## TB01IZ

### Balancing a system matrix corresponding to a triplet (A,B,C) (complex case)

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

Purpose

```  To reduce the 1-norm of a system matrix

S =  ( A  B )
( C  0 )

corresponding to the triple (A,B,C), by balancing. This involves
a diagonal similarity transformation inv(D)*A*D applied
iteratively to A to make the rows and columns of
-1
diag(D,I)  * S * diag(D,I)

as close in norm as possible.

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

S = A,    S = ( A  B )    or    S = ( A )
( C )

```
Specification
```      SUBROUTINE TB01IZ( JOB, N, M, P, MAXRED, A, LDA, B, LDB, C, LDC,
\$                   SCALE, INFO )
C     .. Scalar Arguments ..
CHARACTER          JOB
INTEGER            INFO, LDA, LDB, LDC, M, N, P
DOUBLE PRECISION   MAXRED
C     .. Array Arguments ..
COMPLEX*16         A( LDA, * ), B( LDB, * ), C( LDC, * )
DOUBLE PRECISION   SCALE( * )

```
Arguments

Mode Parameters

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

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrix A, the number of rows of matrix B
and the number of columns of matrix C.
N represents the dimension of the state vector.  N >= 0.

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

P       (input) INTEGER.
The number of rows of matrix C.
P represents the dimension of output vector.  P >= 0.

MAXRED  (input/output) DOUBLE PRECISION
On entry, the maximum allowed reduction in the 1-norm of
S (in an iteration) if zero rows or columns are
encountered.
If MAXRED > 0.0, MAXRED must be larger than one (to enable
the norm reduction).
If MAXRED <= 0.0, then the value 10.0 for MAXRED is
used.
On exit, if the 1-norm of the given matrix S is non-zero,
the ratio between the 1-norm of the given matrix and the
1-norm of the balanced matrix.

A       (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the system state matrix A.
On exit, the leading N-by-N part of this array contains
the balanced matrix inv(D)*A*D.

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

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

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

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

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

SCALE   (output) DOUBLE PRECISION array, dimension (N)
The scaling factors applied to S.  If D(j) is the scaling
factor applied to row and column j, then SCALE(j) = D(j),
for j = 1,...,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(D,I)  * S * diag(D,I)

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

Information about the diagonal matrix D is returned in the vector
SCALE.

```
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.

```
Numerical Aspects
```  None.

```
```  None
```
Example

Program Text

```*     TB01IZ 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
PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
*     .. Local Scalars ..
CHARACTER*1      JOB
INTEGER          I, INFO, J, M, N, P
DOUBLE PRECISION MAXRED
*     .. Local Arrays ..
COMPLEX*16       A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX)
DOUBLE PRECISION SCALE(NMAX)
*     .. External Subroutines ..
EXTERNAL         TB01IZ, UD01MD, UD01MZ
*     .. 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, MAXRED
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
*              Balance system matrix S.
CALL TB01IZ( JOB, N, M, P, MAXRED, A, LDA, B, LDB, C,
\$                      LDC, SCALE, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
CALL UD01MZ( N, N, 3, NOUT, A, LDA,
\$                        'The balanced matrix A', INFO )
IF ( M.GT.0 )
\$               CALL UD01MZ( N, M, 3, NOUT, B, LDB,
\$                            'The balanced matrix B', INFO )
IF ( P.GT.0 )
\$               CALL UD01MZ( P, N, 3, NOUT, C, LDC,
\$                            'The balanced matrix C', INFO )
CALL UD01MD( 1, N, 5, NOUT, SCALE, 1,
\$                        'The scaling vector SCALE', INFO )
WRITE ( NOUT, FMT = 99994 ) MAXRED
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01IZ EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01IZ = ',I2)
99994 FORMAT (/' MAXRED is ',E13.4)
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
END
```
Program Data
``` TB01IZ EXAMPLE PROGRAM DATA
5     2     5       A     0.0
(0.0,0.0)  (1.0000e+000,0.0)          (0.0,0.0)          (0.0,0.0)          (0.0,0.0)
(-1.5800e+006,0.0) (-1.2570e+003,0.0)          (0.0,0.0)          (0.0,0.0)          (0.0,0.0)
(3.5410e+014,0.0)          (0.0,0.0) (-1.4340e+003,0.0)          (0.0,0.0) (-5.3300e+011,0.0)
(0.0,0.0)          (0.0,0.0)          (0.0,0.0)          (0.0,0.0)  (1.0000e+000,0.0)
(0.0,0.0)          (0.0,0.0)          (0.0,0.0) (-1.8630e+004,0.0) (-1.4820e+000,0.0)
(0.0,0.0)          (0.0,0.0)
(1.1030e+002,0.0)          (0.0,0.0)
(0.0,0.0)          (0.0,0.0)
(0.0,0.0)          (0.0,0.0)
(0.0,0.0)  (8.3330e-003,0.0)
(1.0000e+000,0.0)          (0.0,0.0)          (0.0,0.0)          (0.0,0.0)          (0.0,0.0)
(0.0,0.0)          (0.0,0.0)  (1.0000e+000,0.0)          (0.0,0.0)          (0.0,0.0)
(0.0,0.0)          (0.0,0.0)          (0.0,0.0)  (1.0000e+000,0.0)          (0.0,0.0)
(6.6640e-001,0.0)          (0.0,0.0) (-6.2000e-013,0.0)          (0.0,0.0)          (0.0,0.0)
(0.0,0.0)          (0.0,0.0) (-1.0000e-003,0.0)  (1.8960e+006,0.0)  (1.5080e+002,0.0)
```
Program Results
``` TB01IZ EXAMPLE PROGRAM RESULTS

The balanced matrix A (    5X    5)

1                               2                               3
1    0.0000000D+00 +0.0000000D+00i   0.1000000D+05 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
2   -0.1580000D+03 +0.0000000D+00i  -0.1257000D+04 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
3    0.3541000D+05 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i  -0.1434000D+04 +0.0000000D+00i
4    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
5    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i

4                               5
1    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
2    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
3    0.0000000D+00 +0.0000000D+00i  -0.5330000D+03 +0.0000000D+00i
4    0.0000000D+00 +0.0000000D+00i   0.1000000D+03 +0.0000000D+00i
5   -0.1863000D+03 +0.0000000D+00i  -0.1482000D+01 +0.0000000D+00i

The balanced matrix B (    5X    2)

1                               2
1    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
2    0.1103000D+04 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
3    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
4    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
5    0.0000000D+00 +0.0000000D+00i   0.8333000D+02 +0.0000000D+00i

The balanced matrix C (    5X    5)

1                               2                               3
1    0.1000000D-04 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
2    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i   0.1000000D+06 +0.0000000D+00i
3    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
4    0.6664000D-05 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i  -0.6200000D-07 +0.0000000D+00i
5    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i  -0.1000000D+03 +0.0000000D+00i

4                               5
1    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
2    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
3    0.1000000D-05 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
4    0.0000000D+00 +0.0000000D+00i   0.0000000D+00 +0.0000000D+00i
5    0.1896000D+01 +0.0000000D+00i   0.1508000D-01 +0.0000000D+00i

The scaling vector SCALE ( 1X 5)

1              2              3              4              5
1    0.1000000D-04  0.1000000D+00  0.1000000D+06  0.1000000D-05  0.1000000D-03

MAXRED is    0.3488E+10
```