ZGHFDF

Computing the eigenvalues and stable deflating subspaces of a complex skew-Hamiltonian/Hamiltonian
pencil (factored version)

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

Purpose

     To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
     Hamiltonian pencil aS - bH, with

                             (  B  F  )      (  Z11  Z12  )
       S = J Z' J' Z and H = (        ), Z = (            ),
                             (  G -B' )      (  Z21  Z22  )
                                                                   (1)
           (  0  I  )
       J = (        ).
           ( -I  0  )

     The structured Schur form of the embedded real skew-Hamiltonian/
                                                            
     skew-Hamiltonian pencil, aB_S - bB_T, with B_S = J B_Z' J' B_Z,

             (  Re(Z11)  -Im(Z11)  |  Re(Z12)  -Im(Z12)  )
             (                     |                     )
             (  Im(Z11)   Re(Z11)  |  Im(Z12)   Re(Z12)  )
             (                     |                     )
       B_Z = (---------------------+---------------------) ,
             (                     |                     )
             (  Re(Z21)  -Im(Z21)  |  Re(Z22)  -Im(Z22)  )
             (                     |                     )
             (  Im(Z21)   Re(Z21)  |  Im(Z22)   Re(Z22)  )
                                                                    (2)
             ( -Im(B)  -Re(B)  | -Im(F)  -Re(F)  )
             (                 |                 )
             (  Re(B)  -Im(B)  |  Re(F)  -Im(F)  )
             (                 |                 )
       B_T = (-----------------+-----------------) ,  T = i*H,
             (                 |                 )
             ( -Im(G)  -Re(G)  | -Im(B')  Re(B') )
             (                 |                 )
             (  Re(G)  -Im(G)  | -Re(B') -Im(B') )

     is determined and used to compute the eigenvalues. Optionally, if
     COMPQ = 'C', an orthonormal basis of the right deflating subspace,
     Def_-(S, H), of the pencil aS - bH in (1), corresponding to the
     eigenvalues with strictly negative real part, is computed. Namely,
     after transforming aB_S - bB_H, in the factored form, by unitary
     matrices, we have B_Sout = J B_Zout' J' B_Zout,

                ( BA  BD  )              ( BB  BF  )
       B_Zout = (         ) and B_Hout = (         ),               (3)
                (  0  BC  )              (  0 -BB' )

     and the eigenvalues with strictly negative real part of the
     complex pencil aB_Sout - bB_Hout are moved to the top. The 
     notation M' denotes the conjugate transpose of the matrix M.
     Optionally, if COMPU = 'C', an orthonormal basis of the companion
     subspace, range(P_U) [1], which corresponds to the eigenvalues
     with negative real part, is computed. The embedding doubles the
     multiplicities of the eigenvalues of the pencil aS - bH.
Specification
      SUBROUTINE ZGHFDF( COMPQ, COMPU, ORTH, N, Z, LDZ, B, LDB, FG,
     $                   LDFG, NEIG, D, LDD, C, LDC, Q, LDQ, U, LDU,
     $                   ALPHAR, ALPHAI, BETA, IWORK, LIWORK, DWORK,
     $                   LDWORK, ZWORK, LZWORK, BWORK, INFO )
C
C     .. Scalar Arguments ..
      CHARACTER          COMPQ, COMPU, ORTH
      INTEGER            INFO, LDB, LDC, LDD, LDFG, LDQ, LDU, LDWORK,
     $                   LDZ, LIWORK, LZWORK, N, NEIG
C
C     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
      COMPLEX*16         B( LDB, * ), C( LDC, * ), D( LDD, * ),
     $                   FG( LDFG, * ), Q( LDQ, * ), U( LDU, * ),
     $                   Z( LDZ, * ), ZWORK( * )
Arguments

Mode Parameters

     COMPQ   CHARACTER*1
             Specifies whether to compute the right deflating subspace
             corresponding to the eigenvalues of aS - bH with strictly
             negative real part.
             = 'N': do not compute the deflating subspace;
             = 'C': compute the deflating subspace and store it in the
                    leading subarray of Q.

