Beyond unimodular transformations

  • Ramanujam J
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Abstract

This paper presents an approach to modeling loop transformations using linear algebra. Compound transformationsare modeled as integer matrices. Non-singular linear transformations presented here subsumes theclass of unimodular transformations. The loop transformations included are the unimodular transformations-- reversal, skewing, permutation -- and a new transformation, namely stretching. Non-unimodular transformations(with determinant 1) create "holes" in the transformed iteration space, rendering code generationdifficult. We solve this problem by suitably changing the step size of loops in order to "skip" these holeswhen traversing the transformed iteration space. For the class of non-unimodular loop transformations, wepresent algorithms for deriving the loop bounds, the array access expressions and step sizes of loops in thenest. To derive the step sizes, we compute the Hermite Normal Form of the transformation matrix; the stepsizes are the entries on the diagonal of thi...

Author-supplied keywords

  • Hermite normal form
  • Loop transformations
  • code generation
  • integer lattices
  • linear transformations of the iteration space
  • loop stretching
  • nonunimodular transformations

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Authors

  • J. Ramanujam

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