## TG01QD

### Three-domain spectral splitting of a subpencil of a descriptor system

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

Purpose

      To compute orthogonal transformation matrices Q and Z which
      reduce the regular pole pencil A-lambda*E of the descriptor system
      (A-lambda*E,B,C) to the generalized real Schur form with ordered
      generalized eigenvalues. The pair (A,E) is reduced to the form

                 ( A1  *   *  )             ( E1  *   *  )
        Q'*A*Z = ( 0   A2  *  ) ,  Q'*E*Z = ( 0   E2  *  ) ,         (1)
                 ( 0   0   A3 )             ( 0   0   E2 )

      where the subpencils Ak-lambda*Ek, for k = 1, 2, 3, contain the
      generalized eigenvalues which belong to certain domains of
      interest.

Specification

          SUBROUTINE TG01QD( DICO, STDOM, JOBFI, N, M, P, ALPHA, A, LDA,
         $                   E, LDE, B, LDB, C, LDC, N1, N2, N3, ND, NIBLCK,
         $                   IBLCK, Q, LDQ, Z, LDZ, ALPHAR, ALPHAI, BETA,
         $                   TOL, IWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER        DICO, JOBFI, STDOM
          INTEGER          INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M, N,
         $                 N1, N2, N3, ND, NIBLCK, P
          DOUBLE PRECISION ALPHA, TOL
    C     .. Array Arguments ..
          INTEGER          IBLCK( * ), IWORK(*)
          DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
         $                 BETA(*), C(LDC,*), DWORK(*), E(LDE,*), Q(LDQ,*),
         $                 Z(LDZ,*)

Arguments

Mode Parameters

      DICO    CHARACTER*1
              Specifies the type of the descriptor system as follows:
              = 'C':  continuous-time system;
              = 'D':  discrete-time system.

      STDOM   CHARACTER*1
              Specifies the type of the domain of interest for the
              generalized eigenvalues, as follows:
              = 'S':  stability type domain (i.e., left part of complex
                      plane or inside of a circle);
              = 'U':  instability type domain (i.e., right part of complex
                      plane or outside of a circle);
              = 'N':  whole complex domain, excepting infinity.

      JOBFI   CHARACTER*1
              Specifies the type of generalized eigenvalues in the
              leading diagonal block(s) as follows:
              = 'F':  finite generalized eigenvalues are in the
                      leading diagonal blocks (Af,Ef), and the resulting
                      transformed pair has the form

                               ( Af  *  )             ( Ef  *  )
                      Q'*A*Z = (        ) ,  Q'*E*Z = (        ) ;
                               ( 0   Ai )             ( 0   Ei )

              = 'I':  infinite generalized eigenvalues are in the
                      leading diagonal blocks (Ai,Ei), and the resulting
                      transformed pair has the form

                               ( Ai  *  )             ( Ei  *  )
                      Q'*A*Z = (        ) ,  Q'*E*Z = (        ) .
                               ( 0   Af )             ( 0   Ef )

Input/Output Parameters

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

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

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

      ALPHA   (input) DOUBLE PRECISION
              The boundary of the domain of interest for the finite
              generalized eigenvalues of the pair (A,E). For a
              continuous-time system (DICO = 'C'), ALPHA is the boundary
              value for the real parts of the generalized eigenvalues,
              while for a discrete-time system (DICO = 'D'), ALPHA >= 0
              represents the boundary value for the moduli of the
              generalized eigenvalues.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the state dynamics matrix A.
              On exit, the leading N-by-N part of this array contains
              the matrix Q'*A*Z in real Schur form, with the elements
              below the first subdiagonal set to zero.
              If JOBFI = 'I', the N1-by-N1 pair (A1,E1) contains the
              infinite spectrum, the N2-by-N2 pair (A2,E2) contains the
              finite spectrum in the domain of interest, and the
              N3-by-N3 pair (A3,E3) contains the finite spectrum ouside
              of the domain of interest.
              If JOBFI = 'F', the N1-by-N1 pair (A1,E1) contains the
              finite spectrum in the domain of interest, the N2-by-N2
              pair (A2,E2) contains the finite spectrum ouside of the
              domain of interest, and the N3-by-N3 pair (A3,E3) contains
              the infinite spectrum.
              Ai has a block structure as in (2), where A0,0 is ND-by-ND
              and Ai,i is IBLCK(i)-by-IBLCK(i), for i = 1, ..., NIBLCK.
              The domain of interest for the pair (Af,Ef), containing
              the finite generalized eigenvalues, is defined by the
              parameters ALPHA, DICO and STDOM as follows:
                For DICO = 'C':
                   Real(eig(Af,Ef)) < ALPHA if STDOM = 'S';
                   Real(eig(Af,Ef)) > ALPHA if STDOM = 'U'.
                For DICO = 'D':
                   Abs(eig(Af,Ef))  < ALPHA if STDOM = 'S';
                   Abs(eig(Af,Ef))  > ALPHA if STDOM = 'U'.

