Question

4. Let f(x) e. Compare evenly spaced interpolation with Chebyshev interpolation by plotting degree n polynomials of both types on the interval [-1, 1], for n = 10 and 20. Fo evenly spaced interpolation, the left and right interpolation base points should be -1 and 1. By sampling at a 0.01 step size, create the empirical interpolation errors for each type, and plot a comparison. Can the Runge phenomenon be observed in this problem?

- solution must be coded in either matlab or Octave.

- must avoid use of symbolic variables such as "syms x." here we are instructed to use the base GNU Octave, that is, without any external packages.

- best of luck, I am rather stuck

Answer 1

%Matlab code for interpolating polynomials
clear all
close all
%Function for which interpolation have to do
f=@(x) exp(abs(x));
%all n values
n=[10 20];
%loop for creating interpolating function for all n
for i=1:length(n)
%evenly spaced points for -1:1
xinU=linspace(-1,1,n(i)+1);
%corresponding function value for evenly spaced point
yinU=double(f(xinU));
%Chebyshev points for -1:1
xinC=cos(linspace(-pi,0,n(i)+1));
%corresponding function value for Chebyshev points
yinC=double(f(xinC));
xout=-1:0.01:1;
%creating interpolating function for evenly spaced points
p1=polyfit(xinU,yinU,n(i));
ss=0;
%loop for creating polynomial function for evenly spaced points
syms x
for ii=1:length(p1)
ss=ss+p1(ii)*x.^(n(i)+1-ii);
end
%the polynomial function for evenly spaced points
fun1(x)=ss;

%creating interpolating function for Chebyshev points
p2=polyfit(xinC,yinC,n(i));
ss=0;
%loop for creating polynomial function for Chebyshev points
syms x
for ii=1:length(p2)
ss=ss+p2(ii)*x.^(n(i)+1-ii);
end
%the polynomial function for Chebyshev points
fun2(x)=ss;
%Plotting the interpolating function for evenly spaced points
figure(2*i-1)
hold on
plot(xout,f(xout))
%calling of polynomial function for evenly spaced points
plot(xout,fun1(xout))
title(sprintf('Interpolation plot for evenly spaced grid for n=%d',n(i)))
xlabel('XinU')
ylabel('f(x)')
legend('Actual data','evenly spaced interpolated data')
hold off
%Plotting the interpolating function for Chebyshev points
figure(2*i)
hold on
plot(xout,f(xout))
%calling of polynomial function for Chebyshev points
plot(xout,fun2(xout))
title(sprintf('Interpolation plot for Chebyshev grid for n=%d',n(i)))
legend('Actual data','Chebyshev interpolated data')
xlabel('XinC')
ylabel('f(x)')
hold off
%finding error for interpolated data
err1=norm((f(xout)-fun1(xout)),'inf');
fprintf('Error norm for evenly spaced interpolation for n=%d is %f\n',n(i),err1)
err2=norm((f(xout)-fun2(xout)),'inf');
fprintf('Error norm for Chebyshev interpolation for n=%d is %f\n\n',n(i),err2)
end

Matlab code for interpolating polynomials clear all close all Function for which interpolation have to do -e(x) exp (abs (x)); n values al r10 or ereating for i-1:lenqthn nterpolating function for all n tevenly spaced points for -l:1 xinU-linspace(-1,1,n(i)+1); corresponding function value for evenly spaced point yinu-double(f(xinU)) Chebyshev points for -1:1 xinc-cos (linspace-pi,0,n(i)+1)); corresponding function value for Chebyshev points yinc double (f (xinC) ) xout-1:0.01:1 %creating interpolating function for evenly spaced points pl-polyfit (xinU, yinU,n (i)); ss 0; %loop for creating polynomial function for evenly spaced points syms x for ii-1:length (pl) ss-ss+pl (ii) *x.*(n(i)+1-ii); end 8the polynomial function for evenly spaced points funl (x) ss creating interpolating function for Chebyshev points p2-polyfit (xinC, yinC, n (i)) ss 0; loop for creating polynomial function for Chebyshev points syms x for ii-1:1ength (p2) ss-ss+p2 (ii) *x.^(n(i)+1-ii) end the polynomial function for Chebyshev points fun2 (x) ss; Plotting the interpolating function for evenly spaced points figure (2 i-1) hold on plot (xout,f(xout)) %calling of polynomial function for evenly spaced points plot (xout, funl (xout)) title (sprintf('Interpolation plot for evenly spaced grid for n d',ni))) X: ylabel f(x) ' legend Actual data','evenly spaced interpolated data') hold off Plotting the interpolating function for Chebyshev points 1

figure (2*i) hold on plot (xout, f( xout)) tcalling of polynomial function for Chebyshev points plot (xout, fun2 (xout)) title(sprintf('Interpolation plot for Chebyshev grid for n Ad',n(i))) legendActual data','Chebyshev interpolated data') * label('f(x) ') hold off finding error for interpolated data err1-norm((f (xout)-fun 1 ( xout)),'inf'); fprintf'Error norm for evenly spaced interpolation for n=Sd is sf \n',n(i),err1) err2=norm (f (xout)-fun2( xout)),'inf'); fprintf(Error norm for Chebyshev interpolation for n=$d is Sf \n \n',n(i),err2) end Error norm for evenly spaced interpolation for n=10 is 0.660714 Error norm for Chebyshev interpolation for n=10 is 0.058996 Error norm for evenly spaced interpolation for n=20 is 93.164500 Error norm for Chebyshev interpolation for n=20 is 0.029703 2

Interpolation plot for evenly spaced grid for n-10 Actual data evenly spaced interpolated data 2E 2 4 2 2 1 8 1.6 1 4 12 n6 0.8 02 02 -0.4 0.6 0.8 XinU Interpolation plot for Chebyshev grid for n10 28 Actual data Chebyshey interpolated data 2.6 24 22 2 1 8 16 14 12 0.8 0 4 02 0 02 -0.4 0.6 08 XinC 3

Interpolation plot for evenly spaced grid for n-20 100 Actual data evenly spaced interpolated data 80 60 40 20 H n6 0.8 20 1 n 4 02 0 2 -0.4 0.6 0.8 XinU Interpolation plot for Chebyshev grid for n-20 28 Actual data Chebyshev interpolated data 2.6 24 22 1 8 16 14 1ク 0.8 0 4 02 02 -0.4 0.6 08 XinC 4