DGHUEY

Exchanging eigenvalues of a real 2-by-2 or 4-by-4 block upper triangular
skew-Hamiltonian/Hamiltonian pencil (unfactored version)

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

Purpose

     To determine an orthogonal matrix Q, for a real regular 2-by-2 or
     4-by-4 skew-Hamiltonian/Hamiltonian pencil

                     ( A11 A12  )     ( B11  B12  )
         aA - bB = a (          ) - b (           )
                     (  0  A11' )     (  0  -B11' )

     in structured Schur form, such that  J Q' J' (aA - bB) Q  is still
     in structured Schur form but the eigenvalues are exchanged. The
     notation M' denotes the transpose of the matrix M.
Specification
      SUBROUTINE DGHUEY( N, A, LDA, B, LDB, MACPAR, Q, LDQ, DWORK,
     $                   INFO )
C
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LDQ, N
C
C     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DWORK( * ),
     $                   MACPAR( * ), Q( LDQ, * )
Arguments

Input/Output Parameters

     N       (input) INTEGER
             The order of the pencil aA - bB.  N = 2 or N = 4.

     A       (input) DOUBLE PRECISION array, dimension (LDA, N)
             If N = 4, the leading N/2-by-N upper trapezoidal part of
             this array must contain the first block row of the skew-
             Hamiltonian matrix A of the pencil aA - bB in structured
             Schur form. Only the entries (1,1), (1,2), (1,4), and
             (2,2) are referenced.
             If N = 2, this array is not referenced.

     LDA     INTEGER
             The leading dimension of the array A.  LDA >= N/2.

     B       (input) DOUBLE PRECISION array, dimension (LDB, N)
             The leading N/2-by-N part of this array must contain the
             first block row of the Hamiltonian matrix B of the
             pencil aA - bB in structured Schur form. The entry (2,3)
             is not referenced.

     LDB     INTEGER
             The leading dimension of the array B.  LDB >= N/2.

     MACPAR  (input)  DOUBLE PRECISION array, dimension (2)
             Machine parameters:
             MACPAR(1)  (machine precision)*base, DLAMCH( 'P' );
             MACPAR(2)  safe minimum,             DLAMCH( 'S' ).
             This argument is not used for N = 2.

     Q       (output) DOUBLE PRECISION array, dimension (LDQ, N)
             The leading N-by-N part of this array contains the
             orthogonal transformation matrix Q.

     LDQ     INTEGER
             The leading dimension of the array Q.  LDQ >= N.
Workspace
     DWORK   DOUBLE PRECISION array, dimension (24)
             If N = 2, then DWORK is not referenced.
Error Indicator
     INFO    INTEGER
             = 0: succesful exit;
             = 1: the leading N/2-by-N/2 block of the matrix B is
                  numerically singular.
Method
     The algorithm uses orthogonal transformations as described on page
     31 in [2]. The structure is exploited.
References
     [1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
         Numerical computation of deflating subspaces of skew-
         Hamiltonian/Hamiltonian pencils.
         SIAM J. Matrix Anal. Appl., 24 (1), pp. 165-190, 2002.

     [2] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
         Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
         Eigenproblems.
         Tech. Rep., Technical University Chemnitz, Germany,
         Nov. 2007.
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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