Problems fitting a function with a NonlinearLeastSquares model

Hi everybody.

I need to fit some experimental data on the function

y=sqrt(A*((x-x0)/0.04).^-n)

I understand this to be a tricky function, but sadly it's the one that i need to fit. My unknowns are A, x0 and n.

To test the algorithm i prepared a benchmark case where A, x0 and n are know. Sadly the fit does not works (more or less, i'll explain later) not even in this easy way. The script is

    %Initial Data
    M=0.04;
    A=3000;
    n=1.3;
    x0=1;

    %Generate x y coordinates
    x=linspace(x0,100*M,1000);
    y=sqrt(A*((x-x0)/M).^-n)+0.1*randi([-1 1],size(x));
    figure
    loglog(x,y)

    %Select fit range
    minimumFittinX=1.4;
    y=y(x>minimumFittinX);
    x=x(x>minimumFittinX);
    hold on
    loglog(x,y)

    % %%%%%  FIT  %%%%%
    options=fitoptions('Method','NonlinearLeastSquares'); 
    options.Lower      = [500   0          1];
    options.StartPoint = [A     x0         n];  %NB: I start from the correct value!!!
    options.Upper      = [10000 .8*min(x)  2.5]; %Limiting max(x0) to .8*min(x) guarantees to be far enough from the x=x0 asymptote and (x-x0) to be always >0
    options.MaxIter    = 1000;
    type=fittype('sqrt(A*((x-x0)/0.04).^-n)');
    fittedFun = fit(x',y',type,options); %Fitto

    % Results
    IFitted = feval(fittedFun,x);                 
    Afit=fittedFun.A;
    x0fit=fittedFun.x0;
    nfit=fittedFun.n;

    plot(x,IFitted)
    sprintf('Interpolated using equation y=sqrt(A*((x-x0)/M).^-n) with:\nA\t=\t%f\nx0\t=\t%f\nn\t=\t%f',Afit,x0fit,nfit)

As you can see I define a range to be fitted using minimumFittinX.

For value of minimumFittinX above 1.3 there's no problem. If you set minimumFittinX at 1.2 however, fit returns the error:

 Complex value computed by model function, fitting cannot continue.
 Try using or tightening upper and lower bounds on coefficients.

Other times it returns

 Inf computed by model function, fitting cannot continue.
 Try using or tightening upper and lower bounds on coefficients.

Now... using the upper and lower bound I set for the 3 variables the y value should be limited between 0.13 and 57, way far from Inf!! Moreover min(x-x0) = 1.2, therefore where that Complex value comes from??

How can I improve the fit and make it more robust? This is even a case where the starting value where the correct ones. I can't even think about using real experimental data (I tried with obvious results).

[Luca Amerio]

Any guess from anyone?

[Torsten]

Use the fitting function

y=sqrt(A^2*(((x-x0)/0.04).^2).^(-n))

Best wishes

Torsten.

[Torsten]

Or even better (if you start with a positive n)

y=sqrt(A^2*(((x-x0)/0.04).^2).^n)

Best wishes

Torsten.