Summary

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$(I_1,I_2)\cdot {\bf U}$ maps each statement instance (I₁,I₂) to its new position (K₁,K₂) in the transformed loop's exec$^{\rm n}$ sequence

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$(D_1,D_2)\cdot {\bf U}$ gives new dependence distance vector, giving test for validity

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Captures skewing, interchange and reversal

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Compose transformations by matrix multiplication ${\bf U_1 \cdot U_2}$

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Resulting loop's bounds may be a little tricky

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Efficient algorithms exist [#!banerjee:90!#] to maximise parallelism by skewing and loop interchanging

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Efficient algorithms exist to optimise cache performance by finding the combination of blocking, block size, interchange and skewing which leads to the best reuse [#!wolf:91!#]