Numerical Analysis Questions & Answers  
Question by Student 201529190
Professor, in Ch.7 assignment Q.4, the relative of x should also be 0to2.
11.29.17
I don't understand. The relative of $x$? What does this mean? The $x$ interval has been changed to $1\le x\le 2$ everywhere.
Question by Student 201612150
Professor, I think I found something.

We can recall "Formula V" used in the last lecture:
$\phi_{n+1} = \phi_{n} + \Delta tf(t_{n+1/2}, \phi_{n}+\frac{\Delta t}{2}f(t_{n},\phi_{n})+O(\Delta t^{2})) + O(\Delta t^{3})$

Looking closely, We multiply $\Delta t$ by $\phi_{n}+\frac{\Delta t}{2}f(t_{n},\phi_{n})+O(\Delta t^{2})$. Therefore, $\Delta t$ times $O(\Delta t^{2})$ is $O(\Delta t^{3})$.
* Note: I figured out this from the progress to analyze global error.

So due to this, the result of global error analysis is unaffected - since we multiply $\Delta t$ by not only the term $\phi_{n}+\frac{\Delta t}{2}f(t_{n},\phi_{n})$, but also error term $O(\Delta t^{2})$!

Therefore, we now can sure that the modified Euler's method is of order two.
Although I'm not sure if my deduction is correct, but I think this may be an answer.
12.05.17
It's not so simple because you need to show that $\Delta t f(t_{n+1/2}, \phi_{n}+\frac{\Delta t}{2}f(t_{n},\phi_{n}+O(\Delta t^2))$ scales with $O(\Delta t^3)$. Note that you can not simply take $O(\Delta t^2)$ out of $f$ as you did. This needs to be done more carefully.
Question by Student 201327139
Professor. In Chapter.7, we learned about simpson's rule and modified simpson's rule. But, when I was studying Chapter.7 and searched about simpson's rule from googling , I found about simpson's 1/3 rule and simpson's 3/8 rule. What's the difference between what we learned about and these rules?
12.06.17
What we learned in class is the standard Simpson's rule. There are several variations with some (marginal) advantages over the standard form.. You can read about those in the wikipedia, if you are interested.
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