%Newton's Method
%If you modify this file, please indicate here and in the footnote.
%Math263A
\documentclass[12pt]{article}
\usepackage{times}
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\begin{document}
\begin{center}
{\Large
Newton's Method\footnote{Copyright \copyright 2002 Larry Snyder and Todd
Young. All rights reserved. Please address comments to young@math.ohiou.edu.}}
\end{center}
\begin{enumerate}
\item \begin{enumerate}
\item Try the following commands (at the prompt and
press \fbox{Enter}): \\
\verb& syms x & \\
\verb& format long & \dotfill Sets displayed digits to 15.\\
\verb& f = x^3 - 3*x^2 + 1 & \\
\verb& f1 = simplify(diff(f)) & \\
\verb& g = simplify(x - f/f1) & \\
\verb& p = .1 & \\
\verb& p = subs(g, p) &
\item Repeat the command \verb& p = subs(g, p) & until
\verb&p& stops changing.
(Use the up-arrow key to recall the command
instead of typing it.)
\item Assuming the final value is correct, how many
steps did it take to get 7 decimal places
of accuracy? How many steps for 14 decimal places?
\end{enumerate}
\item \begin{enumerate}
\item Type \verb&p = .5& and repeat \verb& p = subs(g,p) & until
\verb&p& stops changing. To what do the approximations
converge this time?
\item Repeat, but start with \verb&p = 3.0&.
\item Why can Newton's method
give three different answers for three different
starting points? (Hint: Use \verb& ezplot(f) & to look at \verb$f(x)$.)
\end{enumerate}
\item Set \verb&p = .11065934333376& and repeat \verb& p = subs(g, p) &
until it converges. How many iterations does it take this time?
\item Repeat the process in (1), starting with \verb&p = .1& for the function
$$
f(x) = \frac{(x-3/4)^{1/3}}{x^{1/3}}
\qquad \verb$(f = ((x-3/4)^(1/3))/(x^(1/3)))$
$$
Write down the first 20 iterations. Do they seem to
be converging to anything? Plot them on the interval \verb$[0,1]$.
Does $f(x) = 0$ have a solution on $[0,1]$? Try that
point as the initial guess and see what happens.
Next, try starting with \verb&p = 0.0&. What is the value of $f$ at $0.0$?
\item Can one always rely on Newton's method?
What are some things to be careful about?
\item Prepare a brief (\verb$< $1 page) written report answering all
the questions. Use complete sentences and standard mathematical notation.
Do {\bf not} get a printout.
\end{enumerate}
\vfill
\noindent
\textsf{The user observes that Newton's method converges very fast for the certain
functions and certain starting points. The convergence can be
slow for other starting points and the final answer can depend on
the starting point. Further, some functions lead to Newton's method
iterations which are actually chaotic (random-like).
}
\end{document}
