## ABSTRACT

An important lesson we have learned from the previous chapters is that computing the product of a series of polynomials is central to the exact analysis of common discrete data models. Often, only some terms from the product are needed. For the purpose of illustrating the main ideas, the sample sizes in the worked examples of these chapters were very small. The multiplications were thereby done with ease. But such an eﬀort grows fast with sample size. Often the data structure may remain suﬃciently sparse to preclude an asymptotic alternative. Barring an eﬃcient method of computation, exact analysis can then be infeasible even on modern computers. Data that are too sparse for the asymptotic method, but in which the numbers are suﬃciently large to make computational feasibility a barrier to exact analysis are not uncommon. An eﬃcient computational strategy is therefore imperative.