Towards Reproducible Blocked LU Factorization

In this article, we address the problem of reproducibility of the blocked LU factorization on GPUs due to cancellations and rounding errors when dealing with floating-point arithmetic. Thanks to the hierarchical structure of linear algebra libraries, the computations carried within this operation can be expressed in terms of the Level-3 BLAS routines as well as the unblocked variant of the factorization, while the latter is correspondingly built upon the Level-1/2 BLAS kernels. In addition, we strengthen numerical stability of the blocked LU factorization via partial row pivoting. Therefore, we propose a double-layer bottom-up approach for ensuring reproducibility of the blocked LU factorization and provide experimental results for its underlying blocks.