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
[1] 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.
SIAM, Philadelphia, 1995.
Numerical Aspects
None.Further Comments
NoneExample
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