SUBROUTINE TF01ND( UPLO, N, M, P, NY, A, LDA, B, LDB, C, LDC, D,
$ LDD, U, LDU, X, Y, LDY, DWORK, INFO )
C
C PURPOSE
C
C To compute the output sequence of a linear time-invariant
C open-loop system given by its discrete-time state-space model
C (A,B,C,D), where A is an N-by-N upper or lower Hessenberg matrix.
C
C The initial state vector x(1) must be supplied by the user.
C
C ARGUMENTS
C
C Mode Parameters
C
C UPLO CHARACTER*1
C Indicates whether the user wishes to use an upper or lower
C Hessenberg matrix as follows:
C = 'U': Upper Hessenberg matrix;
C = 'L': Lower Hessenberg matrix.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrix A. N >= 0.
C
C M (input) INTEGER
C The number of system inputs. M >= 0.
C
C P (input) INTEGER
C The number of system outputs. P >= 0.
C
C NY (input) INTEGER
C The number of output vectors y(k) to be computed.
C NY >= 0.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,N)
C If UPLO = 'U', the leading N-by-N upper Hessenberg part
C of this array must contain the state matrix A of the
C system.
C If UPLO = 'L', the leading N-by-N lower Hessenberg part
C of this array must contain the state matrix A of the
C system.
C The remainder of the leading N-by-N part is not
C referenced.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C B (input) DOUBLE PRECISION array, dimension (LDB,M)
C The leading N-by-M part of this array must contain the
C input matrix B of the system.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C C (input) DOUBLE PRECISION array, dimension (LDC,N)
C The leading P-by-N part of this array must contain the
C output matrix C of the system.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C D (input) DOUBLE PRECISION array, dimension (LDD,M)
C The leading P-by-M part of this array must contain the
C direct link matrix D of the system.
C
C LDD INTEGER
C The leading dimension of array D. LDD >= MAX(1,P).
C
C U (input) DOUBLE PRECISION array, dimension (LDU,NY)
C The leading M-by-NY part of this array must contain the
C input vector sequence u(k), for k = 1,2,...,NY.
C Specifically, the k-th column of U must contain u(k).
C
C LDU INTEGER
C The leading dimension of array U. LDU >= MAX(1,M).
C
C X (input/output) DOUBLE PRECISION array, dimension (N)
C On entry, this array must contain the initial state vector
C x(1) which consists of the N initial states of the system.
C On exit, this array contains the final state vector
C x(NY+1) of the N states of the system at instant NY.
C
C Y (output) DOUBLE PRECISION array, dimension (LDY,NY)
C The leading P-by-NY part of this array contains the output
C vector sequence y(1),y(2),...,y(NY) such that the k-th
C column of Y contains y(k) (the outputs at instant k),
C for k = 1,2,...,NY.
C
C LDY INTEGER
C The leading dimension of array Y. LDY >= MAX(1,P).
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (N)
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 Given an initial state vector x(1), the output vector sequence
C y(1), y(2),..., y(NY) is obtained via the formulae
C
C x(k+1) = A x(k) + B u(k)
C y(k) = C x(k) + D u(k),
C
C where each element y(k) is a vector of length P containing the
C outputs at instant k and k = 1,2,...,NY.
C
C REFERENCES
C
C [1] Luenberger, D.G.
C Introduction to Dynamic Systems: Theory, Models and
C Applications.
C John Wiley & Sons, New York, 1979.
C
C NUMERICAL ASPECTS
C
C The algorithm requires approximately ((N+M)xP + (N/2+M)xN) x NY
C multiplications and additions.
C
C FURTHER COMMENTS
C
C The processing time required by this routine will be approximately
C half that required by the SLICOT Library routine TF01MD, which
C treats A as a general matrix.
C
C CONTRIBUTOR
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, Dec. 1996.
C Supersedes Release 2.0 routine TF01BD by S. Van Huffel, Katholieke
C Univ. Leuven, Belgium.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, Apr. 2003.
C
C KEYWORDS
C
C Discrete-time system, Hessenberg form, multivariable system,
C state-space model, state-space representation, time response.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, LDC, LDD, LDU, LDY, M, N, NY, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), U(LDU,*), X(*), Y(LDY,*)
C .. Local Scalars ..
LOGICAL LUPLO
INTEGER I, IK
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DGEMV, DLASET, DTRMV, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX, MIN
C .. Executable Statements ..
C
INFO = 0
LUPLO = LSAME( UPLO, 'U' )
C
C Test the input scalar arguments.
C
IF( .NOT.LUPLO .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( P.LT.0 ) THEN
INFO = -4
ELSE IF( NY.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -11
ELSE IF( LDD.LT.MAX( 1, P ) ) THEN
INFO = -13
ELSE IF( LDU.LT.MAX( 1, M ) ) THEN
INFO = -15
ELSE IF( LDY.LT.MAX( 1, P ) ) THEN
INFO = -18
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'TF01ND', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( MIN( P, NY ).EQ.0 ) THEN
RETURN
ELSE IF ( N.EQ.0 ) THEN
C
C Non-dynamic system: compute the output vectors.
C
IF ( M.EQ.0 ) THEN
CALL DLASET( 'Full', P, NY, ZERO, ZERO, Y, LDY )
ELSE
CALL DGEMM( 'No transpose', 'No transpose', P, NY, M, ONE,
$ D, LDD, U, LDU, ZERO, Y, LDY )
END IF
RETURN
END IF
C
CALL DCOPY( N, X, 1, DWORK, 1 )
C
DO 30 IK = 1, NY
CALL DGEMV( 'No transpose', P, N, ONE, C, LDC, DWORK, 1, ZERO,
$ Y(1,IK), 1 )
C
CALL DTRMV( UPLO, 'No transpose', 'Non-unit', N, A, LDA,
$ DWORK, 1 )
C
IF ( LUPLO ) THEN
C
DO 10 I = 2, N
DWORK(I) = DWORK(I) + A(I,I-1)*X(I-1)
10 CONTINUE
C
ELSE
C
DO 20 I = 1, N - 1
DWORK(I) = DWORK(I) + A(I,I+1)*X(I+1)
20 CONTINUE
C
END IF
C
CALL DGEMV( 'No transpose', N, M, ONE, B, LDB, U(1,IK), 1, ONE,
$ DWORK, 1 )
C
CALL DCOPY( N, DWORK, 1, X, 1 )
30 CONTINUE
C
CALL DGEMM( 'No transpose', 'No transpose', P, NY, M, ONE, D, LDD,
$ U, LDU, ONE, Y, LDY )
C
RETURN
C *** Last line of TF01ND ***
END