clear

%%
% First consider the integral
% 
% \int_{-1}^1 \frac{1}{1+4x^2} \, dx = \arctan(2).
% 
f = @(x) 1./(1 + 4*x.^2);
exact = atan(2);

%%
% We compare the two spectral integration methods for a range of n values.
n = (8:4:96)';
errCC = 0*n;
errGL = 0*n;
for k = 1:length(n)
    errCC(k) = exact - ccint(f,n(k));
    errGL(k) = exact - glint(f,n(k));
end

figure(1)
semilogy( n, abs(errCC), '.-')
hold on
semilogy( n, abs(errGL), '.-')
axis tight
xlabel('number of nodes')
ylabel('error')
title('Spectral integration of f(x) = 1/(1 + 4x^2)')
legend('CC','GL')
grid on

%%
% (The missing dots are where the error is exactly zero.) Gauss--Legendre
% does converge faster here, but at something less than twice the rate. Now
% we try a more sharply peaked integrand:
% 
% \int_{-1}^1 \frac{1}{1+16x^2} \, dx = \frac{1}{2}\arctan(4).
% 
f = @(x) 1./(1 + 16*x.^2);
exact = atan(4)/2;

%%
n = (8:4:96)';
errCC = 0*n;
errGL = 0*n;
for k = 1:length(n)
    errCC(k) = exact - ccint(f,n(k));
    errGL(k) = exact - glint(f,n(k));
end

figure(2)
semilogy( n, abs(errCC), '.-')
hold on
semilogy( n, abs(errGL), '.-')
axis tight
xlabel('number of nodes')
ylabel('error')
title('Spectral integration of f(x) = 1/(1 + 16x^2)')
legend('CC','GL')
grid on

%%
% The two are very close until about n=40, when the Clenshaw--Curtis
% method slows down.