SUBROUTINE SB08HD( N, M, P, A, LDA, B, LDB, C, LDC, D, LDD, CR,
$ LDCR, DR, LDDR, IWORK, DWORK, INFO )
C
C PURPOSE
C
C To construct the state-space representation for the system
C G = (A,B,C,D) from the factors Q = (AQR,BQR,CQ,DQ) and
C R = (AQR,BQR,CR,DR) of its right coprime factorization
C -1
C G = Q * R ,
C
C where G, Q and R are the corresponding transfer-function matrices.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrix A. Also the number of rows of the
C matrix B and the number of columns of the matrices C and
C CR. N represents the order of the systems Q and R.
C N >= 0.
C
C M (input) INTEGER
C The dimension of input vector. Also the number of columns
C of the matrices B, D and DR and the number of rows of the
C matrices CR and DR. M >= 0.
C
C P (input) INTEGER
C The dimension of output vector. Also the number of rows
C of the matrices C and D. P >= 0.
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the leading N-by-N part of this array must
C contain the state dynamics matrix AQR of the systems
C Q and R.
C On exit, the leading N-by-N part of this array contains
C the state dynamics matrix of the system G.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
C On entry, the leading N-by-M part of this array must
C contain the input/state matrix BQR of the systems Q and R.
C On exit, the leading N-by-M part of this array contains
C the input/state matrix of the system G.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
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 CQ of the system Q.
C On exit, the leading P-by-N part of this array contains
C the state/output matrix of the system G.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C
C D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
C On entry, the leading P-by-M part of this array must
C contain the input/output matrix DQ of the system Q.
C On exit, the leading P-by-M part of this array contains
C the input/output matrix of the system G.
C
C LDD INTEGER
C The leading dimension of array D. LDD >= MAX(1,P).
C
C CR (input) DOUBLE PRECISION array, dimension (LDCR,N)
C The leading M-by-N part of this array must contain the
C state/output matrix CR of the system R.
C
C LDCR INTEGER
C The leading dimension of array CR. LDCR >= MAX(1,M).
C
C DR (input/output) DOUBLE PRECISION array, dimension (LDDR,M)
C On entry, the leading M-by-M part of this array must
C contain the input/output matrix DR of the system R.
C On exit, the leading M-by-M part of this array contains
C the LU factorization of the matrix DR, as computed by
C LAPACK Library routine DGETRF.
C
C LDDR INTEGER
C The leading dimension of array DR. LDDR >= MAX(1,M).
C
C Workspace
C
C IWORK INTEGER array, dimension (M)
C
C DWORK DOUBLE PRECISION array, dimension (MAX(1,4*M))
C On exit, DWORK(1) contains an estimate of the reciprocal
C condition number of the matrix DR.
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 = 1: the matrix DR is singular;
C = 2: the matrix DR is numerically singular (warning);
C the calculations continued.
C
C METHOD
C
C The subroutine computes the matrices of the state-space
C representation G = (A,B,C,D) by using the formulas:
C
C -1 -1
C A = AQR - BQR * DR * CR, B = BQR * DR ,
C -1 -1
C C = CQ - DQ * DR * CR, D = DQ * DR .
C
C REFERENCES
C
C [1] Varga A.
C Coprime factors model reduction method based on
C square-root balancing-free techniques.
C System Analysis, Modelling and Simulation,
C vol. 11, pp. 303-311, 1993.
C
C CONTRIBUTOR
C
C A. Varga, German Aerospace Center, DLR Oberpfaffenhofen,
C July 1998.
C Based on the RASP routine RCFI.
C V. Sima, Research Institute for Informatics, Bucharest, Nov. 1998,
C full BLAS 3 version.
C
C REVISIONS
C
C Dec. 1998, V. Sima, Katholieke Univ. Leuven, Leuven.
C Mar. 2000, V. Sima, Research Institute for Informatics, Bucharest.
C
C KEYWORDS
C
C Coprime factorization, state-space model.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDCR, LDD, LDDR, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), CR(LDCR,*),
$ D(LDD,*), DR(LDDR,*), DWORK(*)
INTEGER IWORK(*)
C .. Local Scalars
DOUBLE PRECISION DRNORM, RCOND
C .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE
C .. External Subroutines ..
EXTERNAL DGECON, DGEMM, DGETRF, DTRSM, MA02GD, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX
C .. Executable Statements ..
C
INFO = 0
C
C Check the scalar input parameters.
C
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( P.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -9
ELSE IF( LDD.LT.MAX( 1, P ) ) THEN
INFO = -11
ELSE IF( LDCR.LT.MAX( 1, M ) ) THEN
INFO = -13
ELSE IF( LDDR.LT.MAX( 1, M ) ) THEN
INFO = -15
END IF
IF( INFO.NE.0 )THEN
C
C Error return.
C
CALL XERBLA( 'SB08HD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( M.EQ.0 )THEN
DWORK(1) = ONE
RETURN
END IF
C
C Factor the matrix DR. First, compute the 1-norm.
C
DRNORM = DLANGE( '1-norm', M, M, DR, LDDR, DWORK )
CALL DGETRF( M, M, DR, LDDR, IWORK, INFO )
IF( INFO.NE.0 ) THEN
INFO = 1
DWORK(1) = ZERO
RETURN
END IF
C -1
C Compute B = BQR * DR , using the factorization P*DR = L*U.
C
CALL DTRSM( 'Right', 'Upper', 'NoTranspose', 'NonUnit', N, M, ONE,
$ DR, LDDR, B, LDB )
CALL DTRSM( 'Right', 'Lower', 'NoTranspose', 'Unit', N, M, ONE,
$ DR, LDDR, B, LDB )
CALL MA02GD( N, B, LDB, 1, M, IWORK, -1 )
C -1
C Compute A = AQR - BQR * DR * CR.
C
CALL DGEMM( 'NoTranspose', 'NoTranspose', N, N, M, -ONE, B, LDB,
$ CR, LDCR, ONE, A, LDA )
C -1
C Compute D = DQ * DR .
C
CALL DTRSM( 'Right', 'Upper', 'NoTranspose', 'NonUnit', P, M, ONE,
$ DR, LDDR, D, LDD )
CALL DTRSM( 'Right', 'Lower', 'NoTranspose', 'Unit', P, M, ONE,
$ DR, LDDR, D, LDD )
CALL MA02GD( P, D, LDD, 1, M, IWORK, -1 )
C -1
C Compute C = CQ - DQ * DR * CR.
C
CALL DGEMM( 'NoTranspose', 'NoTranspose', P, N, M, -ONE, D, LDD,
$ CR, LDCR, ONE, C, LDC )
C
C Estimate the reciprocal condition number of DR.
C Workspace 4*M.
C
CALL DGECON( '1-norm', M, DR, LDDR, DRNORM, RCOND, DWORK, IWORK,
$ INFO )
IF( RCOND.LE.DLAMCH( 'Epsilon' ) )
$ INFO = 2
C
DWORK(1) = RCOND
C
RETURN
C *** Last line of SB08HD ***
END