The conventional error analysis of linear least squares problems goes
as follows .
As above, let 
be the solution to minimizing
computed by
LAPACK using one of the least squares drivers xGELS, xGELSS or xGELSX
(see subsection 2.2.2).
We discuss the most common case, where A is
overdetermined
(i.e., has more rows than columns) and has full rank [45]:
The computed solution, where E and f satisfy
and p(n) is a modestly growing function of n. We take p(n) = 1 in the code fragments above. Let
(approximated by 1/RCOND in the above code fragments),
(= RNORM above), and
(SINT = RNORM / BNORM above). Here,
is the acute angle between the vectors
and
. Then when
is small, the error
is bounded by
where
= COST and
= TANT in the code fragments above.
We avoid overflow by making sure RCOND and COST are both at least
EPSMCH, and by handling the case of a zero B matrix
separately (BNORM = 0).
may be computed directly
from the singular values of A returned by xGELSS (as in the code fragment) or
by xGESVD. It may also be approximated by using xTRCON following calls to
xGELS or xGELSX. xTRCON estimates 
or 
instead
of 
, but these can differ from by at most a factor of n.
If A is rank-deficient, xGELSS and xGELSX can be used to regularize the
problem
by treating all singular values
less than a user-specified threshold
(
) as
exactly zero. The number of singular values treated as nonzero is returned
in RANK. See [45] for error bounds in this case, as well as
[45][19] for the
underdetermined
case.
The solution of the overdetermined, full-rank problem may also be characterized as the solution of the linear system of equations
By solving this linear system using xyyRFS or xyySVX (see section 4.4) componentwise error bounds can also be obtained [7].
