**Purpose**

To compute the output sequence of a linear time-invariant open-loop system given by its discrete-time state-space model (A,B,C,D), where A is an N-by-N upper or lower Hessenberg matrix. The initial state vector x(1) must be supplied by the user.

SUBROUTINE TF01ND( UPLO, N, M, P, NY, A, LDA, B, LDB, C, LDC, D, $ LDD, U, LDU, X, Y, LDY, DWORK, INFO ) 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,*)

**Mode Parameters**

UPLO CHARACTER*1 Indicates whether the user wishes to use an upper or lower Hessenberg matrix as follows: = 'U': Upper Hessenberg matrix; = 'L': Lower Hessenberg matrix.

N (input) INTEGER The order of the matrix A. N >= 0. M (input) INTEGER The number of system inputs. M >= 0. P (input) INTEGER The number of system outputs. P >= 0. NY (input) INTEGER The number of output vectors y(k) to be computed. NY >= 0. A (input) DOUBLE PRECISION array, dimension (LDA,N) If UPLO = 'U', the leading N-by-N upper Hessenberg part of this array must contain the state matrix A of the system. If UPLO = 'L', the leading N-by-N lower Hessenberg part of this array must contain the state matrix A of the system. The remainder of the leading N-by-N part is not referenced. LDA INTEGER The leading dimension of array A. LDA >= MAX(1,N). B (input) DOUBLE PRECISION array, dimension (LDB,M) The leading N-by-M part of this array must contain the input matrix B of the system. LDB INTEGER The leading dimension of array B. LDB >= MAX(1,N). C (input) DOUBLE PRECISION array, dimension (LDC,N) The leading P-by-N part of this array must contain the output matrix C of the system. LDC INTEGER The leading dimension of array C. LDC >= MAX(1,P). D (input) DOUBLE PRECISION array, dimension (LDD,M) The leading P-by-M part of this array must contain the direct link matrix D of the system. LDD INTEGER The leading dimension of array D. LDD >= MAX(1,P). U (input) DOUBLE PRECISION array, dimension (LDU,NY) The leading M-by-NY part of this array must contain the input vector sequence u(k), for k = 1,2,...,NY. Specifically, the k-th column of U must contain u(k). LDU INTEGER The leading dimension of array U. LDU >= MAX(1,M). X (input/output) DOUBLE PRECISION array, dimension (N) On entry, this array must contain the initial state vector x(1) which consists of the N initial states of the system. On exit, this array contains the final state vector x(NY+1) of the N states of the system at instant NY. Y (output) DOUBLE PRECISION array, dimension (LDY,NY) The leading P-by-NY part of this array contains the output vector sequence y(1),y(2),...,y(NY) such that the k-th column of Y contains y(k) (the outputs at instant k), for k = 1,2,...,NY. LDY INTEGER The leading dimension of array Y. LDY >= MAX(1,P).

DWORK DOUBLE PRECISION array, dimension (N)

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

Given an initial state vector x(1), the output vector sequence y(1), y(2),..., y(NY) is obtained via the formulae x(k+1) = A x(k) + B u(k) y(k) = C x(k) + D u(k), where each element y(k) is a vector of length P containing the outputs at instant k and k = 1,2,...,NY.

[1] Luenberger, D.G. Introduction to Dynamic Systems: Theory, Models and Applications. John Wiley & Sons, New York, 1979.

The algorithm requires approximately ((N+M)xP + (N/2+M)xN) x NY multiplications and additions.

The processing time required by this routine will be approximately half that required by the SLICOT Library routine TF01MD, which treats A as a general matrix.

