  We present parallel and sequential dense QR factorization algorithms for tall
and skinny matrices and general rectangular matrices that both minimize
communication, and are as stable as Householder QR. The sequential and parallel
algorithms for tall and skinny matrices lead to significant speedups in
practice over some of the existing algorithms, including LAPACK and ScaLAPACK,
for example up to 6.7x over ScaLAPACK. The parallel algorithm for general
rectangular matrices is estimated to show significant speedups over ScaLAPACK,
up to 22x over ScaLAPACK.
