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%  Homework for the undergraduate course "Numerical Analysus I", 
%  Spring semester, 2010, Wei-Cheng Wang.


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%          The title and header.

\noindent
{\scriptsize Numerical Analysis I, Fall 2010 
(http://www.math.nthu.edu.tw/\~\,wangwc/)} \hfill 

\begin{center}
\large
Quiz 01   
\normalsize
\end{center}
\noindent
Oct 01, 2010. 
% \vspace{.3in}

% \bigskip

%           The questions!

\begin{enumerate}

\item
How many "bits" does it take to
store a floating point number in the range
$$
\pm 1.a_1 a_2 \cdots a_s \times 2^e
$$
with $s=33$, $a_j \in \{0,1\}$, $-511 \leq e \leq 512$?

\item (programming)
 
Find first 15 digits of $\sqrt{100002}-\sqrt{100001}$.
You need to write down your method on the answer sheet.
You can use any program on the computer and then copy your answer
to the answer sheet.

% \item
% Show that
% $$
% \text{Rel}(x_A/y_A) = 
% {\text{Rel}(x_A) - \text{Rel}(y_A) \over 1 - \text{Rel}(y_A) } 
% $$
\item
Give the rate of convergence of the following limits
$$
({\rm a}): \lim_{h\to 0} {\sin h \over h} = 0  \qquad
({\rm b}): \lim_{h\to 0} {\sin h \over h} = 0
$$

\item
Find a numerical answer of $x^3 + x - 3 = 0$.
Make sure to check your method converges. %provide sample code

\end{enumerate}
\vspace{2.0in}


\noindent
{\scriptsize Numerical Analysis I, Fall 2009 
(http://www.math.nthu.edu.tw/\~\,wangwc/)} \hfill 

\begin{center}
\large
Quiz 02   
\normalsize
\end{center}
\noindent
Oct 16, 2009. 
% \vspace{.3in}

% \bigskip

%           The questions!

\begin{enumerate}

\item 
Propose a method of computing $\sin(1.00001)-\sin(1)$
to prevent loss of accuracy.

\item
How many "bits" does it take to
store a floating point number in the range
$$
\pm 1.d_1 d_2 \cdots d_s \times 2^e
$$
with $s=33$, $d_j \in \{0,1\}$, $-509 \leq e \leq 512$?

%% \item
%% Show that
%% $$
%% \text{Rel}(x_A/y_A) = 
%% {\text{Rel}(x_A) - \text{Rel}(y_A) \over 1 - \text{Rel}(y_A) } 
%% $$

\item
Describe the Newton's method for finding $\sqrt[3]{2}$.

\item
Describe the secant method for finding $\sqrt[3]{2}$.

\item
Give a fixed point iteration for solving $x^3 - 3 = 0$.
Make sure to check your method converges.

\end{enumerate}
\end{document}