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An innovative approach is presented for analyzing finite arrays of regularly spaced elements. Our approach is based on coupling an array decomposition technique with a multipole expansion for interacting distant elements. This hybrid technique results in Toeplitz storage for both near-zone matrices and far-zone translation operators, with FFT acceleration for the far-zone element interactions. The matrix storage is of the same order as a single array element, regardless of array size, hence removing the matrix storage bottleneck for large arrays. The total storage requirements of this method are only O(N), where N is the length of the solution vector. Hence, fast and rigorous analysis of very large finite arrays can be accomplished with limited resources.