## MB03ID

### Moving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (factored version)

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

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.

Specification
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, * )

Arguments

Mode Parameters

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.

Input/Output Parameters
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.

Workspace
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).

Error Indicator
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.

Method
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.

References
[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.

Numerical Aspects
3
The algorithm is numerically backward stable and needs O(N ) real
floating point operations.