AG08BZ

Zeros and Kronecker structure of a descriptor system pencil (complex case)

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

Purpose

  To extract from the system pencil

                    ( A-lambda*E B )
        S(lambda) = (              )
                    (      C     D )

  a regular pencil Af-lambda*Ef which has the finite Smith zeros of
  S(lambda) as generalized eigenvalues. The routine also computes
  the orders of the infinite Smith zeros and determines the singular
  and infinite Kronecker structure of system pencil, i.e., the right
  and left Kronecker indices, and the multiplicities of infinite
  eigenvalues.

Specification
      SUBROUTINE AG08BZ( EQUIL, L, N, M, P, A, LDA, E, LDE, B, LDB,
     $                   C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ, NKROR,
     $                   NINFE, NKROL, INFZ, KRONR, INFE, KRONL,
     $                   TOL, IWORK, DWORK, ZWORK, LZWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         EQUIL
      INTEGER           DINFZ, INFO, L, LDA, LDB, LDC, LDD, LDE, LZWORK,
     $                  M, N, NFZ, NINFE, NIZ, NKROL, NKROR, NRANK, P
      DOUBLE PRECISION  TOL
C     .. Array Arguments ..
      INTEGER           INFE(*), INFZ(*), IWORK(*), KRONL(*), KRONR(*)
      DOUBLE PRECISION  DWORK(*)
      COMPLEX*16        A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
     $                  E(LDE,*), ZWORK(*)

Arguments

Mode Parameters

  EQUIL   CHARACTER*1
          Specifies whether the user wishes to balance the system
          matrix as follows:
          = 'S':  Perform balancing (scaling);
          = 'N':  Do not perform balancing.

Input/Output Parameters
  L       (input) INTEGER
          The number of rows of matrices A, B, and E.  L >= 0.

  N       (input) INTEGER
          The number of columns of matrices A, E, and C.  N >= 0.

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

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

  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
          On entry, the leading L-by-N part of this array must
          contain the state dynamics matrix A of the system.
          On exit, the leading NFZ-by-NFZ part of this array
          contains the matrix Af of the reduced pencil.

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

  E       (input/output) COMPLEX*16 array, dimension (LDE,N)
          On entry, the leading L-by-N part of this array must
          contain the descriptor matrix E of the system.
          On exit, the leading NFZ-by-NFZ part of this array
          contains the matrix Ef of the reduced pencil.

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

  B       (input/output) COMPLEX*16 array, dimension (LDB,M)
          On entry, the leading L-by-M part of this array must
          contain the input/state matrix B of the system.
          On exit, this matrix does not contain useful information.

  LDB     INTEGER
          The leading dimension of array B.
          LDB >= MAX(1,L) if M > 0;
          LDB >= 1        if M = 0.

  C       (input/output) COMPLEX*16 array, dimension (LDC,N)
          On entry, the leading P-by-N part of this array must
          contain the state/output matrix C of the system.
          On exit, this matrix does not contain useful information.

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

  D       (input) COMPLEX*16 array, dimension (LDD,M)
          The leading P-by-M part of this array must contain the
          direct transmission matrix D of the system.

  LDD     INTEGER
          The leading dimension of array D.  LDD >= MAX(1,P).

  NFZ     (output) INTEGER
          The number of finite zeros.

  NRANK   (output) INTEGER
          The normal rank of the system pencil.

  NIZ     (output) INTEGER
          The number of infinite zeros.

  DINFZ   (output) INTEGER
          The maximal multiplicity of infinite Smith zeros.

  NKROR   (output) INTEGER
          The number of right Kronecker indices.

  NINFE   (output) INTEGER
          The number of elementary infinite blocks.

  NKROL   (output) INTEGER
          The number of left Kronecker indices.

  INFZ    (output) INTEGER array, dimension (N+1)
          The leading DINFZ elements of INFZ contain information
          on the infinite elementary divisors as follows:
          the system has INFZ(i) infinite elementary divisors of
          degree i in the Smith form, where i = 1,2,...,DINFZ.

  KRONR   (output) INTEGER array, dimension (N+M+1)
          The leading NKROR elements of this array contain the
          right Kronecker (column) indices.

