  This work revisits existing algorithms for the QR factorization of
rectangular matrices composed of p-by-q tiles, where p >= q. Within this
framework, we study the critical paths and performance of algorithms such as
Sameh and Kuck, Modi and Clarke, Greedy, and those found within PLASMA.
Although neither Modi and Clarke nor Greedy is optimal, both are shown to be
asymptotically optimal for all matrices of size p = q^2 f(q), where f is any
function such that \lim_{+\infty} f= 0. This novel and important complexity
result applies to all matrices where p and q are proportional, p = \lambda q,
with \lambda >= 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.
