Purpose
To reduce the pair (B,A) to upper or lower controller Hessenberg form using (and optionally accumulating) unitary state-space transformations.Specification
SUBROUTINE TB01MD( JOBU, UPLO, N, M, A, LDA, B, LDB, U, LDU,
$ DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBU, UPLO
INTEGER INFO, LDA, LDB, LDU, M, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), U(LDU,*)
Arguments
Mode Parameters
JOBU CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix U the unitary state-space transformations for
reducing the system, as follows:
= 'N': Do not form U;
= 'I': U is initialized to the unit matrix and the
unitary transformation matrix U is returned;
= 'U': The given matrix U is updated by the unitary
transformations used in the reduction.
UPLO CHARACTER*1
Indicates whether the user wishes the pair (B,A) to be
reduced to upper or lower controller Hessenberg form as
follows:
= 'U': Upper controller Hessenberg form;
= 'L': Lower controller Hessenberg form.
Input/Output Parameters
N (input) INTEGER
The actual state dimension, i.e. the order of the
matrix A. N >= 0.
M (input) INTEGER
The actual input dimension, i.e. the number of columns of
the matrix B. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state transition matrix A to be transformed.
On exit, the leading N-by-N part of this array contains
the transformed state transition matrix U' * A * U.
The annihilated elements are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= 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 input matrix B to be transformed.
On exit, the leading N-by-M part of this array contains
the transformed input matrix U' * B.
The annihilated elements are set to zero.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
U (input/output) DOUBLE PRECISION array, dimension (LDU,*)
On entry, if JOBU = 'U', then the leading N-by-N part of
this array must contain a given matrix U (e.g. from a
previous call to another SLICOT routine), and on exit, the
leading N-by-N part of this array contains the product of
the input matrix U and the state-space transformation
matrix which reduces the given pair to controller
Hessenberg form.
On exit, if JOBU = 'I', then the leading N-by-N part of
this array contains the matrix of accumulated unitary
similarity transformations which reduces the given pair
to controller Hessenberg form.
If JOBU = 'N', the array U is not referenced and can be
supplied as a dummy array (i.e. set parameter LDU = 1 and
declare this array to be U(1,1) in the calling program).
LDU INTEGER
The leading dimension of array U. If JOBU = 'U' or
JOBU = 'I', LDU >= MAX(1,N); if JOBU = 'N', LDU >= 1.
Workspace
DWORK DOUBLE PRECISION array, dimension (MAX(N,M-1))Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes a unitary state-space transformation U, which
reduces the pair (B,A) to one of the following controller
Hessenberg forms:
|* . . . *|* . . . . . . *|
| . .|. .|
| . .|. .|
| . .|. .|
[U'B|U'AU] = | *|. .| N
| |* .|
| | . .|
| | . .|
| | . .|
| | * . . *|
M N
if UPLO = 'U', or
|* . . * | |
|. . | |
|. . | |
|. . | |
[U'AU|U'B] = |. *| | N
|. .|* |
|. .|. . |
|. .|. . |
|. .|. . |
|* . . . . . . *|* . . . *|
N M
if UPLO = 'L'.
IF M >= N, then the matrix U'B is trapezoidal and U'AU is full.
References
[1] Van Dooren, P. and Verhaegen, M.H.G.
On the use of unitary state-space transformations.
In : Contemporary Mathematics on Linear Algebra and its Role
in Systems Theory, 47, AMS, Providence, 1985.
Numerical Aspects
The algorithm requires O((N + M) x N**2) operations and is backward stable (see [1]).Further Comments
NoneExample
Program Text
* TB01MD EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDU, LDWORK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDU = NMAX,
$ LDWORK = NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, M, N
CHARACTER*1 JOBU, UPLO
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), U(LDU,NMAX),
$ DWORK(LDWORK)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TB01MD
* .. 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, JOBU, UPLO
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( LSAME( JOBU, 'U' ) )
$ READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,N ), I = 1,N )
* Reduce the pair (B,A) to controller Hessenberg form.
CALL TB01MD( JOBU, UPLO, N, M, A, LDA, B, LDB, U, LDU,
$ DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 80 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
80 CONTINUE
IF ( LSAME( JOBU, 'I' ).OR.LSAME( JOBU, 'U' ) ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 100 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( U(I,J), J = 1,N )
100 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01MD = ',I2)
99997 FORMAT (' The transformed state transition matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The transformed input matrix is ')
99994 FORMAT (/' The transformation matrix that reduces (B,A) to contr',
$ 'oller Hessenberg form is ')
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
TB01MD EXAMPLE PROGRAM DATA 6 3 N U 35.0 1.0 6.0 26.0 19.0 24.0 3.0 32.0 7.0 21.0 23.0 25.0 31.0 9.0 2.0 22.0 27.0 20.0 8.0 28.0 33.0 17.0 10.0 15.0 30.0 5.0 34.0 12.0 14.0 16.0 4.0 36.0 29.0 13.0 18.0 11.0 1.0 5.0 11.0 -1.0 4.0 11.0 -5.0 1.0 9.0 -11.0 -4.0 5.0 -19.0 -11.0 -1.0 -29.0 -20.0 -9.0Program Results
TB01MD EXAMPLE PROGRAM RESULTS The transformed state transition matrix is 60.3649 58.8853 5.0480 -5.4406 2.1382 -7.3870 54.5832 33.1865 36.5234 6.3272 -3.1377 8.8154 17.6406 21.4501 -13.5942 0.5417 1.6926 0.0786 -9.0567 10.7202 0.3531 1.5444 -1.2846 24.6407 0.0000 6.8796 -20.1372 -2.6440 2.4983 -21.8071 0.0000 0.0000 0.0000 0.0000 0.0000 27.0000 The transformed input matrix is -16.8819 -8.8260 13.9202 0.0000 13.8240 39.9205 0.0000 0.0000 4.1928 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000