## MB03TD

### Reordering the diagonal blocks of a matrix in (skew-)Hamiltonian Schur form

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

Purpose

```  To reorder a matrix X in skew-Hamiltonian Schur form:

[  A   G  ]          T
X  =  [       T ],   G = -G,
[  0   A  ]

or in Hamiltonian Schur form:

[  A   G  ]          T
X  =  [       T ],   G =  G,
[  0  -A  ]

where A is in upper quasi-triangular form, so that a selected
cluster of eigenvalues appears in the leading diagonal blocks
of the matrix A (in X) and the leading columns of [ U1; -U2 ] form
an orthonormal basis for the corresponding right invariant
subspace.

If X is skew-Hamiltonian, then each eigenvalue appears twice; one
copy corresponds to the j-th diagonal element and the other to the
(n+j)-th diagonal element of X. The logical array LOWER controls
which copy is to be reordered to the leading part of A.

If X is Hamiltonian then the eigenvalues appear in pairs
(lambda,-lambda); lambda corresponds to the j-th diagonal
element and -lambda to the (n+j)-th diagonal element of X.
The logical array LOWER controls whether lambda or -lambda is to
be reordered to the leading part of A.

The matrix A must be in Schur canonical form (as returned by the
LAPACK routine DHSEQR), that is, block upper triangular with
1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has
its diagonal elements equal and its off-diagonal elements of
opposite sign.

```
Specification
```      SUBROUTINE MB03TD( TYP, COMPU, SELECT, LOWER, N, A, LDA, G, LDG,
\$                   U1, LDU1, U2, LDU2, WR, WI, M, DWORK, LDWORK,
\$                   INFO )
C     .. Scalar Arguments ..
CHARACTER         COMPU, TYP
INTEGER           INFO, LDA, LDG, LDU1, LDU2, LDWORK, M, N
C     .. Array Arguments ..
LOGICAL           LOWER(*), SELECT(*)
DOUBLE PRECISION  A(LDA,*), DWORK(*), G(LDG,*), U1(LDU1,*),
\$                  U2(LDU2,*), WI(*), WR(*)

```
Arguments

Mode Parameters

```  TYP     CHARACTER*1
Specifies the type of the input matrix X:
= 'S': X is skew-Hamiltonian;
= 'H': X is Hamiltonian.

COMPU   CHARACTER*1
= 'U': update the matrices U1 and U2 containing the
Schur vectors;
= 'N': do not update U1 and U2.

SELECT  (input/output) LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster.
To select a real eigenvalue w(j), SELECT(j) must be set
to .TRUE.. To select a complex conjugate pair of
eigenvalues w(j) and w(j+1), corresponding to a 2-by-2
diagonal block, both SELECT(j) and SELECT(j+1) must be set
to .TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.

LOWER   (input/output) LOGICAL array, dimension (N)
LOWER controls which copy of a selected eigenvalue is
included in the cluster. If SELECT(j) is set to .TRUE.
for a real eigenvalue w(j); then LOWER(j) must be set to
.TRUE. if the eigenvalue corresponding to the (n+j)-th
diagonal element of X is to be reordered to the leading
part; and LOWER(j) must be set to .FALSE. if the
eigenvalue corresponding to the j-th diagonal element of
X is to be reordered to the leading part. Similarly, for
a complex conjugate pair of eigenvalues w(j) and w(j+1),
both LOWER(j) and LOWER(j+1) must be set to .TRUE. if the
eigenvalues corresponding to the (n+j:n+j+1,n+j:n+j+1)
diagonal block of X are to be reordered to the leading
part.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrix A. N >= 0.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper quasi-triangular matrix A in Schur
canonical form.
