Purpose
To reduce the descriptor system pair (C,A-lambda E) to the RQ-coordinate form by computing an orthogonal transformation matrix Z such that the transformed descriptor system pair (C*Z,A*Z-lambda E*Z) has the descriptor matrix E*Z in an upper trapezoidal form. The right orthogonal transformations performed to reduce E can be optionally accumulated.Specification
SUBROUTINE TG01DD( COMPZ, L, N, P, A, LDA, E, LDE, C, LDC, Z, LDZ,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPZ
INTEGER INFO, L, LDA, LDC, LDE, LDWORK, LDZ, N, P
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ E( LDE, * ), Z( LDZ, * )
Arguments
Mode Parameters
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
L (input) INTEGER
The number of rows of matrices A and E. L >= 0.
N (input) INTEGER
The number of columns of matrices A, E, and C. N >= 0.
P (input) INTEGER
The number of rows of matrix C. P >= 0.
A (input/output) DOUBLE PRECISION 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 transformed matrix A*Z.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION 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 transformed matrix E*Z in upper trapezoidal form,
i.e.
( E11 )
E*Z = ( ) , if L >= N ,
( R )
or
E*Z = ( 0 R ), if L < N ,
where R is an MIN(L,N)-by-MIN(L,N) upper triangular
matrix.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
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 array C. LDC >= MAX(1,P).
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 array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.
Workspace
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 >= MAX(1, MIN(L,N) + MAX(L,N,P)).
For optimum performance
LWORK >= MAX(1, MIN(L,N) + MAX(L,N,P)*NB),
where NB is the optimal blocksize.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The routine computes the RQ factorization of E to reduce it
the upper trapezoidal form.
The transformations are also applied to the rest of system
matrices
A <- A * Z, C <- C * Z.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( L*N*N ) floating point operations.Further Comments
NoneExample
Program Text
* TG01DD EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, PMAX
PARAMETER ( LMAX = 20, NMAX = 20, PMAX = 20)
INTEGER LDA, LDC, LDE, LDZ
PARAMETER ( LDA = LMAX, LDC = PMAX,
$ LDE = LMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MIN(LMAX,NMAX)+MAX(LMAX,NMAX,PMAX) )
* .. Local Scalars ..
CHARACTER*1 COMPZ
INTEGER I, INFO, J, L, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Z(LDZ,NMAX)
* .. External Subroutines ..
EXTERNAL TG01DD
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. 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, P
COMPZ = 'I'
IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) L
ELSE
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) 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 ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find the transformed descriptor system pair
* (A-lambda E,B).
CALL TG01DD( COMPZ, L, N, P, A, LDA, E, LDE, C, LDC,
$ Z, LDZ, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
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 = 99994 )
DO 30 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01DD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01DD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' The transformed input/state matrix C*Z is ')
99993 FORMAT (/' The right transformation matrix Z is ')
99992 FORMAT (/' L is out of range.',/' L = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
END
Program Data
TG01DD EXAMPLE PROGRAM DATA
4 4 2 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 1 0
0 1 -1 1
Program Results
TG01DD EXAMPLE PROGRAM RESULTS The transformed state dynamics matrix A*Z is 0.4082 3.0773 0.6030 0.0000 0.8165 1.7233 0.6030 -1.0000 2.0412 2.8311 2.4121 0.0000 0.0000 0.0000 0.0000 0.0000 The transformed descriptor matrix E*Z is 0.0000 -0.7385 2.1106 0.0000 0.0000 0.7385 1.2060 0.0000 0.0000 0.0000 9.9499 -6.0000 0.0000 0.0000 0.0000 -2.0000 The transformed input/state matrix C*Z is -0.8165 0.4924 -0.3015 -1.0000 0.0000 0.7385 1.2060 1.0000 The right transformation matrix Z is 0.8165 -0.4924 0.3015 0.0000 -0.4082 -0.1231 0.9045 0.0000 0.0000 0.0000 0.0000 -1.0000 0.4082 0.8616 0.3015 0.0000