SUBROUTINE SB10JD( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD, E,
$ LDE, NSYS, DWORK, LDWORK, INFO )
C
C PURPOSE
C
C To convert the descriptor state-space system
C
C E*dx/dt = A*x + B*u
C y = C*x + D*u
C
C into regular state-space form
C
C dx/dt = Ad*x + Bd*u
C y = Cd*x + Dd*u .
C
C ARGUMENTS
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the descriptor system. N >= 0.
C
C M (input) INTEGER
C The column size of the matrix B. M >= 0.
C
C NP (input) INTEGER
C The row size of the matrix C. NP >= 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 matrix A of the descriptor system.
C On exit, the leading NSYS-by-NSYS part of this array
C contains the state matrix Ad of the converted system.
C
C LDA INTEGER
C The leading dimension of the 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 matrix B of the descriptor system.
C On exit, the leading NSYS-by-M part of this array
C contains the input matrix Bd of the converted system.
C
C LDB INTEGER
C The leading dimension of the array B. LDB >= max(1,N).
C
C C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
C On entry, the leading NP-by-N part of this array must
C contain the output matrix C of the descriptor system.
C On exit, the leading NP-by-NSYS part of this array
C contains the output matrix Cd of the converted system.
C
C LDC INTEGER
C The leading dimension of the array C. LDC >= max(1,NP).
C
C D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
C On entry, the leading NP-by-M part of this array must
C contain the matrix D of the descriptor system.
C On exit, the leading NP-by-M part of this array contains
C the matrix Dd of the converted system.
C
C LDD INTEGER
C The leading dimension of the array D. LDD >= max(1,NP).
C
C E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
C On entry, the leading N-by-N part of this array must
C contain the matrix E of the descriptor system.
C On exit, this array contains no useful information.
C
C LDE INTEGER
C The leading dimension of the array E. LDE >= max(1,N).
C
C NSYS (output) INTEGER
C The order of the converted state-space system.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) contains the optimal value
C of LDWORK.
C
C LDWORK INTEGER
C The dimension of the array DWORK.
C LDWORK >= max( 1, 2*N*N + 2*N + N*MAX( 5, N + M + NP ) ).
C For good performance, LDWORK must generally be larger.
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 iteration for computing singular value
C decomposition did not converge.
C
C METHOD
C
C The routine performs the transformations described in [1].
C
C REFERENCES
C
C [1] Chiang, R.Y. and Safonov, M.G.
C Robust Control Toolbox User's Guide.
C The MathWorks Inc., Natick, Mass., 1992.
C
C CONTRIBUTORS
C
C P.Hr. Petkov, D.W. Gu and M.M. Konstantinov, October 1999.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, Oct. 2000,
C Feb. 2001.
C
C KEYWORDS
C
C Descriptor systems, state-space models.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
C ..
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDE, LDWORK, M, N,
$ NP, NSYS
C ..
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), E( LDE, * )
C ..
C .. Local Scalars ..
INTEGER I, IA12, IA21, IB2, IC2, INFO2, IS, ISA, IU,
$ IV, IWRK, J, K, LWA, LWAMAX, MINWRK, NS1
DOUBLE PRECISION EPS, SCALE, TOL
C ..
C .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
C ..
C .. External Subroutines ..
EXTERNAL DGEMM, DGESVD, DLACPY, DLASET, DSCAL, XERBLA
C ..
C .. Intrinsic Functions ..
INTRINSIC DBLE, INT, MAX, SQRT
C ..
C .. Executable Statements ..
C
C Decode and Test input parameters.
C
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( NP.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, NP ) ) THEN
INFO = -9
ELSE IF( LDD.LT.MAX( 1, NP ) ) THEN
INFO = -11
ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
INFO = -13
END IF
C
C Compute workspace.
C
MINWRK = MAX( 1, 2*N*( N + 1 ) + N*MAX( 5, N + M + NP ) )
IF( LDWORK.LT.MINWRK ) THEN
INFO = -16
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SB10JD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( N.EQ.0 ) THEN
NSYS = 0
DWORK( 1 ) = ONE
RETURN
END IF
C
C Set tol.
C
EPS = DLAMCH( 'Epsilon' )
TOL = SQRT( EPS )
C
C Workspace usage.
C
IS = 0
IU = IS + N
IV = IU + N*N
C
IWRK = IV + N*N
C
C Compute the SVD of E.
C Additional workspace: need 5*N; prefer larger.
C
CALL DGESVD( 'S', 'S', N, N, E, LDE, DWORK( IS+1 ), DWORK( IU+1 ),
$ N, DWORK( IV+1 ), N, DWORK( IWRK+1 ), LDWORK-IWRK,
$ INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 1
RETURN
END IF
LWAMAX = MAX( MINWRK, INT( DWORK( IWRK+1 ) + IWRK ) )
C
C Determine the rank of E.
C
NS1 = 0
DO 10 I = 1, N
IF( DWORK( IS+I ).GT.TOL ) NS1 = NS1 + 1
10 CONTINUE
IF( NS1.GT.0 ) THEN
C
C Transform A.
C Additional workspace: need N*max(N,M,NP).
