## MB03XS

### Eigenvalues and real skew-Hamiltonian Schur form of a skew-Hamiltonian matrix

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

Purpose

```  To compute the eigenvalues and real skew-Hamiltonian Schur form of
a skew-Hamiltonian matrix,

[  A   G  ]
W  =  [       T ],
[  Q   A  ]

where A is an N-by-N matrix and G, Q are N-by-N skew-symmetric
matrices. Specifically, an orthogonal symplectic matrix U is
computed so that

T       [  Aout  Gout  ]
U W U =  [            T ] ,
[    0   Aout  ]

where Aout is in Schur canonical form (as returned by the LAPACK
routine DHSEQR). That is, Aout 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.

Optionally, the matrix U is returned in terms of its first N/2
rows

[  U1   U2 ]
U = [          ].
[ -U2   U1 ]

```
Specification
```      SUBROUTINE MB03XS( JOBU, N, A, LDA, QG, LDQG, U1, LDU1, U2, LDU2,
\$                   WR, WI, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         JOBU
INTEGER           INFO, LDA, LDQG, LDU1, LDU2, LDWORK, N
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), DWORK(*), QG(LDQG,*), U1(LDU1,*),
\$                  U2(LDU2,*), WI(*), WR(*)

```
Arguments

Mode Parameters

```  JOBU    CHARACTER*1
Specifies whether matrix U is computed or not, as follows:
= 'N': transformation matrix U is not computed;
= 'U': transformation matrix U is computed.

```
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 matrix A.
On exit, the leading N-by-N part of this array contains
the matrix Aout in Schur canonical form.

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

QG      (input/output) DOUBLE PRECISION array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain in columns 1:N the strictly lower triangular part
of the matrix Q and in columns 2:N+1 the strictly upper
triangular part of the matrix G.
On exit, the leading N-by-N+1 part of this array contains
in columns 2:N+1 the strictly upper triangular part of the
skew-symmetric matrix Gout. The part which contained the
matrix Q is set to zero.
Note that the parts containing the diagonal and the first
superdiagonal of this array are not overwritten by zeros
only if JOBU = 'U' or LDWORK >= 2*N*N - N.

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

U1      (output) DOUBLE PRECISION array, dimension (LDU1,N)
On exit, if JOBU = 'U', the leading N-by-N part of this
array contains the matrix U1.
If JOBU = 'N', this array is not referenced.

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

U2      (output) DOUBLE PRECISION array, dimension (LDU2,N)
On exit, if JOBU = 'U', the leading N-by-N part of this
array contains the matrix U2.
If JOBU = 'N', this array is not referenced.

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

WR      (output) DOUBLE PRECISION array, dimension (N)
WI      (output) DOUBLE PRECISION array, dimension (N)
The real and imaginary parts, respectively, of the
eigenvalues of Aout, which are half of the eigenvalues
of W. The eigenvalues are stored in the same order as on
the diagonal of Aout, with WR(i) = Aout(i,i) and, if
Aout(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0
and WI(i+1) = -WI(i).

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0,  DWORK(1)  returns the optimal value
of LDWORK.
On exit, if  INFO = -14,  DWORK(1)  returns the minimum
value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,(N+5)*N),      if JOBU = 'U';
LDWORK >= MAX(1,5*N,(N+1)*N),  if JOBU = 'N'.

If LDWORK = -1, then 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 related to LDWORK
is issued by XERBLA.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
> 0:  if INFO = i, DHSEQR failed to compute all of the
eigenvalues.  Elements 1:ILO-1 and i+1:N of WR
and WI contain those eigenvalues which have been
successfully computed. The matrix A (and QG) has
been partially reduced; namely, A is upper
Hessenberg in the rows and columns ILO through i.
(See DHSEQR for details.)

```
Method
```  First, using the SLICOT Library routine MB04RB, an orthogonal
symplectic matrix UP is computed so that

T      [  AP  GP  ]
UP W UP =  [        T ]
[  0   AP  ]

is in Paige/Van Loan form. Next, the LAPACK routine DHSEQR is
applied to the matrix AP to compute an orthogonal matrix V so
that Aout = V'*AP*V is in Schur canonical form.
Finally, the transformations

[ V  0 ]
U =  UP * [      ],     Gout = V'*G*V,
[ 0  V ]

using the SLICOT Library routine MB01LD for the latter, are
performed.

```
References
```   Van Loan, C.F.
A symplectic method for approximating all the eigenvalues of
a Hamiltonian matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.

 Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.

```
```  None
```
Example

Program Text

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Program Data
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Program Results
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