## MB04DZ

### Balancing a complex Hamiltonian matrix

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

Purpose

```  To balance a complex Hamiltonian matrix,

[  A   G  ]
H =  [       H ] ,
[  Q  -A  ]

where A is an N-by-N matrix and G, Q are N-by-N Hermitian
matrices. This involves, first, permuting H by a symplectic
similarity transformation to isolate eigenvalues in the first
1:ILO-1 elements on the diagonal of A; and second, applying a
diagonal similarity transformation to rows and columns
ILO:N, N+ILO:2*N to make the rows and columns as close in 1-norm
as possible. Both steps are optional. Assuming ILO = 1, let D be a
diagonal matrix of order N with the scaling factors on the
diagonal. The scaled Hamiltonian is defined by

[  D**-1*A*D   D**-1*G*D**-1  ]
Hs =  [                   H         ] .
[    D*Q*D      -D*A *D**-1   ]

```
Specification
```      SUBROUTINE MB04DZ( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, INFO )
C     .. Scalar Arguments ..
CHARACTER         JOB
INTEGER           ILO, INFO, LDA, LDQG, N
C     .. Array Arguments ..
DOUBLE PRECISION  SCALE(*)
COMPLEX*16        A(LDA,*), QG(LDQG,*)

```
Arguments

Mode Parameters

```  JOB     CHARACTER*1
Specifies the operations to be performed on H:
= 'N':  none, set ILO = 1, SCALE(I) = 1.0, I = 1 .. N;
= 'P':  permute only;
= 'S':  scale only;
= 'B':  both permute and scale.

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

A       (input/output) COMPLEX*16 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 A of the balanced Hamiltonian. In particular,
the strictly lower triangular part of the first ILO-1
columns of A is zero.

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

QG      (input/output) COMPLEX*16 array, dimension
(LDQG,N+1)
On entry, the leading N-by-N+1 part of this array must
contain the lower triangular part of the matrix Q and
the upper triangular part of the matrix G.
On exit, the leading N-by-N+1 part of this array contains
the lower and upper triangular parts of the matrices Q and
G, respectively, of the balanced Hamiltonian. In
particular, the lower triangular part of the first ILO-1
columns of QG is zero.

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

ILO     (output) INTEGER
ILO-1 is the number of deflated eigenvalues in the
balanced Hamiltonian matrix.

SCALE   (output) DOUBLE PRECISION array of dimension (N)
Details of the permutations and scaling factors applied to
H.  For j = 1,...,ILO-1 let P(j) = SCALE(j). If P(j) <= N,
then rows and columns P(j) and P(j)+N are interchanged
with rows and columns j and j+N, respectively. If
P(j) > N, then row and column P(j)-N are interchanged with
row and column j+N by a generalized symplectic
permutation. For j = ILO,...,N the j-th element of SCALE
contains the factor of the scaling applied to row and
column j.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
References
```   Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.

```
```  None
```
Example

Program Text

```*     MB04DZ EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX
PARAMETER        ( NMAX = 100 )
INTEGER          LDA, LDQG
PARAMETER        ( LDA = NMAX, LDQG = NMAX )
*     .. Local Scalars ..
CHARACTER*1      JOB
INTEGER          I, ILO, INFO, J, N
*     .. Local Arrays ..
COMPLEX*16       A(LDA, NMAX), QG(LDQG, NMAX+1)
DOUBLE PRECISION DUMMY(1), SCALE(NMAX)
*     .. External Functions ..
DOUBLE PRECISION DLAPY2, ZLANTR
EXTERNAL         DLAPY2, ZLANTR
*     .. External Subroutines ..
EXTERNAL         MB04DZ
*     .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * )  N, JOB
IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( QG(I,J), J = 1,N+1 ), I = 1,N )
CALL MB04DZ( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10  I = 1, N
WRITE (NOUT, FMT = 99995) ( A(I,J), J = 1,N )
10       CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20  I = 1, N
WRITE (NOUT, FMT = 99995) ( QG(I,J), J = 1,N+1 )
20       CONTINUE
WRITE (NOUT, FMT = 99993)  ILO
IF ( ILO.GT.1 ) THEN
WRITE (NOUT, FMT = 99992) DLAPY2( ZLANTR( 'Frobenius',
\$                 'Lower', 'No Unit', N-1, ILO-1, A(2,1), LDA,
\$                 DUMMY ), ZLANTR( 'Frobenius', 'Lower', 'No Unit',
\$                 N, ILO-1, QG(1,1), LDQG, DUMMY ) )
END IF
END IF
END IF
*
99999 FORMAT (' MB04DZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04DZ = ',I2)
99997 FORMAT (' The balanced matrix A is ')
99996 FORMAT (/' The balanced matrix QG is ')
99995 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' ILO = ',I4)
99992 FORMAT (/' Norm of subdiagonal blocks: ',G7.2)
END
```
Program Data
```MB04DZ EXAMPLE PROGRAM DATA
6       B
(0,0)  (0,0)      (0,0)      (0,0)      (0,0)      (0,0)
(.0994,0)  (0,0)      (0,0)      (0,0)      (0,0)  (.9696,0)
(.3248,0)  (0,0)      (0,0)      (0,0)  (.4372,0)  (.8308,0)
(0,0)  (0,0)      (0,0)  (.0717,0)      (0,0)      (0,0)
(0,0)  (0,0)      (0,0)      (0,0)      (0,0)  (.1976,0)
(0,0)  (0,0)      (0,0)      (0,0)      (0,0)      (0,0)
(0,0)  (0,0)      (0,0)      (0,0)      (0,0)      (0,0)      (0,0)
(0,0)  (0,0)      (0,0)      (0,0)  (.0651,0)      (0,0)      (0,0)
(0,0)  (0,0)      (0,0)      (0,0)      (0,0)      (0,0)      (0,0)
(0,0)  (0,0)  (.0444,0)      (0,0)      (0,0)  (.1957,0)      (0,0)
(.8144,0)  (0,0)      (0,0)      (0,0)  (.3652,0)      (0,0)  (.9121,0)
(.9023,0)  (0,0)      (0,0)      (0,0)      (0,0)      (0,0)  1.0945,0)
```
Program Results
``` MB04DZ EXAMPLE PROGRAM RESULTS

The balanced matrix A is
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.9696  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i    -0.8144  +0.0000i    -0.9023  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.1093  +0.0000i     0.2077  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0717  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.1976  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i

The balanced matrix QG is
0.0000  +0.0000i     0.0000  +0.0000i     0.0994  +0.0000i     0.0000  +0.0000i     0.0651  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0812  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.1776  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.1957  +0.0000i     0.0000  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.3652  +0.0000i     0.0000  +0.0000i     0.9121  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     1.0945  +0.0000i

ILO =    3

Norm of subdiagonal blocks: 0.0
```