     COMPU   CHARACTER*1
             Specifies whether to compute the companion subspace
             corresponding to the eigenvalues of aS - bH with strictly
             negative real part.
             = 'N': do not compute the companion subspace;
             = 'C': compute the companion subspace and store it in the
                    leading subarray of U.

     ORTH    CHARACTER*1
             If COMPQ = 'C' or COMPU = 'C', specifies the technique for
             computing the orthonormal bases of the deflating subspace
             and companion subspace, as follows:
             = 'P':  QR factorization with column pivoting;
             = 'S':  singular value decomposition.
             If COMPQ = 'N' and COMPU = 'N', the ORTH value is not
             used.
Input/Output Parameters
     N       (input) INTEGER
             Order of the pencil aS - bH.  N >= 0, even.

     Z       (input/output) COMPLEX*16 array, dimension (LDZ, N)
             On entry, the leading N-by-N part of this array must
             contain the non-trivial factor Z in the factorization
             S = J Z' J' Z of the skew-Hamiltonian matrix S.
             On exit, if COMPQ = 'C' or COMPU = 'C', the leading
             N-by-N part of this array contains the upper triangular
             matrix BA in (3) (see also METHOD). The strictly lower
             triangular part is not zeroed.
             If COMPQ = 'N' and COMPU = 'N', this array is unchanged
             on exit.

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

     B       (input/output) COMPLEX*16 array, dimension (LDB, N)
             On entry, the leading N/2-by-N/2 part of this array must
             contain the matrix B.
             On exit, if COMPQ = 'C' or COMPU = 'C', the leading
             N-by-N part of this array contains the upper triangular
             matrix BB in (3) (see also METHOD). The strictly lower
             triangular part is not zeroed. 
             If COMPQ = 'N' and COMPU = 'N', this array is unchanged
             on exit.

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

     FG      (input/output) COMPLEX*16 array, dimension (LDFG, N)
             On entry, the leading N/2-by-N/2 lower triangular part of
             this array must contain the lower triangular part of the
             Hermitian matrix G, and the N/2-by-N/2 upper triangular
             part of the submatrix in the columns 2 to N/2+1 of this
             array must contain the upper triangular part of the
             Hermitian matrix F.
             On exit, if COMPQ = 'C' or COMPU = 'C', the leading
             N-by-N part of this array contains the Hermitian matrix
             BF in (3) (see also METHOD). The strictly lower triangular
             part of the input matrix is preserved. The diagonal
             elements might have tiny imaginary parts.
             If COMPQ = 'N' and COMPU = 'N', this array is unchanged
             on exit.

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

     NEIG    (output) INTEGER
             If COMPQ = 'C' or COMPU = 'C', the number of eigenvalues
             in aS - bH with strictly negative real part.

     D       (output) COMPLEX*16 array, dimension (LDD, N)
             If COMPQ = 'C' or COMPU = 'C', the leading N-by-N part of
             this array contains the matrix BD in (3) (see METHOD).
             If COMPQ = 'N' and COMPU = 'N', this array is not
             referenced.

     LDD     INTEGER
             The leading dimension of the array D.
             LDD >= 1,         if COMPQ = 'N' and COMPU = 'N';
             LDD >= MAX(1, N), if COMPQ = 'C' or  COMPU = 'C'.

     C       (output) COMPLEX*16 array, dimension (LDC, N)
             If COMPQ = 'C' or COMPU = 'C', the leading N-by-N part of
             this array contains the lower triangular matrix BC in (3)
             (see also METHOD). The strictly upper triangular part is
             not zeroed. 
             If COMPQ = 'N' and COMPU = 'N', this array is not
             referenced.

     LDC     INTEGER
             The leading dimension of the array C.
             LDC >= 1,         if COMPQ = 'N' and COMPU = 'N';
             LDC >= MAX(1, N), if COMPQ = 'C' or  COMPU = 'C'.

     Q       (output) COMPLEX*16 array, dimension (LDQ, 2*N)
             On exit, if COMPQ = 'C', the leading N-by-NEIG part of
             this array contains an orthonormal basis of the right
             deflating subspace corresponding to the eigenvalues of the
             pencil aS - bH with strictly negative real part.
             The remaining entries are meaningless.
             If COMPQ = 'N', this array is not referenced.

