DGHFDF

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

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

Purpose

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

                                   (  B  F  )      (  0  I  )
       S = T Z = J Z' J' Z and H = (        ), J = (        ),      (1)
                                   (  G -B' )      ( -I  0  )

     where the notation M' denotes the transpose of the matrix M.
     Optionally, if COMPQ = 'C', an orthogonal basis of the right
     deflating subspace of aS - bH corresponding to the eigenvalues
     with strictly negative real part is computed. Optionally, if
     COMPU = 'C', an orthonormal basis of the companion subspace,
     range(P_U) [1], which corresponds to the eigenvalues with strictly
     negative real part, is computed.    
Specification
      SUBROUTINE DGHFDF( COMPQ, COMPU, ORTH, N, Z, LDZ, B, LDB, FG,
     $                   LDFG, NEIG, Q, LDQ, U, LDU, ALPHAR, ALPHAI,
     $                   BETA, IWORK, LIWORK, DWORK, LDWORK, BWORK,
     $                   IWARN, INFO )
C
C     .. Scalar Arguments ..
      CHARACTER          COMPQ, COMPU, ORTH
      INTEGER            INFO, IWARN, LDB, LDFG, LDQ, LDU, LDWORK, LDZ,
     $                   LIWORK, N, NEIG
C
C     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
     $                   BETA( * ), DWORK( * ), FG( LDFG, * ),
     $                   Q( LDQ, * ), U( LDU, * ), Z( LDZ, * )
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' and/or COMPU = 'C', specifies the technique
             for computing the orthogonal basis of the deflating
             subspace, and/or of the 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
             The order of the pencil aS - bH.  N >= 0, even.

     Z       (input/output) DOUBLE PRECISION 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 transformed upper
                               ~
             triangular matrix Z11 (see METHOD), after moving the
             eigenvalues with strictly negative real part to the top
             of the pencil (3). The strictly lower triangular part is
             not zeroed.
             If COMPQ = 'N' and COMPU = 'N', the leading N-by-N part of
             this array contains the matrix Z obtained by the routine
             DGHURV just before the application of the periodic QZ
             algorithm. The elements of the (2,1) block, i.e., in the
             rows N/2+1 to N and in the columns 1 to N/2 are not set to
             zero, but are unchanged on exit.

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

     B       (input) DOUBLE PRECISION array, dimension (LDB, N/2)
             On entry, the leading N/2-by-N/2 part of this array must
             contain the matrix B.

     LDB     INTEGER
             The leading dimension of the array B.  LDB >= MAX(1, N/2).

     FG      (input) DOUBLE PRECISION array, dimension (LDFG, N/2+1)
             On entry, the leading N/2-by-N/2 lower triangular part of
             this array must contain the lower triangular part of the
             symmetric 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
             symmetric matrix F.

     LDFG    INTEGER
             The leading dimension of the array FG.
             LDFG >= MAX(1, N/2).

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

     Q       (output) DOUBLE PRECISION array, dimension (LDQ, 2*N)
             On exit, if COMPQ = 'C', the leading N-by-NEIG part of
             this array contains an orthogonal basis of the right
             deflating subspace corresponding to the eigenvalues of
             aS - bH with strictly negative real part. The remaining
             part of this array is used as workspace.
             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) DOUBLE PRECISION array, dimension (LDU, 2*N)
             On exit, if COMPU = 'C', the leading N-by-NEIG part of
             this array contains an orthogonal basis of the companion
             subspace corresponding to the eigenvalues of aS - bH with
             strictly negative real part. The remaining part of this
             array is used as workspace.
             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/2)
             The real parts of each scalar alpha defining an eigenvalue
             of the pencil aS - bH.

     ALPHAI  (output) DOUBLE PRECISION array, dimension (N/2)
             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/2)
             The scalars beta that define the eigenvalues of the pencil
             aS - bH.
             If INFO = 0, the quantities alpha = (ALPHAR(j),ALPHAI(j)),
             and beta = BETA(j) represent together 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. Due to the skew-Hamiltonian/
             Hamiltonian structure of the pencil, only half of the
             spectrum is saved in ALPHAR, ALPHAI and BETA.
             Specifically, only eigenvalues with imaginary parts
             greater than or equal to zero are stored; their conjugate
             eigenvalues are not stored. If imaginary parts are zero
             (i.e., for real eigenvalues), only positive eigenvalues
             are stored. The remaining eigenvalues have opposite signs.
             If IWARN = 1, one or more BETA(j) is not representable,
             and the eigenvalues are returned as described below (see
             the description of the argument IWARN).
Workspace
     IWORK   INTEGER array, dimension (LIWORK)
             On exit, if INFO = -20, IWORK(1) returns the minimum value
             of LIWORK.
             On exit, if INFO = 0 and IWARN = 1, then IWORK(1), ...,
             IWORK(N/2) return the scaling parameters for the
             eigenvalues of the pencil aS - bH (see IWARN).

