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New algorithms for computing the Discrete Fourier Transform of n points are described. For n in the range of a few tens to a few thousands these algorithms use substantially fewer multiplications than the best algorithm previously known, and about the same number of additions.Computing the Discrete Fourier Transform (DFT) of n points:n-1 2vt Aj = wiiai, j ... .,n -1, w= e[1] =_0 has many applications in scientific and engineering calculation. In 1965 Cooley and Tukey (1) described an algorithm for computing DFT in n/2 log2 n/2 complex multiplications and n log2 n complex additions, when n = 2s is a power of 2. In this note we will describe a new algorithm for computing the DFT. For n in the range of a few tens to a few thousands this algorithm uses substantially fewer multiplications than that described by Cooley and Tukey, and about the same number of additions as theirs. Theoretical background In ref. 2 we considered the following problem: Let Pn = Un + E2 %laiui be a polynomial with coefficients in a field G, let Rn = ZL; xjui and Sn = En=%lyjui be two polynomials with indeterminate coefficients. What is the minimum number of multiplications needed to compute the coefficients of Tp = Rn'Sn mod Pn when multiplications by an element g e C are not counted? Let Tp denote the set of coefficients. It was proved that:THEOREM. If Pn = JV Qi1" where Qj is irreducible (over C) and (Qi, Qj) = 1 for i # j, then the minimum number of multiplications needed to compute Tp is 2n -k.Let p be a prime and consider computing the DFT of p elements. The difficult part of the computation is that of computing Ak =~J~j~d ,=ia,, k = 1,2,..., p-1, w = exp(2iri)/p. The exponent of w is k-j mod p, and since the group of non-zero integers with operation of multiplication modulo p is isomorphic to Zp-j it follows that there exists a permutation H of 11,2, . . , p-1 such that the matrix whose (i,j) element is wvr(i)-(j) is cyclic. Therefore computing A r) = Er=jlwr(i)r(i) aT(j) j = 1,2,... p-1 is the same as computing the coefficients of P-1 p-l E wT)u1'_) a (l) + E:a, p a l-ui) mod uP'-1.[2] According to the theorem this can be done in 2(p-1)-k multiplications, where k is the number of irreducible factors (over the rationals) of uP-1-1.Similar results are obtained when n = pr is a power of a prime. In this case we use the result that the group of integers relatively prime to p with group operation of multiplication modulo pr is Z(p-l)pr for p 5 2 and is Z2 X Z2r-2 for p =2. Summary of resultsAll known algorithms for computing Tp in the minimum number of multiplications require a large number of additions when P has large irreducible factors. Therefore the algorithms for DFT as outlined in the previous section are not of practical interest unless the number of points n is small. Table 1 summarizes the number of multiplications and additions used to compute DFT for small n. We will need later to consider multiplication by w0 = 1 as a multiplication, and we therefore indicate these multiplications as well in the table.For all these al...

A set equations in the quantities a i ( p ), where i = 1, 2, · · ·, m and p ranges over a set R of lattice points in n -space, is called a system of uniform recurrence equations if the following property holds: If p and q are in R and w is an integer n -vector, then a i ( p ) depends directly on a j ( p - w ) if and only if a i ( q ) depends directly on a j ( q - w ). Finite-difference approximations to systems of partial differential equations typically lead to such recurrence equations. The structure of such a system is specified by a dependence graph G having m vertices, in which the directed edges are labeled with integer n -vectors. For certain choices of the set R , necessary and sufficient conditions on G are given for the existence of a schedule to compute all the quantities a i ( p ) explicitly from their defining equations. Properties of such schedules, such as the degree to which computation can proceed “in parallel,” are characterized. These characterizations depend on a certain iterative decomposition of a dependence graph into subgraphs. Analogous results concerning implicit schedules are also given.