## MB03IZ

### Moving eigenvalues with negative real parts of a complex 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 complex skew-Hamiltonian/Hamiltonian pencil aS - bH in
structured Schur form, with

(  0  I  )
S = J Z' J' Z, where J = (        ),
( -I  0  )

to the leading principal subpencil, while keeping the triangular
form. On entry, we have

(  A  D  )      (  B  F  )
Z = (        ), H = (        ),
(  0  C  )      (  0 -B' )

where A and B are upper triangular and C is lower triangular.
Z and H are transformed by a unitary symplectic matrix U and a
unitary 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 conjugate transpose of the matrix M.
Optionally, if COMPQ = 'I' or COMPQ = 'U', the unitary matrix Q
that fulfills (1) is computed.
Optionally, if COMPU = 'I' or COMPU = 'U', the unitary symplectic
matrix

(  U1  U2  )
U = (          )
( -U2  U1  )

that fulfills (1) is computed.

```
Specification
```      SUBROUTINE MB03IZ( COMPQ, COMPU, N, A, LDA, C, LDC, D, LDD, B,
\$                   LDB, F, LDF, Q, LDQ, U1, LDU1, U2, LDU2, NEIG,
\$                   TOL, INFO )
C     .. Scalar Arguments ..
CHARACTER          COMPQ, COMPU
INTEGER            INFO, LDA, LDB, LDC, LDD, LDF, LDQ, LDU1, LDU2,
\$                   N, NEIG
DOUBLE PRECISION   TOL
C     .. Array Arguments ..
COMPLEX*16         A( LDA, * ), B( LDB, * ), C( LDC, * ),
\$                   D( LDD, * ), F( LDF, * ), Q( LDQ, * ),
\$                   U1( LDU1, * ), U2( LDU2, * )

```
Arguments

Mode Parameters

```  COMPQ   CHARACTER*1
Specifies whether or not the unitary 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 unitary matrix Q is returned;
= 'U':  the array Q contains a unitary matrix Q0 on
entry, and the matrix Q0*Q is returned, where Q
is the product of the unitary transformations
that are applied to the pencil aS - bH to reorder
the eigenvalues.

COMPU   CHARACTER*1
Specifies whether or not the unitary 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
unitary symplectic matrix U are returned;
= 'U':  the arrays U1 and U2 contain the corresponding
submatrices of a unitary 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 unitary 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) COMPLEX*16 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.
On exit, the leading  N/2-by-N/2 part of this array
contains the transformed matrix Aout.
The strictly lower triangular part of this array is not
referenced.

LDA     INTEGER
The leading dimension of the array A.  LDA >= MAX(1, N/2).

C       (input/output) COMPLEX*16 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.
On exit, the leading  N/2-by-N/2 part of this array
contains the transformed matrix Cout.
The strictly upper triangular part of this array is not
referenced.

LDC     INTEGER
The leading dimension of the array C.  LDC >= MAX(1, N/2).

D       (input/output) COMPLEX*16 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) COMPLEX*16 array, dimension (LDB, N/2)
On entry, the leading N/2-by-N/2 part of this array must
contain the upper triangular matrix B.
On exit, the leading  N/2-by-N/2 part of this array
contains the transformed matrix Bout.
The strictly lower triangular part of this array is not
referenced.

LDB     INTEGER
The leading dimension of the array B.  LDB >= MAX(1, N/2).

F       (input/output) COMPLEX*16 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 Hermitian matrix
F.
On exit, the leading  N/2-by-N/2 part of this array
contains the transformed matrix Fout.
The strictly lower triangular part of this array is not
referenced.

LDF     INTEGER
The leading dimension of the array F.  LDF >= MAX(1, N/2).

Q       (input/output) COMPLEX*16 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 unitary 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) COMPLEX*16 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 S 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
unitary 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) COMPLEX*16 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 S 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
unitary 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.

```
Tolerances
```  TOL     DOUBLE PRECISION
The tolerance used to decide the sign of the eigenvalues.
If the user sets TOL > 0, then the given value of TOL is
used. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by MIN(N,10)*EPS, is
used instead, where EPS is the machine precision (see
LAPACK Library routine DLAMCH). A larger value might be
needed for pencils with multiple eigenvalues.

```
Error Indicator
```  INFO    INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value.

```
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.
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 in the N/2-th place to the (MM+1)-th
place, where MM denotes the current number of eigenvalues
with negative real parts in aC'*A - bB.

The algorithm uses a sequence of unitary transformations as
described on page 38 in . To achieve those transformations the
elementary SLICOT Library subroutines MB03CZ and MB03GZ are called
for the corresponding matrix structures.

```
References
```   Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.

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

```
```  None
```
Example

Program Text

```  None
```
Program Data
```  None
```
Program Results
```  None
```