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A new algorithm for the solution of linear recurrence systems on parallel or pipelined computers is described. Time bounds, speed-up and efficiency for SIMD and MIMD computers with fixed number of arithmetic elements (AE's), as well as for pipelined computers with fixed number of stages per operation, are obtained. The model of each computer is discussed in detail to explain better performance of the pipelined model. A simple modification in the design of AE's for parallel computers makes parallel model superior.