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We survey some results on linear-time algorithms for systolic arrays. Inparticular, we show how the greatest common divisor (GCD) of two polynomials ofdegree n over a finite field can be computed in time O(n) on a linear systolicarray of O(n) cells; similarly for the GCD of two n-bit binary numbers. We showhow n * n Toeplitz systems of linear equations can be solved in time O(n) on alinear array of O(n) cells, each of which has constant memory size (independentof n). Finally, we outline how a two-dimensional square array of O(n)* O(n)cells can be used to solve (to working accuracy) the eigenvalue problem for asymmetric real n* n matrix in time O(nS(n)). Here S(n) is a slowly growingfunction of n; for practical purposes S(n) can be regarded as a constant. Inaddition to their theoretical interest, these results have potentialapplications in the areas of error-correcting codes, symbolic and algebraiccomputations, signal processing and image processing.

Some linear-time algorithms for systolic arrays