## MB04CD

### Reducing a special real block (anti-)diagonal skew-Hamiltonian/Hamiltonian pencil in factored form to generalized Schur form

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

Purpose

```  To compute the transformed matrices A, B and D, using orthogonal
matrices Q1, Q2 and Q3 for a real N-by-N regular pencil

( A11   0  ) ( B11   0  )     (  0   D12 )
aA*B - bD = a (          ) (          ) - b (          ),    (1)
(  0   A22 ) (  0   B22 )     ( D21   0  )

where A11, A22, B11, B22 and D12 are upper triangular, D21 is
upper quasi-triangular and the generalized matrix product
-1        -1    -1        -1
A11   D12 B22   A22   D21 B11   is upper quasi-triangular, such
that Q3' A Q2, Q2' B Q1 are upper triangular, Q3' D Q1 is upper
quasi-triangular and the transformed pencil
a(Q3' A B Q1) - b(Q3' D Q1) is in generalized Schur form. The
notation M' denotes the transpose of the matrix M.

```
Specification
```      SUBROUTINE MB04CD( COMPQ1, COMPQ2, COMPQ3, N, A, LDA, B, LDB, D,
\$                   LDD, Q1, LDQ1, Q2, LDQ2, Q3, LDQ3, IWORK,
\$                   LIWORK, DWORK, LDWORK, BWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER          COMPQ1, COMPQ2, COMPQ3
INTEGER            INFO, LDA, LDB, LDD, LDQ1, LDQ2, LDQ3, LDWORK,
\$                   LIWORK, N
C     .. Array Arguments ..
LOGICAL            BWORK( * )
INTEGER            IWORK( * )
DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), D( LDD, * ),
\$                   DWORK( * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
\$                   Q3( LDQ3, * )

```
Arguments

Mode Parameters

```  COMPQ1  CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q1, as follows:
= 'N':  Q1 is not computed;
= 'I':  the array Q1 is initialized internally to the unit
matrix, and the orthogonal matrix Q1 is returned;
= 'U':  the array Q1 contains an orthogonal matrix Q01 on
entry, and the matrix Q01*Q1 is returned, where Q1
is the product of the orthogonal transformations
that are applied on the right to the pencil
aA*B - bD in (1).

COMPQ2  CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q2, as follows:
= 'N':  Q2 is not computed;
= 'I':  the array Q2 is initialized internally to the unit
matrix, and the orthogonal matrix Q2 is returned;
= 'U':  the array Q2 contains an orthogonal matrix Q02 on
entry, and the matrix Q02*Q2 is returned, where Q2
is the product of the orthogonal transformations
that are applied on the left to the pencil
aA*B - bD in (1).

COMPQ3  CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q3, as follows:
= 'N':  Q3 is not computed;
= 'I':  the array Q3 is initialized internally to the unit
matrix, and the orthogonal matrix Q3 is returned;
= 'U':  the array Q3 contains an orthogonal matrix Q01 on
entry, and the matrix Q03*Q3 is returned, where Q3
is the product of the orthogonal transformations
that are applied on the right to the pencil
aA*B - bD in (1).

```
Input/Output Parameters
```  N       (input) INTEGER
Order of the pencil aA*B - bD.  N >= 0, even.

A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On entry, the leading N-by-N block diagonal part of this
array must contain the matrix A in (1). The off-diagonal
blocks need not be set to zero.
On exit, the leading N-by-N part of this array contains
the transformed upper triangular matrix.

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

B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the leading N-by-N block diagonal part of this
array must contain the matrix B in (1). The off-diagonal
blocks need not be set to zero.
On exit, the leading N-by-N part of this array contains
the transformed upper triangular matrix.

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

D       (input/output) DOUBLE PRECISION array, dimension (LDD, N)
On entry, the leading N-by-N block anti-diagonal part of
this array must contain the matrix D in (1). The diagonal
blocks need not be set to zero.
On exit, the leading N-by-N part of this array contains
the transformed upper quasi-triangular matrix.

LDD     INTEGER
The leading dimension of the array D.  LDD >= MAX(1, N).

Q1      (input/output) DOUBLE PRECISION array, dimension (LDQ1, N)
On entry, if COMPQ1 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q01, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q01 and the transformation matrix Q1
used to transform the matrices A, B, and D.
On exit, if COMPQ1 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q1.
If COMPQ1 = 'N' this array is not referenced.

LDQ1    INTEGER
LDQ1 >= 1,         if COMPQ1 = 'N';
LDQ1 >= MAX(1, N), if COMPQ1 = 'I' or COMPQ1 = 'U'.

Q2      (input/output) DOUBLE PRECISION array, dimension (LDQ2, N)
On entry, if COMPQ2 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q02, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q02 and the transformation matrix Q2
used to transform the matrices A, B, and D.
On exit, if COMPQ2 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q2.
If COMPQ2 = 'N' this array is not referenced.

LDQ2    INTEGER
The leading dimension of the array Q2.
LDQ2 >= 1,         if COMPQ2 = 'N';
LDQ2 >= MAX(1, N), if COMPQ2 = 'I' or COMPQ2 = 'U'.

Q3      (input/output) DOUBLE PRECISION array, dimension (LDQ3, N)
On entry, if COMPQ3 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q03, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q03 and the transformation matrix Q3
used to transform the matrices A, B and D.
On exit, if COMPQ3 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q3.
If COMPQ3 = 'N' this array is not referenced.

LDQ3    INTEGER
The leading dimension of the array Q3.
LDQ3 >= 1,         if COMPQ3 = 'N';
LDQ3 >= MAX(1, N), if COMPQ3 = 'I' or COMPQ3 = 'U'.

