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<HTML>
<HEAD><TITLE>MB03JP - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03JP">MB03JP</A></H2>
<H3>
Moving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (applying transformations on panels of columns)
</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 move the eigenvalues with strictly negative real parts of an
N-by-N real skew-Hamiltonian/Hamiltonian pencil aS - bH in
structured Schur form,
( A D ) ( B F )
S = ( ), H = ( ),
( 0 A' ) ( 0 -B' )
with A upper triangular and B upper quasi-triangular, to the
leading principal subpencil, while keeping the triangular form.
The notation M' denotes the transpose of the matrix M.
The matrices S and H are transformed by an orthogonal matrix Q
such that
( Aout Dout )
Sout = J Q' J' S Q = ( ),
( 0 Aout' )
(1)
( Bout Fout ) ( 0 I )
Hout = J Q' J' H Q = ( ), with J = ( ),
( 0 -Bout' ) ( -I 0 )
where Aout is upper triangular and Bout is upper quasi-triangular.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the orthogonal matrix Q
that fulfills (1), is computed.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03JP( COMPQ, N, A, LDA, D, LDD, B, LDB, F, LDF, Q,
$ LDQ, NEIG, IWORK, LIWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
INTEGER INFO, LDA, LDB, LDD, LDF, LDQ, LDWORK, LIWORK,
$ N, NEIG
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( LDD, * ),
$ DWORK( * ), F( LDF, * ), Q( LDQ, * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
COMPQ CHARACTER*1
Specifies whether or not the orthogonal transformations
should be accumulated in the array Q, as follows:
= 'N': Q is not computed;
= 'I': the array Q is initialized internally to the unit
matrix, and the orthogonal matrix Q is returned;
= 'U': the array Q contains an orthogonal matrix Q0 on
entry, and the matrix Q0*Q is returned, where Q
is the product of the orthogonal transformations
that are applied to the pencil aS - bH to reorder
the eigenvalues.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix A. The elements of the
strictly lower triangular part of this array are not used.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Aout.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, N/2).
D (input/output) DOUBLE PRECISION array, dimension
(LDD, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the skew-symmetric
matrix D. The diagonal need not be set to zero.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper triangular part of the
matrix Dout.
The strictly lower triangular part of this array is
not referenced, except for the element D(N/2,N/2-1), but
its initial value is preserved.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1, N/2).
B (input/output) DOUBLE PRECISION array, dimension
(LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper quasi-triangular matrix B.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper quasi-triangular part of
the matrix Bout.
The part below the first subdiagonal of this array is
not referenced.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, N/2).
F (input/output) DOUBLE PRECISION array, dimension
(LDF, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular part of the symmetric matrix
F.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed upper triangular part of the
matrix Fout.
The strictly lower triangular part of this array is not
referenced, except for the element F(N/2,N/2-1), but its
initial value is preserved.
LDF INTEGER
The leading dimension of the array F. LDF >= MAX(1, N/2).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
On entry, if COMPQ = 'U', then the leading N-by-N part of
this array must contain a given matrix Q0, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q0 and the transformation matrix Q
used to transform the matrices S and H.
On exit, if COMPQ = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q.
If COMPQ = 'N' this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.
NEIG (output) INTEGER
The number of eigenvalues in aS - bH with strictly
negative real part.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (LIWORK)
LIWORK INTEGER
The dimension of the array IWORK.
LIWORK >= 3*N-3.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The dimension of the array DWORK.
If COMPQ = 'N',
LDWORK >= MAX(2*N+32,108)+5*N/2;
if COMPQ = 'I' or COMPQ = 'U',
LDWORK >= MAX(4*N+32,108)+5*N/2.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: error occured during execution of MB03DD;
= 2: error occured during execution of MB03HD.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The algorithm reorders the eigenvalues like the following scheme:
Step 1: Reorder the eigenvalues in the subpencil aA - bB.
I. Reorder the eigenvalues with negative real parts to the
top.
II. Reorder the eigenvalues with positive real parts to the
bottom.
Step 2: Reorder the remaining eigenvalues with negative real
parts in the pencil aS - bH.
I. Exchange the eigenvalues between the last diagonal block
in aA - bB and the last diagonal block in aS - bH.
II. Move the eigenvalues of the R-th block to the (MM+1)-th
block, where R denotes the number of upper quasi-
triangular blocks in aA - bB and MM denotes the current
number of blocks in aA - bB with eigenvalues with negative
real parts.
The algorithm uses a sequence of orthogonal transformations as
described on page 33 in [1]. To achieve those transformations the
elementary subroutines MB03DD and MB03HD are called for the
corresponding matrix structures.
</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> 3
The algorithm is numerically backward stable and needs O(N ) real
floating point operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
For large values of N, the routine applies the transformations on
panels of columns. The user may specify in INFO the desired number
of columns. If on entry INFO <= 0, then the routine estimates a
suitable value of this number.
</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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