Solving a non linear least square of a sum function with two unknowns

Hello,

i have a Chi-Square merit function with two unknown k1 and k2. Unfortunately, I don't know exactly which syntax to use to minimize this function in order to determine k1 and k2.

are known, each a vector with 1 column and 17 rows. are constant values.

Unfortunately, I am also relatively new with Matlab. I would be very happy about a step by step explanation. I have already found out that lsqnonlin can be used as a solver.

I appreciate your help. Thank you.

[Star Strider]

Note that lsqnonlin fits a function to data.

Do you have a matching vector for ?

[Zehra Ese]

Hi,

you're right, I didn't give enough information. Yes its correct I want to fit a function to data. We have given xdata and ydata. g vector contains all measured values. The first term in parantheses is the function we want to fit to the data. I would like to determine and , while minimizing this Chi-square funktion. All the other coefficients are known, as explained before.

Answer 1

[Matt J]

It would look like this,

kInitial = __________; %initial guess of k

kOptimal = lsqnonlin(@(k) residual(k,am,bm,cm,g,ch,bh,ah) , kInitial);  %do the optimization

function r=residual(k,am,bm,cm,g,ch,bh,ah)


   k1=k(1); %extract k(i) into separate variables, for convenience
   k2=k(2);
   
   numerator=cm+bm*k2+am*k1;  %calculate numerator expression for all m
   denominator=ch+bh*k2+ah*k1; %calculate demonator expression for all m

   r=numerator./denominator - g; %calculate the vector of residuals, for all m

end

[Matt J]

For an initial guess, I would probably do,

kInitial=[ am-g.*ah, bm-g.*bh] \ (g.*ch-cm); %solve the linear system  [ am-g.*ah, bm-g.*bh]  * k =  (g.*ch-cm)

which comes from setting your residuals to zero and re-arranging them as linear equations:

cm+bm*k2+am*k1 = g.*(ch+bh*k2+ah*k1)

[Matt J]

Zehra Ese's answer converted to comment:

Thank u so much!!! It looks like it's working. I have to prove the results and I let you know.

Can I ask you to explain the individual steps to me in comments. I am still a beginner in Matlab and would really like to understand that.

[Matt J]

You're welcome. I've added more comments to the code.