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<HTML>
<HEAD><TITLE>MB03ID - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03ID">MB03ID</A></H2>
<H3>
Moving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (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 move the eigenvalues with strictly negative real parts of an
N-by-N real skew-Hamiltonian/Hamiltonian pencil aS - bH in
structured Schur form, with
( 0 I ) ( A D ) ( B F )
S = J Z' J' Z, J = ( ), Z = ( ), H = ( ),
( -I 0 ) ( 0 C ) ( 0 -B' )
to the leading principal subpencil, while keeping the triangular
form. Above, A is upper triangular, B is upper quasi-triangular,
and C is lower triangular.
The matrices Z and H are transformed by an orthogonal symplectic
matrix U and an orthogonal matrix Q such that
( Aout Dout )
Zout = U' Z Q = ( ), and
( 0 Cout )
(1)
( Bout Fout )
Hout = J Q' J' H Q = ( ),
( 0 -Bout' )
where Aout, Bout and Cout remain in triangular form. The notation
M' denotes the transpose of the matrix M.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the orthogonal matrix Q
that fulfills (1) is computed.
Optionally, if COMPU = 'I' or COMPU = 'U', the orthogonal
symplectic matrix
( U1 U2 )
U = ( )
( -U2 U1 )
that fulfills (1) is computed.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03ID( COMPQ, COMPU, N, A, LDA, C, LDC, D, LDD, B,
$ LDB, F, LDF, Q, LDQ, U1, LDU1, U2, LDU2, NEIG,
$ IWORK, LIWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPU
INTEGER INFO, LDA, LDB, LDC, LDD, LDF, LDQ, LDU1, LDU2,
$ LDWORK, LIWORK, N, NEIG
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), F( LDF, * ),
$ Q( LDQ, * ), U1( LDU1, * ), U2( LDU2, * )
</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.
COMPU CHARACTER*1
Specifies whether or not the orthogonal symplectic
transformations should be accumulated in the arrays U1 and
U2, as follows:
= 'N': U1 and U2 are not computed;
= 'I': the arrays U1 and U2 are initialized internally,
and the submatrices U1 and U2 defining the
orthogonal symplectic matrix U are returned;
= 'U': the arrays U1 and U2 contain the corresponding
submatrices of an orthogonal symplectic matrix U0
on entry, and the updated submatrices U1 and U2
of the matrix product U0*U are returned, where U
is the product of the orthogonal symplectic
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).
C (input/output) DOUBLE PRECISION array, dimension
(LDC, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the lower triangular matrix C. The elements of the
strictly upper 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 Cout.
LDC INTEGER
The leading dimension of the array C. LDC >= 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 matrix D.
On exit, the leading N/2-by-N/2 part of this array
contains the transformed matrix Dout.
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 Z 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'.
U1 (input/output) DOUBLE PRECISION array, dimension
(LDU1, N/2)
On entry, if COMPU = 'U', then the leading N/2-by-N/2 part
of this array must contain the upper left block of a
given matrix U0, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper left block U1 of
the product of the input matrix U0 and the transformation
matrix U used to transform the matrices Z and H.
On exit, if COMPU = 'I', then the leading N/2-by-N/2 part
of this array contains the upper left block U1 of the
orthogonal symplectic transformation matrix U.
If COMPU = 'N' this array is not referenced.
LDU1 INTEGER
The leading dimension of the array U1.
LDU1 >= 1, if COMPU = 'N';
LDU1 >= MAX(1, N/2), if COMPU = 'I' or COMPU = 'U'.
U2 (input/output) DOUBLE PRECISION array, dimension
(LDU2, N/2)
On entry, if COMPU = 'U', then the leading N/2-by-N/2 part
of this array must contain the upper right block of a
given matrix U0, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper right block U2 of
the product of the input matrix U0 and the transformation
matrix U used to transform the matrices Z and H.
On exit, if COMPU = 'I', then the leading N/2-by-N/2 part
of this array contains the upper right block U2 of the
orthogonal symplectic transformation matrix U.
If COMPU = 'N' this array is not referenced.
LDU2 INTEGER
The leading dimension of the array U2.
LDU2 >= 1, if COMPU = 'N';
LDU2 >= MAX(1, N/2), if COMPU = 'I' or COMPU = '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 >= N+1.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
LDWORK INTEGER
The dimension of the array DWORK.
If COMPQ = 'N',
LDWORK >= MAX(2*N+48,171);
if COMPQ = 'I' or COMPQ = 'U',
LDWORK >= MAX(4*N+48,171).
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the periodic QZ algorithm did not converge in SLICOT
Library routine MB03BB;
= 2: an error occured during the execution of MB03CD;
= 3: an error occured during the execution of MB03GD.
</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 aC'*A - 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 aC'*A - 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 aC'*A - bB and MM denotes the current
number of blocks in aC'*A - bB with eigenvalues with
negative real parts.
The algorithm uses a sequence of orthogonal transformations as
described on page 25 in [1]. To achieve those transformations the
elementary subroutines MB03CD and MB03GD 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>
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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