DGHFEY

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

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

Purpose

     To compute an orthogonal matrix Q and an orthogonal symplectic
     matrix U for a real regular 2-by-2 or 4-by-4 skew-Hamiltonian/
     Hamiltonian pencil a J B' J' B - b D with

           ( B11  B12 )      (  D11  D12  )      (  0  I  )
       B = (          ), D = (            ), J = (        ),
           (  0   B22 )      (   0  -D11' )      ( -I  0  )

     such that J Q' J' D Q and U' B Q keep block triangular form, but
     the eigenvalues are reordered. The notation M' denotes the
     transpose of the matrix M.  
Specification
      SUBROUTINE DGHFEY( N, B, LDB, D, LDD, MACPAR, Q, LDQ, U, LDU,
     $                   DWORK, LDWORK, INFO )
C
C     .. Scalar Arguments ..
      INTEGER            INFO, LDB, LDD, LDQ, LDU, LDWORK, N
C
C     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), D( LDD, * ), DWORK( * ),
     $                   MACPAR( * ), Q( LDQ, * ), U( LDU, * )
Arguments

Input/Output Parameters

     N       (input) INTEGER
             The order of the pencil a J B' J' B - b D. N = 2 or N = 4.

     B       (input) DOUBLE PRECISION array, dimension (LDB, N)
             The leading N-by-N part of this array must contain the
             non-trivial factor of the decomposition of the
             skew-Hamiltonian input matrix J B' J' B. The (2,1) block
             is not referenced.

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

     D       (input) DOUBLE PRECISION array, dimension (LDD, N)
             The leading N/2-by-N part of this array must contain the
             first block row of the second matrix of a J B' J' B - b D.
             The matrix D has to be Hamiltonian. The strict lower
             triangle of the (1,2) block is not referenced.

     LDD     INTEGER
             The leading dimension of the array D.  LDD >= 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.

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

     LDU     INTEGER
             The leading dimension of the array U.  LDU >= N.
Workspace
     DWORK   DOUBLE PRECISION array, dimension (LDWORK)
             If N = 2 then DWORK is not referenced.

     LDWORK  INTEGER
             The length of the array DWORK.
             If N = 2 then LDWORK >= 0; if N = 4 then LDWORK >= 12.
Error Indicator
     INFO    INTEGER
             = 0: succesful exit;
             = 1: B11 or B22 is a (numerically) singular matrix.
Method
     The algorithm uses orthogonal transformations as described on page
     22 in [1], but with an improved implementation.
References
     [1] 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.

Return to index