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\begin{center}\textbf{\Large MA497 Numerical Analysis\\Homework 1}\\[2ex]Due: 7 February, 2008\end{center}
\noindent The solution to all problems should be typeset using \LaTeX, and the \LaTeX{} source file should be submitted electronically via email.  The submitted source file will be compiled using \texttt{pdflatex} and the success of this process will represent the initial assessment step.  All source code for numerical algorithms should also be submitted electronically via email.\\[2ex]

\begin{enumerate}
	\item Let $f(x)=2\sqrt{x}-\cos x$.
		\begin{enumerate}
			\item What does the intermediate value theorem imply about $f$ on the interval $[0,1]$?
			\item Show that $f$ on $[0,1]$ satisfies each of the hypotheses of Theorem 2.1 on page 49 of the text.
			\item Theorem 2.1 recalls that the Bisection method generates a sequence $\{p_n\}_{n=1}^\infty$ that converges to a zero $p$ of $f$, and it implicitly provides an upper bound for the number of iterations needed to guarantee $|p_n-p|$ be within a certain amount.  Using this theorem, determine an \textit{a priori} estimate of the number of iterations need to guarantee $|p_n-p|$ be no more than $10^{-4}$ on $[0,1]$.
			\item Write a bisection method algorithm, called \texttt{bisection.m}, which implements this problem, and use this algorithm to determine the \textbf{ACTUAL} number of iterations $N$ needed to guarantee that \[|f(p_N)|\leq 10^{-4}.\]  How does this number compare to the estimate in part (c)?  Explain this discrepancy.
		\end{enumerate}
		\item
			\begin{enumerate}
				\item Determine a contraction mapping $g$ and corresponding Lipschitz constant $k$ on $[1,2]$ with a unique fixed point $x\in[1,2]$ such that $x^3-x-1=0$.
				\item With an inital starting point $p_0=1$, use Corollary 2.4 on page 59 of the text to estimate the number of iterations required to guarantee the approximate fixed point is accuracate to within $10^{-6}$.
				\item Write a fixed point algorithm, called \texttt{fixedpoint.m}, which implements this problem, and determine the \textbf{ACTUAL} number of iterations needed to achieve the accuracy in (b).  How does this compare to the answer in (b)?
			\end{enumerate}
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