MATLAB: Lsqnonlin and stretched exponential function

Hi all,

I am currrently trying to fit my data with lsqnonlin instead lsqcurvefit for a comparative reason and I am facing some troubles.

Starting with the easiest example in https://se.mathworks.com/help/optim/ug/lsqnonlin.html, I modify the function as:

fun = @(r) r(1).*exp(-d.*r(2)).^r(3)-y

This function yields x = [1.0376 3.1962 0.4104].

From here, I calculate my errors in the way:

[x,resnorm,residual,exitflag,output,lambda,J] = lsqnonlin(fun,x0);

N = length(y(:,1));

[Q,R] = qr(J,0);

mse = sum(abs(residual).^2)/(size(J,1)-size(J,2));

Rinv = inv(R);

Sigma_var = Rinv*Rinv'*mse;

x_er = full(sqrt(diag(Sigma_var)));

Here, the values I get are not making any sense to me x_er = [0.02701 6458034.8422 829226.8633]; especially for x(2) and x(3).

Fit and errors are totally fine if I fit similar data in the form: (i) r(1).*exp(-d.*r(2)) or (ii) r(1).*exp(-d.*r(2))+r(3).*exp(-d.*r(4));

Could you please help me? Am I missing something?

Thanks a lot.

Best wishes

Alessandro