Myujth

I have the following set of data

data=0,0,0,0,2,1,0,4,2.247,0,6,3.627,0,8,5.031,1,0,3.346;

where the values are n, L,$varepsilon$ and satisfy the following equations

$E(n,L) = 2n+1 + sqrtL(L+1)-frac34(L)^2 + 1 + beta_0^4$

e[n_, L_] = 2n + 1 + Sqrt[L(L + 1) - 3/4 L^2 + 1 + b0^4]

$varepsilon = fracE(n,L)-E(0,0)E(0,2)-E(0,0)$,

where $beta_0$ should be determined. I don't know how I can use FindFit command of Mathematica to find the best value of $beta_0$ to have the best fit for $varepsilon$.

e[n_, L_] = 2n + 1 + Sqrt[L(L + 1) - 3/4 L^2 + 1 + b0^4]

FindFit[data, (e[n, L] - e[0, 0])/(e[0, 2] - e[0, 0]), b0, n, L]

b0 -> 1.3514967

Which seems reasonable in view of the residuals:

Plot[Evaluate[(e[#, #2] - e[0, 0])/(e[0, 2] - e[0, 0]) - #3 & @@@ data], b0, 0, 3]

The brown and purple residual has bigger slope around the roots in the plots. Hence for Mathematica to minimize the sum of squares in the y-dimension, the mean of the 2 data points that correspond to the big slopes are cared more about than the others. It is purpose specific whether this is appropriate. If it isn't you can add the NormFunction-option to FindFit.

You can also use NMinimize. First we need to write cost function, i.e. residual.

data = 0, 0, 0, 0, 2, 1, 0, 4, 2.247, 0, 6, 3.627, 0, 8, 
 5.031, 1, 0, 3.346;
e[n_, L_] := 2 n + 1 + Sqrt[L (L + 1) - 3/4 L^2 + 1 + b0^4]
cost[b0_] =Sum[(e @@data[[i, 1 ;; 2]] - (data[[i, 3]] (e[0, 2] - e[0, 0]) + 
 e[0, 0]))^2, i, 6];
(*or Total[(e[#1, #2] - (#3 (e[0, 2] - e[0, 0]) + e[0, 0]))^2 & @@@ data]*)

fit = NMinimize[cost[b0] , b0]

0.0196376, b0 -> 1.35462

Since your cost function has only one variable you can also use grid search.

Ordering[val,1] gives position of min value.

b0Val = Range[0, 10, 0.0001];
val = cost[b0Val];
b0Val[[Ordering[val, 1]]]

1.3546

Note that there is another min at b0=-1.3546

b0Val = Range[-1000, 1000, 0.001]; 
val = cost[b0Val]; 
b0Val[[Ordering[val, 2]]]

-1.3546, 1.3546

We can plot cost function

$textcost(b0)=left(-5.031 left(sqrttextb0^4+4-sqrttextb0^4+1right)-sqrttextb0^4+1+sqrttextb0^4+25right)^2\+left(-3.627
left(sqrttextb0^4+4-sqrttextb0^4+1right)-sqrttextb0^4+1+
sqrttextb0^4+16right)^2\+left(2-3.346
left(sqrttextb0^4+4-sqrttextb0^4+1right)right)^2+left(-2.247
left(sqrttextb0^4+4-sqrttextb0^4+1right)-sqrttextb0^4+1+sqrttextb0^4+9right)^2$

Plot[cost[b0], b0, -10, 10]