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<HTML>
<HEAD><TITLE>MB03GD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03GD">MB03GD</A></H2>
<H3>
Exchanging eigenvalues of a real 2-by-2 or 4-by-4 block upper triangular skew-Hamiltonian/Hamiltonian pencil (factored version)
</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 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.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03GD( N, B, LDB, D, LDD, MACPAR, Q, LDQ, U, LDU,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDB, LDD, LDQ, LDU, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), D( LDD, * ), DWORK( * ),
$ MACPAR( * ), Q( LDQ, * ), U( LDU, * )
</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 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.
</PRE>
<B>Workspace</B>
<PRE>
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.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: succesful exit;
= 1: B11 or B22 is a (numerically) singular matrix.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The algorithm uses orthogonal transformations as described on page
22 in [1], but with an improved implementation.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[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.
</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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