## TG01LD

### Finite-infinite decomposition of a descriptor system

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

Purpose

```  To compute orthogonal transformation matrices Q and Z which
reduce the regular pole pencil A-lambda*E of the descriptor system
(A-lambda*E,B,C) to the form (if JOB = 'F')

( Af  *  )             ( Ef  *  )
Q'*A*Z = (        ) ,  Q'*E*Z = (        ) ,                 (1)
( 0   Ai )             ( 0   Ei )

or to the form (if JOB = 'I')

( Ai  *  )             ( Ei  *  )
Q'*A*Z = (        ) ,  Q'*E*Z = (        ) ,                 (2)
( 0   Af )             ( 0   Ef )

where the subpencil Af-lambda*Ef, with Ef nonsingular and upper
triangular, contains the finite eigenvalues, and the subpencil
Ai-lambda*Ei, with Ai nonsingular and upper triangular, contains
the infinite eigenvalues. The subpencil Ai-lambda*Ei is in a
staircase form (see METHOD). If JOBA = 'H', the submatrix Af
is further reduced to an upper Hessenberg form.

```
Specification
```      SUBROUTINE TG01LD( JOB, JOBA, COMPQ, COMPZ, N, M, P, A, LDA,
\$                   E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF, ND,
\$                   NIBLCK, IBLCK, TOL, IWORK, DWORK, LDWORK,
\$                   INFO )
C     .. Scalar Arguments ..
CHARACTER          COMPQ, COMPZ, JOB, JOBA
INTEGER            INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M,
\$                   N, ND, NF, NIBLCK, P
DOUBLE PRECISION   TOL
C     .. Array Arguments ..
INTEGER            IBLCK( * ), IWORK(*)
DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
\$                   DWORK( * ),  E( LDE, * ), Q( LDQ, * ),
\$                   Z( LDZ, * )

```
Arguments

Mode Parameters

```  JOB     CHARACTER*1
= 'F':  perform the finite-infinite separation;
= 'I':  perform the infinite-finite separation.

JOBA    CHARACTER*1
= 'H':  reduce Af further to an upper Hessenberg form;
= 'N':  keep Af unreduced.

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 number of rows of the matrix B, the number of columns
of the matrix C and the order of the square matrices A
and E.  N >= 0.

M       (input) INTEGER
The number of columns of the matrix B.  M >= 0.

P       (input) INTEGER
The number of rows of the 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,

( Af  *  )                 ( Ai  *  )
Q'*A*Z = (        ) ,  or  Q'*A*Z = (        ) ,
( 0   Ai )                 ( 0   Af )

depending on JOB, with Af an NF-by-NF matrix, and Ai an
(N-NF)-by-(N-NF) nonsingular and upper triangular matrix.
If JOBA = 'H', Af is in an upper Hessenberg form.
Otherwise, Af is unreduced.
Ai has a block structure as in (3) or (4), where A0,0 is
ND-by-ND and Ai,i , for i = 1, ..., NIBLCK, is
IBLCK(i)-by-IBLCK(i).

LDA     INTEGER
The leading dimension of the 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,

( Ef  *  )                 ( Ei  *  )
Q'*E*Z = (        ) ,  or  Q'*E*Z = (        ) ,
( 0   Ei )                 ( 0   Ef )

depending on JOB, with Ef an NF-by-NF nonsingular matrix,
and Ei an (N-NF)-by-(N-NF) nilpotent matrix in an upper
block triangular form, as in (3) or (4).

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

B       (input/output) DOUBLE PRECISION array, dimension (LDB,K),
where K = M if JOB = 'F', and K = MAX(M,P) if JOB = 'I'.
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 Q'*B.

LDB     INTEGER
The leading dimension of the 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 the array C.  LDC >= MAX(1,K),
where K = P if JOB = 'F', and K = MAX(M,P) if JOB = 'I'.

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 Householder
transformations 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
Q1;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Q1*Q.

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

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 the array Z.
LDZ >= 1,        if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'I' or 'U'.

NF      (output) INTEGER.
The order of the reduced matrices Af and Ef; also, the
number of finite generalized eigenvalues of the pencil
A-lambda*E.

ND      (output) INTEGER.
The number of non-dynamic infinite eigenvalues of the
pair (A,E). Note: N-ND is the rank of the matrix E.

NIBLCK  (output) INTEGER
If ND > 0, the number of infinite blocks minus one.
If ND = 0, then NIBLCK = 0.

IBLCK   (output) INTEGER array, dimension (N)
IBLCK(i) contains the dimension of the i-th block in the
staircase form (3) or (4), with i = 1,2, ..., NIBLCK.

```
Tolerances
```  TOL     DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR factorization with column pivoting whose estimated
condition number is less than 1/TOL. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
TOLDEF = N**2*EPS,  is used instead, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
TOL < 1.

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

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 >= 1, and if N > 0,
LDWORK >= N + MAX(3*N,M,P).

If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  the pencil A-lambda*E is not regular.