  INFE    (output) INTEGER array, dimension (1+MIN(L+P,N+M))
          The leading NINFE elements of INFE contain the
          multiplicities of infinite eigenvalues.

  KRONL   (output) INTEGER array, dimension (L+P+1)
          The leading NKROL elements of this array contain the
          left Kronecker (row) indices.

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 (or RQ) factorization with column (or row) pivoting
          whose estimated condition number is less than 1/TOL.
          If the user sets TOL <= 0, then default tolerances are
          used instead, as follows: TOLDEF = L*N*EPS in TG01FZ
          (to determine the rank of E) and TOLDEF = (L+P)*(N+M)*EPS
          in the rest, where EPS is the machine precision
          (see LAPACK Library routine DLAMCH).  TOL < 1.

Workspace
  IWORK   INTEGER array, dimension (N+max(1,M))
          On output, IWORK(1) contains the normal rank of the
          transfer function matrix.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          LDWORK >= max(4*(L+N), 2*max(L+P,M+N))), if EQUIL = 'S',
          LDWORK >= 2*max(L+P,M+N)),               if EQUIL = 'N'.

  ZWORK   COMPLEX*16 array, dimension (LZWORK)
          On exit, if INFO = 0, ZWORK(1) returns the optimal value
          of LZWORK.

  LZWORK  INTEGER
          The length of the array ZWORK.
          LZWORK >= max( max(L+P,M+N)*(M+N) +
                         max(min(L+P,M+N) + max(min(L,N),3*(M+N)-1),
                             3*(L+P), 1))
          For optimum performance LZWORK should be larger.

          If LZWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          ZWORK array, returns this value as the first entry of
          the ZWORK array, and no error message related to LZWORK
          is issued by XERBLA.

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

Method
  The routine extracts from the system matrix of a descriptor
  system (A-lambda*E,B,C,D) a regular pencil Af-lambda*Ef which
  has the finite zeros of the system as generalized eigenvalues.
  The procedure has the following main computational steps:

     (a) construct the (L+P)-by-(N+M) system pencil

          S(lambda) = ( B  A )-lambda*( 0  E );
                      ( D  C )        ( 0  0 )

     (b) reduce S(lambda) to S1(lambda) with the same finite
         zeros and right Kronecker structure but with E
         upper triangular and nonsingular;

     (c) reduce S1(lambda) to S2(lambda) with the same finite
         zeros and right Kronecker structure but with D of
         full row rank;

     (d) reduce S2(lambda) to S3(lambda) with the same finite zeros
         and with D square invertible;

     (e) perform a unitary transformation on the columns of

         S3(lambda) = (A-lambda*E   B) in order to reduce it to
                      (     C       D)

         (Af-lambda*Ef   X), with Y and Ef square invertible;
         (     0         Y)

     (f) compute the right and left Kronecker indices of the system
         matrix, which together with the multiplicities of the
         finite and infinite eigenvalues constitute the
         complete set of structural invariants under strict
         equivalence transformations of a linear system.

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

Numerical Aspects
  The algorithm is backward stable (see [1]).

Further Comments
  In order to compute the finite Smith zeros of the system
  explicitly, a call to this routine may be followed by a
  call to the LAPACK Library routines ZGEGV or ZGGEV.

