clear all;
% Define the integrand function
f = @(z) 1./(1+z.^2);
% Exact value of the integral
exact_val = pi/4;
% Initialize vectors for storing errors and step sizes
h = zeros(1, 16);
simp_err = zeros(1, 16);
% Compute approximations and errors for each step size
for k = 1:16
h(k) = 2.^(-k);
% Simpson's rule
x = 0 : h(k) : 1;
simp_approx = h(k)/3 * (f(x(1)) + 4*sum(f(x(2:2:end-1))) + 2*sum(f(x(3:2:end-1))) + f(x(end)));
le(k)=simp_approx;
simp_err(k) = abs(simp_approx - exact_val);
end
% Plot errors against step sizes in log-log scale
loglog(h, simp_err, '.-');
hold on;
loglog(h,h.^4,'-r');
hold off
xlabel('$Step\;size\;(h)$','Interpreter','latex');
ylabel('$Absolute\;error$','Interpreter','latex');
legend('$Simpsons\;rule$','$Line\;of\;O(h^4)$','Interpreter','latex');
grid on;
