## MB03GZ

### 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 MB03GZ( Z11, Z12, Z22, H11, H12, CO1, SI1, CO2, SI2 )
    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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