## MB04RB

### Reduction of a skew-Hamiltonian matrix to Paige/Van Loan (PVL) form (blocked version)

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

Purpose

```  To reduce 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, to Paige/Van Loan (PVL) form. That is, an orthogonal
symplectic matrix U is computed so that

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

where Aout is in upper Hessenberg form.
Blocked version.

```
Specification
```      SUBROUTINE MB04RB( N, ILO, A, LDA, QG, LDQG, CS, TAU, DWORK,
\$                   LDWORK, INFO )
C     .. Scalar Arguments ..
INTEGER           ILO, INFO, LDA, LDQG, LDWORK, N
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), CS(*), DWORK(*), QG(LDQG,*), TAU(*)

```
Arguments

Input/Output Parameters

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

ILO     (input) INTEGER
It is assumed that A is already upper triangular and Q is
zero in rows and columns 1:ILO-1. ILO is normally set by a
previous call to the SLICOT Library routine MB04DS;
otherwise it should be set to 1.
1 <= ILO <= N+1, if N > 0; ILO = 1, if 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 and, in the zero part of Aout,
information about the elementary reflectors used to
compute the PVL factorization.

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. The parts containing the
diagonal and the first superdiagonal of this array are not
referenced.
On exit, the leading N-by-N+1 part of this array contains
in its first N-1 columns information about the elementary
reflectors used to compute the PVL factorization and in
its last N columns the strictly upper triangular part of
the matrix Gout.

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

CS      (output) DOUBLE PRECISION array, dimension (2N-2)
On exit, the first 2N-2 elements of this array contain the
cosines and sines of the symplectic Givens rotations used
to compute the PVL factorization.

TAU     (output) DOUBLE PRECISION array, dimension (N-1)
On exit, the first N-1 elements of this array contain the
scalar factors of some of the elementary reflectors.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0,  DWORK(1)  returns the optimal value
of LDWORK, 8*N*NB + 3*NB, where NB is the optimal
block size.
On exit, if  INFO = -10,  DWORK(1)  returns the minimum
value of LDWORK.

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

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.

```
Method
```  An algorithm similar to the block algorithm for the symplectic
URV factorization described in [2] is used.

The matrix U is represented 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) ).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i) = 0
and v(i+1) = 1; v(i+2:n) is stored on exit in QG(i+2:n,i), and
tau in QG(i+1,i).

Each F(i) has the form

F(i) = I - nu * w * w'

where nu is a real scalar, and w is a real vector with w(1:i) = 0
and w(i+1) = 1; w(i+2:n) is stored on exit in A(i+2:n,i), and
nu in TAU(i).

Each G(i) is a Givens rotation acting on rows i+1 and n+i+1, where
the cosine is stored in CS(2*i-1) and the sine in CS(2*i).

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

```
References
```  [1] 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.

[2] Kressner, D.
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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