## TG01AZ

### Balancing the matrices of the system pencil corresponding to a descriptor triple (A-lambda E,B,C) (complex case)

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

Purpose

```  To balance the matrices of the system pencil

S =  ( A  B ) - lambda ( E  0 ) :=  Q - lambda Z,
( C  0 )          ( 0  0 )

corresponding to the descriptor triple (A-lambda E,B,C),
by balancing. This involves diagonal similarity transformations
(Dl*A*Dr - lambda Dl*E*Dr, Dl*B, C*Dr) applied to the system
(A-lambda E,B,C) to make the rows and columns of system pencil
matrices

diag(Dl,I) * S * diag(Dr,I)

as close in norm as possible. Balancing may reduce the 1-norms
of the matrices of the system pencil S.

The balancing can be performed optionally on the following
particular system pencils

S = A-lambda E,

S = ( A-lambda E  B ),    or

S = ( A-lambda E ).
(     C      )

```
Specification
```      SUBROUTINE TG01AZ( JOB, L, N, M, P, THRESH, A, LDA, E, LDE,
\$                   B, LDB, C, LDC, LSCALE, RSCALE, DWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER          JOB
INTEGER            INFO, L, LDA, LDB, LDC, LDE, M, N, P
DOUBLE PRECISION   THRESH
C     .. Array Arguments ..
COMPLEX*16         A( LDA, * ), B( LDB, * ), C( LDC, * ),
\$                   E( LDE, * )
DOUBLE PRECISION   DWORK( * ), LSCALE( * ), RSCALE( * )

```
Arguments

Mode Parameters

```  JOB     CHARACTER*1
Indicates which matrices are involved in balancing, as
follows:
= 'A':  All matrices are involved in balancing;
= 'B':  B, A and E matrices are involved in balancing;
= 'C':  C, A and E matrices are involved in balancing;
= 'N':  B and C matrices are not involved in balancing.

```
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, E, and C.  N >= 0.

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

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

THRESH  (input) DOUBLE PRECISION
Threshold value for magnitude of elements:
elements with magnitude less than or equal to
THRESH are ignored for balancing. THRESH >= 0.
The magnitude is computed as the sum of the absolute
values of the real and imaginary parts.

A       (input/output) COMPLEX*16 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 balanced matrix Dl*A*Dr.

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

E       (input/output) COMPLEX*16 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 balanced matrix Dl*E*Dr.

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

B       (input/output) COMPLEX*16 array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B.
On exit, if M > 0, the leading L-by-M part of this array
contains the balanced matrix Dl*B.
The array B is not referenced if M = 0.

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

C       (input/output) COMPLEX*16 array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, if P > 0, the leading P-by-N part of this array
contains the balanced matrix C*Dr.
The array C is not referenced if P = 0.

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

LSCALE  (output) DOUBLE PRECISION array, dimension (L)
The scaling factors applied to S from left.  If Dl(j) is
the scaling factor applied to row j, then
SCALE(j) = Dl(j), for j = 1,...,L.

RSCALE  (output) DOUBLE PRECISION array, dimension (N)
The scaling factors applied to S from right.  If Dr(j) is
the scaling factor applied to column j, then
SCALE(j) = Dr(j), for j = 1,...,N.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (3*(L+N))

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

```
Method
```  Balancing consists of applying a diagonal similarity
transformation
-1
diag(Dl,I)  * S * diag(Dr,I)

to make the 1-norms of each row of the first L rows of S and its
corresponding N columns nearly equal.

Information about the diagonal matrices Dl and Dr are returned in
the vectors LSCALE and RSCALE, respectively.

```
References
```   Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.

 R.C. Ward, R. C.
Balancing the generalized eigenvalue problem.
SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

```
Numerical Aspects
```  None.

```
```  None
```
Example

Program Text

