## TG01HD

### Orthogonal reduction of a descriptor system to the controllability staircase form

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

Purpose

```  To compute orthogonal transformation matrices Q and Z which
reduce the N-th order descriptor system (A-lambda*E,B,C)
to the form

( Ac  *  )             ( Ec  *  )           ( Bc )
Q'*A*Z = (        ) ,  Q'*E*Z = (        ) ,  Q'*B = (    ) ,
( 0  Anc )             ( 0  Enc )           ( 0  )

C*Z = ( Cc Cnc ) ,

where the NCONT-th order descriptor system (Ac-lambda*Ec,Bc,Cc)
is a finite and/or infinite controllable. The pencil
Anc - lambda*Enc is regular of order N-NCONT and contains the
uncontrollable finite and/or infinite eigenvalues of the pencil
A-lambda*E.

For JOBCON = 'C' or 'I', the pencil ( Bc Ec-lambda*Ac ) has full
row rank NCONT for all finite lambda and is in a staircase form
with
_      _          _        _
( E1,0   E1,1  ...  E1,k-1   E1,k  )
(        _          _        _     )
( Bc Ec ) = (  0     E2,1  ...  E2,k-1   E2,k  ) ,  (1)
(              ...  _        _     )
(  0       0   ...  Ek,k-1   Ek,k  )

_          _        _
( A1,1  ...  A1,k-1   A1,k  )
(            _        _     )
Ac      = (   0   ...  A2,k-1   A2,k  ) ,         (2)
(       ...           _     )
(   0   ...    0      Ak,k  )
_
where Ei,i-1 is an rtau(i)-by-rtau(i-1) full row rank matrix
_
(with rtau(0) = M) and Ai,i is an rtau(i)-by-rtau(i)
upper triangular matrix.

For JOBCON = 'F', the pencil ( Bc Ac-lambda*Ec ) has full
row rank NCONT for all finite lambda and is in a staircase form
with
_     _          _        _
( A1,0  A1,1  ...  A1,k-1   A1,k  )
(       _          _        _     )
( Bc Ac ) = (  0    A2,1  ...  A2,k-1   A2,k  ) ,   (3)
(             ...  _        _     )
(  0      0   ...  Ak,k-1   Ak,k  )

_          _        _
( E1,1  ...  E1,k-1   E1,k  )
(            _        _     )
Ec      = (   0   ...  E2,k-1   E2,k  ) ,         (4)
(       ...           _     )
(   0   ...    0      Ek,k  )
_
where Ai,i-1 is an rtau(i)-by-rtau(i-1) full row rank matrix
_
(with rtau(0) = M) and Ei,i is an rtau(i)-by-rtau(i)
upper triangular matrix.

For JOBCON = 'C', the (N-NCONT)-by-(N-NCONT) regular pencil
Anc - lambda*Enc has the form

( Ainc - lambda*Einc         *          )
Anc - lambda*Enc = (                                       ) ,
(        0           Afnc - lambda*Efnc )

where:
1) the NIUCON-by-NIUCON regular pencil Ainc - lambda*Einc,
with Ainc upper triangular and nonsingular, contains the
uncontrollable infinite eigenvalues of A - lambda*E;
2) the (N-NCONT-NIUCON)-by-(N-NCONT-NIUCON) regular pencil
Afnc - lambda*Efnc, with Efnc upper triangular and
nonsingular, contains the uncontrollable finite
eigenvalues of A - lambda*E.

Note: The significance of the two diagonal blocks can be
interchanged by calling the routine with the
arguments A and E interchanged. In this case,
Ainc - lambda*Einc contains the uncontrollable zero
eigenvalues of A - lambda*E, while Afnc - lambda*Efnc
contains the uncontrollable nonzero finite and infinite
eigenvalues of A - lambda*E.

For JOBCON = 'F', the pencil Anc - lambda*Enc has the form

Anc - lambda*Enc = Afnc - lambda*Efnc ,

where the regular pencil Afnc - lambda*Efnc, with Efnc
upper triangular and nonsingular, contains the uncontrollable
finite eigenvalues of A - lambda*E.

For JOBCON = 'I', the pencil Anc - lambda*Enc has the form

Anc - lambda*Enc = Ainc - lambda*Einc ,

where the regular pencil Ainc - lambda*Einc, with Ainc
upper triangular and nonsingular, contains the uncontrollable
nonzero finite and infinite eigenvalues of A - lambda*E.

The left and/or right orthogonal transformations Q and Z
performed to reduce the system matrices can be optionally
accumulated.

The reduced order descriptor system (Ac-lambda*Ec,Bc,Cc) has
the same transfer-function matrix as the original system
(A-lambda*E,B,C).

