I have found a pretty nice description of the polyhedra model in this paper. They are describing the math behind iterating through for-loops in a program (in last figure below), which forms a polyhedron of affine inequalities. Earlier they give definitions of affine functions and polyhedra.
I am having trouble understanding the meaning of the matrices below (first figure), which I assume is somehow related to the polyhedron of affine inequalities. I was wondering if one could explain the elements of the matrices below.
The following passage describes the matrix / first figure:
Iteration domains capture the dynamic instances of all statements — all possible values of surrounding loop iterators — through a set of affine inequalities. For example, statement R in Figure 2.1 is executed for every value of the pair of surrounding loop counters, called the iteration vector: the iteration vector of statement $R$ is $\vec{x_R} = (i, j)$. Hence, the iteration domain of R is defined by its enclosing loop bounds:
$$D_R = \{i, j \mid 1 \leq i \leq lg−1 \land 1 \leq j \leq wt−1\}$$
which forms a polyhedron (a space bounded by inequalities, a.k.a. hyperplanes or faces). To model iteration domains whose size are known only symbolically at compile-time, we resort to parametric polyhedra. Program constants which are unknown at compile-time are named global parameters and the parameters vector is noted $\vec{n}$. For instance for the Ring-Roberts example lg and wt are global parameters and $D_R$ is a parametric form of $\vec{n} = (lg\ wt)$.
Each integral point inside this polyhedron corresponds to exactly one execution of statement $R$, and its coordinates in $D_R$ matches the values of the loop iterators at the execution of this instance. This model let the compiler manipulate statement execution and iteration ordering at the most precise level. In the remainder of this thesis, we use matrix form in homogeneous coordinates to express polyhedra. For instance, for the iteration domain of $R$ is written:
