# Proving convergence of a difference method

• Oct 16th 2013, 09:13 AM
phys251
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. :(
• Oct 16th 2013, 04:53 PM
chiro
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.
• Oct 16th 2013, 05:03 PM
LimpSpider
Re: Proving convergence of a difference method
Quote:

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?
• Oct 16th 2013, 08:50 PM
phys251
Re: Proving convergence of a difference method
Quote:

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?

$\displaystyle w_{p} - w_{p-4} = \frac{h}{3}(8f_{p-1} - 4f_{p-2} + 8f_{p-3})$

Sorry, was supposed to be w's, not y's. w's are approximate solutions.
• Oct 16th 2013, 09:58 PM
SlipEternal
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?
• Oct 17th 2013, 08:58 AM
phys251
Re: Proving convergence of a difference method
Quote:

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.)
• Oct 17th 2013, 09:33 AM
SlipEternal
Re: Proving convergence of a difference method
Quote:

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 $\displaystyle (f_p), (w_p)$ are two sequences of functions such that if $\displaystyle \lim_{p\to \infty} f_p(x) = f(x)$ and $\displaystyle \lim_{p \to \infty} w_p(x) = w(x)$ then $\displaystyle f(x) = w'(x)$? And are you trying to show that the sequences $\displaystyle (f_p)$ and $\displaystyle (w_p)$ both converge?
• Oct 17th 2013, 02:54 PM
phys251
Re: Proving convergence of a difference method
Quote:

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

I'm just trying to figure this out. I thought that f(x) = w'. Now I'm confused. :(
• Oct 17th 2013, 02:59 PM
SlipEternal
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.