Double Loops

Abstract

In Chapter 2, we introduced the main concepts and techniques of dependence analysis in terms of a single loop. In Chapter 4, we develop a formal structure for doing dependence analysis in a perfect loop nest, and then go on to study general programs and general methods in later chapters. The current chapter forms a bridge between Chapter 2 and the rest of the book. Here, we study the major changes that happen as we move from a single loop to a more general program: index and iteration variables become index and iteration vectors; dependence distances become dependence (distance) vectors; and there is usually more than one unknown integer parameter in the general solution of the dependence equation(s). The main difficulty we face now is that we are required to seek integer solutions to a system of linear inequalities in more than one variable (Step 5, Algorithm 1.1).