MB03GZ

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

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.

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.

  The algorithm uses unitary and unitary symplectic transformations
  as described on page 37 in [1].

  [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.

  The algorithm is numerically backward stable.

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Program Text

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