     LDQ     INTEGER
             The leading dimension of the array Q.
             LDQ >= 1,           if COMPQ = 'N';
             LDQ >= MAX(1, 2*N), if COMPQ = 'C'.

     U       (output) COMPLEX*16 array, dimension (LDU, 2*N)
             On exit, if COMPU = 'C', the leading N-by-NEIG part of
             this array contains an orthonormal basis of the companion
             subspace corresponding to the eigenvalues of the
             pencil aS - bH with strictly negative real part. The
             remaining entries are meaningless.
             If COMPU = 'N', this array is not referenced.

     LDU     INTEGER
             The leading dimension of the array U.
             LDU >= 1,         if COMPU = 'N';
             LDU >= MAX(1, N), if COMPU = 'C'.

     ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
             The real parts of each scalar alpha defining an eigenvalue
             of the pencil aS - bH.

     ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
             The imaginary parts of each scalar alpha defining an
             eigenvalue of the pencil aS - bH.
             If ALPHAI(j) is zero, then the j-th eigenvalue is real.

     BETA    (output) DOUBLE PRECISION array, dimension (N)
             The scalars beta that define the eigenvalues of the pencil
             aS - bH.
             Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
             beta = BETA(j) represent the j-th eigenvalue of the pencil
             aS - bH, in the form lambda = alpha/beta. Since lambda may
             overflow, the ratios should not, in general, be computed.
Workspace
     IWORK   INTEGER array, dimension (LIWORK)

     LIWORK  INTEGER
             The dimension of the array IWORK.  LIWORK >= 2*N+9.

     DWORK   DOUBLE PRECISION array, dimension (LDWORK)
             On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
             On exit, if INFO = -26, DWORK(1) returns the minimum
             value of LDWORK.

     LDWORK  INTEGER
             The dimension of the array DWORK.
             LDWORK >= c*N**2 + N + MAX(2*N, 24) + 3, where
                       c = 18, if                 COMPU = 'C';
                       c = 16, if COMPQ = 'C' and COMPU = 'N';
                       c = 13, if COMPQ = 'N' and COMPU = 'N'.
             For good performance LDWORK should be generally 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.

     ZWORK   COMPLEX*16 array, dimension (LZWORK)
             On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
             On exit, if INFO = -28, ZWORK(1) returns the minimum
             value of LZWORK.

     LZWORK  INTEGER
             The dimension of the array ZWORK.
             LZWORK >= 8*N + 28, if COMPQ = 'C';
             LZWORK >= 6*N + 28, if COMPQ = 'N' and COMPU = 'C';
             LZWORK >= 1,        if COMPQ = 'N' and COMPU = 'N'.
             For good performance LZWORK should be generally 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.

     BWORK   LOGICAL array, dimension (LBWORK)
             LBWORK >= 0, if COMPQ = 'N' and COMPU = 'N';
             LBWORK >= N, if COMPQ = 'C' or  COMPU = 'C'.
Error Indicator
     INFO    INTEGER
             = 0: succesful exit;
             < 0: if INFO = -i, the i-th argument had an illegal value;
             = 1: the algorithm was not able to reveal information
                  about the eigenvalues from the 2-by-2 blocks in the
                  SLICOT Library routine MB03BD (called by DGHFST);
             = 2: periodic QZ iteration failed in the SLICOT Library
                  routines MB03BD or MB03BZ when trying to
                  triangularize the 2-by-2 blocks;
             = 3: the singular value decomposition failed in the LAPACK
                  routine ZGESVD (for ORTH = 'S').
Method
     First T = i*H is set. Then, the embeddings, B_Z and B_T, of the
     matrices S and T, are determined and, subsequently, the routine
     DGHFST is applied to compute the structured Schur form, i.e.,
     the factorizations

     ~                (  BZ11  BZ12  )
     B_Z = U' B_Z Q = (              ) and
                      (    0   BZ22  )

     ~                     (  T11  T12  )
     B_T = J Q' J' B_T Q = (            ),
                           (   0   T11' )

     where Q is real orthogonal, U is real orthogonal symplectic, BZ11,
     BZ22' are upper triangular and T11 is upper quasi-triangular.

