Discrete Derivatives: Types & Uses

[yetar]

What types of discrete derative are there?

Thanks in advance.

 

[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?

 

[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?

 

[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.

 

[yetar]

What is Differential Quadratures?

 

[arildno]

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

 

[Tantoblin]

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?

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.

 

[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.