Curve Fitting for equation with two parameters

Question

I have two arrays:

E= [6656400;
    13322500;
    19980900;
    26625600;
    33292900;
    39942400;
    46648900;
    53290000]

and

J=[0.0000000021;
    0.0000000047;
    0.0000000128;
    0.0000000201;
    0.0000000659;
    0.0000000748;
    0.0000001143;
    0.0000001397]

I want to find the appropriate curve fitting for the above data by applying this equation:

J=A0.*(298).^2.*exp(-(W-((((1.6e-19)^3)/(4*pi*2.3*8.854e-12))^0.5).*E.^0.5)./((1.38e-23).*298))

I want to select the starting value of W from 1e-19

I have tried the curve fitting tools but it is not helping me to solve it!

Then, I selected some random values of A0=1.2e9 and W=2.243e-19, it gave me a better results. But I want to find the right values by using the code (not the curve fitting Apps)

Can you help me please?

You can use this post for inspiration: stackoverflow.com/questions/28112559/… - Essentially, you should take the log of the equation so that it becomes linear in log space, then calculate the coefficients by linear regression. — rayryeng

TroyHaskin TroyHaskin · Accepted Answer · 2015-01-29T19:15:07

A quick (and potentially easy) solution method would be to pose the curve fit as a minimization problem.

Define a correlation function that takes the fit parameters as an argument:

% x(1) == A0; x(2) == W
Jfunc = @(x) x(1).*(298).^2.*exp(-(x(2)-((((1.6e-19)^3)/(4*pi*2.3*8.854e-12))^0.5).*E.^0.5)./((1.38e-23).*298));

Then a objective function to minimize. Since you have data J we'll minimize the sum-of-squares of the difference between the data and the correlation:

Objective = @(x) sum((Jfunc(x) - J).^2);

And then attempt to minimize the objective using fminsearch:

x0  = [1.2E9;2.243E-19];
sol = fminsearch(Objective,x0);

I used the guesses you gave. For nonlinear solutions, a good first guess is often important for convergence.

If you have the Optimization Toolbox, you can also try lsqcurvefit or lsqnonlin (fminsearch is vanilla MATLAB).

Curve Fitting for equation with two parameters

1 Answers