Example

Program Text

*     AG08BZ EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          LMAX, MMAX, NMAX, PMAX
      PARAMETER        ( LMAX = 20, MMAX = 20, NMAX = 20, PMAX = 20 )
      INTEGER          LDA, LDAEMX, LDB, LDC, LDD, LDE, LDQ, LDZ
      PARAMETER        ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
     $                   LDD = PMAX, LDE = LMAX, LDQ = 1, LDZ = 1,
     $                   LDAEMX = MAX( PMAX + LMAX, NMAX + MMAX ) )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = MAX( 4*( LMAX + NMAX ), 2*LDAEMX,
     $                                 8*NMAX ) )
      INTEGER          LZWORK
      PARAMETER        ( LZWORK = MAX( 1, LDAEMX*LDAEMX +
     $                                 MAX( MIN( LMAX+PMAX, MMAX+NMAX )+
     $                                      MAX( MIN( LMAX, NMAX ),
     $                                           3*( MMAX+NMAX )-1 ),
     $                                      3*( LMAX+PMAX ) ) ) )
*     .. Local Scalars ..
      DOUBLE PRECISION TOL
      INTEGER          DINFZ, I, INFO, J, L, M, N, NFZ, NINFE, NIZ,
     $                 NKROL, NKROR, NRANK, P
      CHARACTER*1      EQUIL
*     .. Local Arrays ..
      COMPLEX*16       A(LDA,NMAX), ALPHA(NMAX), ASAVE(LDA,NMAX),
     $                 B(LDB,MMAX),  BETA(NMAX), BSAVE(LDB,MMAX),
     $                 C(LDC,NMAX), CSAVE(LDC,NMAX),
     $                 D(LDD,MMAX), DSAVE(LDD,MMAX),
     $                 E(LDE,NMAX), ESAVE(LDE,NMAX), Q(LDQ,1), Z(LDZ,1),
     $                 ZWORK(LZWORK)
      DOUBLE PRECISION DWORK(LDWORK)
      INTEGER          INFE(1+LMAX+PMAX), INFZ(NMAX+1),
     $                 IWORK(NMAX+MMAX), KRONL(LMAX+PMAX+1),
     $                 KRONR(NMAX+MMAX+1)
*     .. External Subroutines ..
      EXTERNAL         AG08BZ, ZGEGV, ZLACPY
*     .. 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 = * ) L, N, M, P, TOL, EQUIL
      IF( ( L.LT.0 .OR. L.GT.LMAX ) .OR. ( N.LT.0 .OR. N.GT.NMAX ) )
     $   THEN
         WRITE ( NOUT, FMT = 99972 ) L, N
      ELSE
         IF( M.LT.0 .OR. M.GT.MMAX ) THEN
            WRITE ( NOUT, FMT = 99971 ) M
         ELSE
            IF( P.LT.0 .OR. P.GT.PMAX ) THEN
               WRITE ( NOUT, FMT = 99970 ) P
            ELSE
               READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
               READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
               READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
               READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
               READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
               CALL ZLACPY( 'F', L, N, A, LDA, ASAVE, LDA )
               CALL ZLACPY( 'F', L, N, E, LDE, ESAVE, LDE )
               CALL ZLACPY( 'F', L, M, B, LDB, BSAVE, LDB )
               CALL ZLACPY( 'F', P, N, C, LDC, CSAVE, LDC )
               CALL ZLACPY( 'F', P, M, D, LDD, DSAVE, LDD )
*              Compute poles (call the routine with M = 0, P = 0).
               CALL AG08BZ( EQUIL, L, N, 0, 0, A, LDA, E, LDE, B, LDB,
     $                      C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
     $                      NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
     $                      KRONL, TOL, IWORK, DWORK, ZWORK, LZWORK,
     $                      INFO )
*
               IF( INFO.NE.0 ) THEN
                  WRITE ( NOUT, FMT = 99998 ) INFO
               ELSE
                  WRITE ( NOUT, FMT = 99968 ) NIZ
                  DO 10 I = 1, DINFZ
                     WRITE ( NOUT, FMT = 99967 ) INFZ(I), I
   10             CONTINUE
                  WRITE ( NOUT, FMT = 99962 ) NINFE
                  IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99958 )
     $                                      ( INFE(I), I = 1,NINFE )
                  IF( NFZ.EQ.0 ) THEN
                     WRITE ( NOUT, FMT = 99965 )
                  ELSE
                     WRITE ( NOUT, FMT = 99966 )
                     WRITE ( NOUT, FMT = 99990 )
                     DO 20 I = 1, NFZ
                        WRITE ( NOUT, FMT = 99989 )
     $                        ( A(I,J), J = 1,NFZ )
   20                CONTINUE
                     WRITE ( NOUT, FMT = 99995 )
                     DO 30 I = 1, NFZ
                        WRITE ( NOUT, FMT = 99989 )
     $                        ( E(I,J), J = 1,NFZ )
   30                CONTINUE
