Tiling is a well-known loop transformation that can be used to exploit data reuse at the register level and to improve a program's ILP. Previous work on tiling and also commercial compilers are able to perform tiling for the register level in more than one dimension when the iteration space is rectangular. However, they either cannot handle or can only handle limited cases of non-rectangular iteration spaces. Non-rectangular iteration spaces are commonly found in linear algebra algorithms or can arise as a result of applying previous transformations such as loop skewing. In this paper we present a new general algorithm to perform tiling for the register level in more than one dimension in both rectangular and non-rectangular iteration spaces. Our method uses index set splitting to distinguish loop nests that traverse boundary tiles of the tiled iteration space from loop nests that traverse non-boundary tiles. We evaluate our method using as benchmarks typical linear algebra algorithms having non-rectangular iteration spaces. Results measured on both ALPHA 21064 and MIPS R10000 machines show that our method achieves speedups in the range of 1.11 to 5.96 over commercial compilers and preprocessors able to perform optimizing code transformations.
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