     LIWORK  INTEGER
             The dimension of the array IWORK.
             LIWORK >= N + 18,      if COMPQ = 'N' and COMPU = 'N';
             LIWORK >= MAX( N + 18, N/2 + 48, 5*N/2 + 1 ), otherwise.

     DWORK   DOUBLE PRECISION array, dimension (LDWORK)
             On exit, if INFO = 0, DWORK(1) returns the optimal value
             of LDWORK, and DWORK(2) returns the machine base, b.
             On exit, if INFO = -22, DWORK(1) returns the minimum value
             of LDWORK.

     LDWORK  INTEGER
             The dimension of the array DWORK.
             LDWORK >= c*N**2 + max( N*N + MAX( N/2+252, 432 ),
                                     MAX(8*N+48,171) ), where
                       c = a,   if COMPU = 'N',
                       c = a+1, if COMPU = 'C', and
                       a = 6,   if COMPQ = 'N',
                       a = 9,   if COMPQ = 'C'.
             For good performance LDWORK should be generally larger.

             If LDWORK = -1  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 is issued by XERBLA.

     BWORK   LOGICAL array, dimension (N/2)
Warning Indicator
     IWARN   INTEGER
             = 0: no warning;
             = 1: the eigenvalues will under- or overflow if evaluated;
                  therefore, the j-th eigenvalue is represented by
                  the quantities alpha = (ALPHAR(j),ALPHAI(j)),
                  beta = BETA(j), and gamma = IWORK(j) in the form
                  lambda = (alpha/beta) * b**gamma, where b is the
                  machine base (often 2.0), returned in DWORK(2).
Error Indicator
     INFO    INTEGER
             = 0: succesful exit;
             < 0: if INFO = -i, the i-th argument had an illegal value;
             = 1: periodic QZ iteration failed in the routines DGHURV,
                  DGHFYR or the SLICOT libaray routine MB03BB (QZ
                  iteration did not converge or computation of the
                  shifts failed);
             = 2: standard QZ iteration failed in the routines DGHFYR
                  or DGHFEX (called by DGHFXC);
             = 3: a numerically singular matrix was found in the routine
                  DGHFEY (called by DGHFXC);
             = 4: the singular value decomposition failed in the LAPACK
                  routine DGESVD (for ORTH = 'S').
Method
     First, the decompositions of S and H are computed via orthogonal
     matrices Q1 and Q2 and orthogonal symplectic matrices U1 and U2,
     such that

                                   ( T11  T12 )
       Q1' T U1 = Q1' J Z' J' U1 = (          ),
                                   (  0   T22 )

                  ( Z11  Z12 )
       U2' Z Q2 = (          ),                                     (2)
                  (  0   Z22 )

                  ( H11  H12 )
       Q1' H Q2 = (          ),
                  (  0   H22 )

     where T11, T22', Z11, Z22', H11 are upper triangular and H22' is
     upper quasi-triangular.

     Then, orthogonal matrices Q3, Q4 and U3 are found, for the
     matrices

       ~     ( T22'  0  )  ~     ( T11'  0  )  ~   (   0   H11 )
       Z11 = (          ), Z22 = (          ), H = (           ),
             (  0   Z11 )        (  0   Z22 )      ( -H22'  0  )

               ~          ~       ~          ~
     such that Z11 := U3' Z11 Q4, Z22 := U3' Z22 Q3 are upper
                    ~          ~
     triangular and H11 := Q3' H Q4 is upper quasi-triangular. The
     following matrices are computed:

       ~          ( -T12'  0  )        ~          (  0   H12 )
       Z12 := U3' (           ) Q3 and H12 := Q3' (          ) Q3.
                  (  0    Z12 )                   ( H12'  0  )

     Then, an orthogonal matrix Q and an orthogonal symplectic matrix U
     are found such that the eigenvalues with strictly negative real
     parts of the pencil

             ~    ~          ~    ~           ~    ~
           ( Z11  Z12 )'   ( Z11  Z12 )     ( H11  H12  )
       a J (      ~   ) J' (      ~   ) - b (      ~    )           (3)
           (  0   Z22 )    (  0   Z22 )     (  0  -H11' )

     are moved to the top of this pencil.

