A Linear Algebraic View of Loop Transformations and Their Interaction

A Linear Algebraic View of Loop

Although optimizing transformations have been studied for over two decades, the interactions between them is not well understood. This is particularly important for the success of parallelizing compilers. In order to deal with interactions, we view loop transformations as multiplication by a suitable matrix. The transformations considered are loop interchange, permutation, reversal, hyperplane ...

On One-Sided Ideals of Rings of Linear Transformations

Addendum to: "Infinite-dimensional versions of the primary, cyclic and Jordan decompositions", by M. Radjabalipour

In his paper mentioned in the title, which appears in the same issue of this journal, Mehdi Radjabalipour derives the cyclic decomposition of an algebraic linear transformation. A more general structure theory for linear transformations appears in Irving Kaplansky's lovely 1954 book on infinite abelian groups. We present a translation of Kaplansky's results for abelian groups into the terminolo...

Linear Loop Transformations in Optimising Compilers for Parallel Machines

We present the linear loop transformation framework which is the formal basis for state of the art optimization techniques in restructuring compilers for parallel machines. The framework uniies most existing transformations and provides a systematic set of code generation techniques for arbitrary compound loop transformations. The algebraic representation of the loop structure and its transform...

Non - unimodular Transformations of Nested

This paper presents a linear algebraic approach to modeling loop transformations. The approach uniies apparently unrelated recent developments in super-compiler technology. Speciically we show the relationship between the dependence abstraction called dependence cones, and fully permutable loop nests. Compound transformations are modeled as matrices. Non-singular linear transformations presente...