```
Specification
```      SUBROUTINE TG01HD( JOBCON, COMPQ, COMPZ, N, M, P, A, LDA, E, LDE,
\$                   B, LDB, C, LDC, Q, LDQ, Z, LDZ, NCONT, NIUCON,
\$                   NRBLCK, RTAU, TOL, IWORK, DWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER          COMPQ, COMPZ, JOBCON
INTEGER            INFO, LDA, LDB, LDC, LDE, LDQ, LDZ,
\$                   M, N, NCONT, NIUCON, NRBLCK, P
DOUBLE PRECISION   TOL
C     .. Array Arguments ..
INTEGER            IWORK( * ), RTAU( * )
DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, *  ),
\$                   DWORK( * ),  E( LDE, * ), Q( LDQ, * ),
\$                   Z( LDZ, * )

```
Arguments

Mode Parameters

```  JOBCON  CHARACTER*1
= 'C':  separate both finite and infinite uncontrollable
eigenvalues;
= 'F':  separate only finite uncontrollable eigenvalues:
= 'I':  separate only nonzero finite and infinite
uncontrollable eigenvalues.

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.

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
```  N       (input) INTEGER
The dimension of the descriptor state vector; also the
order of square matrices A and E, the number of rows of
matrix B, and the number of columns of matrix C.  N >= 0.

M       (input) INTEGER
The dimension of descriptor system input vector; also the
number of columns of matrix B.  M >= 0.

P       (input) INTEGER
The dimension of descriptor system output vector; also the
number of rows of matrix C.  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 N-by-N state matrix A.
On exit, the leading N-by-N part of this array contains
the transformed state matrix Q'*A*Z,

( Ac   *  )
Q'*A*Z = (         ) ,
( 0   Anc )

where Ac is NCONT-by-NCONT and Anc is
(N-NCONT)-by-(N-NCONT).
If JOBCON = 'F', the matrix ( Bc Ac ) is in the
controllability staircase form (3).
If JOBCON = 'C' or 'I', the submatrix Ac is upper
triangular.
If JOBCON = 'C', the Anc matrix has the form

( Ainc   *  )
Anc = (           ) ,
(  0   Afnc )

where the NIUCON-by-NIUCON matrix Ainc is nonsingular and
upper triangular.
If JOBCON = 'I', Anc is nonsingular and upper triangular.

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

E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the N-by-N descriptor matrix E.
On exit, the leading N-by-N part of this array contains
the transformed descriptor matrix Q'*E*Z,

( Ec   *  )
Q'*E*Z = (         ) ,
( 0   Enc )

where Ec is NCONT-by-NCONT and Enc is
(N-NCONT)-by-(N-NCONT).
If JOBCON = 'C' or 'I', the matrix ( Bc Ec ) is in the
controllability staircase form (1).
If JOBCON = 'F', the submatrix Ec is upper triangular.
If JOBCON = 'C', the Enc matrix has the form

( Einc   *  )
Enc = (           ) ,
(  0   Efnc )

where the NIUCON-by-NIUCON matrix Einc is nilpotent
and the (N-NCONT-NIUCON)-by-(N-NCONT-NIUCON) matrix Efnc
is nonsingular and upper triangular.
If JOBCON = 'F', Enc is nonsingular and upper triangular.

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

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

( Bc )
Q'*B = (    ) ,
( 0  )

where Bc is NCONT-by-M.
For JOBCON = 'C' or 'I', the matrix ( Bc Ec ) is in the
controllability staircase form (1).
For JOBCON = 'F', the matrix ( Bc Ac ) is in the
controllability staircase form (3).