```*     TG01AZ 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
PARAMETER        ( LDA = LMAX, LDB = LMAX, LDC = PMAX,
\$                   LDE = LMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = MAX( 1, 3*(LMAX+NMAX ) ) )
*     .. Local Scalars ..
CHARACTER*1      JOBS
INTEGER          I, INFO, J, L, M, N, P
DOUBLE PRECISION ABCNRM, ENORM, SABCNM, SENORM, THRESH
*     .. Local Arrays ..
COMPLEX*16       A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
\$                 E(LDE,NMAX)
DOUBLE PRECISION DWORK(LDWORK), LSCALE(LMAX), RSCALE(NMAX)
*     .. External Functions ..
DOUBLE PRECISION ZLANGE
EXTERNAL         ZLANGE
*     .. External Subroutines ..
EXTERNAL         TG01AZ
*     .. 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 = * ) L, N, M, P, JOBS, THRESH
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) L
ELSE
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,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 = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
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 )
*                 Compute norms before scaling
ABCNRM = MAX( ZLANGE( '1', L, N, A, LDA, DWORK ),
\$                          ZLANGE( '1', L, M, B, LDB, DWORK ),
\$                          ZLANGE( '1', P, N, C, LDC, DWORK ) )
ENORM = ZLANGE( '1', L, N, E, LDE, DWORK )
*                 Find the transformed descriptor system
*                 (A-lambda E,B,C).
CALL TG01AZ( JOBS, L, N, M, P, THRESH, A, LDA, E, LDE,
\$                         B, LDB, C, LDC, LSCALE, RSCALE, DWORK,
\$                         INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
SABCNM = MAX( ZLANGE( '1', L, N, A, LDA, DWORK ),
\$                             ZLANGE( '1', L, M, B, LDB, DWORK ),
\$                             ZLANGE( '1', P, N, C, LDC, DWORK ) )
SENORM = ZLANGE( '1', L, N, E, LDE, DWORK )
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 = 99993 )
DO 30 I = 1, L
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 )
WRITE ( NOUT, FMT = 99985 ) ( LSCALE(I), I = 1,L )
WRITE ( NOUT, FMT = 99990 )
WRITE ( NOUT, FMT = 99985 ) ( RSCALE(J), J = 1,N )
WRITE ( NOUT, FMT = 99994 )
\$                       ABCNRM, SABCNM, ENORM, SENORM
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01AZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01AZ = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Dl*A*Dr is ')
99996 FORMAT (/' The transformed descriptor matrix Dl*E*Dr is ')
99995 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99994 FORMAT (/' Norm of [ A B; C 0]         =', 1PD10.3/
\$         ' Norm of scaled [ A B; C 0]  =', 1PD10.3/
\$         ' Norm of E                   =', 1PD10.3/
\$         ' Norm of scaled E            =', 1PD10.3)
99993 FORMAT (/' The transformed input/state matrix Dl*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Dr is ')
99991 FORMAT (/' The diagonal of left scaling matrix Dl is ')
99990 FORMAT (/' The diagonal of right scaling matrix Dr 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 (20(1X,F9.4))
END
```
Program Data
```TG01AZ EXAMPLE PROGRAM DATA
4    4     2     2     A   0.0
(-1,0)       (0,0)       (0,0)  (0.003,0)
(0,0)       (0,0)  (0.1000,0)   (0.02,0)
(100,0)      (10,0)       (0,0)    (0.4,0)
(0,0)       (0,0)       (0,0)    (0.0,0)
(1,0)     (0.2,0)       (0,0)    (0.0,0)
(0,0)       (1,0)       (0,0)  ( 0.01,0)
(300,0)      (90,0)       (6,0)    (0.3,0)
(0,0)       (0,0)      (20,0)    (0.0,0)
(10,0)       (0,0)
(0,0)       (0,0)
(0,0)    (1000,0)
(10000,0)   (10000,0)
(-0.1,0)     (0.0,0)   (0.001,0)    (0.0,0)
(0.0,0)    (0.01,0)  (-0.001,0) (0.0001,0)

```
Program Results
``` TG01AZ EXAMPLE PROGRAM RESULTS

The transformed state dynamics matrix Dl*A*Dr is
-1.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.3000  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     1.0000  +0.0000i     2.0000  +0.0000i
1.0000  +0.0000i     0.1000  +0.0000i     0.0000  +0.0000i     0.4000  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i

The transformed descriptor matrix Dl*E*Dr is
1.0000  +0.0000i     0.2000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i
0.0000  +0.0000i     1.0000  +0.0000i     0.0000  +0.0000i     1.0000  +0.0000i
3.0000  +0.0000i     0.9000  +0.0000i     0.6000  +0.0000i     0.3000  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i     0.2000  +0.0000i     0.0000  +0.0000i

The transformed input/state matrix Dl*B is
100.0000  +0.0000i     0.0000  +0.0000i
0.0000  +0.0000i     0.0000  +0.0000i
0.0000  +0.0000i   100.0000  +0.0000i
100.0000  +0.0000i   100.0000  +0.0000i

The transformed state/output matrix C*Dr is
-0.0100  +0.0000i     0.0000  +0.0000i     0.0010  +0.0000i     0.0000  +0.0000i
0.0000  +0.0000i     0.0010  +0.0000i    -0.0010  +0.0000i     0.0010  +0.0000i

The diagonal of left scaling matrix Dl is
10.0000   10.0000    0.1000    0.0100

The diagonal of right scaling matrix Dr is
0.1000    0.1000    1.0000   10.0000

Norm of [ A B; C 0]         = 1.100D+04
Norm of scaled [ A B; C 0]  = 2.000D+02
Norm of E                   = 3.010D+02
Norm of scaled E            = 4.000D+00
```