## MB03HD

### Exchanging eigenvalues of a real 2-by-2 or 4-by-4 skew-Hamiltonian/Hamiltonian pencil in structured Schur form

[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 MB03HD( N, A, LDA, B, LDB, MACPAR, Q, LDQ, DWORK,
\$                   INFO )
C     .. Scalar Arguments ..
INTEGER            INFO, LDA, LDB, LDQ, N
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, but slightly perturbed values
have been used. This is a warning.

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.