I'm trying to replace "t" in the code with ti=a + (b−a)*((xi+1)/2) to plot the Lagrange polynomial interpolation with Chebyshev points.

clear
n = 8; % the order of the polynomial
a = 1; % left end of the interval
b = -1; % right end of the interval
h = (b - a)/n; % interpolation grid size
t = a:h:b; % interpolation points
f = 1./(1 + 25*t.^2); % f(x) = 1./(1 + 25*t.^2), This is the function evaluated at interpolation points
%%%% pn(x) = \sum f(t_i)l_i(x)
hh = 0.01; % grid to plot the function both f and p
x = a:hh:b;
fexact = 1./(1 + 25*x.^2); %exact function f at x
l = zeros(n+1, length(x)); %%%% l(1,:): l_0(x), ..., l(n+1): l_n(x)
nn = ones(n+1, length(x));
d = ones(n + 1, length(x));
for i = 1:n+1
    for j = 1:length(x)
        nn(i,j) = 1;     
        d(i,j) = 1;     
        for k = 1:n+1
            if i ~= k
                nn(i,j) = nn(i,j) * (x(j) - t(k));
                d(i,j) = d(i,j) * (t(i) - t(k));  
            end      
        end     
        l(i,j) = nn(i,j)/d(i,j);  
    end
end
fapp = zeros(length(x),1);
for j = 1:length(x)
   for i=1:n+1 
       fapp(j) = fapp(j) + f(i)*l(i,j);  
   end
end 
En = 0;
Ed = 0;
for i = 1:length(x) 
     Ed = Ed + fexact(i)^2; 
     En = En + (fexact(i) - fapp(i))^2;
end
Ed = sqrt(Ed);
En = sqrt(En);
E = En/Ed;
display(E)
plot(x,fexact,'b*-')
hold on
plot(x,fapp,'ro-' )

[ebtisam almehmadi]

I have no clue how to code xi to make the code works, where xi=−cosθi, θi=ih,h=π/n, i= 0:n.

the plot for n=8 should be the same as above figure.

code must give nine chebychev points for n=8

Answer 1

[Ashutosh Singh Baghel]

Hi ebtisam,

I understand you faced issue with 'plot', please update the variable 'hh' with proper value as a decrement, instead of increment.

example =

hh = -0.01;             %provide a negative sign as a>b

Also, from the comment you provided, try to write the equation into a code format.

The reference guide for creating a linear, equally spaced vector can be found here.