Invertastic: Large-scale Dense Matrix Inversion

Linear algebraic techniques are widely used in scientific computing, often requiring large-scale parallel resources such as those provided by the ARCHER service. Libraries exist to facilitate the development of appropriate parallel software, but use of these involves intricacies in decomposition of the problem, managing parallel input and output, passing messages and the execution of the linear algebra operations themselves. In this paper a relatively simple application, Invertastic, is presented. This is designed to perform a real operation: the inversion of a dense symmetric positive definite matrix using multiple processors in parallel. The inversion of arbitrarily large matrices is demonstrated, with the only constraint being the size of compute resource available. Inversion cost is known to have O(N3) complexity, which the results confirm allowing for some parallel communication overhead. Inversion of a 2,097,152 x 2,097,152 matrix (of size 32TB) took 6.4 hours on 2,048 compute nodes (49,152 cores). The Invertastic software is freely available online and installed as a central package on ARCHER. It can be used directly (e.g. for genomic studies where the matrix represents the genetic relationships between multiple individuals), or instead as a reference or template for the development of more