One-sided Dense Matrix Factorizations on a Multicore with Multiple GPU Accelerators^*

Abstract

One-sided dense matrix factorizations are important computational kernels in many scientific and engineering simulations. In this paper, we propose two extensions of both right-looking (LU and QR) and left-looking (Cholesky) one-sided factorization algorithms to utilize the computing power of current heterogeneous architectures. We first describe a new class of non-GPU-resident algorithms that factorize only a submatrix of a coefficient matrix on a GPU at a time. We then extend the algorithms to use multiple GPUs attached to a multicore. These extensions not only enable the factorization of a matrix that does not fit in the aggregated memory of the multiple GPUs at once, but also provide potential of fully utilizing the computing power of the architectures. Since data movement is expensive on the current architectures, these algorithms are designed to minimize the data movement at multiple levels. To demonstrate the effectiveness of these algorithms, we present their performance on a single compute node of the Keeneland system, which consists of twelve Intel Xeon processors and three NVIDIA GPUs. The performance results show both negligible overheads and scalable performance of our non-GPU-resident and multi-GPU algorithms, respectively. These extensions are now parts of the MAGMA software package, a set of the state-of-the-art dense linear algebra routines for a multicore with GPUs.