ZGHFEY

Exchanging eigenvalues of a complex 2-by-2 skew-Hamiltonian/Hamiltonian pencil in structured Schur
form (factored version)

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

Purpose

     To compute a unitary matrix Q and a unitary symplectic matrix U
     for a complex regular 2-by-2 skew-Hamiltonian/Hamiltonian pencil
     aS - bH with S = J Z' J' Z, where

            (  Z11  Z12  )         (  H11  H12  )
        Z = (            ) and H = (            ),
            (   0   Z22  )         (   0  -H11' )

     such that U' Z Q, (J Q J' )' H Q are both upper triangular, but the  
     eigenvalues of (J Q J')' ( aS - bH ) Q are in reversed order.
     The matrices Q and U are represented by

            (  CO1  SI1  )         (  CO2  SI2  )
        Q = (            ) and U = (            ), respectively.
            ( -SI1' CO1  )         ( -SI2' CO2  )

     The notation M' denotes the conjugate transpose of the matrix M.
Specification
      SUBROUTINE ZGHFEY( Z11, Z12, Z22, H11, H12, CO1, SI1, CO2, SI2 )
C
C     .. Scalar Arguments ..
      DOUBLE PRECISION   CO1, CO2
      COMPLEX*16         H11, H12, SI1, SI2, Z11, Z12, Z22
Arguments

Input/Output Parameters

     Z11     (input) COMPLEX*16
             Upper left element of the non-trivial factor Z in the
             factorization of S.

     Z12     (input) COMPLEX*16
             Upper right element of the non-trivial factor Z in the
             factorization of S.

     Z22     (input) COMPLEX*16
             Lower right element of the non-trivial factor Z in the
             factorization of S.

     H11     (input) COMPLEX*16
             Upper left element of the Hamiltonian matrix H.

     H12     (input) COMPLEX*16
             Upper right element of the Hamiltonian matrix H.

     CO1     (output) DOUBLE PRECISION
             Upper left element of Q.

     SI1     (output) COMPLEX*16
             Upper right element of Q.

     CO2     (output) DOUBLE PRECISION
             Upper left element of U.

     SI2     (output) COMPLEX*16
             Upper right element of U.
Method
     The algorithm uses unitary and unitary symplectic transformations
     as described on page 37 in [1].
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
     
     The algorithm is numerically backward stable.
Further Comments
   
     None.
Example

Program Text

     None.
Program Data
     None.   
Program Results
     None.

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