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I want to modify the code to plot the Lagrange polynomial interpolation with Chebyshev points. Map the n+ 1 Chebyshev interpolation points from [-1,1] to [2,3]

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ebtisam almehmadi
ebtisam almehmadi 2021-7-21
回答： Abhinaya Kennedy 2024-6-5
clear
n = 3; % the order of the polynomial
a = 2.0; % left end of the interval
b = 3.0; % right end of the interval
h = (b - a)/n; % interpolation grid size
t = a:h:b; % interpolation points
f = 1./t; % f(x) = 1./x, 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./x; %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-' )

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回答（1 个）

Abhinaya Kennedy
Abhinaya Kennedy 2024-6-5
Hi Ebtisam,
To use Chebyshev points, replace the line "t = a:h:b;" with this:
t_cheb = cos(linspace(0, pi, n+1));
t = (a + b)/2 + (b - a)/2 * t_cheb;
This generates Chebyshev points in [-1, 1] and maps them to the interval [2, 3]. The rest of the code remains unchanged.

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