The Organization of Computations for Uniform Recurrence Equations

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.