Solving Euler's Method: Accuracy of y_100 = 3.1515

[bl4ke360]

Homework Statement

http://img853.imageshack.us/img853/5578/euler.png

Homework Equations

y_(n+1)=y_n+f(x_n,y_n)h

The Attempt at a Solution

With n = 100 and step size h = 0.01, I got y_100 = 3.1515

I don't see how this is right, because the directions imply that it's supposed to be accurate to pi within 4 decimal places, but this last value is higher than pi. Also what does it mean by using different values of h, and still keeping n = 100? That doesn't make any sense because then it wouldn't evaluate up to y(1) = pi.

 

[Dick]

Euler's method isn't all that accurate. It doesn't give you pi that accurately with 100 steps as you found. I don't know what they mean by using different values of h either. That won't give you an approximation to pi, it will give you an approximation to 4*arctan(h*n). May just be a badly written question. They might just mean how small does h have to be to give you a good appoximation to pi after 1/h steps.

 

[bl4ke360]

This problem is 10% of my final grade (and due tomorrow) so I want to make sure I'm doing it correctly. From the directions it says it's supposed to converge, but it doesn't look like it does unless I'm missing something. It converges at around 6.299, with n = 100,000. I also verified it by solving the function directly after solving for it from the D.E. which is y(x)=4arctan(x). I don't see what I'm supposed to do from here.

 

[Dick]

This problem is 10% of my final grade (and due tomorrow) so I want to make sure I'm doing it correctly. From the directions it says it's supposed to converge, but it doesn't look like it does unless I'm missing something. It converges at around 6.299, with n = 100,000. I also verified it by solving the function directly after solving for it from the D.E. which is y(x)=4arctan(x). I don't see what I'm supposed to do from here.

Euler's method will converge if you use a small enough step size. Did you use n=100000 with h=1/(100000)? Though round-off errors and stability might start taking their toll. I'm not exactly sure what it expects you to do from the wording either.

 

[bl4ke360]

I see what you mean now, but since I only have to go to n=100 I guess I don't have to worry about that here. But I don't know if I'm supposed to change the step size partially into it, or do a separate run of euler's method with a different step size. Since the directions says to use only one table, I would assume the I have to change the step size before n reaches 100, but that doesn't make any sense to me.

 

[Dick]

I see what you mean now, but since I only have to go to n=100 I guess I don't have to worry about that here. But I don't know if I'm supposed to change the step size partially into it, or do a separate run of euler's method with a different step size. Since the directions says to use only one table, I would assume the I have to change the step size before n reaches 100, but that doesn't make any sense to me.

That's the problem with the question. You won't get to x=1 and pi if you only change the step size and not n.