Hands-on-Chebyshev Polynomials; Approximation of Functions with MATLAB I

1. The Chebyshev series and Maclaurin series for $e^x$ are given as the following;
   
   $$\begin{array}{l} e^x=0.9946+0.9973x+0.5430x^2+0.1772x^3 \\ e^x=1+x+0.5x^2+0.1667x^3\\ \end{array}$$
   
   
   • Tabulate the error values for the interval [-1,1].
   • Plot the error values for the interval.
   Solution:
   

   function week9litem1(ll,ul,s)
   format short;
   %format long;
   disp('     x        e^x    Chebyshev   Error   Maclaurin   Error')
   x =(ll:s:ul)';
   taylor=exp(x);
   max=(ul-ll)/s+1;
   for i=1:max
   chebyshev=(0.9946 + 0.9973*x(i) + 0.5430*x(i)^2 + 0.1772*x(i)^3);
   errorchebyshev(i)=taylor(i)-chebyshev;
   maclaurin=(1+x(i)+0.5*x(i)^2+0.1667*x(i)^3);
   errormaclaurin(i)=taylor(i)- maclaurin;
   D=[x(i),taylor(i),chebyshev,errorchebyshev(i),maclaurin,
                                                 errormaclaurin(i)];  
   disp(D); 
   end 
   plot(x,errorchebyshev,'o',x,errormaclaurin,'-')
   

   save with the name week9litem1.m. Then;

   >> week9litem1(-1,1,0.1)
   

   Figure 1: plot(x,errorchebyshev,'o',x,errormaclaurin,'-').

   
2. Write a MATLAB program using Chebyshev polynomials to economize a Maclaurin series for $e^x$ in the interval [0,1] with a precision of $0.0001$.
   • Tabulate the error values.
   • Plot the behaviors of the exact and approximated functions and errors for the interval.
   • Utilize the following code segment;
   
   

   
   
   
3. Exercise: Modify the program so that it uses Chebyshev polynomials to economize a Maclaurin series for $e^{2x}$ in the interval [0,1] with a precision of $0.008$.
   • Tabulate the error values.
   • Plot the behaviors of the exact and approximated functions and errors for the interval.