Loop optimization
In compiler theory, loop optimization is the process of increasing execution speed and reducing the overheads associated of loops. It plays an important role in improving cache performance and making effective use of parallel processing capabilities. Most execution time of a scientific program is spent on loops; as such, many compiler optimization techniques have been developed to make them faster.
Since instructions inside loops can be executed repeatedly, it is frequently not possible to give a bound on the number of instruction executions that will be impacted by a loop optimization. This presents challenges when reasoning about the correctness and benefits of a loop optimization, specifically the representations of the computation being optimized and the optimization(s) being performed.
Loop optimization can be viewed as the application of a sequence of specific loop transformations (listed below or in) to the source code or intermediate representation, with each transformation having an associated test for legality. A transformation (or sequence of transformations) generally must preserve the temporal sequence of all dependencies if it is to preserve the result of the program (i.e., be a legal transformation). Evaluating the benefit of a transformation or sequence of transformations can be quite difficult within this approach, as the application of one beneficial transformation may require the prior use of one or more other transformations that, by themselves, would result in reduced performance.
The unimodular transformation approach uses a single unimodular matrix to describe the combined result of a sequence of many of the above transformations. Central to this approach is the view of the set of all executions of a statement within n loops as a set of integer points in an n-dimensional space, with the points being executed in lexicographical order. For example, the executions of a statement nested inside an outer loop with index i and an inner loop with index j can be associated with the pairs of integers . The application of a unimodular transformation corresponds to the multiplication of the points within this space by the matrix. For example, the interchange of two loops corresponds to the matrix .
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