Exploiting Fine-Grain Parallelism in Recursive LU Factorization

The LU factorization is an important numerical algorithm for solving system of linear equations in science and engineering and is characteristic of many dense linear algebra computations. It has even become the de facto numerical algorithm implemented within the LINPACK benchmark to rank the most powerful supercomputers in the world, collected in the TOP500 website. In this context, the challenge in developing new algorithms for the scientific community resides in the combination of two goals: achieving high performance and maintaining the accuracy of the numerical algorithm. This paper proposes a novel approach for computing the LU factorization in parallel on multicore architectures, which not only improves the overall performance, but also sustains the numerical quality of the standard LU factorization algorithm with partial pivoting. While the update of the trailing submatrix is computationally intensive and highly parallel, the inherently problematic portion of the LU factorization is the panel factorization due to its memory-bound characteristic as well as the atomicity of selecting the appropriate pivots. We present a new approach to LU factorization of (narrow and tall) panel submatrices. It uses a parallel fine-grained recursive formulation of the factorization. It is based on conflict-free partitioning of the data and lockless synchronization mechanisms. As a result, our implementation lets the overall computation naturally flow without contention. Our recursive panel factorization provides the necessary performance increase for the inherently problematic portion of the LU factorization of square matrices. The reason is that even though the panel