## AB07MD

### Dual of a given state-space representation

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

Purpose

```  To find the dual of a given state-space representation.

```
Specification
```      SUBROUTINE AB07MD( JOBD, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
\$                   INFO )
C     .. Scalar Arguments ..
CHARACTER         JOBD
INTEGER           INFO, LDA, LDB, LDC, LDD, M, N, P
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*)

```
Arguments

Mode Parameters

```  JOBD    CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state space model:
= 'D':  D is present;
= 'Z':  D is assumed a zero matrix.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the state-space representation.  N >= 0.

M       (input) INTEGER
The number of system inputs.  M >= 0.

P       (input) INTEGER
The number of system outputs.  P >= 0.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the dual state dynamics matrix A'.

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

B       (input/output) DOUBLE PRECISION array, dimension
(LDB,MAX(M,P))
On entry, the leading N-by-M part of this array must
contain the original input/state matrix B.
On exit, the leading N-by-P part of this array contains
the dual input/state matrix C'.

LDB     INTEGER
The leading dimension of array B.  LDB >= MAX(1,N).

C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, the leading M-by-N part of this array contains
the dual state/output matrix B'.

LDC     INTEGER
The leading dimension of array C.
LDC >= MAX(1,M,P) if N > 0.
LDC >= 1 if N = 0.

D       (input/output) DOUBLE PRECISION array, dimension
(LDD,MAX(M,P))
On entry, if JOBD = 'D', the leading P-by-M part of this
array must contain the original direct transmission
matrix D.
On exit, if JOBD = 'D', the leading M-by-P part of this
array contains the dual direct transmission matrix D'.
The array D is not referenced if JOBD = 'Z'.

LDD     INTEGER
The leading dimension of array D.
LDD >= MAX(1,M,P) if JOBD = 'D'.
LDD >= 1 if JOBD = 'Z'.

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

```
Method
```  If the given state-space representation is the M-input/P-output
(A,B,C,D), its dual is simply the P-input/M-output (A',C',B',D').

```
References
```  None

```
Numerical Aspects
```  None

```
```  None
```
Example

Program Text

```*     AB07MD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX, MMAX, PMAX
PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER          MAXMP
PARAMETER        ( MAXMP = MAX( MMAX, PMAX ) )
INTEGER          LDA, LDB, LDC, LDD
PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = MAXMP,
\$                   LDD = MAXMP )
*     .. Local Scalars ..
CHARACTER*1      JOBD
INTEGER          I, INFO, J, M, N, P
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
\$                 D(LDD,MAXMP)
*     .. External functions ..
LOGICAL          LSAME
EXTERNAL         LSAME
*     .. External Subroutines ..
EXTERNAL         AB07MD
*     .. Intrinsic Functions ..
INTRINSIC        MAX
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, JOBD
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
*              Find the dual of the ssr (A,B,C,D).
CALL AB07MD( JOBD, N, M, P, A, LDA, B, LDB, C, LDC, D,
\$                      LDD, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20             CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,P )
40             CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,N )
60             CONTINUE
IF ( LSAME( JOBD, 'D' ) ) THEN
WRITE ( NOUT, FMT = 99993 )
DO 80 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( D(I,J), J = 1,P )
80                CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB07MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB07MD = ',I2)
99997 FORMAT (' The dual state dynamics matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The dual input/state matrix is ')
99994 FORMAT (/' The dual state/output matrix is ')
99993 FORMAT (/' The dual direct transmission matrix is ')
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' M is out of range.',/' M = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
END
```
Program Data
``` AB07MD EXAMPLE PROGRAM DATA
3     1     2     D
1.0   2.0   0.0
4.0  -1.0   0.0
0.0   0.0   1.0
1.0   0.0   1.0
0.0   1.0  -1.0
0.0   0.0   1.0
0.0   1.0
```
Program Results
``` AB07MD EXAMPLE PROGRAM RESULTS

The dual state dynamics matrix is
1.0000   4.0000   0.0000
2.0000  -1.0000   0.0000
0.0000   0.0000   1.0000

The dual input/state matrix is
0.0000   0.0000
1.0000   0.0000
-1.0000   1.0000

The dual state/output matrix is
1.0000   0.0000   1.0000

The dual direct transmission matrix is
0.0000   1.0000
```