## 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 . The structure is exploited.

```
References
```   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.

 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.

```
```  None
```
Example

Program Text

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Program Data
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Program Results
```  None
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