Assignment 3 - Interpolation and Curve Fitting

1. Fitting noisy data to $y=5xe^{-3x}$.
   • The following MATLAB code segment creates a synthetic data set by adding noise to $g(x)=5xe^{-3x}$. You should supply the value of $n$, which is the number of data points to be generated.

     % --- Generate synthetic data
     x0 = 0.01;         % Starting point ~= 0 avoids log(0)
     noise = 0.05;      % Magnitude of noise
     x = linspace(x0,2,n);
     y = 5*x.*exp(-3*x);% Create the true function y = g(x)
     yn = y + noise*(rand(size(x))-0.5);  %Noise multiplies 
                        % random values v  in the range 
     		   % -0.5 <= v <= 0.5 
     yn = abs(yn);      % Make sure all data are positive
     

   • Now, we have a table (assume that obtained from an experiment). It is proposed that an unknown function has the form
     
     $$f(x)=c_1xe^{c_2x}$$
     
     

   • Write a MATLAB program to solve the normal equations to find the values of $c_1$ and $c_2$ for the given data set.
     Hint: Nonlinear function is transformed to linear function as $\upsilon=\alpha u+ \beta$ and
     
     $$\begin{array}{ll} \upsilon=ln(x/y) & u=x \\ \beta=ln c_1 & \alpha=c_2\\ \end{array}$$
     
     

   • Your program should generate the data set, fit the data by using least-square approxmiation and plot the fitted and original functions.
   • Study with different $n$ values (say 50,100,200). Discuss the effect of increasing $n$ on the fitted curve.
2. Write a MATLAB program to fit a cubic to the data which is generated in the previous question. Evaluate the cubic on the data. Compare this polynomial with the functions ($f(x)$ and $g(x)$) used in the previous question by plotting all these functions on the same graph.