Ordinary least-squares fit

1. (6 points) Ordinary least-squares fit by QR-decomposition

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

      The parameters to the function should be the data to fit and the set of functions the linear combination of which should fit the data. The function must calculate the vector of the coefficients, c_k .

   2. Fit the following data,

      x  =   0.100   0.145   0.211   0.307   0.447   0.649   0.944   1.372   1.995   2.900
      y  =  12.644   9.235   7.377   6.460   5.555   5.896   5.673   6.964   8.896  11.355
      dy =   0.858   0.359   0.505   0.403   0.683   0.605   0.856   0.351   1.083   1.002
      

      with the linear combination of functions {1/x, 1, x}.

   The fitting functions {f_k(x)} can be implemented, e.g., as

   • a vector of functions (C++11),
     

     std::vector<std::function<double(double)>> fitfunctions = {f1,...,fm};
     

   • a list of functions (Python),
     

     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/x; break;
        case 1: return 1.0;   break;
        case 2: return x;     break;
        default: {fprintf(stderr,"funs: wrong i:%d",i); return NAN;}
        }
     }
     

2. (3 points) Uncertainties of the fitting coefficients

   • Modify you least-squares fitting function such that it also calculates the covariance matrix and the uncertainties of the fitting coefficients.

   • Make up some interesting fits and check that the reported uncertainties of the fitting coefficients are reasonable. For example,

     

3. (1 points) Ordinary least-squares solution by thin singular-value decomposition

   1. Implement a function which makes a singular value decomposition of a (tall) matrix A by diagonalising A^TA (with your implementation of the Jacobi eigenvalue algorithm), A^TA=VDV^T (where D is a diagonal matrix with eigenvalues of A^TA and V is the matrix of the corresponding eigenvectors) and then using A=UΣV^T where U=AVD^-1/2 and Σ=D^1/2.
   2. Implement a function which solves the linear least squares problem Ax=b using you implementation of singular-value decomposition.
   3. Implement a function which makes ordinary least-squares fit using your implementation of singular-value decomposition.

4. (0 points) Extra

   Make up an interesting exercise here.