@Article{Demmel:2008:EPI,
  author =       "James Demmel and Yozo Hida and Xiaoye S. Li and
                  E. Jason Riedy",
  title =        "Extra-precise Iterative Refinement for Overdetermined 
		  Least Squares Problems",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume =       "35",
  number =       "4",
  month =        feb,
  year =         2009,
  pages =        "28:1--28:32",
  URL =          "http://doi.acm.org/10.1145/1462173.1462177",
  accepted =     "25 June 2008",
  abstract =     "
		 We present the algorithm, error bounds, and numerical
		 results for extra-precise iterative refinement
		 applied to overdetermined linear least squares (LLS)
		 problems. We apply our linear system refinement algorithm
		 to Bj\"{o}rck's augmented linear system formulation of an
		 LLS problem. Our algorithm reduces the forward normwise
		 and componentwise errors to $\ord(\workingprec)$, where
		 $\workingprec$ is the working precision, unless the
		 system is too ill conditioned.  In contrast to linear
		 systems, we provide two separate error bounds for
		 the solution $x$ and the residual $r$. The refinement
		 algorithm requires only limited use of extra precision
		 and adds only $\ord(m n)$ work to the $\ord(m n^2)$ cost
		 of QR factorization for problems of size $m$-by-$n$. The
		 extra precision calculation is facilitated by the new
		 extended-precision BLAS standard in a portable way,
		 and the refinement algorithm will be included in a
		 future release of LAPACK and can be extended to the
		 other types of least squares problems.",
}