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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.

**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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