We approximate the integral of the function over the interval .
f = @(x) exp(sin(7*x));
a = 0; b = 2;
In lieu of the exact value, we will use the built-in integral function to find an accurate result.
I = integral(f,a,b,'abstol',1e-14,'reltol',1e-14);
fprintf('Integral = %.15f\n',I)
Integral = 2.663219782761539
Here is the error at .
T = trapezoid(f,a,b,40);
err = I - T
err =
9.1685e-04
In order to check the order of accuracy, we double n a few times and observe how the error decreases.
n = 40*2.^(0:5)';
err = zeros(size(n));
for k = 1:length(n)
T = trapezoid(f,a,b,n(k));
err(k) = I - T;
end
table(n,err)
ans = 6×2 table
nerr
1409.1685e-04
2802.3006e-04
31605.7568e-05
43201.4395e-05
56403.5990e-06
612808.9975e-07
Each doubling of n cuts the error by a factor of about 4, which is consistent with second-order convergence. Another check: the slope on a log-log graph should be .
loglog(n,abs(err),'.-')
hold on, loglog(n,3e-3*(n/n(1)).^(-2),'--')
xlabel('n'), ylabel('error'), axis tight % ignore this line
title('Convergence of trapezoidal quadrature') % ignore this line
legend('observed','2nd order') % ignore this line