lsqcurvefit does not work for very small parameter fitting

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Xiaodi
Xiaodi 2019-5-8
I use lsqcurvefit to fit a very small parameter (smaller than 1e-15), but this function lsqcurvefit cannot work well, unless I give a very accurate initial (guess) value. Are there some solutions to fit a very small paremter? I put my example code in the following
clear
%% Data
xdata = [0.44;0.45;0.46;0.47;0.48;0.49;0.50;0.51;0.52;0.53;0.54;0.55;0.56;0.57;0.58;0.59;0.60;0.61;0.62;0.63];
ydata = [1.2e-09;1.7e-09;2.5e-09;3.7e-09;5.5e-09;8.1e-09;1.2e-08;1.8e-08;2.6e-08;3.8e-08;5.6e-08;8.3e-08;1.2e-07;1.8e-07;2.6e-07;3.9e-07;5.7e-07;8.4e-07;1.2e-06;1.8e-06];
x0_1 = 3e-17;
x0_2 = 5e-17;
%% Option
options = optimoptions(@lsqcurvefit, ...
'StepTolerance', 1e-150, ...
'MaxFunctionEvaluations', 2e3, ...
'MaxIterations', 2e3);
%% function
fun = @(p,xdata)p(1).*exp(xdata/0.026);
%% fit
coef1 = lsqcurvefit(fun,x0_1,xdata,ydata,0,1,options);
coef2 = lsqcurvefit(fun,x0_2,xdata,ydata,0,1,options);
%% Plot
figure
semilogy(xdata, ydata, 'o', 'DisplayName', 'data');hold on;
semilogy(xdata, fun(coef1,xdata), 'r-', 'DisplayName', 'fit with bad start');
semilogy(xdata, fun(coef2,xdata), 'b--', 'DisplayName','fit with good start');
legend('show')
untitled.jpg
From above plot, we can know that
  1. When start point x0 is good, the fitting result is good;
  2. however, the lsqcurvefit seems not to work, because the fiited results are always the same as initial point (coef1=x0_1,coef2=x0_2).
I have tried following solutions:
1. transform the form function to log. I would not like that, because the example function is the simplified function, the converting would be very hard; besides, I have at least 10 equations like it, if I can find a way to fit the original function, I would not like to transform them;
2. Set options of optimization, I have tried them like the above codes, but it does not help, maybe some other options to be set?

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