Tiled QR factorization algorithms

This work revisits existing algorithms for the QR factorization of rectangular matrices composed of p × q tiles, where p ≥ q. Within this framework, we study the critical paths and performance of algorithms such as SAMEH-KUCK, FI BONACCI, GREEDY, and those found within PLASMA. Al though neither FIBONACCI nor GREEDY is optimal, both are shown to be asymptotically optimal for all matrices of size p = q² f(q), where f is any function such that lim₊∞ f = 0. This novel and important complexity result applies to all matrices where p and q are proportional, p = λq, with λ ≥ 1, thereby encompassing many important situations in practice (least squares). We provide an extensive set of experiments that show the superiority of the new algorithms for tall matrices.