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.