I was plotting some data in loglog scale "Errors against step sizes" but my curve is not connected , and i can not fix it, what is the problem?

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;

Answer 1

[Walter Roberson]

Some of your simp_err are exactly 0 but log plots cannot draw exact 0 (or negative) so the plot disconnects there.

This happens at smaller step sizes that go along with larger k, but smaller values are more towards the left so the gaps appear more towards the left instead of more towards the right (increasing k)