@Article{Drmac:2008:,
  author =       "Zlatko Drma\v{c} and Zvonimir Bujanovi\'{c}",
  title =        "On the Failure of Rank Revealing QR Factorization Software --- A Case Study",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume =       "35",
  number =       "2",
  month =        jul,
  year =         "2008",
  pages =        "12:1--12:28",
  URL =          "http://doi.acm.org/10.1145/1377612.1377616",
  abstract =     "This paper reports an unexpected and rather erratic behavior of the LAPACK software 
                 implementation of the QR factorization with Businger{Golub column pivoting. It is shown that, 
                 due to finite precision arithmetic, the software implementation of the factorization can 
                 catastrophically fail to produce properly structured triangular factor, thus leading to 
                 potentially severe underestimate of a matrix's numerical rank. The 30 year old problem, dating 
                 back to LINPACK, has (undetectedly) badly affected many computational routines and software 
                 packages, as well as the study of rank revealing QR factorizations. We combine computer 
                 experiments and numerical analysis to isolate, analyze and fix the problem. Our modification 
                 of the current LAPACK xGEQP3 routine is already included in the LAPACK 3.1.0 release. The 
                 modified routine is numerically more robust and with a negligible overhead. We also provide a 
                 new, equally efficient and provably numerically safe, partial column norm updating strategy.",
}