Discrete derative?

1. Apr 28, 2006

yetar

What types of discrete derative are there?

2. Apr 29, 2006

arildno

Well, given a sequence $a_{n}$ the expression $\frac{a_{n+1}-a_{n}}{(n+1)-n}=a_{n+1}-a_{n}$ ought to count as one, agreed?

3. Apr 29, 2006

yetar

Yes, I agree.
However, I want to know if there are more sophisticated discrete derative which are commonly used.
Maybe:
$(0.25*a_{n+2}+0.75*a_{n+1}-0.75*a_{n-1}-0.25*a_{n-1})/2$
Are there any other models of discrete deratives?

Last edited: Apr 29, 2006
4. Apr 29, 2006

arildno

Well, if you are interested in discretization schemes like leap-frog and that sort of thing, you should look into numerical maths/computational mechanics books.

Alternatively, I'm sure our army of PF'ers will come along soon enough to supply you with more info.

5. Apr 29, 2006

yetar

6. Apr 29, 2006

arildno

It is a numerical method I haven't learnt about.

7. Apr 29, 2006

Tantoblin

Well, by comparing Taylor expansions you can prove that the sequence you posted is an approximation to the first derivative that is correct to at least second order, contrary to $(a_{n+1}-a_{n})/h$, which is only first-order.

With a little more insight you can convince yourself that any derivative of a smooth function can be approximated to any order, if only you have access to the values of the function at sufficiently many points.

8. Apr 30, 2006

benorin

Do a search for "finite calculus" also called "umbral calculus" the forward/backward difference operators and rising/falling factorial powers are also akin to the topic, an excellect reference is the book "Concrete Mathematics" by Graham, Knuth, and Patashnik.