Finding Optimal h for Truncation Error O(h2)

[ver_mathstats]

Homework Statement

Part (A) Find the truncation error of the approximation f'(x) by (f(x+h)-f(x-h))/2h.

Part (B) When evaluated on a computer, for what value of h the error of this approximation is the smallest?

Part (C) For the function f(x)=sinxe^cosx plot the error |(f'(x)-(f(x+h)-f(x-h))/2h)| versus h fot the appropriate values of h.

Relevant Equations

f'(x) by (f(x+h)-f(x-h))/2h

Part (A) I have already determined the truncation error of order O(h²).

Part (B) I'm struggling with how to approach this part. I do not really understand where to begin to figure out what value of h is the approximation the smallest. Is this looking for a particular range? Or are we isolating for h? And I do understand that h is the step count, just not entirely positive on how to find the ideal value of h.

Part (C) For plotting the function I am using matlab, I just want to know if I am understanding correctly, I have to substitute my function into the error? And that is what I am plotting?

Thank you.

 

[pbuk]

Oh dear, I am not sure the question is worded properly: where is it from? Taking it at face value:

Part (A) Find the truncation error of the approximation f'(x) by (f(x+h)-f(x-h))/2h.

Assuming the question really does mean truncation error and not roundoff (or rounding) error then you need to write the Taylor expansions for $f(x + h)$ and $f(x - h)$ (hint: you can end the expansions at $\dots + \mathcal{O}(h^4)$), plug them into $\frac{f(x+h)-f(x-h)}{2h}$ and gather terms to find an expression in the form $\frac{ah^n}{b}\frac{d^kf}{dx}(x) + \mathcal{O}(h^m)$. This material is usually introduced by working through a similar exercise for the approximation $D_{f'} = \frac{f(x+h)-f(x)}{h}$ in class, did you do this?

Part (B) When evaluated on a computer, for what value of h the error of this approximation is the smallest?

This is where the question starts to worry me: when you introduce "evaluation on a computer" you are looking at the trade-off between truncation error and rounding error, and this is not trivial. Did you consider this for $D_{f'} = \frac{f(x+h)-f(x)}{h}$ in class? On the other hand if we are still only talking about truncation error then the fact that we are "evaluating on a computer" is irrelevant and there is no value of $h$ for which the approximation is smallest.

Part (C) For the function f(x)=sinxe^cosx plot the error |(f'(x)-(f(x+h)-f(x-h))/2h)| versus h fot the appropriate values of h.

So now I'm really confused: are we talking about total error or just truncation error? Both of these depend on $x$ as well as $h$ so how do we deal with that? What are the "appropriate values of $h$"? Or is the question actually "For the function f(x)=sinxe^cosx plot the error |(f'(x)-(f(x+h)-f(x-h))/2h)| versus h x fot for the appropriate values of h value of h given in Part (B)?"

 

[ver_mathstats]

Oh dear, I am not sure the question is worded properly: where is it from? Taking it at face value:Assuming the question really does mean truncation error and not roundoff (or rounding) error then you need to write the Taylor expansions for $f(x + h)$ and $f(x - h)$ (hint: you can end the expansions at $\dots + \mathcal{O}(h^4)$), plug them into $\frac{f(x+h)-f(x-h)}{2h}$ and gather terms to find an expression in the form $\frac{ah^n}{b}\frac{d^kf}{dx}(x) + \mathcal{O}(h^m)$. This material is usually introduced by working through a similar exercise for the approximation $D_{f'} = \frac{f(x+h)-f(x)}{h}$ in class, did you do this?This is where the question starts to worry me: when you introduce "evaluation on a computer" you are looking at the trade-off between truncation error and rounding error, and this is not trivial. Did you consider this for $D_{f'} = \frac{f(x+h)-f(x)}{h}$ in class? On the other hand if we are still only talking about truncation error then the fact that we are "evaluating on a computer" is irrelevant and there is no value of $h$ for which the approximation is smallest.So now I'm really confused: are we talking about total error or just truncation error? Both of these depend on $x$ as well as $h$ so how do we deal with that? What are the "appropriate values of $h$"? Or is the question actually "For the function f(x)=sinxe^cosx plot the error |(f'(x)-(f(x+h)-f(x-h))/2h)| versus h x fot for the appropriate values of h value of h given in Part (B)?"

I ended up getting the answers for part a and for part b, my issue is part c, my teacher said to choose my own values of h and I believe I am supposed to be graphing the central difference formula given for f(x)=sinxe^cosx.

 

[pbuk]

Sorry I didn't spot that this is a different function but I still don't understand what the teacher is saying. I would tackle this by writing some code to calculate and plot the difference between the exact value and the central difference formula over a range of values of x for a few values of h: experiment to see what works well.

 

[xDocMath]

I ended up getting the answers for part a and for part b, my issue is part c, my teacher said to choose my own values of h and I believe I am supposed to be graphing the central difference formula given for f(x)=sinxe^cosx.

For part b, is the answer when h equals machine epsilon?

 

[ver_mathstats]

For part b, is the answer when h equals machine epsilon?

Yes it is.

 

[pbuk]

Yes it is.

That may be the answer your lecturer expects, but it is not the right answer.

For instance assume $\epsilon_M = 2^{-53}$. If we want to approximate $f'(x)$ at $x = 10^{17}$ using $h =\epsilon_M$ then $10^{17} \pm 2^{-53}$ both evaluate to $10^{17}$.

The value of $h$ for maximising precision depends on both $x$ and $f(x)$.