Interpolation and Curve Fitting

1. The MATLAB procedure for polynomial least-squares is polyfit. Study the following example;

   x =(0:0.1:5)'; 					% x from 0 to 5 in steps of 0.1
   y = sin(x); 						% get y values
   p = polyfit(x,y,3); 		% fit a cubic to the data
   f = polyval(p,x); 			% evaluate the cubic on the x data
   plot(x,y,'o',x,f,'-') 	% plot y and its approximation f
   

2. For the given data points;
   
   $$\begin{array}{rr} x & Y \\ \hline 0.000 & 1.500 \\ 0.142 & ... ...714 & 0.821 \\ 0.857 & 0.442 \\ 1.000 & 0.552 \\ \end{array}$$
   
   
   • to which we will fit $y(x)=\alpha e^{\beta x}$
     Hint: First, we should compute a new table with $z(x)=ln y(x)$
     
     $$\begin{array}{rr} x & z \\ \hline 0.000 & \\ 0.142 & \\ 0.... ...\\ 0.571 & \\ 0.714 & \\ 0.857 & \\ 1.000 & \\ \end{array}$$
     
     

   • Construct the normal equations
     Hints: $A=ln \alpha$ and $C=\beta$
   • Solve these normal equations to find $A$ and $C$
   • Convert back to the original variables
   • Plot $Y$ vs $x$ and $y$ vs $x$ then compare them.
   • Soln: $y(x)=1.561e^{-1.132x}$
3. Apply the procedure given in the first item by using the data set in the previous item.
   
   • Hints:
     • fit a cubic to the data
     • evaluate the cubic on the x data
   • Plot by $plot(x,Y,'o',x,f,'-')$
   • Compare this least-square polynomial with the function used in the previous item.