## TG01MD

### Finite-infinite generalized real Schur form decomposition 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 form (if JOB = 'F')

                 ( Af  *  )             ( Ef  *  )
        Q'*A*Z = (        ) ,  Q'*E*Z = (        ) ,                 (1)
                 ( 0   Ai )             ( 0   Ei )

      or to the form (if JOB = 'I')

                 ( Ai  *  )             ( Ei  *  )
        Q'*A*Z = (        ) ,  Q'*E*Z = (        ) ,                 (2)
                 ( 0   Af )             ( 0   Ef )

      where the pair (Af,Ef) is in a generalized real Schur form, with
      Ef nonsingular and upper triangular and Af in real Schur form.
      The subpencil Af-lambda*Ef contains the finite eigenvalues.
      The pair (Ai,Ei) is in a generalized real Schur form with
      both Ai and Ei upper triangular. The subpencil Ai-lambda*Ei,
      with Ai nonsingular and Ei nilpotent contains the infinite
      eigenvalues and is in a block staircase form (see METHOD).

Specification

          SUBROUTINE TG01MD( JOB, N, M, P, A, LDA, E, LDE, B, LDB, C, LDC,
         $                   ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, NF, ND,
         $                   NIBLCK, IBLCK, TOL, IWORK, DWORK, LDWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER          JOB
          INTEGER            INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M,
         $                   N, ND, NF, NIBLCK, P
          DOUBLE PRECISION   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

      JOB     CHARACTER*1
              = 'F':  perform the finite-infinite separation;
              = 'I':  perform the infinite-finite separation.

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.

      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 in the form

              ( Af  *  )                    ( Ai  *  )
              (        ) for JOB = 'F', or  (        )  for JOB = 'I',
              ( 0   Ai )                    ( 0   Af )

              where Af is an NF-by-NF matrix in real Schur form, and Ai
              is an (N-NF)-by-(N-NF) nonsingular and upper triangular
              matrix. Ai has a block structure as in (3) or (4), where
              A0,0 is ND-by-ND and Ai,i , for i = 1, ..., NIBLCK, is
              IBLCK(i)-by-IBLCK(i). (See METHOD.)

      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 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 in the form

              ( Ef  *  )                    ( Ei  *  )
              (        ) for JOB = 'F', or  (        )  for JOB = 'I',
              ( 0   Ei )                    ( 0   Ef )

              where Ef is an NF-by-NF nonsingular and upper triangular
              matrix, and Ei is an (N-NF)-by-(N-NF) nilpotent matrix in
              an upper triangular block form as in (3) or (4).

      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 N-by-M 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 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 the array C.  LDC >= MAX(1,P).

      ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
              ALPHAR(1:NF) will be set to the real parts of the diagonal
              elements of Af that would result from reducing A and E to
              the Schur form, and then further reducing both of them to
              triangular form using unitary transformations, subject to
              having the diagonal of E positive real.  Thus, if Af(j,j)
              is in a 1-by-1 block (i.e., Af(j+1,j) = Af(j,j+1) = 0),
              then ALPHAR(j) = Af(j,j). Note that the (real or complex)
              values (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,NF, are
              the finite generalized eigenvalues of the matrix pencil
              A - lambda*E.

      ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
              ALPHAI(1:NF) will be set to the imaginary parts of the
              diagonal elements of Af that would result from reducing A
              and E to Schur form, and then further reducing both of
              them to triangular form using unitary transformations,
              subject to having the diagonal of E positive real. Thus,
              if Af(j,j) is in a 1-by-1 block (see above), then
              ALPHAI(j) = 0. Note that the (real or complex) values
              (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,NF, are the
              finite generalized eigenvalues of the matrix pencil
              A - lambda*E.