```
Method
```  The subroutine is based on the reduction algorithm of .
If JOB = 'F', the matrices Ai and Ei have the form

( A0,0  A0,k ... A0,1 )         ( 0  E0,k ... E0,1 )
Ai = (  0    Ak,k ... Ak,1 ) ,  Ei = ( 0   0   ... Ek,1 ) ;   (3)
(  :     :    .    :  )         ( :   :    .    :  )
(  0     0   ... A1,1 )         ( 0   0   ...   0  )

if JOB = 'I', the matrices Ai and Ei have the form

( A1,1 ... A1,k  A1,0 )         ( 0 ... E1,k  E1,0 )
Ai = (  :    .    :    :   ) ,  Ei = ( :  .    :    :   ) ,   (4)
(  :   ... Ak,k  Ak,0 )         ( : ...   0   Ek,0 )
(  0   ...   0   A0,0 )         ( 0 ...   0     0  )

where Ai,i , for i = 0, 1, ..., k, are nonsingular upper
triangular matrices. A0,0 corresponds to the non-dynamic infinite
modes of the system.

```
References
```   Misra, P., Van Dooren, P., and Varga, A.
Computation of structural invariants of generalized
state-space systems.
Automatica, 30, pp. 1921-1936, 1994.

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

```
```  The number of infinite poles is computed as

NIBLCK
NINFP =     Sum  IBLCK(i) = N - ND - NF.
i=1

The multiplicities of infinite poles can be computed as follows:
there are IBLCK(k)-IBLCK(k+1) infinite poles of multiplicity
k, for k = 1, ..., NIBLCK, where IBLCK(NIBLCK+1) = 0.
Note that each infinite pole of multiplicity k corresponds to
an infinite eigenvalue of multiplicity k+1.

```
Example

Program Text

```*     TG01LD 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          LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
\$                   LDE = NMAX, LDQ = NMAX, LDZ = NMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = NMAX+MAX( 3*NMAX, MMAX, PMAX ) )
*     .. Local Scalars ..
CHARACTER*1      COMPQ, COMPZ, JOB, JOBA
INTEGER          I, INFO, J, M, N, ND, NF, NIBLCK, P
DOUBLE PRECISION TOL
*     .. Local Arrays ..
INTEGER          IBLCK(NMAX), IWORK(NMAX)
DOUBLE PRECISION A(LDA,NMAX),   B(LDB,MMAX), C(LDC,NMAX),
\$                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
\$                 Z(LDZ,NMAX)
*     .. External Subroutines ..
EXTERNAL         TG01LD
*     .. 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, JOB, JOBA, TOL
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 reduced descriptor system
*              (A-lambda E,B,C).
CALL TG01LD( JOB, JOBA, COMPQ, COMPZ, N, M, P, A, LDA,
\$                      E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF,
\$                      ND, NIBLCK, IBLCK, TOL, IWORK, DWORK,
\$                      LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NF, ND
WRITE ( NOUT, FMT = 99989 ) NIBLCK + 1
IF ( NIBLCK.GT.0 ) THEN
WRITE ( NOUT, FMT = 99985 )
\$                     ( IBLCK(I), I = 1, NIBLCK )
END IF
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 (' TG01LD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01LD = ',I2)
99997 FORMAT (/' The reduced state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The reduced descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Order of reduced system =', I5/
\$        ' Number of non-dynamic infinite eigenvalues =', I5)
99993 FORMAT (/' The reduced input/state matrix Q''*B is ')
99992 FORMAT (/' The reduced state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT ( ' Number of infinite blocks = ',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 ( ' Dimension of the blocks'/20I5)
END
```
Program Data
```TG01LD EXAMPLE PROGRAM DATA
4     2     2     F     N     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
-1     0     1     0
0     1    -1     1
```
Program Results
``` TG01LD EXAMPLE PROGRAM RESULTS

Order of reduced system =    3
Number of non-dynamic infinite eigenvalues =    1
Number of infinite blocks =     1

The reduced state dynamics matrix Q'*A*Z is
2.4497  -1.3995   0.2397  -4.0023
-0.0680  -0.0030   0.1739  -1.6225
0.3707   0.0161  -0.9482   0.1049
0.0000   0.0000   0.0000   2.2913

The reduced descriptor matrix Q'*E*Z is
9.9139   4.7725  -3.4725  -2.3836
0.0000  -1.2024   2.0137   0.7926
0.0000   0.0000   0.2929  -0.9914
0.0000   0.0000   0.0000   0.0000

The reduced input/state matrix Q'*B is
-0.2157  -0.9705
0.3015   0.9516
0.7595   0.0991
1.1339   0.3780

The reduced state/output matrix C*Z is
0.5345  -1.1134   0.3758   0.5774
-1.0690   0.2784  -1.2026   0.5774

The left transformation matrix Q is
-0.2157  -0.5088   0.6109   0.5669
-0.1078  -0.2544  -0.7760   0.5669
-0.9705   0.1413  -0.0495  -0.1890
0.0000   0.8102   0.1486   0.5669

The right transformation matrix Z is
-0.5345   0.6263   0.4617  -0.3299
-0.8018  -0.5219  -0.2792  -0.0825
0.0000  -0.4871   0.8375   0.2474
-0.2673   0.3132  -0.0859   0.9073
```