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<HTML>
<HEAD><TITLE>TG01CD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="TG01CD">TG01CD</A></H2>
<H3>
Orthogonal reduction of a descriptor system pair (A-lambda E,B) to the QR-coordinate form
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To reduce the descriptor system pair (A-lambda E,B) to the
QR-coordinate form by computing an orthogonal transformation
matrix Q such that the transformed descriptor system pair
(Q'*A-lambda Q'*E, Q'*B) has the descriptor matrix Q'*E
in an upper trapezoidal form.
The left orthogonal transformations performed to reduce E
can be optionally accumulated.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE TG01CD( COMPQ, L, N, M, A, LDA, E, LDE, B, LDB, Q, LDQ,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ
INTEGER INFO, L, LDA, LDB, LDE, LDQ, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ),
$ E( LDE, * ), Q( LDQ, * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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 and E. N >= 0.
M (input) INTEGER
The number of columns of matrix B. M >= 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 Q'*A.
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 Q'*E in upper trapezoidal form,
i.e.
( E11 )
Q'*E = ( ) , if L >= N ,
( 0 )
or
Q'*E = ( E11 E12 ), if L < N ,
where E11 is an MIN(L,N)-by-MIN(L,N) upper triangular
matrix.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the input/state matrix B.
On exit, the leading L-by-M part of this array contains
the transformed matrix Q'*B.
LDB INTEGER
The leading dimension of array B.
LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,L)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix Q,
where Q' is the product of Householder
transformations which are applied to A,
E, and B on the left.
If COMPQ = 'U': on entry, the leading L-by-L part of this
array must contain an orthogonal matrix
Q1;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix
Q1*Q.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,L), if COMPQ = 'U' or 'I'.
</PRE>
<B>Workspace</B>
<PRE>
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,M)).
For optimum performance
LWORK >= MAX(1, MIN(L,N) + MAX(L,N,M)*NB),
where NB is the optimal blocksize.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine computes the QR factorization of E to reduce it
to the upper trapezoidal form.
The transformations are also applied to the rest of system
matrices
A <- Q' * A , B <- Q' * B.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm is numerically backward stable and requires
0( L*L*N ) floating point operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* TG01CD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER LMAX, NMAX, MMAX
PARAMETER ( LMAX = 20, NMAX = 20, MMAX = 20)
INTEGER LDA, LDB, LDE, LDQ
PARAMETER ( LDA = LMAX, LDB = LMAX,
$ LDE = LMAX, LDQ = LMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MIN(LMAX,NMAX)+MAX(LMAX,NMAX,MMAX) )
* .. Local Scalars ..
CHARACTER*1 COMPQ
INTEGER I, INFO, J, L, M, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX)
* .. External Subroutines ..
EXTERNAL TG01CD
* .. 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, M
COMPQ = '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 ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
* Find the transformed descriptor system pair
* (A-lambda E,B).
CALL TG01CD( COMPQ, L, N, M, A, LDA, E, LDE, B, LDB,
$ Q, LDQ, 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, L
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, L
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,L )
40 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01CD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01CD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' The transformed input/state matrix Q''*B is ')
99993 FORMAT (/' The left transformation matrix Q is ')
99992 FORMAT (/' L is out of range.',/' L = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' M is out of range.',/' M = ',I5)
END
</PRE>
<B>Program Data</B>
<PRE>
TG01CD 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
0 0
0 1
1 1
</PRE>
<B>Program Results</B>
<PRE>
TG01CD EXAMPLE PROGRAM RESULTS
The transformed state dynamics matrix Q'*A is
-0.6325 -0.9487 0.0000 -4.7434
-0.8706 -0.2176 -0.7255 -0.3627
-0.5203 -0.1301 0.3902 1.4307
-0.7559 -0.1890 0.5669 2.0788
The transformed descriptor matrix Q'*E is
-3.1623 -9.1706 -5.6921 -2.8460
0.0000 -1.3784 -1.3059 -1.3784
0.0000 0.0000 -2.4279 0.0000
0.0000 0.0000 0.0000 0.0000
The transformed input/state matrix Q'*B is
-0.3162 -0.9487
0.6529 -0.2176
-0.4336 -0.9538
1.1339 0.3780
The left transformation matrix Q is
-0.3162 0.6529 0.3902 0.5669
0.0000 -0.7255 0.3902 0.5669
-0.9487 -0.2176 -0.1301 -0.1890
0.0000 0.0000 -0.8238 0.5669
</PRE>
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