      BETA    (output) DOUBLE PRECISION array, dimension (N)
              BETA(1:NF) will be set to the (real) diagonal elements of
              Ef that would result from reducing A and E to Schur form,
              and then further reducing both of them to triangular form
              using unitary transformations, subject to having the
              diagonal of E positive real. Thus, if Af(j,j) is in a
              1-by-1 block (see above), then BETA(j) = Ef(j,j).
              Note that the (real or complex) values
              (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,NF, are the
              finite generalized eigenvalues of the matrix pencil
              A - lambda*E.

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

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

      NF      (output) INTEGER
              The order of the reduced matrices Af and Ef; also, the
              number of finite generalized eigenvalues of the pencil
              A-lambda*E.

      ND      (output) INTEGER
              The number of non-dynamic infinite eigenvalues of the
              matrix 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 (3), where i = 1,2,...,NIBLCK.

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 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 iteration did not converge.

Method

      For the separation of infinite structure, the reduction algorithm
      of [1] is employed.
      If JOB = 'F', the matrices Ai and Ei 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 ) ;   (3)
            (  :     :    .    :  )         ( :   :    .    :  )
            (  0     0   ... A1,1 )         ( 0   0   ...   0  )

      if JOB = 'I' the matrices Ai and Ei have the form

            ( A1,1 ... A1,k  A1,0 )         ( 0 ... E1,k  E1,0 )
       Ai = (  :    .    :    :   ) ,  Ei = ( :  .    :    :   ) ,   (4)
            (  :   ... 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. 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

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

Further Comments

      The number of infinite poles is computed as

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

    *     TG01MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          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 )
    *     .. Local Scalars ..
          CHARACTER*1      JOB
          INTEGER          I, INFO, J, M, N, ND, NF, NIBLCK, P
          DOUBLE PRECISION 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 Subroutines ..
          EXTERNAL         TG01MD
    *     .. 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, JOB, 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 TG01MD( JOB, N, M, P, A, LDA, E, LDE, B, LDB, C,
         $                      LDC, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ,
         $                      NF, ND, NIBLCK, IBLCK, TOL, IWORK, DWORK,
         $                      LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99994 ) NF, 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, NF
                         WRITE ( NOUT, FMT = 99984 )
         $                  DCMPLX( ALPHAR(I), ALPHAI(I) )/BETA(I)
       70             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TG01MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TG01MD = ',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 (' Order of reduced system =', I5/
         $        ' Number of non-dynamic 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 ( ' 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 ')
          END

Program Data

    TG01MD EXAMPLE PROGRAM DATA
      4     2     2     F     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

     TG01MD EXAMPLE PROGRAM RESULTS

     Order of reduced system =    3
     Number of non-dynamic infinite eigenvalues =    1
     Number of infinite blocks =     1

     The transformed state dynamics matrix Q'*A*Z is
       1.2803  -2.3613  -0.9025  -3.9982
       0.0000  -0.5796   0.8504   0.4350
       0.0000   0.0000   0.0000   1.5770
       0.0000   0.0000   0.0000   2.2913

     The transformed descriptor matrix Q'*E*Z is
       9.3142  -4.1463   5.4026  -2.3944
       0.0000   0.1594   0.1212  -1.0948
       0.0000   0.0000   2.3524  -0.6008
       0.0000   0.0000   0.0000   0.0000

     The transformed input/state matrix Q'*B is
      -0.2089  -0.9712
       0.6948  -0.0647
      -0.4336  -0.9538
       1.1339   0.3780

     The transformed state/output matrix C*Z is
      -0.0469  -0.9391  -0.8847   0.5774
      -1.0697   0.3620   1.1795   0.5774

     The left transformation matrix Q is
      -0.2089   0.6948   0.3902   0.5669
      -0.1148  -0.7163   0.3902   0.5669
      -0.9712  -0.0647  -0.1301  -0.1890
       0.0000   0.0000  -0.8238   0.5669

     The right transformation matrix Z is
       0.0469   0.9391  -0.0843  -0.3299
      -0.9962   0.0189  -0.0211  -0.0825
       0.0000  -0.0000  -0.9689   0.2474
      -0.0735   0.3432   0.2317   0.9073

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

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