## MB04WD

### Generating an orthogonal basis spanning an isotropic subspace

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

Purpose

```  To generate a matrix Q with orthogonal columns (spanning an
isotropic subspace), which is defined as the first n columns
of a product of symplectic reflectors and Givens rotations,

Q = 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(k),H(k) ) G(k) diag( F(k),F(k) ).

The matrix Q is returned in terms of its first 2*M rows

[  op( Q1 )   op( Q2 ) ]
Q = [                      ].
[ -op( Q2 )   op( Q1 ) ]

Blocked version of the SLICOT Library routine MB04WU.

```
Specification
```      SUBROUTINE MB04WD( TRANQ1, TRANQ2, M, N, K, Q1, LDQ1, Q2, LDQ2,
\$                   CS, TAU, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         TRANQ1, TRANQ2
INTEGER           INFO, K, LDQ1, LDQ2, LDWORK, M, N
C     .. Array Arguments ..
DOUBLE PRECISION  CS(*), DWORK(*), Q1(LDQ1,*), Q2(LDQ2,*), TAU(*)

```
Arguments

Mode Parameters

```  TRANQ1  CHARACTER*1
Specifies the form of op( Q1 ) as follows:
= 'N':  op( Q1 ) = Q1;
= 'T':  op( Q1 ) = Q1';
= 'C':  op( Q1 ) = Q1'.

TRANQ2  CHARACTER*1
Specifies the form of op( Q2 ) as follows:
= 'N':  op( Q2 ) = Q2;
= 'T':  op( Q2 ) = Q2';
= 'C':  op( Q2 ) = Q2'.

```
Input/Output Parameters
```  M       (input) INTEGER
The number of rows of the matrices Q1 and Q2. M >= 0.

N       (input) INTEGER
The number of columns of the matrices Q1 and Q2.
M >= N >= 0.

K       (input) INTEGER
The number of symplectic Givens rotations whose product
partly defines the matrix Q. N >= K >= 0.

Q1      (input/output) DOUBLE PRECISION array, dimension
(LDQ1,N) if TRANQ1 = 'N',
(LDQ1,M) if TRANQ1 = 'T' or TRANQ1 = 'C'
On entry with TRANQ1 = 'N', the leading M-by-K part of
this array must contain in its i-th column the vector
which defines the elementary reflector F(i).
On entry with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
K-by-M part of this array must contain in its i-th row
the vector which defines the elementary reflector F(i).
On exit with TRANQ1 = 'N', the leading M-by-N part of this
array contains the matrix Q1.
On exit with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
N-by-M part of this array contains the matrix Q1'.

LDQ1    INTEGER
The leading dimension of the array Q1.
LDQ1 >= MAX(1,M),  if TRANQ1 = 'N';
LDQ1 >= MAX(1,N),  if TRANQ1 = 'T' or TRANQ1 = 'C'.

Q2      (input/output) DOUBLE PRECISION array, dimension
(LDQ2,N) if TRANQ2 = 'N',
(LDQ2,M) if TRANQ2 = 'T' or TRANQ2 = 'C'
On entry with TRANQ2 = 'N', the leading M-by-K part of
this array must contain in its i-th column the vector
which defines the elementary reflector H(i) and, on the
diagonal, the scalar factor of H(i).
On entry with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
K-by-M part of this array must contain in its i-th row the
vector which defines the elementary reflector H(i) and, on
the diagonal, the scalar factor of H(i).
On exit with TRANQ2 = 'N', the leading M-by-N part of this
array contains the matrix Q2.
On exit with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
N-by-M part of this array contains the matrix Q2'.

LDQ2    INTEGER
The leading dimension of the array Q2.
LDQ2 >= MAX(1,M),  if TRANQ2 = 'N';
LDQ2 >= MAX(1,N),  if TRANQ2 = 'T' or TRANQ2 = 'C'.

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

TAU     (input) DOUBLE PRECISION array, dimension (K)
On entry, the first K 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, MAX(M+N,8*N*NB + 15*NB*NB), where NB is
the optimal block size determined by the function UE01MD.
On exit, if  INFO = -13,  DWORK(1)  returns the minimum
value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.  LDWORK >= MAX(1,M+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.

```
References
```  [1] 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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