The Tall-Skinny QR (TSQR) algorithm is more communication effi_cient than the standard
Householder algorithm for QR decomposition of matrices with many more rows than columns.
However, TSQR produces a different representation of the orthogonal factor and therefore
requires more software development to support the new representation. Further, implicitly
applying the orthogonal factor to the trailing matrix in the context of factoring a square matrix
is more complicated and costly than with the Householder representation.
We show how to perform TSQR and then reconstruct the Householder vector representation with
the same asymptotic communication effi_ciency and little extra computational cost. We demonstrate
the high performance and numerical stability of this algorithm both theoretically and empirically. The
new Householder reconstruction algorithm allows us to design more effi_cient parallel QR algorithms,
with significantly lower latency cost compared to Householder QR and lower bandwidth and latency
costs compared with Communication-Avoiding QR (CAQR) algorithm. As a result, our final parallel
QR algorithm outperforms ScaLAPACK and Elemental implementations of Householder QR and our
implementation of CAQR on the Hopper Cray XE6 NERSC system.