Linear Equations and Inequalities

Abstract

Equations and inequalities in integer variables arise naturally in the dependence analysis of a program, when we try to analyze the mem- ory reference pattern. The equations are created by matching the corresponding subscripts of instances of two subscripted variables that are elements of the same array. The variables in these equations are integer variables, since they come from the index variables of loops, which take integer values. A polynomial equation in integer variables is called a diophantine equation.1 The array subscripts are usually linear (affine) functions of the index variables, with integer coefficients; consequently, they lead to linear diophantine equations with integer coefficients.