# Thread: Proving convergence of a difference method

1. ## Proving convergence of a difference method

"Find whether y_p - y_(p-4) = h/3 * (8f_(p-1) - 4f_(p-2) + 8f_(p-3)) is convergent."

I really have no idea how to get this going. I need help. I was told by a classmate to expand this in Taylor's, but I have no clue how Taylor's could possibly enter the picture here. Please help.

2. ## Re: Proving convergence of a difference method

Hey phys251.

What do mean by Taylors? If you can provide a definition for whatever "Taylors" is then perhaps we can make some progress.

3. ## Re: Proving convergence of a difference method

Originally Posted by chiro
Hey phys251.

What do mean by Taylors? If you can provide a definition for whatever "Taylors" is then perhaps we can make some progress.
I think he refers to the Taylor series. I can't decipher the equation itself.

@OP, could you write it in TeX?

4. ## Re: Proving convergence of a difference method

Originally Posted by LimpSpider
I think he refers to the Taylor series. I can't decipher the equation itself.

@OP, could you write it in TeX?
$w_{p} - w_{p-4} = \frac{h}{3}(8f_{p-1} - 4f_{p-2} + 8f_{p-3})$

Sorry, was supposed to be w's, not y's. w's are approximate solutions.

5. ## Re: Proving convergence of a difference method

What are the w's and f's supposed to be? And what is h? What are constants and what are variables? What is p? By that, I mean for what values of p is the equation valid?

6. ## Re: Proving convergence of a difference method

Originally Posted by SlipEternal
What are the w's and f's supposed to be? And what is h? What are constants and what are variables? What is p? By that, I mean for what values of p is the equation valid?
f = w'(x). No other information is given--this is a general proof. (Of course, we do get to make the usual assumptions such as the function's being differentiable over all real numbers.)

7. ## Re: Proving convergence of a difference method

Originally Posted by phys251
f = w'(x). No other information is given--this is a general proof. (Of course, we do get to make the usual assumptions such as the function's being differentiable over all real numbers.)
There needs to be a basic understanding of the notation. Are you saying that $(f_p), (w_p)$ are two sequences of functions such that if $\lim_{p\to \infty} f_p(x) = f(x)$ and $\lim_{p \to \infty} w_p(x) = w(x)$ then $f(x) = w'(x)$? And are you trying to show that the sequences $(f_p)$ and $(w_p)$ both converge?

8. ## Re: Proving convergence of a difference method

Originally Posted by SlipEternal
There needs to be a basic understanding of the notation. Are you saying that $(f_p), (w_p)$ are two sequences of functions such that if $\lim_{p\to \infty} f_p(x) = f(x)$ and $\lim_{p \to \infty} w_p(x) = w(x)$ then $f(x) = w'(x)$? And are you trying to show that the sequences $(f_p)$ and $(w_p)$ both converge?
I'm just trying to figure this out. I thought that f(x) = w'. Now I'm confused.

9. ## Re: Proving convergence of a difference method

I'm afraid that I don't understand the notation you are using. Unless you can provide some context that will help determine what the notation means, I don't know how to help.