Not to my knowledge. This is tough problem. Least squares by their very nature are approximations. There is a lot of literature out there. Some of them are given here:

http://research.microsoft.com/en-us/um/people/yangliu/publication/a%20revisit%20to%20least%20squares%20orthogonal%20distance%20fitting%20of%20parametric%20curves%20and%20surfaces.pdf

http://igpphome.ucsd.edu/~cathy/Classes/SIO223A/sio223a.chap8.pdf

http://jips-k.org/dlibrary/JIPS_v04_no4_paper4.pdf

yes, you can with certain erorr that can be defined based on the used method (methods like newton or Gauss with only the first drevative of the fn). However, you should set a parameter constrains as it got reflection on the physical meaning of the parameter.

Another convergence problem could also raise up if the variations are too big

for example : Nonlinear Fitting Technology of 7-Parameter Cross-WLF Viscosity Model

http://www.scientific.net/AMR.189-193.2103

Please clarify your question. Is it about fitting a parametric curve to data using a least sum of squares of the residuals?

And what is 'Total least squares' as opposed to just 'least square'?

Chi squared minimization generally works to find the best least squares fit,

but as pointed out previously, least squares may not be the best approach.

(google on "maximum likelihood techniques" for an alternative approach)