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From: W. eWatson on 26 Jun 2010 13:17
Let my simplify matters here, and just concentrate on the central point.

I'm looking at a paper that deals with 5 NL (nonlinear) equations and 8
unknown parameters.

A. a=a0+arctan((y-y0)/(x-x0)
B. z=V*r+S*e**(D*r)
r=sqrt((x-x0)**2+(y-y0)**2)
and
C. cos(z)=cos(u)*cos(z)-sin(u)*sin(ep)*cos(b)
sin(a-E) = sin(b)*sin(u)/sin(z)

a0, xo, y0, V, S, D, ep, and E are the parameters. Data is available for
the variables.

What I've presented is probably a bit difficult to understand without a
good optics understanding, but my question is something like is this
commonly done to solve a system of NLLSQ? It looks a bit wild. It seems
to me he first solves for parameters in A, then uses them in B, then
those in C. I guess it makes sense, but is such a method often
successful? Comments?





From: John D'Errico on 26 Jun 2010 13:43
"W. eWatson" <wolftracks(a)invalid.com> wrote in message <i05co3$mi4$1(a)news.eternal-september.org>...
> Let my simplify matters here, and just concentrate on the central point.
>
> I'm looking at a paper that deals with 5 NL (nonlinear) equations and 8
> unknown parameters.
>
> A. a=a0+arctan((y-y0)/(x-x0)
> B. z=V*r+S*e**(D*r)
> r=sqrt((x-x0)**2+(y-y0)**2)
> and
> C. cos(z)=cos(u)*cos(z)-sin(u)*sin(ep)*cos(b)
> sin(a-E) = sin(b)*sin(u)/sin(z)
>
> a0, xo, y0, V, S, D, ep, and E are the parameters. Data is available for
> the variables.
>
> What I've presented is probably a bit difficult to understand without a
> good optics understanding, but my question is something like is this
> commonly done to solve a system of NLLSQ? It looks a bit wild. It seems
> to me he first solves for parameters in A, then uses them in B, then
> those in C. I guess it makes sense, but is such a method often
> successful? Comments?

It may work. If this author has used the scheme,
then apparently he did make it work, at least on
one of his problems.

Is it a stable procedure? Probably not terribly so.
You would need to verify the results from each step
to decide that convergence to a reasonable solution
was obtained for that step. And remember that each
step returns only a result that has a convergence
tolerance on it. So the second step will be a bit
fuzzier than the first, and the third step the fuzziest
of all.

But done with care, with good starting values, it
might work.

John
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