Numerical Analysis Questions & Answers  
Question by Student 201427122
Professor, When I solved Question #1 (a), I appointed $$X_{min}=π/2$$ $$X_{max}=3π/2$$ Then I used the bisection method. Consequently The answer is $X_{13}$=3.1412... So $X_{13}$ is 3.141 for least 4 significant digits. Continuously, $$X_{14}=3.14140,X_{15}=3.14149, X_{16}=3.1415$$ Then I think $X_{16}$=3.142 for least 4 significant digits is answer. Because π is 3.142 for least 4 significant digits. I don't understand Why Your answer is 13. Did I equate wrong equation in this Question?
Here's a hint. Simply obtaining the 4th digit is not enough. You need to make sure the 4th significant digit is actually valid.. The best way to solve this problem is to specify what is the maximum allowable value of the relative error $\epsilon$ needed here and to determine analytically how many iterations is needed to reach this level. In the exam, I may ask you a similar question but instead of 4 significant digits, I may ask you for 100 significant digits. Solving the problem by hand one iteration at a time will take too long. Thus, you should aim to find here an analytical expression giving you number of iterations for a certain $\epsilon$ sought. 1 point bonus.
Question by Student 201627131
Professor, I have a question about radian in C code. If I use radian in C code, Can compiler read radian? And, PI is infinite decimals, so how can I express PI? I think if I declare PI = 3.14, result will include big error.
Just define $\pi$ in the preamble of your code with
#define pi 3.141592920
In C, the functions sin, cos, etc must always be used with radians.
Question by Student 201627148
professor, in Q#4(b), when I was solving that, I found something strange. I guess root is π because $x_1,x_2,x_3$ are convergent to π. so I made this equation ε=x-π, because of ε=x-r.then I got $ ε_1,ε_2,ε_3$ and p. But p is different every trying. I think x is infinite number, so it has error. Is answer p perfectly same 1.618?
Yes $p$ varies as the iteration count varies. You should explain why this is. There is one reason for the discrepancy in the first few iterations and another reason when approaching machine accuracy. Explain both.
Question by Student 201427122
Professor, In Question #5 1, $$ |\epsilon_{n}|_{Newton}/|\epsilon_{n}|_{secant}=(|1/2*f^{\prime\prime}(r)/f^{\prime}(r)| |\epsilon_{0}|)^{(2^{n}-1.618^{n})} $$ When I have written down my note, This equation was proportional to each other. which one do i choose between two equation?
Proportional to does not exclude equal to..
Question by Student 201427129
professor i wonder about Q#2 for Q#4-2
to solve Q#4, i must find interactions of Q#2 by secant method.
when i try, it give only condition $x_0$
accroding to lecture book, that method have 2 conditions
so i guess $x_1$ by orders of convergence to yield the at least itercations
(bisection problems are solved in #Q1 with same way
$\epsilon_{n+1}=\frac{1}{2}\epsilon_n\;\; \epsilon_k=\frac{\pi}{2^k}\;\; 3.142-\pi=\frac{\pi}{2^k} $ )

but in #Q2 problem is too hard
because it's convergence is superliner.
($k ^{p} = \frac{1} {2} *|\frac{sin(\pi)}{cos(\pi)}|=0, \epsilon =0,x_{n+1}=root) $
it means that $x_{n+1}=cost?!?!?$ so confused..
so i can't find proper and the smallest interactions
can you give some ways?
You should not determine $k$ or $p$ analytically from the function here. Rather you should find them from the error obtained at each iteration only.
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