     Second, the routine ZGHFXC is applied, to compute a
                    ~                                 ~
     unitary matrix Q and a unitary symplectic matrix U, such that

                   ~    ~
     ~  ~   ~   (  Z11  Z12  )
     U' B_Z Q = (       ~    ) =: B_Zout,
                (   0   Z22  )

       ~        ~    ~   (  H11  H12  )
     J Q' J'(-i*B_T) Q = (            ) =: B_Hout,
                         (   0  -H11' )
          ~    ~   
     with Z11, Z22', H11 upper triangular, and such that the spectrum

              ~       ~       ~
     Spec_-(J B_Z' J' B_Z, -i*B_T) is contained in the spectrum of the
                                                   ~    ~
     2*NEIG-by-2*NEIG leading principal subpencil aZ22'*Z11 - bH11.

     Finally, the right deflating subspace and the companion subspace
     are computed. See also page 21 in [1] for more details.
References
     [1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
         Numerical Computation of Deflating Subspaces of Embedded
         Hamiltonian Pencils.
         Tech. Rep. SFB393/99-15, Technical University Chemnitz,
         Germany, June 1999.
Numerical Aspects
     
                                                               3
     The algorithm is numerically backward stable and needs O(N )
     complex floating point operations.
Further Comments
   
     This routine does not perform any scaling of the matrices. Scaling
     might sometimes be useful, and it should be done externally.
Example

