Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination
Résumé
We present block algorithms and their implementation for the parallelization of Gaussian elimination over a finite field on shared memory architectures. Specificities of exact computations over a finite field include the use of sub-cubic matrix arithmetic and of costly modular reductions. As a consequence coarse grain block algorithms perform more efficiently than fine grain ones and recursive algorithms are preferred. We incrementally build efficient kernels, for matrix multiplication first, then triangular system solving, on top of which a recursive PLUQ decomposition algorithm is built. We study the parallelization of these kernels using several algorithmic variants: either iterative or recursive and using different splitting strategies. Experiments show that recursive adaptive methods for matrix multiplication, hybrid recursive-iterative methods for triangular system solve and recursive versions of PLUQ decompositions, together with various data mapping policies, provide the best performance on a 32 cores NUMA architecture. Overall, we show that the overhead of modular reductions is compensated by the fast linear algebra algorithms and that exact dense linear algebra matches the performance of full rank reference numerical software even in the presence of rank deficiencies.
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