**Program Text**

* TF01ND EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2017 NICONET e.V. * * .. Parameters .. INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER NMAX, MMAX, PMAX, NYMAX PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20, NYMAX = 20 ) INTEGER LDA, LDB, LDC, LDD, LDU, LDY PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, $ LDD = PMAX, LDU = MMAX, LDY = PMAX ) INTEGER LDWORK PARAMETER ( LDWORK = NMAX ) * .. Local Scalars .. CHARACTER*1 UPLO INTEGER I, INFO, J, K, M, N, NY, P * .. Local Arrays .. DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX), $ D(LDD,MMAX), DWORK(LDWORK), U(LDU,NYMAX), $ X(NMAX), Y(LDY,NYMAX) * .. External Subroutines .. EXTERNAL TF01ND * .. Executable Statements .. * WRITE ( NOUT, FMT = 99999 ) * Skip the heading in the data file and read the data. READ ( NIN, FMT = '()' ) READ ( NIN, FMT = * ) N, M, P, NY, UPLO IF ( N.LE.0 .OR. N.GT.NMAX ) THEN WRITE ( NOUT, FMT = 99994 ) N ELSE READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N ) IF ( M.LE.0 .OR. M.GT.MMAX ) THEN WRITE ( NOUT, FMT = 99993 ) M ELSE READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M ) IF ( P.LE.0 .OR. P.GT.PMAX ) THEN WRITE ( NOUT, FMT = 99992 ) P ELSE READ ( NIN, FMT = * ) ( ( C(I,J), I = 1,P ), J = 1,N ) READ ( NIN, FMT = * ) ( ( D(I,J), I = 1,P ), J = 1,M ) READ ( NIN, FMT = * ) ( X(I), I = 1,N ) IF ( NY.LE.0 .OR. NY.GT.NYMAX ) THEN WRITE ( NOUT, FMT = 99991 ) NY ELSE READ ( NIN, FMT = * ) $ ( ( U(I,J), I = 1,M ), J = 1,NY ) * Compute y(1),...,y(NY) of the given system. CALL TF01ND( UPLO, N, M, P, NY, A, LDA, B, LDB, C, $ LDC, D, LDD, U, LDU, X, Y, LDY, DWORK, $ INFO ) * IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE WRITE ( NOUT, FMT = 99997 ) NY DO 20 K = 1, NY WRITE ( NOUT, FMT = 99996 ) K, Y(1,K) WRITE ( NOUT, FMT = 99995 ) ( Y(J,K), J = 2,P ) 20 CONTINUE END IF END IF END IF END IF END IF STOP * 99999 FORMAT (' TF01ND EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from TF01ND = ',I2) 99997 FORMAT (' The output sequence Y(1),...,Y(',I2,') is',/) 99996 FORMAT (' Y(',I2,') : ',F8.4) 99995 FORMAT (9X,F8.4,/) 99994 FORMAT (/' N is out of range.',/' N = ',I5) 99993 FORMAT (/' M is out of range.',/' M = ',I5) 99992 FORMAT (/' P is out of range.',/' P = ',I5) 99991 FORMAT (/' NY is out of range.',/' NY = ',I5) END

TF01ND EXAMPLE PROGRAM DATA 3 2 2 10 U 0.0000 -0.0700 0.0000 1.0000 0.8000 -0.1500 0.0000 0.0000 0.5000 0.0000 2.0000 1.0000 -1.0000 -0.1000 1.0000 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 0.5000 0.0000 0.5000 1.0000 1.0000 1.0000 -0.6922 -1.4934 0.3081 -2.7726 2.0039 0.2614 -0.9160 -0.6030 1.2556 0.2951 -1.5734 1.5639 -0.9942 1.8957 0.8988 0.4118 -1.4893 -0.9344 1.2506 -0.0701

TF01ND EXAMPLE PROGRAM RESULTS The output sequence Y(1),...,Y(10) is Y( 1) : 0.3078 -0.0928 Y( 2) : -1.5275 1.2611 Y( 3) : -1.3026 3.4002 Y( 4) : -0.3512 -0.7060 Y( 5) : -0.5922 5.4532 Y( 6) : -1.1693 1.1846 Y( 7) : -1.3029 2.2286 Y( 8) : 1.7529 -1.9534 Y( 9) : 0.6793 -4.4965 Y(10) : -0.0349 1.1654