We stated four versions of the linear dependence problem at the end of Chapter 3, and then spent the last two chapters developing tools needed to solve that problem. Through some of the informal examples on dependence tests, the reader has already been given a glimpse of what to expect in this chapter. The problem is always to solve a system of linear diophantine equations subject to a set of linear constraints. The equations and constraints may take certain forms based on the program and the characteristics of the particular dependence under consideration. There are two major types of dependence tests: exact and approximate. In an exact test, we actually find the general (integer) solution to the system of equations and test to see if a solution exists that fits all the constraints. In an approximate test, we check if there is an integer solution to the system or to each individual equation, and then test certain necessary conditions for the existence of a solution to the system or to each individual equation, subject to the constraints. As we saw in Chapter 5, there are a number of undetermined integer parameters in the general solution to a system of equations. If the number of parameters is 0 or 1, we may apply the exact test, since then the inequalities (constraints) can be easily solved.
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