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The adaptation of the Cooley-Tukey, the Pease and the Stockham FFT's to vector computers is discussed. Each of these algorithms computes the same result namely, the discrete Fourier transform. They differ only in the way that intermediate computations are stored. Yet it is this difference that makes one or the other more appropriate depending on the application. This difference also influences the computational efficiency on a vector computer and motivates the development of methods to improve efficiency. Each of the FFT's is defined rigorously by a short expository FORTRAN program which provides the basis for discussions about vectorization. Several methods for lengthening vectors are discussed, including the case of multiple and multi-dimensional transforms where M sequences of length N can be transformed as a single sequence of length MN using a 'truncated' FFT. The implementation of an in place FFT on a computer with memory-to-memory architecture is made possible by in place matrix-vector multiplication.