                     CALL ZGEGV( 'No vectors', 'No vectors', NFZ, A,
     $                           LDA, E, LDE, ALPHA, BETA, Q, LDQ,
     $                           Z, LDZ, ZWORK, LZWORK, DWORK, INFO )
*
                     IF( INFO.NE.0 ) THEN
                        WRITE ( NOUT, FMT = 99997 ) INFO
                     ELSE
                        WRITE ( NOUT, FMT = 99996 )
                        DO 40 I = 1, NFZ
                           WRITE ( NOUT, FMT = 99979 ) ALPHA(I)/BETA(I)
   40                   CONTINUE
                     END IF
                  END IF
               END IF
               CALL ZLACPY( 'F', L, N, ASAVE, LDA, A, LDA )
               CALL ZLACPY( 'F', L, N, ESAVE, LDE, E, LDE )
*              Check the observability and compute the ordered set of
*              the observability indices (call the routine with M = 0).
               CALL AG08BZ( EQUIL, L, N, 0, P, A, LDA, E, LDE, B, LDB,
     $                      C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
     $                      NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
     $                      KRONL, TOL, IWORK, DWORK, ZWORK, LZWORK,
     $                      INFO )
*
               IF( INFO.NE.0 ) THEN
                  WRITE ( NOUT, FMT = 99998 ) INFO
               ELSE
                  WRITE ( NOUT, FMT = 99964 ) NIZ
                  DO 50 I = 1, DINFZ
                     WRITE ( NOUT, FMT = 99967 ) INFZ(I), I
   50             CONTINUE
                  WRITE ( NOUT, FMT = 99962 ) NINFE
                  IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99960 )
     $                                     ( INFE(I), I = 1,NINFE )
                  WRITE ( NOUT, FMT = 99994 ) ( KRONL(I), I = 1,NKROL )
                  IF( NFZ+NINFE.EQ.0 ) WRITE ( NOUT, FMT = 99993 )
                  IF( NFZ.EQ.0 ) THEN
                     WRITE ( NOUT, FMT = 99957 )
                  ELSE
                     WRITE ( NOUT, FMT = 99991 )
                     WRITE ( NOUT, FMT = 99990 )
                     DO 60 I = 1, NFZ
                        WRITE ( NOUT, FMT = 99989 )
     $                        ( A(I,J), J = 1,NFZ )
   60                CONTINUE
                     WRITE ( NOUT, FMT = 99995 )
                     DO 70 I = 1, NFZ
                        WRITE ( NOUT, FMT = 99989 )
     $                        ( E(I,J), J = 1,NFZ )
   70                CONTINUE
                     CALL ZGEGV( 'No vectors', 'No vectors', NFZ, A,
     $                           LDA, E, LDE, ALPHA, BETA, Q, LDQ,
     $                           Z, LDZ, ZWORK, LZWORK, DWORK, INFO )
*
                     IF( INFO.NE.0 ) THEN
                        WRITE ( NOUT, FMT = 99997 ) INFO
                     ELSE
                        WRITE ( NOUT, FMT = 99996 )
                        DO 80 I = 1, NFZ
                           WRITE ( NOUT, FMT = 99979 ) ALPHA(I)/BETA(I)
   80                   CONTINUE
                     END IF
                  END IF
               END IF
               CALL ZLACPY( 'F', L, N, ASAVE, LDA, A, LDA )
               CALL ZLACPY( 'F', L, N, ESAVE, LDE, E, LDE )
               CALL ZLACPY( 'F', P, N, CSAVE, LDC, C, LDC )
*              Check the controllability and compute the ordered set of
*              the controllability indices (call the routine with P = 0)
               CALL AG08BZ( EQUIL, L, N, M, 0, A, LDA, E, LDE, B, LDB,
     $                      C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
     $                      NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
     $                      KRONL, TOL, IWORK, DWORK, ZWORK, LZWORK,
     $                      INFO )
*
               IF( INFO.NE.0 ) THEN
                  WRITE ( NOUT, FMT = 99998 ) INFO
               ELSE
                  WRITE ( NOUT, FMT = 99963 ) NIZ
                  DO 90  I = 1, DINFZ
                     WRITE ( NOUT, FMT = 99967 ) INFZ(I), I
   90             CONTINUE
                  WRITE ( NOUT, FMT = 99962 ) NINFE
                  IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99959 )
     $                                     ( INFE(I), I = 1,NINFE )
                  WRITE ( NOUT, FMT = 99988 ) ( KRONR(I), I = 1,NKROR )
                  IF( NFZ+NINFE.EQ.0 ) WRITE ( NOUT, FMT = 99987 )
                  IF( NFZ.EQ.0 ) THEN
