## TG01DD

### Orthogonal reduction of a descriptor system pair (C,A-lambda E) to the RQ-coordinate form

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

Purpose

```  To reduce the descriptor system pair (C,A-lambda E) to the
RQ-coordinate form by computing an orthogonal transformation
matrix Z such that the transformed descriptor system pair
(C*Z,A*Z-lambda E*Z) has the descriptor matrix E*Z in an upper
trapezoidal form.
The right orthogonal transformations performed to reduce E can
be optionally accumulated.

```
Specification
```      SUBROUTINE TG01DD( COMPZ, L, N, P, A, LDA, E, LDE, C, LDC, Z, LDZ,
\$                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER          COMPZ
INTEGER            INFO, L, LDA, LDC, LDE, LDWORK, LDZ, N, P
C     .. Array Arguments ..
DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), DWORK( * ),
\$                   E( LDE, * ), Z( LDZ, * )

```
Arguments

Mode Parameters

```  COMPZ   CHARACTER*1
= 'N':  do not compute Z;
= 'I':  Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U':  Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.

```
Input/Output Parameters
```  L       (input) INTEGER
The number of rows of matrices A and E.  L >= 0.

N       (input) INTEGER
The number of columns of matrices A, E, and C.  N >= 0.

P       (input) INTEGER
The number of rows of matrix C.  P >= 0.

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

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

E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the descriptor matrix E.
On exit, the leading L-by-N part of this array contains
the transformed matrix E*Z in upper trapezoidal form,
i.e.

( E11 )
E*Z = (     ) ,  if L >= N ,
(  R  )
or

E*Z = ( 0  R ),  if L < N ,

where R is an MIN(L,N)-by-MIN(L,N) upper triangular
matrix.

LDE     INTEGER
The leading dimension of array E.  LDE >= MAX(1,L).

C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Z.

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

Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N':  Z is not referenced.
If COMPZ = 'I':  on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Z,
which is the product of Householder
transformations applied to A, E, and C
on the right.
If COMPZ = 'U':  on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Z1;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Z1*Z.

LDZ     INTEGER
The leading dimension of array Z.
LDZ >= 1,        if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= MAX(1, MIN(L,N) + MAX(L,N,P)).
For optimum performance
LWORK >= MAX(1, MIN(L,N) + MAX(L,N,P)*NB),
where NB is the optimal blocksize.

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

```
Method
```  The routine computes the RQ factorization of E to reduce it
the upper trapezoidal form.

The transformations are also applied to the rest of system
matrices

A <- A * Z,  C <- C * Z.

```
Numerical Aspects
```  The algorithm is numerically backward stable and requires
0( L*N*N )  floating point operations.

```
```  None
```
Example

Program Text

```*     TG01DD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          LMAX, NMAX, PMAX
PARAMETER        ( LMAX = 20, NMAX = 20, PMAX = 20)
INTEGER          LDA, LDC, LDE, LDZ
PARAMETER        ( LDA = LMAX, LDC = PMAX,
\$                   LDE = LMAX, LDZ = NMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = MIN(LMAX,NMAX)+MAX(LMAX,NMAX,PMAX) )
*     .. Local Scalars ..
CHARACTER*1      COMPZ
INTEGER          I, INFO, J, L, N, P
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX),
\$                 DWORK(LDWORK), E(LDE,NMAX), Z(LDZ,NMAX)
*     .. External Subroutines ..
EXTERNAL         TG01DD
*     .. Intrinsic Functions ..
INTRINSIC        MAX, MIN
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) L, N, P
COMPZ = 'I'
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) L
ELSE
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
*              Find the transformed descriptor system pair
*              (A-lambda E,B).
CALL TG01DD( COMPZ, L, N, P, A, LDA, E, LDE, C, LDC,
\$                      Z, LDZ, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10             CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20             CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 30 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
30             CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
40             CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01DD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' The transformed input/state matrix C*Z is ')
99993 FORMAT (/' The right transformation matrix Z is ')
99992 FORMAT (/' L is out of range.',/' L = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
END
```
Program Data
```TG01DD EXAMPLE PROGRAM DATA
4    4     2    0.0
-1     0     0     3
0     0     1     2
1     1     0     4
0     0     0     0
1     2     0     0
0     1     0     1
3     9     6     3
0     0     2     0
-1     0     1     0
0     1    -1     1
```
Program Results
``` TG01DD EXAMPLE PROGRAM RESULTS

The transformed state dynamics matrix A*Z is
0.4082   3.0773   0.6030   0.0000
0.8165   1.7233   0.6030  -1.0000
2.0412   2.8311   2.4121   0.0000
0.0000   0.0000   0.0000   0.0000

The transformed descriptor matrix E*Z is
0.0000  -0.7385   2.1106   0.0000
0.0000   0.7385   1.2060   0.0000
0.0000   0.0000   9.9499  -6.0000
0.0000   0.0000   0.0000  -2.0000

The transformed input/state matrix C*Z is
-0.8165   0.4924  -0.3015  -1.0000
0.0000   0.7385   1.2060   1.0000

The right transformation matrix Z is
0.8165  -0.4924   0.3015   0.0000
-0.4082  -0.1231   0.9045   0.0000
0.0000   0.0000   0.0000  -1.0000
0.4082   0.8616   0.3015   0.0000
```