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 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).

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

LDQ     INTEGER
The leading dimension of array Q.
LDQ >= 1,        if COMPQ = 'N';
LDQ >= MAX(1,N), if COMPQ = 'U' or 'I'.

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 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
Zc;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Zc*Z.

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

NCONT   (output) INTEGER
The order of the reduced matrices Ac and Ec, and the
number of rows of reduced matrix Bc; also the order of
the controllable part of the pair (A-lambda*E,B).

NIUCON  (output) INTEGER
For JOBCON = 'C', the order of the reduced matrices
Ainc and Einc; also the number of uncontrollable
infinite eigenvalues of the pencil A - lambda*E.
For JOBCON = 'F' or 'I', NIUCON has no significance
and is set to zero.

NRBLCK  (output) INTEGER
For JOBCON = 'C' or 'I', the number k, of full row rank
_
blocks Ei,i in the staircase form of the pencil
(Bc Ec-lambda*Ac) (see (1) and (2)).
For JOBCON = 'F', the number k, of full row rank blocks
_
Ai,i in the staircase form of the pencil (Bc Ac-lambda*Ec)
(see (3) and (4)).

RTAU    (output) INTEGER array, dimension (N)
RTAU(i), for i = 1, ..., NRBLCK, is the row dimension of
_         _
the full row rank block Ei,i-1 or Ai,i-1 in the staircase
form (1) or (3) for JOBCON = 'C' or 'I', or
for JOBCON = 'F', respectively.

```
Tolerances
```  TOL     DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A-lambda*E, B). If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
reciprocal condition numbers in rank determinations; a
(sub)matrix whose estimated condition number is less than
1/TOL is considered to be of full rank.  If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by  TOLDEF = N*N*EPS,  is used instead, where EPS
is the machine precision (see LAPACK Library routine
DLAMCH).  TOL < 1.

```
Workspace
```  IWORK   INTEGER array, dimension (M)

DWORK   DOUBLE PRECISION array, dimension (MAX(N,2*M))

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

```
Method
```  The subroutine is based on the reduction algorithms of .

```
References
```   A. Varga
Computation of Irreducible Generalized State-Space
Realizations.
Kybernetika, vol. 26, pp. 89-106, 1990.

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

```
```  If the system matrices A, E and B are badly scaled, it is
generally recommendable to scale them with the SLICOT routine
TG01AD, before calling TG01HD.

```
Example

Program Text

```*     TG01HD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          LMAX, NMAX, MMAX, PMAX
PARAMETER        ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER          LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER        ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
\$                   LDE = LMAX, LDQ = LMAX, LDZ = NMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = MAX( 1, NMAX, 2*MMAX ) )
*     .. Local Scalars ..
CHARACTER*1      COMPQ, COMPZ, JOBCO
INTEGER          I, INFO, J, M, N, NCONT, NIUCON, NRBLCK, P
DOUBLE PRECISION TOL
*     .. Local Arrays ..
INTEGER          IWORK(MMAX), RTAU(NMAX)
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
\$                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX),
\$                 Z(LDZ,NMAX)
*     .. External Subroutines ..
EXTERNAL         TG01HD
*     .. Intrinsic Functions ..
INTRINSIC        MAX
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, TOL, JOBCO
COMPQ = 'I'
COMPZ = 'I'
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
*              Find the transformed descriptor system (A-lambda E,B,C).
CALL TG01HD( JOBCO, COMPQ, COMPZ, N, M, P, A, LDA,
\$                      E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ,
\$                      NCONT, NIUCON, NRBLCK, RTAU, TOL, IWORK,
\$                      DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NCONT, NIUCON
WRITE ( NOUT, FMT = 99985 )
WRITE ( NOUT, FMT = 99984 ) ( RTAU(I), I = 1,NRBLCK )
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 ) ( E(I,J), J = 1,N )
20             CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30             CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40             CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
50             CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60             CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01HD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01HD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Dimension of controllable part   =', I5/
\$        ' Number of uncontrollable infinite eigenvalues =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT (/' L is out of range.',/' L = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT (/' The staircase form row dimensions are ' )
99984 FORMAT (10I5)
END
```
Program Data
```TG01HD EXAMPLE PROGRAM DATA
7    3     2     0.0    C
2     0     2     0    -1     3     1
0     1     0     0     1     0     0
0     0     0     1     0     0     1
0     0     2     0    -1     3     1
0     0     0     1     0     0     1
0     1     0     0     1     0     0
0     0     0     1     0     0     1
0     0     1     0     0     0     0
0     0     0     0     0     1     0
0     0     0     0     0     0     1
0     0     0     0     0     0     1
0     0     0     1     0     0     0
0     0     1     0    -1     0     0
1     3     0     2     0     0     0
2     1     0
0     0     0
0     0     0
0     0     0
0     0     0
0     0     0
1     2     3
1     0     0     1     0     0     1
0    -1     1     0    -1     1     0

```
Program Results
``` TG01HD EXAMPLE PROGRAM RESULTS

