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From: John H Meyers on 5 May 2008 14:17
On Sat, 03 May 2008 22:19:04 -0500, Rodger Rosenbaum wrote:

> [Eliminating the word "solution" from a discussion is] an unworkable solution.
> Linear algebra texts are going to continue to use the word "solution".

The AUR and/or these newsgroup discussions are certainly free
to offer discussions that can explain what's going on
without using the word, or which can include,
either before or after such an explanation,
that in connection with LSQ, the word "solution" is used
to mean "any X for which the norm of the residual 'B-A*X'
attains its minimum possible value,"
to make completely clear that this is not necessarily
a "solution" to the _equations_ A*X=B.

What I see as the heart of the differing "takes" on this subject
is that the word "solution" was never properly defined _anywhere_
(when you gave a definition, it was in the form of defining
a longer phrase containing qualifying words in front of
the word "solution," which makes it seem that your definition
is about a special kind of "solution," which would then
make it seem that "solution" must be some larger class,
and what larger class is there than "all X"?).

The AUR also really, IMO, steers the reader (who may not be
an expert in linear algebra, but just an ordinary calculator user)
into believing that "solution" refers to a solution
of the _equations_ A*X=B -- the AUR actually says that,
even though it also contradicts itself elsewhere.

LSQ can be used for other purposes, even to refine
the accuracy of solutions to the equations, I believe,
so I think it deserves an explanation that clarifies this.

I just made analogy between LSQ and LR, and I like the idea
of expressing LSQ as something looking for a "best fit"
for a set of equations, just as LR looks for a "best fit" line,
even to a set of points that don't lie on a line
(LSQ thus also acting upon equation sets that have no
"exact-fit" solutions) -- this can be pointed out,
whether or not linear algebra texts do so, and I think
it will help to add something for general understanding.

Even the numeric solver, when there is a local minimum
but no exact solution, tries to locate that local minimum instead,
just as one more slightly analogous function that exists in the calculator.

Some of the "back and forth" in this thread seems to have come from
taking statements to an extreme interpretation; for example,
my saying that there is a correct one-sentence general definition
for the action of LSQ did not mean objection to elaborating
all the particular cases, and all the other material that could
be presented, and my saying that we can also explain LSQ
even without having to use or define the word "solution"
does not mean that we can't also include a definition --
just that if we are going to explain it using _only_
the word "solution," then we (the AUR, say) had better
actually say what we really mean, as well as not
saying incorrect (and confusing) things.

"A rose by any other name would smell as sweet,"
but I guess this doesn't work the same for "solution" :)

Language must be the fundamental sin,
because as soon as human beings got that knowledge,
that's where they've gotten into trouble ever since :)

--
From: John H Meyers on 5 May 2008 23:41
I wrote:

> LSQ can be used for other purposes, even to refine
> the accuracy of solutions to the equations, I believe.

My bad memory -- that was RSD (calculates the residual,
B-A*X from B, A, and X, hopefully with increased precision)

...where B and X can be either one vector,
or a matrix of several vectors, just as with LSQ.

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