What is the best numerical method of approximation of the derivative?

Question

If I want to compute an approximation of the derivative of a simple function like $\cos$ and I can choose between finite differences and the derivative of the polynomial interpolating some given points, what is the best method to do it?

EDIT: By the best method, I mean the method by which I can obtain the most precise approximation of the derivative.

You'd better say "smooth". Numerically, the cosine function is neither "simple" nor "complicated", it is just a numerical function. — , Apr 08 '20 at 08:22
@YvesDaoust In fact, comparing the error trend obtained using the methods mentioned with cosine, I noticed that there are not many differences. Probably the interpolating polynomial allows having a slightly lower error and a slightly better approximation by carefully choosing the interpolation nodes. That's why I was asking what was, in general, the best way to get an approximation by choosing between these methods. — , Apr 08 '20 at 08:39
See also my previous answers on similar topics increasing numerical precision does not converge numerical solution, 'Numerical Differentiation' how to compute for the round off error, and Central Difference to Approximate $f''(x)$. — Lutz Lehmann, Apr 08 '20 at 09:30

score 3 · Answer 1 · edited Apr 08 '20 at 09:18

This is an uneasy topic, numerical differentiation is an ill-posed problem.

The finite difference methods of different orders have an error term proportional to a derivative of some order of the function, so that higher order are well suited for smoother functions.

An important factor is the choice of the step, because for too small steps, numerical errors become dominant and the computed derivative becomes meaningless.

Read the Wikipedia article on Numerical differentiation.

Even worse if the data values are inaccurate or noisy. In such a case, you combine with a lowpass filters, such as Savitzky–Golay filter (Wikipedia) .

"Best" depends on the function and how it is computed.

What is the best numerical method of approximation of the derivative?

1 Answers1