On exit, the leading N-by-N part of this array contains
the reordered matrix A, again in Schur canonical form,
with the selected eigenvalues in the diagonal blocks.

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

G       (input/output) DOUBLE PRECISION array, dimension (LDG,N)
On entry, if TYP = 'S', the leading N-by-N part of this
array must contain the strictly upper triangular part of
the skew-symmetric matrix G. The rest of this array is not
referenced.
On entry, if TYP = 'H', the leading N-by-N part of this
array must contain the upper triangular part of the
symmetric matrix G. The rest of this array is not
referenced.
On exit, if TYP = 'S', the leading N-by-N part of this
array contains the strictly upper triangular part of the
skew-symmetric matrix G, updated by the orthogonal
symplectic transformation which reorders X.
On exit, if TYP = 'H', the leading N-by-N part of this
array contains the upper triangular part of the symmetric
matrix G, updated by the orthogonal symplectic
transformation which reorders X.

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

U1      (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
On entry, if COMPU = 'U', the leading N-by-N part of this
array must contain U1, the (1,1) block of an orthogonal
symplectic matrix U = [ U1, U2; -U2, U1 ].
On exit, if COMPU = 'U', the leading N-by-N part of this
array contains the (1,1) block of the matrix U,
postmultiplied by the orthogonal symplectic transformation
which reorders X. The leading M columns of U form an
orthonormal basis for the specified invariant subspace.
If COMPU = 'N', this array is not referenced.

LDU1    INTEGER
The leading dimension of the array U1.
LDU1 >= MAX(1,N),  if COMPU = 'U';
LDU1 >= 1,         otherwise.

U2      (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
On entry, if COMPU = 'U', the leading N-by-N part of this
array must contain U2, the (1,2) block of an orthogonal
symplectic matrix U = [ U1, U2; -U2, U1 ].
On exit, if COMPU = 'U', the leading N-by-N part of this
array contains the (1,2) block of the matrix U,
postmultiplied by the orthogonal symplectic transformation
which reorders X.
If COMPU = 'N', this array is not referenced.

LDU2    INTEGER
The leading dimension of the array U2.
LDU2 >= MAX(1,N),  if COMPU = 'U';
LDU2 >= 1,         otherwise.

WR      (output) DOUBLE PRECISION array, dimension (N)
WI      (output) DOUBLE PRECISION array, dimension (N)
The real and imaginary parts, respectively, of the
reordered eigenvalues of A. The eigenvalues are stored
in the same order as on the diagonal of A, with
WR(i) = A(i,i) and, if A(i:i+1,i:i+1) is a 2-by-2 diagonal
block, WI(i) > 0 and WI(i+1) = -WI(i). Note that if an
eigenvalue is sufficiently ill-conditioned, then its value
may differ significantly from its value before reordering.

M       (output) INTEGER
The dimension of the specified invariant subspace.
0 <= M <= N.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0,  DWORK(1)  returns the optimal
value of LDWORK.
On exit, if  INFO = -18,  DWORK(1)  returns the minimum
value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.  LDWORK >= MAX(1,N).

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.
= 1:  reordering of X failed because some eigenvalue pairs
are too close to separate (the problem is very
ill-conditioned); X may have been partially
reordered, and WR and WI contain the eigenvalues in
the same order as in X.

