Ordinary least-squares fit

1. (6 points)

   Implement a routine which makes a least-squares fit of a given data-set, {x_i, y_i, δy_i | i=1...n}, with a linear combination of given functions, F(x) =∑ _k=1..m c_k f_k(x).

   The subroutine must calculate the vector of the coefficients, c_k, and the covariance matrix. The fitting functions {f_k(x)} can be programmed, e.g., as

   • a vector of functions (C++11,JavaScript,Python),
     std::vector<std::function<double(double)>> fitfunctions = {f1,...,fm};

   • a function of two parameters (C,Fortran,C++,Python),
     double fitfunctions(int i, double x){return fi(x);}
     For example:

     #include<math.h> // NAN is here
     double funs(int i, double x){
        switch(i){
        case 0: return 1.0; break;
        case 1: return x;   break;
        case 2: return x*x; break;
        default: {fprintf(stderr,"funs: wrong i:%d",i); return NAN;}
        }
     }
     

   • you can simply precalculate the matrix A_ik=f_k(x_i) and pass it as a parameter.
   Make up an interesting fitting example as an illustration of how your subroutine works.
2. (3 points)
   Make up some interesting fits and check that the reported errors of the fitting coefficients are reasonable. For example,

   

3. (1 point)
   Make up two experimental points with error bars, {x_i, y_i ± Δ y_i | i=1,2}. Assume that the theoretical curve is a straight line. Make an ordinary least-squares fit of your experimental data with a linear function y(x)=a+bx. Make a plot of your results. On the plot illustrate the probable distribution of the variable y as function of x (in the assumption of normal distribution of y_i, a, b if you need that).

   Hint: the variation of y with respect to variations of a and b as function of x is given as
   δy = δa + δb x.
   Therefore
   δy² = δa² + δb² x² +2δa δb x.
   For normally distributed a and b this translates into
   Δy²(x) = Δa² + Δb²x² + 2<δa δb>x
   where <δa δb> is the covariance.

4. (0 points)

   Make up an interesting exercise here.