C
CALL DGEMM( 'T', 'N', N, N, N, ONE, DWORK( IU+1 ), N, A, LDA,
$ ZERO, DWORK( IWRK+1 ), N )
CALL DGEMM( 'N', 'T', N, N, N, ONE, DWORK( IWRK+1 ), N,
$ DWORK( IV+1 ), N, ZERO, A, LDA )
C
C Transform B.
C
CALL DLACPY( 'Full', N, M, B, LDB, DWORK( IWRK+1 ), N )
CALL DGEMM( 'T', 'N', N, M, N, ONE, DWORK( IU+1 ), N,
$ DWORK( IWRK+1 ), N, ZERO, B, LDB )
C
C Transform C.
C
CALL DLACPY( 'Full', NP, N, C, LDC, DWORK( IWRK+1 ), NP )
CALL DGEMM( 'N', 'T', NP, N, N, ONE, DWORK( IWRK+1 ), NP,
$ DWORK( IV+1 ), N, ZERO, C, LDC )
C
K = N - NS1
IF( K.GT.0 ) THEN
ISA = IU + K*K
IV = ISA + K
IWRK = IV + K*MAX( K, NS1 )
C
C Compute the SVD of A22.
C Additional workspace: need 5*K; prefer larger.
C
CALL DGESVD( 'S', 'S', K, K, A( NS1+1, NS1+1 ), LDA,
$ DWORK( ISA+1 ), DWORK( IU+1 ), K,
$ DWORK( IV+1 ), K, DWORK( IWRK+1 ), LDWORK-IWRK,
$ INFO2 )
IF( INFO2.NE.0 ) THEN
INFO = 1
RETURN
END IF
IA12 = IWRK
IB2 = IA12 + NS1*K
IC2 = IB2 + K*M
C
LWA = INT( DWORK( IWRK+1 ) ) + IWRK
LWAMAX = MAX( LWA, LWAMAX, IC2 + K*NP )
C
C Compute the transformed A12.
C
CALL DGEMM( 'N', 'T', NS1, K, K, ONE, A( 1, NS1+1 ), LDA,
$ DWORK( IV+1 ), K, ZERO, DWORK( IA12+1 ), NS1 )
C
C Compute CC2.
C
CALL DGEMM( 'N', 'T', NP, K, K, ONE, C( 1, NS1+1 ), LDC,
$ DWORK( IV+1 ), K, ZERO, DWORK( IC2+1 ), NP )
C
C Compute the transformed A21.
C
IA21 = IV
CALL DGEMM( 'T', 'N', K, NS1, K, ONE, DWORK( IU+1 ), K,
$ A( NS1+1, 1 ), LDA, ZERO, DWORK( IA21+1 ), K )
C
C Compute BB2.
C
CALL DGEMM( 'T', 'N', K, M, K, ONE, DWORK( IU+1 ), K,
$ B( NS1+1, 1 ), LDB, ZERO, DWORK( IB2+1 ), K )
C
C Compute A12*pinv(A22) and CC2*pinv(A22).
C
DO 20 J = 1, K
SCALE = ZERO
IF( DWORK( ISA+J ).GT.TOL ) SCALE = ONE/DWORK( ISA+J )
CALL DSCAL( NS1, SCALE, DWORK( IA12+(J-1)*NS1+1 ), 1 )
CALL DSCAL( NP, SCALE, DWORK( IC2+(J-1)*NP+1 ), 1 )
20 CONTINUE
C
C Compute Ad.
C
CALL DGEMM( 'N', 'N', NS1, NS1, K, -ONE, DWORK( IA12+1 ),
$ NS1, DWORK( IA21+1 ), K, ONE, A, LDA )
C
C Compute Bd.
C
CALL DGEMM( 'N', 'N', NS1, M, K, -ONE, DWORK( IA12+1 ), NS1,
$ DWORK( IB2+1 ), K, ONE, B, LDB )
C
C Compute Cd.
C
CALL DGEMM( 'N', 'N', NP, NS1, K, -ONE, DWORK( IC2+1 ), NP,
$ DWORK( IA21+1 ), K, ONE, C, LDC )
C
C Compute Dd.
C
CALL DGEMM( 'N', 'N', NP, M, K, -ONE, DWORK( IC2+1 ), NP,
$ DWORK( IB2+1 ), K, ONE, D, LDD )
END IF
DO 30 I = 1, NS1
SCALE = ONE/SQRT( DWORK( IS+I ) )
CALL DSCAL( NS1, SCALE, A( I, 1 ), LDA )
CALL DSCAL( M, SCALE, B( I, 1 ), LDB )
30 CONTINUE
DO 40 J = 1, NS1
SCALE = ONE/SQRT( DWORK( IS+J ) )
CALL DSCAL( NS1, SCALE, A( 1, J ), 1 )
CALL DSCAL( NP, SCALE, C( 1, J ), 1 )
40 CONTINUE
NSYS = NS1
ELSE
CALL DLASET( 'F', N, N, ZERO, -ONE/EPS, A, LDA )
CALL DLASET( 'F', N, M, ZERO, ZERO, B, LDB )
CALL DLASET( 'F', NP, N, ZERO, ZERO, C, LDC )
NSYS = N
END IF
DWORK( 1 ) = DBLE( LWAMAX )
RETURN
C *** Last line of SB10JD ***
END