```
References
```   Bai, Z. and Demmel, J.W.
On Swapping Diagonal Blocks in Real Schur Form.
Linear Algebra Appl., 186, pp. 73-95, 1993.

 Benner, P., Kressner, D., and Mehrmann, V.
Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory,
Algorithms and Applications. Techn. Report, TU Berlin, 2003.

```
```  None
```
Example

Program Text

```*     MB03TD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER        ( ZERO = 0.0D0 )
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX
PARAMETER        ( NMAX = 100 )
INTEGER          LDA, LDG, LDRES, LDU1, LDU2, LDWORK
PARAMETER        ( LDA  = NMAX, LDG  = NMAX, LDRES  = NMAX,
\$                   LDU1 = NMAX, LDU2 = NMAX, LDWORK = 8*NMAX )
*     .. Local Scalars ..
CHARACTER*1      COMPU, TYP
INTEGER          I, INFO, J, N, M
*     .. Local Arrays ..
LOGICAL          LOWER(NMAX), SELECT(NMAX)
DOUBLE PRECISION A(LDA, NMAX), DWORK(LDWORK), G(LDG, NMAX),
\$                 RES(LDRES,NMAX), U1(LDU1,NMAX), U2(LDU2,NMAX),
\$                 WR(NMAX), WI(NMAX)
*     .. External Functions ..
LOGICAL          LSAME
DOUBLE PRECISION MA02JD
EXTERNAL         LSAME, MA02JD
*     .. External Subroutines ..
EXTERNAL         MB03TD
*     .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * )  N, TYP, COMPU
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( SELECT(J), J = 1,N )
READ ( NIN, FMT = * ) ( LOWER(J), J = 1,N )
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( COMPU, 'U' ) ) THEN
READ ( NIN, FMT = * ) ( ( U1(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( U2(I,J), J = 1,N ), I = 1,N )
END IF
CALL MB03TD( TYP, COMPU, SELECT, LOWER, N, A, LDA, G, LDG, U1,
\$                LDU1, U2, LDU2, WR, WI, M, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( COMPU, 'U' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10  I = 1, N
WRITE ( NOUT, FMT = 99994 )
\$                  ( U1(I,J), J = 1,N ), ( U2(I,J), J = 1,N )
10             CONTINUE
DO 20  I = 1, N
WRITE ( NOUT, FMT = 99994 )
\$                  ( -U2(I,J), J = 1,N ), ( U1(I,J), J = 1,N )
20             CONTINUE
WRITE ( NOUT, FMT = 99992 ) MA02JD( .FALSE., .FALSE., N,
\$                 U1, LDU1, U2, LDU2, RES, LDRES )
END IF
*
WRITE ( NOUT, FMT = 99996 )
DO 30  I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( A(I,J), J = 1,N )
30          CONTINUE
*
WRITE ( NOUT, FMT = 99995 )
IF ( LSAME( TYP, 'S' ) ) THEN
DO 40  I = 1, N
WRITE ( NOUT, FMT = 99994 )
\$               ( -G(J,I), J = 1,I-1 ), ZERO, ( G(I,J), J = I+1,N )
40             CONTINUE
ELSE
DO 50  I = 1, N
WRITE ( NOUT, FMT = 99994 )
\$               ( G(J,I), J = 1,I-1 ), ( G(I,J), J = I,N )
50             CONTINUE
END IF
END IF
END IF
*
99999 FORMAT (' MB03TD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03TD = ',I2)
99997 FORMAT (' The orthogonal symplectic factor U is ')
99996 FORMAT (/' The matrix A in reordered Schur canonical form is ')
99995 FORMAT (/' The matrix G is ')
99994 FORMAT (20(1X,F9.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' Orthogonality of U: || U''*U - I ||_F = ',G7.2)
END
```
Program Data
```MB03TD EXAMPLE PROGRAM DATA
5	S	U
.F. .T. .T. .F. .F.
.F. .T. .T. .F. .F.
0.9501    0.7621    0.6154    0.4057    0.0579
0    0.4565    0.7919    0.9355    0.3529
0   -0.6822    0.4565    0.9169    0.8132
0         0         0    0.4103    0.0099
0         0         0         0    0.1389
0   -0.1834   -0.1851    0.5659    0.3040
0         0    0.4011   -0.9122    0.2435
0         0         0    0.4786   -0.2432
0         0         0         0   -0.5272
0         0         0         0         0
1     0     0     0     0
0     1     0     0     0
0     0     1     0     0
0     0     0     1     0
0     0     0     0     1
0     0     0     0     0
0     0     0     0     0
0     0     0     0     0
0     0     0     0     0
0     0     0     0     0
```
Program Results
```MB03TD EXAMPLE PROGRAM RESULTS

The orthogonal symplectic factor U is
0.0407    0.4847    0.8737    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000
0.1245   -0.3866    0.2087    0.4509   -0.1047    0.3229    0.1248   -0.0843    0.1967    0.6415
-0.0933    0.4089   -0.2225   -0.4085    0.0709   -0.2171    0.2156   -0.1095    0.4348    0.5551
-0.1059   -0.5250    0.2962   -0.0295    0.2207   -0.6789    0.1133   -0.0312    0.2979   -0.1112
0.3937    0.3071   -0.1887    0.5332   -0.4351   -0.4423    0.0600   -0.0127    0.1679   -0.1179
0.0000    0.0000    0.0000    0.0000    0.0000    0.0407    0.4847    0.8737    0.0000    0.0000
-0.3229   -0.1248    0.0843   -0.1967   -0.6415    0.1245   -0.3866    0.2087    0.4509   -0.1047
0.2171   -0.2156    0.1095   -0.4348   -0.5551   -0.0933    0.4089   -0.2225   -0.4085    0.0709
0.6789   -0.1133    0.0312   -0.2979    0.1112   -0.1059   -0.5250    0.2962   -0.0295    0.2207
0.4423   -0.0600    0.0127   -0.1679    0.1179    0.3937    0.3071   -0.1887    0.5332   -0.4351

Orthogonality of U: || U'*U - I ||_F = .21E-14

The matrix A in reordered Schur canonical form is
0.4565   -0.4554    0.2756   -0.8651   -1.2050
1.1863    0.4565    0.2186   -0.0233    0.8293
0.0000    0.0000    0.9501    0.0625   -0.0064
0.0000    0.0000    0.0000    0.4103    0.5597
0.0000    0.0000    0.0000    0.0000    0.1389

The matrix G is
0.0000    0.3298   -0.0292   -0.1571    0.1751
-0.3298    0.0000   -0.0633   -0.2951    0.2396
0.0292    0.0633    0.0000    0.9567    0.7485
0.1571    0.2951   -0.9567    0.0000    0.2960
-0.1751   -0.2396   -0.7485   -0.2960    0.0000
```