## MB04WP

### Generating an orthogonal symplectic matrix which performed the reduction in MB04PU

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

Purpose

```  To generate an orthogonal symplectic matrix U, which is defined as
a product of symplectic reflectors and Givens rotations

U = diag( H(1),H(1) )      G(1)  diag( F(1),F(1) )
diag( H(2),H(2) )      G(2)  diag( F(2),F(2) )
....
diag( H(n-1),H(n-1) ) G(n-1) diag( F(n-1),F(n-1) ).

as returned by MB04PU. The matrix U is returned in terms of its
first N rows

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

```
Specification
```      SUBROUTINE MB04WP( N, ILO, U1, LDU1, U2, LDU2, CS, TAU, DWORK,
\$                   LDWORK, INFO )
C     .. Scalar Arguments ..
INTEGER           ILO, INFO, LDU1, LDU2, LDWORK, N
C     .. Array Arguments ..
DOUBLE PRECISION  CS(*), DWORK(*), U1(LDU1,*), U2(LDU2,*), TAU(*)

```
Arguments

Input/Output Parameters

```  N       (input) INTEGER
The order of the matrices U1 and U2.  N >= 0.

ILO     (input) INTEGER
ILO must have the same value as in the previous call of
MB04PU. U is equal to the unit matrix except in the
submatrix
U([ilo+1:n n+ilo+1:2*n], [ilo+1:n n+ilo+1:2*n]).
1 <= ILO <= N, if N > 0; ILO = 1, if N = 0.

U1      (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
On entry, the leading N-by-N part of this array must
contain in its i-th column the vector which defines the
elementary reflector F(i).
On exit, the leading N-by-N part of this array contains
the matrix U1.

LDU1    INTEGER
The leading dimension of the array U1.  LDU1 >= MAX(1,N).

U2      (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
On entry, the leading N-by-N part of this array must
contain in its i-th column the vector which defines the
elementary reflector H(i) and, on the subdiagonal, the
scalar factor of H(i).
On exit, the leading N-by-N part of this array contains
the matrix U2.

LDU2    INTEGER
The leading dimension of the array U2.  LDU2 >= MAX(1,N).

CS      (input) DOUBLE PRECISION array, dimension (2N-2)
On entry, the first 2N-2 elements of this array must
contain the cosines and sines of the symplectic Givens
rotations G(i).

TAU     (input) DOUBLE PRECISION array, dimension (N-1)
On entry, the first N-1 elements of this array must
contain the scalar factors of the elementary reflectors
F(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 = -10,  DWORK(1)  returns the minimum
value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK. LDWORK >= MAX(1,2*(N-ILO)).
For optimum performance LDWORK should be larger. (See
SLICOT Library routine MB04WD).

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.

```
Numerical Aspects
```  The algorithm requires O(N**3) floating point operations and is
strongly backward stable.

```
References
```  [1] C. F. VAN LOAN:
A symplectic method for approximating all the eigenvalues of
a Hamiltonian matrix.
Linear Algebra and its Applications, 61, pp. 233-251, 1984.

[2] D. KRESSNER:
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.

```
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Example

Program Text

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Program Data
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Program Results
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