Program Text

*     ZGHFDF EXAMPLE PROGRAM TEXT
*
*     .. Parameters ..
      INTEGER            DKIND
      PARAMETER          ( DKIND = KIND( 0.0D+0 ) )
      COMPLEX(DKIND) ::  ZERO
      PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ) )
      INTEGER            NIN, NOUT
      PARAMETER          ( NIN = 5, NOUT = 6 )
      INTEGER            NMAX
      PARAMETER          ( NMAX = 50 )
      INTEGER            LDB, LDC, LDD, LDFG, LDQ, LDU, LDWORK, LDZ,
     $                   LIWORK, LZWORK
      PARAMETER          ( LDB  = NMAX, LDC =   NMAX, LDD = NMAX,
     $                     LDFG = NMAX, LDQ = 2*NMAX, LDU = NMAX,
     $                     LDWORK = 18*NMAX*NMAX + NMAX +
     $                              MAX( 2*NMAX, 24 ), LDZ  = NMAX,
     $                     LIWORK = NMAX + 9, LZWORK = 8*NMAX + 28 )
*
*     .. Local Scalars ..
      CHARACTER          COMPQ, COMPU, ORTH
      INTEGER            I, INFO, J, M, N, NEIG
*
*     .. Local Arrays ..
      COMPLEX(DKIND) ::  B( LDB, NMAX ), C( LDC, NMAX ), D( LDD, NMAX ),
     $                   FG( LDFG, NMAX ), Q( LDQ, 2*NMAX ),
     $                   U( LDU, 2*NMAX ), Z( LDZ, NMAX ),
     $                   ZWORK( LZWORK )
      DOUBLE PRECISION   ALPHAI( NMAX ),  ALPHAR( NMAX ), BETA( NMAX ),
     $                   DWORK( LDWORK )
      INTEGER            IWORK( LIWORK )
      LOGICAL            BWORK( NMAX )
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*
*     .. External Subroutines ..
      EXTERNAL           ZGHFDF, ZLASET
*
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MOD
*
*     .. Executable Statements ..
*
      WRITE( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read in the data.
      READ( NIN, FMT = * )
      READ( NIN, FMT = * ) COMPQ, COMPU, ORTH, N
      IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
         WRITE( NOUT, FMT = 99998 ) N
      ELSE
         M = N/2
         READ( NIN, FMT = * ) ( (  Z( I, J ), J = 1, N   ), I = 1, N )
         READ( NIN, FMT = * ) ( (  B( I, J ), J = 1, M   ), I = 1, M )
         READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, M+1 ), I = 1, M )
*        Compute the eigenvalues and orthogonal bases of the right
*        deflating subspace and companion subspace of a complex
*        skew-Hamiltonian/Hamiltonian pencil, corresponding to the
*        eigenvalues with strictly negative real part.
         CALL ZGHFDF( COMPQ, COMPU, ORTH, N, Z, LDZ, B, LDB, FG, LDFG,
     $                NEIG, D, LDD, C, LDC, Q, LDQ, U, LDU, ALPHAR,
     $                ALPHAI, BETA, IWORK, LIWORK, DWORK, LDWORK, ZWORK,
     $                LZWORK, BWORK, INFO )
*
         IF( INFO.NE.0 ) THEN
            WRITE( NOUT, FMT = 99997 ) INFO
         ELSE
*           Set to zero the strict lower triangles of B and FG, and the
*           strict upper triangles of C, to avoid references to
*           undefined entries.
            IF( M.GT.1 ) THEN
               CALL ZLASET( 'L', M-1, M-1, ZERO, ZERO,  B( 2, 1 ), LDB )
               CALL ZLASET( 'L', M-1, M-1, ZERO, ZERO, FG( 2, 1 ), LDFG)
               CALL ZLASET( 'U', M-1, M-1, ZERO, ZERO,  C( 1, 2 ), LDC )
            END IF
*
            WRITE( NOUT, FMT = 99996 )
            DO 10 I = 1, N
               WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N )
   10       CONTINUE
            IF( LSAME( COMPQ, 'C' ) .OR. LSAME( COMPU, 'C' ) ) THEN
               WRITE( NOUT, FMT = 99994 )
               DO 20 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( D( I, J ), J = 1, N )
   20          CONTINUE
               WRITE( NOUT, FMT = 99993 )
               DO 30 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( C( I, J ), J = 1, N )
   30          CONTINUE
               WRITE( NOUT, FMT = 99992 )
               DO 40 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N )
   40          CONTINUE
               WRITE( NOUT, FMT = 99991 )
               DO 50 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N )
   50          CONTINUE
            END IF
            WRITE( NOUT, FMT = 99990 )
            WRITE( NOUT, FMT = 99989 ) ( ALPHAR( I ), I = 1, N )
            WRITE( NOUT, FMT = 99988 )
            WRITE( NOUT, FMT = 99989 ) ( ALPHAI( I ), I = 1, N )
            WRITE( NOUT, FMT = 99987 )
            WRITE( NOUT, FMT = 99989 ) (   BETA( I ), I = 1, N )
            IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
