## MB03JD

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

[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,

(  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.

Specification
SUBROUTINE MB03JD( 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, * )

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.

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).

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.

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+32,108);
if COMPQ = 'I' or COMPQ = 'U',
LDWORK >= MAX(4*N+32,108).

Error Indicator
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.

Method
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.

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.