Communication-avoiding Cholesky-QR2 for rectangular matrices

Scalable algorithms for solving least squares and eigenvalue problems are critical given the increasing parallelism within modern machines. We address this concern by presenting a new scalable QR factorization algorithm intended to accelerate these problems for rectangular matrices. Our contribution is a communication-avoiding distributed-memory parallelization of an existing Cholesky-based QR factorization algorithm called CholeskyQR2. Our algorithm executes on a 3D processor grid, the dimensions of which can be tuned to trade-off costs in synchronization, interprocessor communication, computational work, and memory footprint. It improves the communication cost complexity with respect to state-of-the-art parallel QR implementations by $\Theta(P^{1/6})$. We implement the new 3D CholeskyQR2 algorithm and study its performance relative to ScaLAPACK on Stampede 2, an Intel Knights Landing cluster, demonstrating improvements in parallel scalability and absolute performance.