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This paper develops a numerical algorithm for calculating the time-responses of a linear feedback system comprising a finite dimensional controller and a distributed-parameter plant described by a heat conduction equation. By approximating the spatial derivatives of the heat equation with finite differences, the feedback system is represented as a large system of linear differential-algebraic equations (DAEs). The IMN recursions, which are suitable for stiff differential equations, are employed in solving such DAEs. The sparse structure in the associated linear algebraic equations is exploited by developing a special LU-factorization scheme. It is found that the algorithm with the new LU-factorization scheme requires only O(ncirc) arithmetic operations where ncirc is the systempsilas dimension, as opposed to O(ncirc2) when there is no sparsity exploitation. Hence, substantial computational economy is further achieved.