numeric integrals

[mindstalk]

So there are the left and right Riemann sums, and the much better midpoint Riemann sum. Recently I wondered about integrals that took the average of the endpoint of each strip: (f(x)+f(x+step))/2. The thing is that most of the halves combine, so you can just add up f(x) for each whole step, plus f(start)/2 and f(end)/2. How's that compare?

Much better than the left and right sums, but not quite as good as the standard midpoint one. E.g. the integrals of sin(x) over 0 to pi/2 are

left_: 0.9992143962198378
right: 1.0007851925466327
mid__: 1.0000001028083885
mine_: 0.9999997943832352

All the other integrals I tried show a similar pattern: x, x^2, x^3, 1/x, e(x)... the two are close, but midpoint is just a bit closer to the correct answer. Or looked at another way, has close to 1/2 the error... hmm, that factor is consistent too. I should look into that.

Or: if I just recalled my terminology correctly, midpoint Riemann sums have half the error of trapezoidal Riemann sums. Which is not what I would have expected.

Comments

[mtbc]

I don't have an answer but it can certainly be illuminating to investigate surprises!

[mindstalk]

I looked up numerical integration on WP, and saw fairly well defined error formulas, which yes favor the midpoint. More regular formulas than I observe but that could be rounding error.

I haven't comprehended Simpson's Rule yet, but one version of it is (2M+T)/3, weighted average of the two sums, which cuts the error massively. Like, on exp(10)-exp(0), the left and right have error magnitude 110, mid and trap magnitude 0.1-0.2, and the last 7e-8. !