In the context of model-based failure detection and diagnosis, there is a residual generation which can be obtained through a generalized parity space concept. This paper presents a new algorithm involving the generation of redundancy or parity equations for dynamic linear time-invariant systems. The proposed scheme involves the successive elimination of state sub-vectors and their replacement by output values as well as the processing of an output done one at a time to avoid matrix inversion. Some examples point out the numerical efficiency of the proposed procedure.