Here is a code that does the trapezoid rule applied to a function on [0, 100] and where I use a random vector with 1000 entries. The integration is repeated 1 million times. The output gives my computer 3 seconds for the first method, 9 seconds for the second, and 11.5 seconds for the third.
But the only difference between the third and the first is that the third method puts the calculation into a function. The difference between the second and the first is the vector notation. I don't understand why these are leading to such different efficiencies. Moreover, I am worried about the fact that putting an identical script into a function multiples the time by a factor of three!
Here is the setup:
x = linspace(0, 100, 1000);
dx = x(2) - x(1);
y = rand(size(x));
First method:
G = 0;
tic
for j = 1:1000000
g = 2*y;
g(1) = g(1)/2;
g(end) = g(end)/2;
G = G + dx/2*sum(g);
end
toc
disp(['RUN 1: G = ', num2str(G)]);
Second method
G = 0;
tic
for j = 1:1000000
G = G + dx/2*sum([y(1) 2*y(2:end-1) y(end)]);
end
toc
disp(['RUN 2: G = ', num2str(G)]);
Third method:
G = 0;
tic
for j = 1:1000000
G = G + mytrapz_equal(dx, y);
end
toc
disp(['RUN 3: G = ', num2str(G)]);
The same function as Run 1:
function F = mytrapz_equal(dx, y)
g = 2*y;
g(1) = g(1)/2;
g(end) = g(end)/2;
F = dx/2*sum(g);
end
end
Can anybody shed some light?
