Question

6. We want to study the effect of different choices of interpolation points {X0,X1,..., 2n} on the function wn(x) = (x – xo)(x – 21)... (x – In) in the formula for the error in interpolation polynomials. In particular, we want to study evenly spaced points and Chebyshev points in the interval (-1, 1). Con- sider the following choices: 2i (a) x₂ = –1 + i = 0.....n n 7T (b) X; = - cos i = 0,...,n. n +1 2 In each case, plot wn(2) in the interval (-1,1] for n = 10 using fine enough resolution (with the number of points more than n). Also plot wn(x) with n = 20 and 30. Discuss the results.

Answer 1

%%Matlab code for plotting interpolation polynomial function wn(x) clear all close all %Different choice of interpolation points (n) nn=110 20 301; Loop for ploting different choice of interpolation points (n) for j=1:length(nn) a=nn(i): %For a particular n zl=1; %initialize the function wn(s) for evenly spaced points z2=1; %initialize the function wn(x) for Chebyshev points %creating the function wn(x) syms x for i=1:n+1 %creating evenly spaced points xx1=1+(2*(1-1)/n); %creating Chebyshev points Xx2=-cos((pi/(n+1))*(0.5+1-1)); %Creating the function zl=(x-xx1)*zl; z2=(x-1x2)*22; end %functions using evenly spaced point f1(x)=z1; %function using Chebyshev points f2()=z2;

%Fine resolution points for ploting the function xl=linspace(-1,1,1000); % Ploting wn(x) function figure(j) hold on plot(x1,double(f1(x1))) plot(x1,double(12(x1))) title(l'plotting of w_o(x) in the interval of 1-1 11 for ' num2str(nn(j)) "D) xlabel('x') ylabel('w_n(x)") legend('evenly spaced points', 'Chebyshev points") hold off end

plotting of w/(x) in the interval of [-1 1] for n=10 0.01 0.008 evenly spaced points Chebyshev points 0.006 0.004 0.002 w.() -0.002 -0.004 -0.006 -0.008 -0.01 -1 -0.8 -0.6 -0.2 0 02 0.4 0.6 08 1 x plotting of w(x) in the interval of [-1 1] for n=20 25*10-4 evenly spaced points Chebyshev points N 1.5 1 0.5 (x)"M 0 -05 -1 -15 -25 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 X

10-5 plotting of w,(x) in the interval of [-1 11 for n=30 0.8 evenly spaced polnts Chebyshev points 0.6 0.4 0.2 w (X) -02 -0.4 -0.6 -08 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 06 08 х From the plots we can see that for evenly spaced points there will be some error for low n value as well as high n value, but for Chebyshev points such edge effect is not there and error will decrese for higher n vakues. From the result we can conclude Chebyshev points are more useful than evenly spaced point for interpolation.