ZGHUDP

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

           (  A  D  )         (  B  F  )
       S = (        ) and H = (        ).                           (1)
           (  E  A' )         (  G -B' )

     The structured Schur form of the embedded real skew-Hamiltonian/
     skew-Hamiltonian pencil aB_S - bB_T, defined as


             (  Re(A)  -Im(A)  |  Re(D)  -Im(D)  )
             (                 |                 )
             (  Im(A)   Re(A)  |  Im(D)   Re(D)  )
             (                 |                 )
       B_S = (-----------------+-----------------) , and
             (                 |                 )
             (  Re(E)  -Im(E)  |  Re(A')  Im(A') )
             (                 |                 )
             (  Im(E)   Re(E)  | -Im(A')  Re(A') )
                                                                    (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. The notation M'
     denotes the conjugate transpose of the matrix M. Optionally,
     if COMPQ = 'C', an orthonormal basis of the right deflating
     subspace of the pencil aS - bH, corresponding to the eigenvalues
     with strictly negative real part, is computed. Namely, after
     transforming aB_S - bB_H by unitary matrices, we have

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

     and the eigenvalues with strictly negative real part of the
     complex pencil aB_Sout - bB_Hout are moved to the top. The
     embedding doubles the multiplicities of the eigenvalues of the
     pencil aS - bH.  
Specification
      SUBROUTINE ZGHUDP( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
     $                   LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
     $                   IWORK, DWORK, LDWORK, ZWORK, LZWORK, BWORK,
     $                   INFO )
C
C     .. Scalar Arguments ..
      CHARACTER          COMPQ, ORTH
      INTEGER            INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
     $                   LZWORK, N, NEIG
C
C     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
      COMPLEX*16         A( LDA, * ), B( LDB, * ), DE( LDDE, * ),
     $                   FG( LDFG, * ), Q( LDQ, * ), ZWORK( * )
Arguments

Mode Parameters

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

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

     A       (input/output) COMPLEX*16 array, dimension (LDA, N)
             On entry, the leading N/2-by-N/2 part of this array must
             contain the matrix A.
             On exit, if COMPQ = '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, but it is preserved.
             If COMPQ = 'N', this array is unchanged on exit.

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

     DE      (input/output) COMPLEX*16 array, dimension (LDDE, N)
             On entry, the leading N/2-by-N/2 lower triangular part of
             this array must contain the lower triangular part of the
             skew-Hermitian matrix E, 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 skew-Hermitian matrix D.
             On exit, if COMPQ = 'C', the leading N-by-N part of this
             array contains the skew-Hermitian matrix BD in (3) (see
             also METHOD). The strictly lower triangular part of the
             input matrix is preserved.
             If COMPQ = 'N', this array is unchanged on exit.

     LDDE    INTEGER
             The leading dimension of the array DE.  LDDE >= 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', 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; the elements below the first subdiagonal of the
             input matrix are preserved.
             If COMPQ = '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', 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', 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', the number of eigenvalues in aS - bH with
             strictly negative real part.

     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'.

     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 >= N, if ORTH =  'P';
             LIWORK >= 0, if ORTH <> 'P'.

     DWORK   DOUBLE PRECISION array, dimension (LDWORK)
             On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
             On exit, if INFO = -20, DWORK(1) returns the minimum value
             of LDWORK.

     LDWORK  INTEGER
             The dimension of the array DWORK.
             LDWORK >= MAX( 1,  5*N*N + 3*N ), if COMPQ = 'N';
             LDWORK >= MAX( 1, 11*N*N + 2*N ), if COMPQ = 'C'.
             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 = -22, ZWORK(1) returns the minimum
             value of LZWORK.

     LZWORK  INTEGER
             The dimension of the array ZWORK.
             LZWORK >= 1,       if COMPQ = 'N';
             LZWORK >= 8*N + 4, if COMPQ = 'C'.
             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';
             LBWORK >= N, if COMPQ = 'C'.
Error Indicator
     INFO    INTEGER
             = 0: succesful exit;
             < 0: if INFO = -i, the i-th argument had an illegal value;
             = 1: QZ iteration failed in the routine DGHUSP (QZ
                  iteration did not converge or computation of the
                  shifts failed);
             = 2: QZ iteration failed in the LAPACK routine ZHGEQZ 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_S and B_T, of the
     matrices S and T, are determined and, subsequently, the routine
     DGHUSP is applied to compute the structured Schur form, i.e., the
     factorizations

     ~                     (  S11  S12  )
     B_S = J Q' J' B_S Q = (            ) and
                           (   0   S11' )

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

     where Q is real orthogonal, S11 is upper triangular, and T11 is
     upper quasi-triangular.

     Second, the routine ZGHUXP is applied, to compute a unitary matrix
     ~
     Q, such that

                        ~    ~
       ~     ~   ~   (  S11  S12  )
     J Q' J' B_S Q = (       ~    ) =: B_Sout,
                     (   0   S11' )

       ~        ~    ~   (  H11  H12  )
     J Q' J'(-i*B_T) Q = (            ) =: B_Hout,
                         (   0  -H11' )
          ~                                               ~       ~
     with S11, H11 upper triangular, and such that Spec_-(B_S, -i*B_T)
     is contained in the spectrum of the 2*NEIG-by-2*NEIG leading
                          ~
     principal subpencil aS11 - bH11.

     Finally, the right deflating subspace is computed.
     See also page 22 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.
     For large values of N, the routine applies the transformations on
     panels of columns. The user may specify in INFO the desired number
     of columns. If on entry INFO <= 0, then the routine estimates a
     suitable value of this number.
Example

Program Text

     None.
Program Data
     None.
Program Results
     None.

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