     Finally, an orthogonal basis of the right deflating subspace
     and an orthogonal basis of the companion subspace corresponding to
     the eigenvalues with strictly negative real part are computed.
     See also page 11 in [1] for more details.
References
     [1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
         Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
         Eigenproblems.
         Tech. Rep., Technical University Chemnitz, Germany,
         Nov. 2007.
Numerical Aspects
     
                                                               3
     The algorithm is numerically backward stable and needs O(N )
     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

*     DGHFDF EXAMPLE PROGRAM TEXT
*
*     .. Parameters ..
      INTEGER            NIN, NOUT
      PARAMETER          ( NIN = 5, NOUT = 6 )
      INTEGER            NMAX
      PARAMETER          ( NMAX = 50 )
      INTEGER            LDB, LDFG, LDQ, LDU, LDWORK, LDZ, LIWORK
      PARAMETER          ( LDB = NMAX/2, LDFG = NMAX/2, LDQ = 2*NMAX,
     $                     LDU = NMAX,   LDZ  = NMAX,
     $                     LDWORK = 10*NMAX*NMAX +
     $                              MAX( NMAX*NMAX +
     $                                   MAX( NMAX/2 + 252, 432 ),
     $                                   MAX( 8*NMAX +  48, 171 ) ),
     $                     LIWORK = MAX( NMAX + 18, NMAX/2 + 48,
     $                                   5*NMAX/2 + 1 ) )
*
*     .. Local Scalars ..
      CHARACTER          COMPQ, COMPU, ORTH
      INTEGER            I, INFO, IWARN, J, M, N, NEIG
*
*     .. Local Arrays ..
      LOGICAL            BWORK( NMAX/2 )
      INTEGER            IWORK( LIWORK )
      DOUBLE PRECISION   ALPHAI( NMAX/2 ),  ALPHAR( NMAX/2 ),
     $                   B( LDB, NMAX/2 ),    BETA( NMAX/2 ),
     $                   DWORK( LDWORK ), FG( LDFG, NMAX/2+1 ),
     $                   Q( LDQ, 2*NMAX ),  U( LDU, 2*NMAX ),
     $                   Z( LDZ, NMAX )
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*
*     .. External Subroutines ..
      EXTERNAL           DGHFDF
*
*     .. 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 real
*        skew-Hamiltonian/Hamiltonian pencil, corresponding to the
*        eigenvalues with strictly negative real part.
         CALL DGHFDF( COMPQ, COMPU, ORTH, N, Z, LDZ, B, LDB, FG, LDFG,
     $                NEIG, Q, LDQ, U, LDU, ALPHAR, ALPHAI, BETA, IWORK,
     $                LIWORK, DWORK, LDWORK, BWORK, IWARN, INFO )
*
         IF( INFO.NE.0 ) THEN
            WRITE( NOUT, FMT = 99997 ) INFO
         ELSE
            WRITE( NOUT, FMT = 99996 )
            DO 10 I = 1, N
               WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N )
   10       CONTINUE
            WRITE( NOUT, FMT = 99994 )
            WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
            WRITE( NOUT, FMT = 99993 )
            WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
            WRITE( NOUT, FMT = 99992 )
            WRITE( NOUT, FMT = 99995 ) (   BETA( I ), I = 1, M )
            IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
               WRITE( NOUT, FMT = 99991 )
               DO 20 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
   20          CONTINUE
            END IF
            IF( LSAME( COMPU, 'C' ) .AND. NEIG.GT.0 ) THEN
               WRITE( NOUT, FMT = 99990 )
               DO 30 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( U( I, J ), J = 1, NEIG )
   30          CONTINUE
            END IF
            IF( LSAME( COMPQ, 'C' ) .OR. LSAME( COMPU, 'C' ) )
     $         WRITE( NOUT, FMT = 99989 ) NEIG
         END IF
      END IF
      STOP
*
99999 FORMAT( 'DGHFDF EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from DGHFDF = ', I2 )
99996 FORMAT( 'The matrix Z on exit is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The vector ALPHAR is ' )
99993 FORMAT( 'The vector ALPHAI is ' )
99992 FORMAT( 'The vector BETA is ' )
99991 FORMAT( 'The deflating subspace corresponding to the ',
     $        'eigenvalues with negative real part is ' )
99990 FORMAT( 'The companion subspace corresponding to the ',
     $        'eigenvalues with negative real part is ' )
99989 FORMAT( 'The number of eigenvalues in the initial pencil with ',
     $        'negative real part is ', I2 )
      END
Program Data
DGHFDF EXAMPLE PROGRAM DATA
   C   C   P   8
   3.1472   4.5751  -0.7824   1.7874  -2.2308  -0.6126   2.0936   4.5974
   4.0579   4.6489   4.1574   2.5774  -4.5383  -1.1844   2.5469  -1.5961
  -3.7301  -3.4239   2.9221   2.4313  -4.0287   2.6552  -2.2397   0.8527
   4.1338   4.7059   4.5949  -1.0777   3.2346   2.9520   1.7970  -2.7619
   1.3236   4.5717   1.5574   1.5548   1.9483  -3.1313   1.5510   2.5127
  -4.0246  -0.1462  -4.6429  -3.2881  -1.8290  -0.1024  -3.3739  -2.4490
  -2.2150   3.0028   3.4913   2.0605   4.5022  -0.5441  -3.8100   0.0596
   0.4688  -3.5811   4.3399  -4.6817  -4.6555   1.4631  -0.0164   1.9908
   0.6882  -3.3782  -3.3435   1.8921
  -0.3061   2.9428   1.0198   2.4815
  -4.8810  -1.8878  -2.3703  -0.4946
  -1.6288   0.2853   1.5408  -4.1618
  -2.4013  -2.7102   0.3834  -3.9335   3.1730
  -3.1815  -2.3620   4.9613   4.6190   3.6869
   3.6929   0.7970   0.4986  -4.9537  -4.1556
   3.5303   1.2206  -1.4905   0.1325  -1.0022  
Program Results
DGHFDF EXAMPLE PROGRAM RESULTS
The matrix Z on exit is 
   4.4128   0.1059  -1.8709  -1.2963   4.3448  -2.7633  -2.3580  -2.1931
   0.0000  10.0337  -1.9797  -1.8052   1.0112  -1.1335  -1.2374  -0.3107
   0.0000   0.0000   8.9476  -1.8523   1.8578   0.5807   1.4157  -1.3007
   0.0000  -0.0000   0.0000   7.0889   2.1193   2.1634   2.4393  -0.1148
   0.0765   1.0139   0.0000  -1.5390   8.3187   5.0172  -0.7738   2.8626
   1.1884  -0.9225   0.0000   0.2905  -0.0000  -6.4090  -2.1994   2.5933
  -0.5446   0.1502   0.0000  -0.5299   0.0000   0.0000  -4.7155  -2.3817
   1.8739  -1.8461   0.0000  -0.0670   0.0000   0.0000   0.0000  -5.3153
The vector ALPHAR is 
   0.7353   0.0000   0.5168  -0.5168
The vector ALPHAI is 
   0.0000   0.7190   0.5610   0.5610
The vector BETA is 
   2.0000   2.8284  11.3137  11.3137
The deflating subspace corresponding to the eigenvalues with negative real part is 
  -0.2509   0.3670   0.0416
  -0.3267  -0.7968  -0.1019
   0.0263   0.0338  -0.5795
  -0.0139  -0.0491  -0.5217
  -0.4637   0.2992  -0.4403
  -0.1345   0.3071  -0.0917
  -0.1364   0.2013   0.3447
  -0.7601  -0.0495   0.2426
The companion subspace corresponding to the eigenvalues with negative real part is 
  -0.3219   0.6590   0.1693
  -0.5216  -0.1829  -0.0689
  -0.0413  -0.4664  -0.1359
   0.1310  -0.1702   0.4543
  -0.3598   0.2660   0.3355
  -0.5082  -0.0512  -0.6035
  -0.3582  -0.4513   0.4649
   0.2991   0.0932  -0.2207
The number of eigenvalues in the initial pencil with negative real part is  3

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