Automatically Computing Unimodular Transforming Matrix to Parallelize Nested Sequential Loops
自动寻找使多重串行循环并行化的幺模变换

Lacking an effective and feasible algorithm to compute the valid unimodular matrix for parallelizing of the outer loops, previous parallelizing researches can not automatically reveal the parallelism in such sequential nested-loops as have n-dimension distance matrix. In this paper, the authors discuss a general outer-loop parallelizing method by valid unimodular transformations, prove the existence of such a valid unimodular transformation, and suggest several practical computing algorithms through the constructive proofs. This discovered unimodular transformation can have the maximal number of parallelizable outer-loops transformed. Thus, the application scope of the algorithms can be enlarged to non-perfect or non-constant dependence distance loops.