We use uppercase letters for matrices. The notation means that X has columns . The notation means that X has elements .
In general, we use lowercase Greek letters for scalars.
The following symbols have the indicated meaning
The index space -- the set of values of the loop index vector
A The matrix that transforms natural to new loop indices
The matrix A with its columns scaled
to have euclidean length one
D The matrix of dependences
C The matrix of data fluxes
The ratio of the volume of a tile to its surface area
The vector of block size parameters
A normal vector to a tiling hyperplane; one of the columns of A\
A bound on the size of local memory.
The time required to perform the computation at a point
in the index space.
The time required to move data across one unit of area
in the hyperplane normal to .
We shall make considerable use of determinants. If
is a real, square matrix, then the real-valued
function is the volume of the parallelepiped subtended by the columns of X:
Thus, . Also . If Y is also , then . If is a triangular matrix, then .
Let denote the one-dimensional subspace spanned by the vector z, and let denote its orthogonal complement.
Proof: Let be a k-1-vector chosen so that for each , is orthogonal to . Construct the matrix
Then, since C is triangular and has unit diagonal, . Since is a vector of length one, . Thus,