how to perform Numerical differentation

Suppose I can generate infinitely many values of x and f(x). Now I need the value of f'(x) at x=1. What is the best way to find it with satisfactory accuracy.

Answer 1

[John D'Errico]

I would use a tool that is designed to solve that problem - derivest . Of course, since I wrote the tool, that makes it obvious. It is on the File Exchange. Download the code to use it.

A nice feature of derivest is it also gives you an estimate of its uncertainty in that estimate.

So, for example...

[dydx,err] = derivest(@sin,2)
dydx =
        -0.416146836547146
err =
       1.5487174604626e-14

Was it correct? Looks ok to me.

cos(2)
ans =
        -0.416146836547142

abs(dydx - cos(2))
ans =
       3.5527136788005e-15

[suvadip paul]

But I dont know the functional form, only numerical values of f(x) are known

[Torsten]

With or without strong noise ?

Maybe you could include a graph ?

Best wishes

Torsten.

[John D'Errico]

Nothing in that tool presumes the "functional form" is known. It merely assumes that you give it a general function, and that it can evaluate the function, then computing the derivative at your location.

It requires nothing more than that your function be smooth and moderately well-behaved. Of course, if your function contains noise, then you must use a tool that is capable of dealing with noise. In that case, I would suggest fitting a simple low order polynomial model in the vicinity of the point in question. Polyfit would then suffice.