                     WRITE ( NOUT, FMT = 99956 )
                  ELSE
                     WRITE ( NOUT, FMT = 99985 )
                     WRITE ( NOUT, FMT = 99990 )
                     DO 100 I = 1, NFZ
                        WRITE ( NOUT, FMT = 99989 )
     $                        ( A(I,J), J = 1,NFZ )
  100                CONTINUE
                     WRITE ( NOUT, FMT = 99995 )
                     DO 110 I = 1, NFZ
                        WRITE ( NOUT, FMT = 99989 )
     $                        ( E(I,J), J = 1,NFZ )
  110                CONTINUE
                     CALL ZGEGV( 'No vectors', 'No vectors', NFZ, A,
     $                           LDA, E, LDE, ALPHA, BETA, Q, LDQ,
     $                           Z, LDZ, ZWORK, LZWORK, DWORK, INFO )
*
                     IF( INFO.NE.0 ) THEN
                        WRITE ( NOUT, FMT = 99997 ) INFO
                     ELSE
                        WRITE ( NOUT, FMT = 99982 )
                        DO 120 I = 1, NFZ
                           WRITE ( NOUT, FMT = 99979 ) ALPHA(I)/BETA(I)
  120                   CONTINUE
                     END IF
                  END IF
               END IF
               CALL ZLACPY( 'F', L, N, ASAVE, LDA, A, LDA )
               CALL ZLACPY( 'F', L, N, ESAVE, LDE, E, LDE )
               CALL ZLACPY( 'F', L, M, BSAVE, LDB, B, LDB )
               CALL ZLACPY( 'F', P, N, CSAVE, LDC, C, LDC )
               CALL ZLACPY( 'F', P, M, DSAVE, LDD, D, LDD )
*              Compute the structural invariants of the given system.
               CALL AG08BZ( EQUIL, L, N, M, P, A, LDA, E, LDE, B, LDB,
     $                      C, LDC, D, LDD, NFZ, NRANK, NIZ, DINFZ,
     $                      NKROR, NINFE, NKROL, INFZ, KRONR, INFE,
     $                      KRONL, TOL, IWORK, DWORK, ZWORK, LZWORK,
     $                      INFO )
*
               IF( INFO.NE.0 ) THEN
                  WRITE ( NOUT, FMT = 99998 ) INFO
               ELSE
                  IF( L.EQ.N ) THEN
                     WRITE ( NOUT, FMT = 99969 ) NRANK - N
                  ELSE
                     WRITE ( NOUT, FMT = 99955 ) NRANK
                  END IF
                  WRITE ( NOUT, FMT = 99984 ) NFZ
                  IF( NFZ.GT.0 ) THEN
*                    Compute the finite zeros of the given system.
*                    Workspace: need 8*NFZ.
                     WRITE ( NOUT, FMT = 99983 )
                     WRITE ( NOUT, FMT = 99990 )
                     DO 130 I = 1, NFZ
                        WRITE ( NOUT, FMT = 99989 )
     $                        ( A(I,J), J = 1,NFZ )
  130                CONTINUE
                     WRITE ( NOUT, FMT = 99995 )
                     DO 140 I = 1, NFZ
                        WRITE ( NOUT, FMT = 99989 )
     $                        ( E(I,J), J = 1,NFZ )
  140                CONTINUE
                     CALL ZGEGV( 'No vectors', 'No vectors', NFZ, A,
     $                           LDA, E, LDE, ALPHA, BETA, Q, LDQ,
     $                           Z, LDZ, ZWORK, LZWORK, DWORK, INFO )
*
                     IF( INFO.NE.0 ) THEN
                        WRITE ( NOUT, FMT = 99997 ) INFO
                     ELSE
                        WRITE ( NOUT, FMT = 99981 )
                        DO 150 I = 1, NFZ
                           WRITE ( NOUT, FMT = 99979 ) ALPHA(I)/BETA(I)
  150                   CONTINUE
                     END IF
                  END IF
                  WRITE ( NOUT, FMT = 99978 ) NIZ
                  DO 160 I = 1, DINFZ
                     WRITE ( NOUT, FMT = 99977 ) INFZ(I), I
  160             CONTINUE
                  WRITE ( NOUT, FMT = 99962 ) NINFE
                  IF( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99961 )
     $                                     ( INFE(I), I = 1,NINFE )
                  WRITE ( NOUT, FMT = 99976 ) NKROR
                  IF( NKROR.GT.0 ) WRITE ( NOUT, FMT = 99975 )
     $                                     ( KRONR(I), I = 1,NKROR )
                  WRITE ( NOUT, FMT = 99974 ) NKROL
                  IF( NKROL.GT.0 ) WRITE ( NOUT, FMT = 99973 )
     $                                     ( KRONL(I), I = 1,NKROL )
               END IF
            END IF
         END IF
      END IF
*
      STOP
*
99999 FORMAT (' AG08BZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AG08BZ = ',I2)
99997 FORMAT (' INFO on exit from ZGEGV = ',I2)
