Re: Choosing parameters.

[Daniel Carrera <dcarrera math toronto edu>]

[snip]
>    If the initial guess is (a,b,c) = (1,1,1), the program arrives at:
>
>    (a,b,c) ==  (16.3355999, -0.03256876, 1.68292832)
>
>    This is the wrong answer.
>
> Yes.  In fact it went outside the range I specified for a.
>
> Now pick a new random starting point and retry.  Sometimes the algorithm
> converges, sometimes it does not.  But we can compare the cases where
> it does converge and pick the best (in the least squares sense).
>
> How does that sound?

I'm not an expert on this, but it sounds good to me.

The LM method tries to minimize the Chi Squared by (roughly speaking)
moving the parameters in the direction of steepest descent.

Notice, the algorithm always converges.  Problem is that some times
it converges to a local minimum and gets trapped there.

I suppose that it makes sense to try many points and see which one arrives
at the lowest Chi Squared.

Won't that be expensive though?  Also, how do we know how many points to
try?  That number depends on the user-defined function and the size of the
intervals chosen.

Furthermore, wouldn't it still be good to help the user pick good ranges
by seeing a plot?  Is this difficult?

Daniel.