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We consider the problem of analysis and control of spatially invariant discretely distributed systems. It is well known that for certain types of subsystem models, the interconnected systems can be represented by infinite dimensional Laurent operators with rational symbols. Using Fourier techniques, the resulting analysis and control problems can be written as finite dimensional eigenvalue inequalities, Lyapunov equations, and Riccati equations rationally parametric over the unit circle. However, exploiting this paradigm for efficient analysis and synthesis computations has hitherto been difficult. In this paper, we develop computationally efficient iterative methods for finding rational approximations to the solutions of such problems to arbitrary accuracy.