               WRITE( NOUT, FMT = 99986 )
               DO 60 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
   60          CONTINUE
            END IF
            IF( LSAME( COMPU, 'C' ) .AND. NEIG.GT.0 ) THEN
               WRITE( NOUT, FMT = 99985 )
               DO 70 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( U( I, J ), J = 1, NEIG )
   70          CONTINUE
            END IF
            IF( LSAME( COMPQ, 'C' ) .OR. LSAME( COMPU, 'C' ) )
     $         WRITE( NOUT, FMT = 99984 ) NEIG
         END IF
      END IF
      STOP
*
99999 FORMAT( 'ZGHFDF EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from ZGHFDF = ', I2 )
99996 FORMAT( 'The matrix Z on exit is ' )
99995 FORMAT( 20( 1X, F9.4, SP, F9.4, S, 'i ') )
99994 FORMAT( 'The matrix D is ' )
99993 FORMAT( 'The matrix C is ' )
99992 FORMAT( 'The matrix B on exit is ' )
99991 FORMAT( 'The matrix F on exit is ' )
99990 FORMAT( 'The vector ALPHAR is ' )
99989 FORMAT( 50( 1X, F8.4 ) )
99988 FORMAT( 'The vector ALPHAI is ' )
99987 FORMAT( 'The vector BETA is ' )
99986 FORMAT( 'The deflating subspace corresponding to the ',
     $        'eigenvalues with negative real part is ' )
99985 FORMAT( 'The companion subspace corresponding to the ',
     $        'eigenvalues with negative real part is ' )
99984 FORMAT( 'The number of eigenvalues in the initial pencil with ',
     $        'negative real part is ', I2 )
      END
Program Data
ZGHFDF EXAMPLE PROGRAM DATA
   C   C   P   4
   (0.0328,0.9611)   (0.6428,0.2585)   (0.7033,0.4254)   (0.2552,0.7053)
   (0.0501,0.2510)   (0.2827,0.8865)   (0.4719,0.5387)   (0.0389,0.5676)
   (0.5551,0.4242)   (0.0643,0.2716)   (0.1165,0.7875)   (0.9144,0.3891)
   (0.0539,0.7931)   (0.0408,0.2654)   (0.9912,0.0989)   (0.0991,0.6585)
   (0.0547,0.8726)   (0.4008,0.8722)
   (0.7423,0.6166)   (0.2631,0.5872)
   (0.8740,0)        (0.3697,0)        (0.9178,0.6418)
   (0.7748,0.5358)   (0.1652,0)        (0.2441,0)  
Program Results
ZGHFDF EXAMPLE PROGRAM RESULTS
The matrix Z on exit is 
    1.1347  -0.1694i     0.0920  -0.0894i     0.5253  +0.0280i    -0.0597  +0.1098i 
    0.0000  +0.0000i    -0.9874  -0.6015i     0.2523  -0.0600i     0.3178  -0.0902i 
    0.5551  +0.4242i     0.0643  +0.2716i     0.7553  -0.3356i     0.4772  -0.3177i 
    0.0539  +0.7931i     0.0408  +0.2654i     0.9912  +0.0989i     0.9064  -0.1055i 
The matrix D is 
   -0.7634  -0.2773i    -0.8466  -0.9586i    -0.0308  -0.0175i    -0.2754  -0.0715i 
    1.2612  -0.2643i    -0.7291  -0.3165i     0.0282  -0.1748i     0.4091  +0.0233i 
    0.3773  -0.1536i    -0.3937  -0.0480i    -0.1635  +0.1617i    -0.1775  +0.1277i 
    0.7540  -0.0280i    -0.6860  -0.8306i    -0.2446  +0.0943i    -0.0722  +0.0517i 
The matrix C is 
    0.5063  +0.1548i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i 
   -0.0046  +0.1049i     0.3884  +0.3420i     0.0000  +0.0000i     0.0000  +0.0000i 
   -1.1206  +0.1313i    -0.2270  -0.1753i     0.4300  -0.6107i     0.0000  +0.0000i 
   -0.6127  -0.1939i    -0.5713  -0.7913i     0.3739  -0.2943i    -1.1501  -0.0850i 
The matrix B on exit is 
    0.3322  +1.9093i    -0.1216  -0.1193i    -0.0030  +0.0330i     0.0405  +0.0592i 
    0.0000  +0.0000i     0.1863  -1.8998i     0.2983  +0.2974i     0.6636  +0.5916i 
    0.0000  +0.0000i     0.0000  +0.0000i     0.4459  -0.7452i    -0.0625  +0.2197i 
    0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.1418  +0.7392i 
The matrix F on exit is 
    0.0258  -0.0000i    -0.0878  +0.1090i     0.3547  +0.5306i    -0.0138  -0.8770i 
    0.0000  +0.0000i     0.0864  +0.0000i    -0.3788  -0.2829i    -0.3303  -0.0415i 
    0.0000  +0.0000i     0.0000  +0.0000i    -0.0184  +0.0000i     0.1077  -0.0795i 
    0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i    -0.0938  +0.0000i 
The vector ALPHAR is 
   0.4295  -0.4295   0.0000   0.0000
The vector ALPHAI is 
   1.5363   1.5363  -1.4069  -0.7153
The vector BETA is 
   0.5000   0.5000   1.0000   1.0000
The deflating subspace corresponding to the eigenvalues with negative real part is 
   -0.2249  +0.4158i 
   -0.1984  -0.3100i 
    0.7286  -0.0427i 
    0.3282  -0.0251i 
The companion subspace corresponding to the eigenvalues with negative real part is 
   -0.1542  -0.0712i 
   -0.4162  -0.3021i 
   -0.0806  -0.6946i 
   -0.4580  -0.0889i 
The number of eigenvalues in the initial pencil with negative real part is  1

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