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###

Further Details: Error Bounds for General Linear Model Problems

In this subsection, we will summarize the available error bounds.
The reader may also refer to [2,13,50,80]
for further details.

Let
and
be the solutions
by the driver routine xGGGLM (see subsection
4.6). Then
is normwise
backward stable and
is stable
in a mixed forward/backward sense [13]. Specifically,
we have
and
,
where
and
solve
,
and

and *q*(*m*,*n*,*p*) is a modestly growing function of *m*, *n*, and *p*.
We take *q*(*m*,*n*,*p*) = 1 in the code fragment above.
Let
denote the Moore-Penrose pseudo-inverse of *X*.
Let
( = `CNDAB` above) and
( = `CNDBA` above)
where
and
.
When
is small, the errors
and
are bounded by

When *B* = *I*, the GLM problem is the standard LS problem.
*y* is the residual vector *y* = *d* - *Ax*, and we have

and

where
and
.
The error bound of
is the same
as in the LSE problem (see section 4.6.1.1),
which is essentially the same as given in section 4.5.1.
The bound on the error in
is the same as that provided
in [55, section 5.3.7].

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*Susan Blackford*

*1999-10-01*