Dimension of controllable part   =    3
Number of uncontrollable infinite eigenvalues =    1

The staircase form row dimensions are
2    1

The transformed state dynamics matrix Q'*A*Z is
0.0000   0.0000   0.0000   0.0000  -1.2627   0.4334   0.4666
0.0000   2.0000   0.0000  -3.7417  -0.8520   0.2924  -0.4342
0.0000   0.0000   1.7862   0.3780  -0.2651  -0.7723   0.0000
0.0000   0.0000   0.0000   3.7417   0.8520  -0.2924   0.4342
0.0000   0.0000   0.0000   0.0000  -1.5540   0.5334   0.5742
0.0000   0.0000   0.0000   0.0000  -0.6533   0.2242   0.2414
0.0000   0.0000   0.0000   0.0000  -0.5892   0.2022   0.2177

The transformed descriptor matrix Q'*E*Z is
-1.8325   1.0000   2.3752   0.0000  -0.8214   0.2819   1.8016
0.4887   0.0000   0.3770  -0.5345   0.1874   0.5461   0.0000
-0.1728   0.0000  -0.1333  -1.1339   0.1325   0.3861   0.0000
0.0000   0.0000   0.0000   0.0000   0.8520  -0.2924   0.4342
0.0000   0.0000   0.0000   0.0000  -1.0260  -0.1496   0.0000
0.0000   0.0000   0.0000   0.0000   0.0000   1.1937   0.0000
0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.0000

The transformed input/state matrix Q'*B is
1.0000   2.0000   3.0000
2.0000   1.0000   0.0000
0.0000   0.0000   0.0000
0.0000   0.0000   0.0000
0.0000   0.0000   0.0000
0.0000   0.0000   0.0000
0.0000   0.0000   0.0000

The transformed state/output matrix C*Z is
0.0000   1.0000   0.0000   0.0000  -1.2627   0.4334   0.4666
0.3665   0.0000  -0.9803  -1.6036   0.1874   0.5461   0.0000

The left transformation matrix Q is
0.0000   1.0000   0.0000   0.0000   0.0000   0.0000   0.0000
0.0000   0.0000   0.7071   0.0000   0.2740  -0.6519   0.0000
0.0000   0.0000   0.0000   0.0000   0.8304   0.3491  -0.4342
0.0000   0.0000   0.0000  -1.0000   0.0000   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000   0.4003   0.1683   0.9008
0.0000   0.0000   0.7071   0.0000  -0.2740   0.6519   0.0000
1.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000

The right transformation matrix Z is
0.0000   1.0000   0.0000   0.0000   0.0000   0.0000   0.0000
-0.6108   0.0000   0.7917   0.0000   0.0000   0.0000   0.0000
0.4887   0.0000   0.3770  -0.5345   0.1874   0.5461   0.0000
0.0000   0.0000   0.0000   0.0000  -0.4107   0.1410   0.9008
0.6108   0.0000   0.4713   0.2673  -0.1874  -0.5461   0.0000
-0.1222   0.0000  -0.0943  -0.8018  -0.1874  -0.5461   0.0000
0.0000   0.0000   0.0000   0.0000  -0.8520   0.2924  -0.4342
```