Asynchronous Task-Based Polar Decomposition on Single Node Manycore Architectures

This paper introduces the first asynchronous, task-based formulation of the polar decomposition and its corresponding implementation on manycore architectures. Based on a formulation of the iterative QR dynamically-weighted Halley algorithm (QDWH) for the calculation of the polar decomposition, the proposed implementation replaces the original LU factorization for the condition number estimator by the more adequate QR factorization to enable software portability across various architectures. Relying on fine-grained computations, the novel task-based implementation is capable of taking advantage of the identity structure of the matrix involved during the QDWH iterations, which decreases the overall algorithmic complexity. Furthermore, the artifactual synchronization points have been weakened compared to previous implementations, unveiling look-ahead opportunities for better hardware occupancy. The overall QDWH-based polar decomposition can then be represented as a directed acyclic graph (DAG), where nodes represent computational tasks and edges define the inter-task data dependencies. The StarPU dynamic runtime system is employed to traverse the DAG, to track the various data dependencies and to asynchronously schedule the computational tasks on the underlying hardware resources, resulting in an out-of-order task scheduling. Benchmarking experiments show significant improvements against existing state-of-the-art high performance implementations for the polar decomposition on latest shared-memory vendors' systems, while maintaining numerical accuracy.