## TG01CD

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

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

Purpose

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

```
Specification
```      SUBROUTINE TG01CD( COMPQ, L, N, M, A, LDA, E, LDE, B, LDB, Q, LDQ,
\$                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER          COMPQ
INTEGER            INFO, L, LDA, LDB, LDE, LDQ, LDWORK, M, N
C     .. Array Arguments ..
DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DWORK( * ),
\$                   E( LDE, * ), Q( LDQ, * )

```
Arguments

Mode Parameters

```  COMPQ   CHARACTER*1
= 'N':  do not compute Q;
= 'I':  Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U':  Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.

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

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

M       (input) INTEGER
The number of columns of matrix B.  M >= 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 Q'*A.

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 Q'*E in upper trapezoidal form,
i.e.

( E11 )
Q'*E = (     ) ,     if L >= N ,
(  0  )
or

Q'*E = ( E11 E12 ),  if L < N ,

where E11 is an MIN(L,N)-by-MIN(L,N) upper triangular
matrix.

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

B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B.
On exit, the leading L-by-M part of this array contains
the transformed matrix Q'*B.

LDB     INTEGER
The leading dimension of array B.
LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.

Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,L)
If COMPQ = 'N':  Q is not referenced.
If COMPQ = 'I':  on entry, Q need not be set;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix Q,
where Q' is the product of Householder
transformations which are applied to A,
E, and B on the left.
If COMPQ = 'U':  on entry, the leading L-by-L part of this
array must contain an orthogonal matrix
Q1;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix
Q1*Q.

LDQ     INTEGER
The leading dimension of array Q.
LDQ >= 1,        if COMPQ = 'N';
LDQ >= MAX(1,L), if COMPQ = '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,M)).
For optimum performance
LWORK >= MAX(1, MIN(L,N) + MAX(L,N,M)*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 QR factorization of E to reduce it
to the upper trapezoidal form.

The transformations are also applied to the rest of system
matrices

A <- Q' * A ,  B <- Q' * B.

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

```
```  None
```
Example

Program Text

```*     TG01CD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          LMAX, NMAX, MMAX
PARAMETER        ( LMAX = 20, NMAX = 20, MMAX = 20)
INTEGER          LDA, LDB, LDE, LDQ
PARAMETER        ( LDA = LMAX, LDB = LMAX,
\$                   LDE = LMAX, LDQ = LMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = MIN(LMAX,NMAX)+MAX(LMAX,NMAX,MMAX) )
*     .. Local Scalars ..
CHARACTER*1      COMPQ
INTEGER          I, INFO, J, L, M, N
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX),
\$                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX)
*     .. External Subroutines ..
EXTERNAL         TG01CD
*     .. 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, M
COMPQ = '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 ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
*              Find the transformed descriptor system pair
*              (A-lambda E,B).
CALL TG01CD( COMPQ, L, N, M, A, LDA, E, LDE, B, LDB,
\$                      Q, LDQ, 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, L
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30             CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,L )
40             CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01CD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01CD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' The transformed input/state matrix Q''*B is ')
99993 FORMAT (/' The left transformation matrix Q is ')
99992 FORMAT (/' L is out of range.',/' L = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' M is out of range.',/' M = ',I5)
END
```
Program Data
```TG01CD 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
0     0
0     1
1     1
```
Program Results
``` TG01CD EXAMPLE PROGRAM RESULTS

The transformed state dynamics matrix Q'*A is
-0.6325  -0.9487   0.0000  -4.7434
-0.8706  -0.2176  -0.7255  -0.3627
-0.5203  -0.1301   0.3902   1.4307
-0.7559  -0.1890   0.5669   2.0788

The transformed descriptor matrix Q'*E is
-3.1623  -9.1706  -5.6921  -2.8460
0.0000  -1.3784  -1.3059  -1.3784
0.0000   0.0000  -2.4279   0.0000
0.0000   0.0000   0.0000   0.0000

The transformed input/state matrix Q'*B is
-0.3162  -0.9487
0.6529  -0.2176
-0.4336  -0.9538
1.1339   0.3780

The left transformation matrix Q is
-0.3162   0.6529   0.3902   0.5669
0.0000  -0.7255   0.3902   0.5669
-0.9487  -0.2176  -0.1301  -0.1890
0.0000   0.0000  -0.8238   0.5669
```