```
Workspace
```  IWORK   INTEGER array, dimension (LIWORK)

LIWORK  INTEGER
The dimension of the array IWORK.
LIWORK >= MAX( N/2+1, 48 ).

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -20, DWORK(1) returns the minimum value
of LDWORK.

LDWORK  INTEGER
The dimension of the array DWORK.
LDWORK >= 3*N*N + MAX( N/2 + 252, 432 ).
For good performance LDWORK should be generally larger.

If LDWORK = -1  a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.

BWORK   LOGICAL array, dimension (N/2)

```
Error Indicator
```  INFO    INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the periodic QZ algorithm failed to reorder the
eigenvalues (the problem is very ill-conditioned) in
the SLICOT Library routine MB03KD;
= 2: the standard QZ algorithm failed in the LAPACK
routine DGGEV, called by the SLICOT routine MB03CD;
= 3: the standard QZ algorithm failed in the LAPACK
routines DGGES, called by the SLICOT routines MB03CD
or MB03ED;
= 4: the standard QZ algorithm failed to reorder the
eigenvalues in the LAPACK routine DTGSEN, called by
the SLICOT routine MB03CD.

```
Method
```  First, the periodic QZ algorithm (see also  and ) is applied
-1        -1    -1        -1
to the formal matrix product A11   D12 B22   A22   D21 B11   to
reorder the eigenvalues, i.e., orthogonal matrices V1, V2, V3, V4,
V5 and V6 are computed such that V2' A11 V1, V2' D12 V3,
V4' B22 V3, V5' A22 V4, V5' D21 V6 and V1' B11 V6 keep the
triangular form, but they can be partitioned into 2-by-2 block
forms and the last diagonal blocks correspond to all nonpositive
real eigenvalues of the formal product, and the first diagonal
blocks correspond to the remaining eigenvalues.

Second, Q1 = diag(V6, V3), Q2 = diag(V1, V4), Q3 = diag(V2, V5)
and

( AA11 AA12   0    0  )
(                     )
(   0  AA22   0    0  )
A := Q3' A Q2 =: (                     ),
(   0    0  AA33 AA34 )
(                     )
(   0    0    0  AA44 )

( BB11 BB12   0    0  )
(                     )
(   0  BB22   0    0  )
B := Q2' B Q1 =: (                     ),
(   0    0  BB33 BB34 )
(                     )
(   0    0    0  BB44 )

(   0    0  DD13 DD14 )
(                     )
(   0    0    0  DD24 )
D := Q3' D Q1 =: (                     ),
( DD31 DD32   0    0  )
(                     )
(   0  DD42   0    0  )

-1          -1     -1          -1
are set, such that AA22   DD24 BB44   AA44   DD42 BB22   has only
nonpositive real eigenvalues.

Third, the permutation matrix

(  I  0  0  0  )
(              )
(  0  0  I  0  )
P = (              ),
(  0  I  0  0  )
(              )
(  0  0  0  I  )

where I denotes the identity matrix of appropriate size is used to
transform aA*B - bD to block upper triangular form

( AA11   0  | AA12   0  )
(           |           )
(   0  AA33 |   0  AA34 )   ( AA1  *  )
A := P' A P = (-----------+-----------) = (         ),
(   0    0  | AA22   0  )   (  0  AA2 )
(           |           )
(   0    0  |   0  AA44 )

( BB11   0  | BB12   0  )
(           |           )
(   0  BB33 |   0  BB34 )   ( BB1  *  )
B := P' B P = (-----------+-----------) = (         ),
(   0    0  | BB22   0  )   (  0  BB2 )
(           |           )
(   0    0  |   0  BB44 )

(   0  DD13 |   0  DD14 )
(           |           )
( DD31   0  | DD32   0  )   ( DD1  *  )
D := P' D P = (-----------+-----------) = (         ).
(   0    0  |   0  DD24 )   (  0  DD2 )
(           |           )
(   0    0  | DD42   0  )

Then, further orthogonal transformations that are provided by the
SLICOT Library routines MB03ED and MB03CD are used to
triangularize the subpencil aAA1 BB1 - bDD1.

Finally, the subpencil aAA2 BB2 - bDD2 is triangularized by
applying a special permutation matrix.

```
References
```   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.

 Bojanczyk, A., Golub, G. H. and Van Dooren, P.
The periodic Schur decomposition: algorithms and applications.
In F.T. Luk (editor), Advanced Signal Processing Algorithms,
Architectures, and Implementations III, Proc. SPIE Conference,
vol. 1770, pp. 31-42, 1992.

 Hench, J. J. and Laub, A. J.
Numerical Solution of the discrete-time periodic Riccati
equation. IEEE Trans. Automat. Control, 39, 1197-1210, 1994.

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

```
```  None
```
Example

Program Text

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