Numerical Analysis Questions & Answers  
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.
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?
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.
Question by Student 201427127
Professor. I want to check my answer sheet. How can I check?
You can come to my office in the afternoon. I'll be here tomorrow and friday, and next week from thursday.
Question by Student 201627143
Professor, can I check my answer sheet too?
Sure, you can come. I'm a bit busy now thus because I have to prepare slides for a conference I'll be going to next week. So if possible, come to see me after next Wednesday.
Question by Student 201427127
Professor, I can't distinct what is prod1 and prod2 at C++ programing code that you made which was PI and SIGMA at your note on blackboard.
You need to use to write all mathematics. PI and SIGMA should be written using the same mathematical symbols I used in class. Check out the mini HOWTO in the Skylounge. Ask your question again below and I'll answer it if it is correctly typeset.
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