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<HTML>
<HEAD><TITLE>MB03HD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03HD">MB03HD</A></H2>
<H3>
Exchanging eigenvalues of a real 2-by-2 or 4-by-4 skew-Hamiltonian/Hamiltonian pencil in structured Schur form
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
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.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
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, * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (24)
If N = 2, then DWORK is not referenced.
</PRE>
<B>Error Indicator</B>
<PRE>
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.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The algorithm uses orthogonal transformations as described on page
31 in [2]. The structure is exploited.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm is numerically backward stable.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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