( 0   0   * )

              where Ec is NR-by-NR and Enc is (N1-NR)-by-(N1-NR).
              Both Ec and Enc are upper triangular and Enc is
              nonsingular.

      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 L-by-M input matrix B in the partitioned
              form
                       ( B1 )
                   B = (    ) ,
                       ( 0  )

              where B1 is N1-by-M.
              On exit, the leading L-by-M part of this array contains
              the transformed input matrix

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

              where Bc is NR-by-M.
              The matrix ( Bc Ac ) is in the controllability
              staircase form (1).

      LDB     INTEGER
              The leading dimension of 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 state/output matrix C.
              On exit, the leading P-by-N part of this array contains
              the transformed matrix C*Zc.

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

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

      LDQ     INTEGER
              The leading dimension of array Q.
              LDQ >= 1,        if COMPQ = 'N';
              LDQ >= MAX(1,L), 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 Zc,
                              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 Z;
                              on exit, the leading N-by-N part of this
                              array contains the orthogonal matrix
                              Z*Zc.

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

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

      NRBLCK  (output) INTEGER                      _
              The number k, of full row rank blocks Ai,i in the
              staircase form of the pencil (Bc Ac-lambda*Ec) (see (1)
              and (2)).

      RTAU    (output) INTEGER array, dimension (N1)
              RTAU(i), for i = 1, ..., NRBLCK, is the row dimension of
                                      _
              the full row rank block Ai,i-1 in the staircase form (1).

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 = L*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,L,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 algorithm of [1].

References

      [1] Varga, A.
          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*N1**2 )  floating point operations.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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