SUBROUTINE TG01DD( COMPZ, L, N, P, A, LDA, E, LDE, C, LDC, Z, LDZ,
$ DWORK, LDWORK, INFO )
C
C PURPOSE
C
C To reduce the descriptor system pair (C,A-lambda E) to the
C RQ-coordinate form by computing an orthogonal transformation
C matrix Z such that the transformed descriptor system pair
C (C*Z,A*Z-lambda E*Z) has the descriptor matrix E*Z in an upper
C trapezoidal form.
C The right orthogonal transformations performed to reduce E can
C be optionally accumulated.
C
C ARGUMENTS
C
C Mode Parameters
C
C COMPZ CHARACTER*1
C = 'N': do not compute Z;
C = 'I': Z is initialized to the unit matrix, and the
C orthogonal matrix Z is returned;
C = 'U': Z must contain an orthogonal matrix Z1 on entry,
C and the product Z1*Z is returned.
C
C Input/Output Parameters
C
C L (input) INTEGER
C The number of rows of matrices A and E. L >= 0.
C
C N (input) INTEGER
C The number of columns of matrices A, E, and C. N >= 0.
C
C P (input) INTEGER
C The number of rows of matrix C. P >= 0.
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the leading L-by-N part of this array must
C contain the state dynamics matrix A.
C On exit, the leading L-by-N part of this array contains
C the transformed matrix A*Z.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,L).
C
C E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
C On entry, the leading L-by-N part of this array must
C contain the descriptor matrix E.
C On exit, the leading L-by-N part of this array contains
C the transformed matrix E*Z in upper trapezoidal form,
C i.e.
C
C ( E11 )
C E*Z = ( ) , if L >= N ,
C ( R )
C or
C
C E*Z = ( 0 R ), if L < N ,
C
C where R is an MIN(L,N)-by-MIN(L,N) upper triangular
C matrix.
C
C LDE INTEGER
C The leading dimension of array E. LDE >= MAX(1,L).
C
C C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
C On entry, the leading P-by-N part of this array must
C contain the state/output matrix C.
C On exit, the leading P-by-N part of this array contains
C the transformed matrix C*Z.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
C If COMPZ = 'N': Z is not referenced.
C If COMPZ = 'I': on entry, Z need not be set;
C on exit, the leading N-by-N part of this
C array contains the orthogonal matrix Z,
C which is the product of Householder
C transformations applied to A, E, and C
C on the right.
C If COMPZ = 'U': on entry, the leading N-by-N part of this
C array must contain an orthogonal matrix
C Z1;
C on exit, the leading N-by-N part of this
C array contains the orthogonal matrix
C Z1*Z.
C
C LDZ INTEGER
C The leading dimension of array Z.
C LDZ >= 1, if COMPZ = 'N';
C LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= MAX(1, MIN(L,N) + MAX(L,N,P)).
C For optimum performance
C LWORK >= MAX(1, MIN(L,N) + MAX(L,N,P)*NB),
C where NB is the optimal blocksize.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C METHOD
C
C The routine computes the RQ factorization of E to reduce it
C the upper trapezoidal form.
C
C The transformations are also applied to the rest of system
C matrices
C
C A <- A * Z, C <- C * Z.
C
C NUMERICAL ASPECTS
C
C The algorithm is numerically backward stable and requires
C 0( L*N*N ) floating point operations.
C
C CONTRIBUTOR
C
C A. Varga, German Aerospace Center, DLR Oberpfaffenhofen.
C March 1999. Based on the RASP routine RPDSRQ.
C
C REVISIONS
C
C July 1999, V. Sima, Research Institute for Informatics, Bucharest.
C
C KEYWORDS
C
C Descriptor system, matrix algebra, matrix operations,
C orthogonal transformation.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
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, * )
C .. Local Scalars ..
LOGICAL ILZ
INTEGER ICOMPZ, LN, WRKOPT
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DGERQF, DLASET, DORMRQ, XERBLA
C .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
C
C .. Executable Statements ..
C
C Decode COMPZ.
C
IF( LSAME( COMPZ, 'N' ) ) THEN
ILZ = .FALSE.
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'U' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 2
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 3
ELSE
ICOMPZ = 0
END IF
C
C Test the input parameters.
C
INFO = 0
WRKOPT = MAX( 1, MIN( L, N ) + MAX( L, N, P ) )
IF( ICOMPZ.EQ.0 ) THEN
INFO = -1
ELSE IF( L.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( P.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, L ) ) THEN
INFO = -6
ELSE IF( LDE.LT.MAX( 1, L ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -10
ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN
INFO = -12
ELSE IF( LDWORK.LT.WRKOPT ) THEN
INFO = -14
END IF
IF( INFO .NE. 0 ) THEN
CALL XERBLA( 'TG01DD', -INFO )
RETURN
END IF
C
C Initialize Q if necessary.
C
IF( ICOMPZ.EQ.3 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
C
C Quick return if possible.
C
IF( L.EQ.0 .OR. N.EQ.0 ) THEN
DWORK( 1 ) = ONE
RETURN
END IF
C
LN = MIN( L, N )
C
C Compute the RQ decomposition of E, E = R*Z.
C
C Workspace: need MIN(L,N) + L;
C prefer MIN(L,N) + L*NB.
C
CALL DGERQF( L, N, E, LDE, DWORK, DWORK( LN+1 ), LDWORK-LN, INFO )
WRKOPT = MAX( WRKOPT, INT( DWORK( LN+1 ) ) + LN )
C
C Apply transformation on the rest of matrices.
C
C A <-- A * Z'.
C Workspace: need MIN(L,N) + L;
C prefer MIN(L,N) + L*NB.
C
CALL DORMRQ( 'Right', 'Transpose', L, N, LN, E( L-LN+1,1 ), LDE,
$ DWORK, A, LDA, DWORK( LN+1 ), LDWORK-LN, INFO )
WRKOPT = MAX( WRKOPT, INT( DWORK( LN+1 ) ) + LN )
C
C C <-- C * Z'.
C Workspace: need MIN(L,N) + P;
C prefer MIN(L,N) + P*NB.
C
CALL DORMRQ( 'Right', 'Transpose', P, N, LN, E( L-LN+1,1 ), LDE,
$ DWORK, C, LDC, DWORK( LN+1 ), LDWORK-LN, INFO )
WRKOPT = MAX( WRKOPT, INT( DWORK( LN+1 ) ) + LN )
C
C Z <-- Z1 * Z'.
C Workspace: need MIN(L,N) + N;
C prefer MIN(L,N) + N*NB.
C
IF( ILZ ) THEN
CALL DORMRQ( 'Right', 'Transpose', N, N, LN, E( L-LN+1,1 ),
$ LDE, DWORK, Z, LDZ, DWORK( LN+1 ), LDWORK-LN,
$ INFO )
WRKOPT = MAX( WRKOPT, INT( DWORK( LN+1 ) ) + LN )
END IF
C
C Set lower triangle of E to zero.
C
IF( L.LT.N ) THEN
CALL DLASET( 'Full', L, N-L, ZERO, ZERO, E, LDE )
IF( L.GE.2 ) CALL DLASET( 'Lower', L-1, L, ZERO, ZERO,
$ E( 2, N-L+1 ), LDE )
ELSE
IF( N.GE.2 ) CALL DLASET( 'Lower', N-1, N, ZERO, ZERO,
$ E( L-N+2, 1 ), LDE )
END IF
C
DWORK(1) = WRKOPT
C
RETURN
C *** Last line of TG01DD ***
END