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This paper presents an algorithm for computing the fast Fourier transform, based on a method proposed by Cooley and Tukey. As in their algorithm, the dimension of the transform is factored (if possible), and elementary transforms of dimension are computed for each factor of . An improved method of computing a transform step corresponding to an odd factor of is given; with this method, the number of complex multiplications for an elementary transform of dimension is reduced from to for odd . The fast Fourier transform, when computed in place, requires a final permutation step to arrange the results in normal order. This algorithm includes an efficient method for permuting the results in place. The algorithm is described mathematically and illustrated by a FORTRAN subroutine.