## TG01LD

### Finite-infinite 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 subpencil Af-lambda*Ef, with Ef nonsingular and upper
      triangular, contains the finite eigenvalues, and the subpencil
      Ai-lambda*Ei, with Ai nonsingular and upper triangular, contains
      the infinite eigenvalues. The subpencil Ai-lambda*Ei is in a
      staircase form (see METHOD). If JOBA = 'H', the submatrix Af
      is further reduced to an upper Hessenberg form.

Specification

          SUBROUTINE TG01LD( JOB, JOBA, COMPQ, COMPZ, N, M, P, A, LDA,
         $                   E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF, ND,
         $                   NIBLCK, IBLCK, TOL, IWORK, DWORK, LDWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ, COMPZ, JOB, JOBA
          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, * ), B( LDB, * ), 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.

      JOBA    CHARACTER*1
              = 'H':  reduce Af further to an upper Hessenberg form;
              = 'N':  keep Af unreduced.

      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 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,

                                 ( Af  *  )                 ( Ai  *  )
                        Q'*A*Z = (        ) ,  or  Q'*A*Z = (        ) ,
                                 ( 0   Ai )                 ( 0   Af )

              depending on JOB, with Af an NF-by-NF matrix, and Ai an
              (N-NF)-by-(N-NF) nonsingular and upper triangular matrix.
              If JOBA = 'H', Af is in an upper Hessenberg form.
              Otherwise, Af is unreduced.
              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).

      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,

                                 ( Ef  *  )                 ( Ei  *  )
                        Q'*E*Z = (        ) ,  or  Q'*E*Z = (        ) ,
                                 ( 0   Ei )                 ( 0   Ef )

              depending on JOB, with Ef an NF-by-NF nonsingular matrix,
              and Ei an (N-NF)-by-(N-NF) nilpotent matrix in an upper
              block triangular 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,K),
              where K = M if JOB = 'F', and K = MAX(M,P) if JOB = 'I'.
              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,K),
              where K = P if JOB = 'F', and K = MAX(M,P) if JOB = 'I'.

      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 Householder
                               transformations 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
                               Q1;
                               on exit, the leading N-by-N part of this
                               array contains the orthogonal matrix
                               Q1*Q.

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

      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 Householder
                               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
                               Z1;
                               on exit, the leading N-by-N part of this
                               array contains the orthogonal matrix
                               Z1*Z.

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

      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
              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) or (4), with 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 >= N + MAX(3*N,M,P).

              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.

Method

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

    *     TG01LD 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 = NMAX+MAX( 3*NMAX, MMAX, PMAX ) )
    *     .. Local Scalars ..
          CHARACTER*1      COMPQ, COMPZ, JOB, JOBA
          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),   B(LDB,MMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
         $                 Z(LDZ,NMAX)
    *     .. External Subroutines ..
          EXTERNAL         TG01LD
    *     .. 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, JOB, JOBA, TOL
          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 reduced descriptor system
    *              (A-lambda E,B,C).
                   CALL TG01LD( JOB, JOBA, COMPQ, COMPZ, N, M, P, A, LDA,
         $                      E, LDE, B, LDB, C, LDC, 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
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TG01LD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TG01LD = ',I2)
    99997 FORMAT (/' The reduced state dynamics matrix Q''*A*Z is ')
    99996 FORMAT (/' The reduced 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 reduced input/state matrix Q''*B is ')
    99992 FORMAT (/' The reduced 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 ( ' Dimension of the blocks'/20I5)
          END

Program Data

    TG01LD EXAMPLE PROGRAM DATA
      4     2     2     F     N     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

     TG01LD EXAMPLE PROGRAM RESULTS

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

     The reduced state dynamics matrix Q'*A*Z is
       2.4497  -1.3995   0.2397  -4.0023
      -0.0680  -0.0030   0.1739  -1.6225
       0.3707   0.0161  -0.9482   0.1049
       0.0000   0.0000   0.0000   2.2913

     The reduced descriptor matrix Q'*E*Z is
       9.9139   4.7725  -3.4725  -2.3836
       0.0000  -1.2024   2.0137   0.7926
       0.0000   0.0000   0.2929  -0.9914
       0.0000   0.0000   0.0000   0.0000

     The reduced input/state matrix Q'*B is
      -0.2157  -0.9705
       0.3015   0.9516
       0.7595   0.0991
       1.1339   0.3780

     The reduced state/output matrix C*Z is
       0.5345  -1.1134   0.3758   0.5774
      -1.0690   0.2784  -1.2026   0.5774

     The left transformation matrix Q is
      -0.2157  -0.5088   0.6109   0.5669
      -0.1078  -0.2544  -0.7760   0.5669
      -0.9705   0.1413  -0.0495  -0.1890
       0.0000   0.8102   0.1486   0.5669

     The right transformation matrix Z is
      -0.5345   0.6263   0.4617  -0.3299
      -0.8018  -0.5219  -0.2792  -0.0825
       0.0000  -0.4871   0.8375   0.2474
      -0.2673   0.3132  -0.0859   0.9073

Return to index