Please can people advise me on this

[a.mlw.walker]

Hi I have attached an image of part of a freely available paper I was reading. It shows the equations for least squares minimization of some equations based on empirical data.

I am not completely confident I understand the required steps, and therefore just wanted to talk it through with others, see what you say and see if it sparks any ideas to solve these.

As I understand it, the first equation (eqn 40) is minimizes using levenberg marquardt for a, b and c0. k is 1,2,3,4... t_k is times stored, the rest of the equation is trying to model the time it will take (whatever that time may be).

Ok so using levenberg marquardt estimate a, b and c0.
But the the next equation (eqn 41) says that he freezes beta to be ab^2, and uses golden section method to 'refine' a and b?

Did the leveneberg marquardt not do a good enough job because I thought we found and ab that way?

the author also give the sub equations for c0. Why? I thought we estimated c0?

Ok so however it has been done, we have a good estimate for a, b and c0.
Eqn 42. Same thing again except now for nu, phi, omega^2.

How is it explaining to solve this?
"robus linear estimation"? "Directly minimizing"? Is that how? Why can't we use levenberg marquardt again?

I just would like discussion, I am not after just the answer, I would like to understand when to use what, and why...

Thanks guys

By the way, that is the whole chapter I haven't left anything out except the chapter title

 

[a.mlw.walker]

Is this in the wrong forum?