      LDA     INTEGER
              The leading dimension of the 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 descriptor matrix E.
              On exit, the leading N-by-N part of this array contains
              the matrix Q'*E*Z in upper triangular form, with the
              elements below the diagonal set to zero. Its structure
              corresponds to the block structure of the matrix Q'*A*Z
              (see description of A).

      LDE     INTEGER
              The leading dimension of the 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 input matrix B.
              On exit, the leading N-by-M part of this array contains
              the transformed input matrix Q'*B.

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

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

      N1      (output) INTEGER
      N2      (output) INTEGER
      N3      (output) INTEGER
              The number of the generalized eigenvalues of the pairs
              (A1,E1), (A2,E2) and (A3,E3), respectively.

      ND      (output) INTEGER.
              The number of non-dynamic infinite eigenvalues of the
              pair (A,E). Note: N-ND is the rank of the matrix E.

      NIBLCK  (output) INTEGER
              If ND > 0, the number of infinite blocks minus one.
              If ND = 0, then NIBLCK = 0.

      IBLCK   (output) INTEGER array, dimension (N)
              IBLCK(i) contains the dimension of the i-th block in the
              staircase form (2), where i = 1,2,...,NIBLCK.

      Q       (output) DOUBLE PRECISION array, dimension (LDQ,N)
              The leading N-by-N part of this array contains the
              orthogonal matrix Q, where Q' is the product of orthogonal
              transformations applied to A, E, and B on the left.

      LDQ     INTEGER
              The leading dimension of the array Q.  LDQ >= MAX(1,N).

      Z       (output) DOUBLE PRECISION array, dimension (LDZ,N)
              The leading N-by-N part of this array contains the
              orthogonal matrix Z, which is the product of orthogonal
              transformations applied to A, E, and C on the right.

      LDZ     INTEGER
              The leading dimension of the array Z.  LDZ >= MAX(1,N).

      ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
      ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
      BETA    (output) DOUBLE PRECISION array, dimension (N)
              On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j = 1, ..., N,
              are the generalized eigenvalues.
              ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j = 1, ..., N, are
              the diagonals of the complex Schur form (S,T) that would
              result if the 2-by-2 diagonal blocks of the real Schur
              form of (A,E) were further reduced to triangular form
              using 2-by-2 complex unitary transformations.
              If ALPHAI(j) is zero, then the j-th eigenvalue is real;
              if positive, then the j-th and (j+1)-st eigenvalues are a
              complex conjugate pair, with ALPHAI(j+1) negative.

Tolerances

      TOL     DOUBLE PRECISION
              A tolerance used in rank decisions to determine the
              effective rank, which is defined as the order of the
              largest leading (or trailing) triangular submatrix in the
              QR factorization with column pivoting whose estimated
              condition number is less than 1/TOL.  If the user sets
              TOL <= 0, then an implicitly computed, default tolerance
              TOLDEF = N**2*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, and if N = 0,
              LDWORK >= 4*N,    if STDOM = 'N';
              LDWORK >= 4*N+16, if STDOM = 'S' or 'U'.
              For optimum performance LDWORK should be larger.

              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.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the pencil A-lambda*E is not regular;
              = 2:  the QZ algorithm failed to compute all generalized
                    eigenvalues of the pair (A,E);
              = 3:  a failure occured during the ordering of the
                    generalized real Schur form of the pair (A,E).

Method

      The separation of the finite and infinite parts is based on the
      reduction algorithm of [1].
      If JOBFI = 'F', the matrices of the pair (Ai,Ei), containing the
      infinite generalized eigenvalues, have the form

            ( A0,0  A0,k ... A0,1 )         ( 0  E0,k ... E0,1 )
       Ai = (  0    Ak,k ... Ak,1 ) ,  Ei = ( 0   0   ... Ek,1 ) ;   (2)
            (  :     :    .    :  )         ( :   :    .    :  )
            (  0     0   ... A1,1 )         ( 0   0   ...   0  )

      if JOBFI = 'I', the matrices Ai and Ei have the form

            ( A1,1 ... A1,k  A1,0 )         ( 0 ... E1,k  E1,0 )
       Ai = (  :    .    :    :   ) ,  Ei = ( :  .    :    :   ) ,   (3)
            (  :   ... Ak,k  Ak,0 )         ( : ...   0   Ek,0 )
            (  0   ...   0   A0,0 )         ( 0 ...   0     0  )

      where Ai,i , for i = 0, 1, ..., k, are nonsingular upper
      triangular matrices, and A0,0 corresponds to the non-dynamic
      infinite modes of the system.

References

      [1] Misra, P., Van Dooren, P., and Varga, A.
          Computation of structural invariants of generalized
          state-space systems.
          Automatica, 30, pp. 1921-1936, 1994.

Numerical Aspects

                                      3
      The algorithm requires about 25N  floating point operations.

Further Comments

      The number of infinite poles is computed as

                    NIBLCK
         NINFP =     Sum  IBLCK(i) = N - ND - NF,
                     i=1

      where NF is the number of finite generalized eigenvalues.
      The multiplicities of infinite poles can be computed as follows:
      there are IBLCK(k)-IBLCK(k+1) infinite poles of multiplicity k,
      for k = 1, ..., NIBLCK, where IBLCK(NIBLCK+1) = 0.
      Note that each infinite pole of multiplicity k corresponds to an
      infinite eigenvalue of multiplicity k+1.

