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 Björck’s augmented linear
system formulation of an LLS problem. Our algorithm
reduces the forward normwise and componentwise
errors to $O(\varepsilon)$ 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 $O(mn)$ work to the $O(mn^2)$ 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.
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