**Purpose**

To compute the output sequence of a linear time-invariant open-loop system given by its discrete-time state-space model with an (N+P)-by-(N+M) general system matrix S, ( A B ) S = ( ) . ( C D ) The initial state vector x(1) must be supplied by the user. The input and output trajectories are stored as in the SLICOT Library routine TF01MY.

SUBROUTINE TF01MX( N, M, P, NY, S, LDS, U, LDU, X, Y, LDY, $ DWORK, LDWORK, INFO ) C .. Scalar Arguments .. INTEGER INFO, LDS, LDU, LDWORK, LDY, M, N, NY, P C .. Array Arguments .. DOUBLE PRECISION DWORK(*), S(LDS,*), U(LDU,*), X(*), Y(LDY,*)

**Input/Output Parameters**

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. S (input) DOUBLE PRECISION array, dimension (LDS,N+M) The leading (N+P)-by-(N+M) part of this array must contain the system matrix S. LDS INTEGER The leading dimension of array S. LDS >= MAX(1,N+P). U (input) DOUBLE PRECISION array, dimension (LDU,M) The leading NY-by-M part of this array must contain the input vector sequence u(k), for k = 1,2,...,NY. Specifically, the k-th row of U must contain u(k)'. LDU INTEGER The leading dimension of array U. LDU >= MAX(1,NY). 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+1. Y (output) DOUBLE PRECISION array, dimension (LDY,P) The leading NY-by-P part of this array contains the output vector sequence y(1),y(2),...,y(NY) such that the k-th row 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,NY).

DWORK DOUBLE PRECISION array, dimension (LDWORK) LDWORK INTEGER The length of the array DWORK. LDWORK >= 0, if MIN(N,P,NY) = 0; otherwise, LDWORK >= N+P, if M = 0; LDWORK >= 2*N+M+P, if M > 0. For better performance, LDWORK should be larger.

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) ) ( x(k) ) ( ) = S ( ) , ( y(k) ) ( 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) x (N + P) x NY multiplications and additions.

The implementation exploits data locality as much as possible, given the workspace length.

**Program Text**

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