In my work we fit a parabola to some data in order to determine three parameters. I recently talked to someone who pointed out that the ISO standard related to the fit equation had changed. The claim is that there is some subtlety with the fit which requires using the new form. I would like a second opinion.
The older standard lists the equation as
d^2 = a + b*z + c*z^2
The fit of the data to this curve can be accomplished using a linear least squares method.
The new form of the equation is
d = SQRT(a + b*z + c*z^2)
The fit of the data to this equation is done using the Levenberg-Marquardt algorithm.
My question is, what is the difference? During an early version of our software we estimated the parameters in the first equation using the linear least squares method followed by fine tuning using the Levenberg-Marquardt method. Seems to me that the results should be identical if both fits are performed using the LM method. Perhaps the “subtlety” is comparing a linear least squares method result in the first form to an LM result in the second form.
Do you have any insight on this question?