\documentstyle[12pt]{article}
%\documentstyle[12pt,ncs]{article}
\topmargin=-1.25in
\oddsidemargin=0in
%\evensidemargin=0in
\textwidth=6.5in
\textheight=10.5in
\parindent=0in
%\parskip=10pt
\pagestyle{empty}

%\newfont{\cmss}{cmss12.tfm}
\tolerance=3000
%\setcounter{chapter}{1}

\begin{document}

\input 370top

\centerline{{\it Class 12}: {\bf Wednesday September 30}}

\begin{description}
\item[SUMMARY] Analyzing Root Finding Algorithms: Bisection 
\vspace*{-12pt}
\item[READING] Burden \& Faires, 47--54
\end{description}

{\large\bf General Root-Finding Algorithm}\\
1. Plot the function, in order to get an initial guess for the root and to check for problems 
2. Select an initial guess [or bracket ]
3. Iteratively refine your initial guess
4. Decide you are "converged"
5. Stop

{\large\bf demobisect.m}\\
There is another implementation of Bisection Algorithm in {\bf q:/mfiles/math370/demobise.m} Modify this m-file to find the root of $f(d)=2552-30d^2+d^3$\\
How many steps does it take to converge? Using what initial bracket?
\vfill

{\large\bf Analyzing Convergence of Bisection}\\
Write down an expression for the size of $|b_n-a_n|$ which depends on $b-a$ and 
the $n$-th iterate (note: $|b_0 - a_0| = b-a$)
\vfill
 
Solve this formula for $n$.
\vfill
Try and predict how many iterations it will take Bisection to find the zero of\\
 $f(x) = \log(x) - 5 + x$ on the interval [1,9] to 5 decimal places
\vfill
Go to the computer and see how many iterations {\bf demobisect.m} actually takes to converge.\\
Explain.
\vfill
\newpage
{\large\bf Convergence Criteria}\\
There are a number of different ways to consider that a method has ``converged''

There is convergence criteria on $f(x)$ and convergence criteria on $x$
\vfill
\vfill
\vfill
{\bf Question}\\
There is also relative convergence versus absolute convergence. Which do you think is the ``best'' method of assessessimg convergence?
\vfill
 


\end{document}
