clear
format compact

%%
f = @(x) 1./(1 + x.^2);
h = logspace(-3, -1, 60);
M = logspace(3, 16, 60);

for i = 1:length(h)
    for j = 1:length(M)
        I(i,j) = intde(f,h(i),M(j));
    end
end
err = abs(I-pi);
figure(1)
contourf(h,M,-log10(err)')
set(gca,'xsc','log','xdir','rev','ysc','log')
colorbar
xlabel('h')
ylabel('M')
title('Number of accurate digits')

%%
% As predicted, the error can't much beat 1/M. Even for very large M,
% however, not many nodes are needed, as seen in the very weak dependence
% of the accuracy on h. For instance,
[I,x] = intde(f,0.1,1e10);
err = abs(pi-I)
number_of_nodes = length(x)

xpos = (x>0);
figure(2)
semilogx(x(xpos),f(x(xpos)),'.-')
axis tight
ylim([0 1])
xlabel('x')
ylabel('f(x)')
title('Positive nodes used for integration')
grid on