Numerical Analysis Questions & Answers  
Question by Student 201627108
Professor, I have a question about a cubic spline. When we use the cubic spline, Why we assume the only third polynomial like that $$f(x) = ax^{3}+bx^{2}+cx^{1}+d$$ Can we use any different expression such as the exponential function with the natural constant, log function and etc... ?
11.28.17
This may lead to some issues when joining the different polynomials with each other so that the first, second, and third derivatives match. But there may be a way.. Hmm, this is giving me an idea... Maybe I'll ask you a similar question in the final exam..
Question by Student 201529190
Dear professor, when we use the Simpson method every f(xi) need obey 1,4,2,4,2,4,1. There must be odd terms so i think that the problem for the programmer.
11.29.17
You're on the right track but it's not as well explained as it could be. 2 points bonus.
Question by Student 201627131
Professor, I think when we use simpson rule, we must use odd number. Because interval count N-1, and we bound two interval like $$Interval\,I_1 = Between\,i_1,i_2,i_3$$ So, We need even number interval(N-1), and N must odd number.
Yes, very good explanation. 3 points bonus.
Question by Student 201327139
Professor, in Ch.7 assignment Q.3, we have to find

\( \int_{x=0}^{x=2} e^{x^{2}} dx \)

but, your question is ' in the interval \( 1< x < 2 \)'. I'm so confused.
Fixed. Good observation, 2 points bonus.
Question by Student 201529190
Professor, in Ch.7 assignment Q.4, the relative of x should also be 0to2.
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.
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