Constructing Fast Algorithms by Expanding a Set of Matrices into Rank-1 Matrices

This paper introduces the notion of numerical basis for a numerical space and uses it to establish a relation between a fast algorithm for computing a discrete linear transform and the problem of expanding a given finite set of matrices as a linear combination of rank-1 matrices. It is shown that the number of multiplications of the algorithm is given by the number of rank-1 matrices in the expansion. Applying this approach, an algorithm for computing three components of the nine-point discrete Fourier transform (DFT) and an algorithm to compute the seven-point DFT with the least possible number of multiplications are shown.