How to perform a minimization between two spectra using a nonlinear least square function?

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Dear All,
I have two spectra together for comparison and I want to minimize the difference between them. The solver of the objective function is nonliear least square (in this filed, Levenberg-Marquardt is the best method). Here is the graph dipecting the simulated and measured data. The data is attached and here is the lines that call them:
load Observed.txt;
wavelength1=Observed(:,1);
total1=Observed(:,2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
load Measured.txt;
wavelength2=Measured(:,1);
total2=Measured(:,2);
The way I understand it is that a nonlinear least square provides me with the best fit. How does someone generate the fit? Please execuse my ingnorance over this issue as I am stumbling with it for the past month and I am realy keen to understand it. I am using this data to get familiar and really understand the minimization concept and how is it implemented in Matlab so that I can apply it in the rest of the data.
Thank you very much for the help, guidance and time in advance
Cheers,

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