99996 FORMAT (/' Unobservable finite eigenvalues'/
     $         ' real  part     imag  part ')
99995 FORMAT (/' The matrix Ef is ')
99994 FORMAT (/' The left Kronecker indices of [A-lambda*E;C] are ',
     $          /(20(I3,2X)))
99993 FORMAT (/' The system (A-lambda*E,C) is completely observable ')
99991 FORMAT (/' The finite output decoupling zeros are the eigenvalues'
     $       , ' of the pair (Af,Ef). ')
99990 FORMAT (/' The matrix Af is ')
99989 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99988 FORMAT (/' The right Kronecker indices of [A-lambda*E,B] are ',
     $        /( 20(I3,2X) ) )
99987 FORMAT (/' The system (A-lambda*E,B) is completely controllable ')
99985 FORMAT (/' The input decoupling zeros are the eigenvalues of the',
     $         ' pair (Af,Ef). ')
99984 FORMAT (/' The number of finite zeros = ',I3)
99983 FORMAT (/' The finite zeros are the eigenvalues ',
     $         'of the pair (Af,Ef)')
99982 FORMAT (/' Uncontrollable finite eigenvalues'/
     $         ' real  part     imag  part ')
99981 FORMAT (/' Finite zeros'/' real  part     imag  part ')
99979 FORMAT (1X,F9.4,SP,F9.4,S,'i ')
99978 FORMAT (//' The number of infinite zeros = ',I3)
99977 FORMAT ( I4,' infinite zero(s) of order ',I3)
99976 FORMAT (/' The number of right Kronecker indices = ',I3)
99975 FORMAT (/' Right Kronecker indices of [A-lambda*E,B;C,D]'
     $         ,' are ', /(20(I3,2X)))
99974 FORMAT (/' The number of left Kronecker indices = ',I3)
99973 FORMAT (/' The left Kronecker indices of [A-lambda*E,B;C,D]'
     $         ,' are ',  /(20(I3,2X)))
99972 FORMAT (/' L or N is out of range.',/' L = ', I5, '  N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
99970 FORMAT (/' P is out of range.',/' P = ',I5)
99969 FORMAT (/' Normal rank  of transfer function matrix = ',I3)
99968 FORMAT (//' The number of infinite poles = ',I3)
99967 FORMAT ( I4,' infinite pole(s) of order ',I3)
99966 FORMAT (/' The finite poles are the eigenvalues',
     $         ' of the pair (Af,Ef). ')
99965 FORMAT (/' The system has no finite poles ')
99964 FORMAT (//' The number of unobservable infinite poles = ',I3)
99963 FORMAT (//' The number of uncontrollable infinite poles = ',I3)
99962 FORMAT (/' The number of infinite Kronecker blocks = ',I3)
99961 FORMAT (/' Multiplicities of infinite eigenvalues of '
     $         ,'[A-lambda*E,B;C,D] are ', /(20(I3,2X)))
99960 FORMAT (/' Multiplicities of infinite eigenvalues of '
     $         ,'[A-lambda*E;C] are ', /(20(I3,2X)))
99959 FORMAT (/' Multiplicities of infinite eigenvalues of '
     $         ,'[A-lambda*E,B] are ', /(20(I3,2X)))
99958 FORMAT (/' Multiplicities of infinite eigenvalues of A-lambda*E'
     $         ,' are ', /(20(I3,2X)))
99957 FORMAT (/' The system (A-lambda*E,C) has no finite output',
     $         ' decoupling zeros ')
99956 FORMAT (/' The system (A-lambda*E,B) has no finite input',
     $         ' decoupling zeros ')
99955 FORMAT (/' Normal rank  of system pencil = ',I3)
      END
Program Data
 AG08BZ EXAMPLE PROGRAM DATA
   9    9    3     3     1.e-7     N
    (1,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (1,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (1,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (1,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (1,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (1,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (1,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (1,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (1,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (1,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (1,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (1,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (1,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (1,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (0,0)   (1,0)   (0,0)
   (-1,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)
    (0,0)  (-1,0)   (0,0)
    (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)  (-1,0)
    (0,0)   (0,0)   (0,0)
    (0,0)   (0,0)   (0,0)
    (0,0)   (1,0)   (1,0)   (0,0)   (3,0)    (4,0)    (0,0)   (0,0)    (2,0)
    (0,0)   (1,0)   (0,0)   (0,0)   (4,0)    (0,0)    (0,0)   (2,0)    (0,0)
    (0,0)   (0,0)   (1,0)   (0,0)  (-1,0)    (4,0)    (0,0)  (-2,0)    (2,0)
    (1,0)   (2,0)  (-2,0)
    (0,0)  (-1,0)  (-2,0)
    (0,0)   (0,0)   (0,0)