Example

Program Text

    *     TG01QD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          DOUBLE PRECISION ZERO
          PARAMETER        ( ZERO = 0.0D0 )
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          LDA, LDB, LDC, LDE, LDQ, LDZ
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDE = NMAX, LDQ = NMAX, LDZ = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 4*NMAX+16 )
    *     .. Local Scalars ..
          CHARACTER*1      DICO, JOBFI, STDOM
          INTEGER          I, INFO, J, M, N, N1, N2, N3, ND, NIBLCK, P
          DOUBLE PRECISION ALPHA, TOL
    *     .. Local Arrays ..
          INTEGER          IBLCK(NMAX), IWORK(NMAX)
          DOUBLE PRECISION A(LDA,NMAX),  ALPHAI(NMAX), ALPHAR(NMAX),
         $                 B(LDB,MMAX),    BETA(NMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
         $                 Z(LDZ,NMAX)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         TG01QD
    *     .. Intrinsic Functions ..
          INTRINSIC        DCMPLX
    *     .. 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, DICO, STDOM, JOBFI, ALPHA, TOL
          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 reduced descriptor system
    *              (A-lambda E,B,C).
                   CALL TG01QD( DICO, STDOM, JOBFI, N, M, P, ALPHA, A, LDA,
         $                      E, LDE, B, LDB, C, LDC, N1, N2, N3, ND,
         $                      NIBLCK, IBLCK, Q, LDQ, Z, LDZ, ALPHAR,
         $                      ALPHAI, BETA, TOL, IWORK, DWORK, LDWORK,
         $                      INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99994 ) N1, N2, N3
                      WRITE ( NOUT, FMT = 99983 ) ND
                      WRITE ( NOUT, FMT = 99989 ) NIBLCK + 1
                      IF ( NIBLCK.GT.0 ) THEN
                         WRITE ( NOUT, FMT = 99985 )
         $                     ( IBLCK(I), I = 1, NIBLCK )
                      END IF
                      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
                      WRITE ( NOUT, FMT = 99985 )
                      DO 70 I = 1, N
                         IF ( BETA(I).EQ.ZERO .OR. ALPHAI(I).EQ.ZERO ) THEN
                            WRITE ( NOUT, FMT = 99984 )
         $                     ALPHAR(I)/BETA(I)
                         ELSE
                            WRITE ( NOUT, FMT = 99984 )
         $                     DCMPLX( ALPHAR(I), ALPHAI(I) )/BETA(I)
                         END IF
       70             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TG01QD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TG01QD = ',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 (' Number of eigenvalues of the three blocks =', 3I5)
    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 ( ' Number of infinite blocks = ',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 finite generalized eigenvalues are '/
         $         ' real  part     imag  part ')
    99984 FORMAT (1X,F9.4,SP,F9.4,S,'i ')
    99983 FORMAT ( ' Number of non-dynamic infinite eigenvalues = ',I5)
          END

Program Data

    TG01QD EXAMPLE PROGRAM DATA
      4     2     2     C     S     F     -1.E-7     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

Program Results

     TG01QD EXAMPLE PROGRAM RESULTS

     Number of eigenvalues of the three blocks =    1    2    1
     Number of non-dynamic infinite eigenvalues =     1
     Number of infinite blocks =     1

     The transformed state dynamics matrix Q'*A*Z is
       1.6311   2.1641  -0.5848  -3.6517
       0.0000  -0.4550   1.0935   1.6851
       0.0000   0.0000   0.0000   1.5770
       0.0000   0.0000   0.0000   2.2913

     The transformed descriptor matrix Q'*E*Z is
      -0.4484   9.6340   5.1601  -2.6183
       0.0000  -3.3099  -1.6050  -0.2756
       0.0000   0.0000   2.3524  -0.6008
       0.0000   0.0000   0.0000   0.0000

     The transformed input/state matrix Q'*B is
       0.0232  -0.9413
       0.7251   0.2478
      -0.4336  -0.9538
       1.1339   0.3780

     The transformed state/output matrix C*Z is
       0.8621   0.3754  -0.8847   0.5774
       0.1511  -1.1192   1.1795   0.5774

     The left transformation matrix Q is
       0.0232   0.7251   0.3902   0.5669
      -0.3369  -0.6425   0.3902   0.5669
      -0.9413   0.2478  -0.1301  -0.1890
       0.0000   0.0000  -0.8238   0.5669

     The right transformation matrix Z is
      -0.8621  -0.3754  -0.0843  -0.3299
       0.4258  -0.9008  -0.0211  -0.0825
       0.0000   0.0000  -0.9689   0.2474
      -0.2748  -0.2184   0.2317   0.9073

     The finite generalized eigenvalues are
     real  part     imag  part
       -3.6375
        0.1375
        0.0000
      Infinity

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