Question

In Matlab (diary on), do the following: 

1. Generate N+1=11 equi-spaced nodes Xi in the interval [−5,5]:
         X = [-5:1:5];    (to see values, omit the ;)
   or    X = linspace(-5,5,11);
   and evaluate f(x) at these nodes:
         Y = 1./(1+X.^2);
   The N+1 points (Xi, Yi) are the data points to be interpolated by various methods. 
   Plot them to see where they are:
         plot(X,Y,'o')
         title('N+1 = 11 equi-spaced points of Runge function')
         pause 

   Also generate lots of points xi at which to evaluate f(x) and the interpolants for plotting:
         x = [-5:0.1:5];   (this is a lower case x, not X)
   Evaluate f(x) at these xi's and plot y=f(x) and the data points:
         plot(x,y,'-', X,Y,'o')
         title('Runge f(x) and data pts')
         pause 

Now, we use the data points (Xi, Yi) to construct various interpolants.
A good interpolant should "reproduce" the function f(x) as close as possible. 
Let's try a few.


2. Nth degree interpolating polynomial:
   Use Matlab's polyinterp to construct (the coefficients of) the Nth degree interpolating polynomial (here N=10):
         pN = polyfit( X,Y, N); 
   Now this can be evaluated anywhere in the interval [-5,5] with polyval, e.g. at the xi's:
         v = polyval( pN, x); 
   Find the Inf-norm error ∥y-v∥∞:
         err = norm(y-v, inf) 
   and plot both f(x) and pN(x) on the same plot:
         plot(x,y,'-b', x,v,'*-r')
         title('f(x) and pN(x) at plotting pts')
         pause 

   Is this a good interpolant ?  Why ?
        

   
3. Interpolation at Chebychev nodes:
   Generate N+1=11 Chebychev points (Xchebi, Ychebi) in [a,b]:
         fprintf('------ chebychev nodes ------\n')
         K = N+1;
         a=−5;  b=5;
         for i=1:K
             Xcheb(i)=(a+b)/2 + (b−a)/2 *cos( (i−0.5)*pi/K );
         end
         Ycheb = 1./(1+Xcheb.^2);

   Follow the steps in 2. to produce the Nth degree interpolating polynomial pNcheb 
   based on the Chebychev nodes, its values vcheb at the xi's, the error ∥y − vcheb∥∞,
   and plot both f(x) and pNcheb(x) on the same plot.

   Compare the error and plot with those from 2.
   Which one is better ?  why ?
        


4. Piecewise linear interpolation:
   Use Matlab's interp1 to construct the linear interpolant:
         lin = interp1(X,Y, x, 'linear'); 
   Repeat the steps of 2. 

   Compare errors and plots.
        

5. Piecewise cubic interpolation:
   Use Matlab's interp1 to construct the cubic interpolant:
         cub = interp1(X,Y, x, 'cubic'); 
   Repeat the steps of 2. 

   Compare errors and plots.
        


6. Cubic spline interpolation:
   Use Matlab's interp1 to construct the spline interpolant:
         spl = interp1(X,Y, x, 'spline'); 
   Repeat the steps of 2. 

   Compare errors and plots.
        


7. To see that the error gets worse for bigger N for equi-spaced nodes but not for Chebychev nodes 
   (for this f(x) at least), repeat 2. and 3. with N = 20.

Answer 1