Program Results
 AG08BZ EXAMPLE PROGRAM RESULTS



 The number of infinite poles =   6
   0 infinite pole(s) of order   1
   3 infinite pole(s) of order   2

 The number of infinite Kronecker blocks =   3

 Multiplicities of infinite eigenvalues of A-lambda*E are 
  3    3    3

 The system has no finite poles 


 The number of unobservable infinite poles =   4
   0 infinite pole(s) of order   1
   2 infinite pole(s) of order   2

 The number of infinite Kronecker blocks =   3

 Multiplicities of infinite eigenvalues of [A-lambda*E;C] are 
  1    3    3

 The left Kronecker indices of [A-lambda*E;C] are 
  0    1    1

 The system (A-lambda*E,C) has no finite output decoupling zeros 


 The number of uncontrollable infinite poles =   0

 The number of infinite Kronecker blocks =   3

 Multiplicities of infinite eigenvalues of [A-lambda*E,B] are 
  1    1    1

 The right Kronecker indices of [A-lambda*E,B] are 
  2    2    2

 The system (A-lambda*E,B) has no finite input decoupling zeros 

 Normal rank  of transfer function matrix =   2

 The number of finite zeros =   1

 The finite zeros are the eigenvalues of the pair (Af,Ef)

 The matrix Af is 
   -0.7705  +0.0000i 

 The matrix Ef is 
   -0.7705  +0.0000i 

Finite zeros
 real  part     imag  part 
    1.0000  +0.0000i 


 The number of infinite zeros =   2
   0 infinite zero(s) of order   1
   1 infinite zero(s) of order   2

 The number of infinite Kronecker blocks =   5

 Multiplicities of infinite eigenvalues of [A-lambda*E,B;C,D] are 
  1    1    1    1    3

 The number of right Kronecker indices =   1

 Right Kronecker indices of [A-lambda*E,B;C,D] are 
  2

 The number of left Kronecker indices =   1

 The left Kronecker indices of [A-lambda*E,B;C,D] are 
  1

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