%%  File:  rootsGuide.tex
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%%  Study Guide for Numerical Methods with MATLAB
%%  Root-Finding Chapter


\documentclass{article}

\usepackage{studyGuide}
\renewcommand{\guideName}{Root-Finding Study Guide}


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\title{Study Guide for Root Finding}
\author{Gerald Recktenwald}

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\begin{document}

%\maketitle

% -------------------------
\section*{Bare Essentials}
At the end of this chapter you should be able to
\begin{enumerate}
	\item	Write any scalar equation in the form $f(x)=0$
	\item	Give a graphical interpretation of the location of a root on
			the $x$ axis (when equation is written as $f(x)=0$).
	\item	Explain the role of bracketing.
			%  purpose is to find intervals that are likely to contain roots,
			%  not find an close estimate of the root
	\item	Write a simple equation that expresses the condition for finding a root
			in a bracket interval.
			%  f(a)*f(b)<0
	\item	Manually perform a few steps of the bisection method
	\item	Identify the one situation where bisection will return an incorrect
			value for $x$ as a root.
			%  when $x$ is at a singularity
	\item	Manually perform a few steps of Secant method
	\item	Identify situtations that cause Netwon's method to fail.
			% when $f'(x)=0$ or $f'(x)\approx 0$
	\item	Manually perform a few steps of Newton's method
	\item	Identify situtations that cause Netwon's method to fail.
			% when $f'(x)=0$ or $f'(x)\approx 0$
	\item	Describe in words two complimentary convergence criteria.
			% small $|f(x)|$ and root contained in a small interval
	\item	Write expressions for two complimentary convergence criteria.
			% $|f(x)|<\delta_f$, and
			% $|x_\mathrm{new}-x\mathrm{old}|< \delta_x$
	\item	List the methods used by the built in \texttt{fzero} command.
	\item	List reasons why simple root-finding schemes are not recommended
			to search for roots of polynomials.
	\item	Name the procedure used by \texttt{roots} to find the roots of a
			polynomial.
\end{enumerate}

% -------------------------
% \subsection{Working with \MATLAB}
\noindent To perform basic root-finding with \MATLAB\ you will need to
\begin{enumerate}
	\item	Plot any $f(x)$ as a means of graphically identifying the location of
			roots.
	\item	Write an m-file that evaluates $y=f(x)$ for use with \texttt{bisect},
			\texttt{secant}, and \texttt{fzero}
	\item	Write an m-file that evaluates $f(x)$ and $f'(x)$ for use with 
			the \texttt{newton} function 
	\item	Find zeros of a function with the \texttt{bisect}, \texttt{newton},
			and \texttt{fzero}.
	\item	Find roots of polynomials with the \texttt{roots} command.
\end{enumerate}


% -------------------------
\section*{An Expanded Core of Knowledge}
After mastering the bare essentials you should move on to a deeper understanding
of the fundamentals.  Doing so involves being able to
\begin{enumerate}
	\item	Qualitatively compare the convergence rates of bisection, secant and
			Newton's method
	\item	Describe the \texttt{fzero} command, and how it relates to bisection,
			secant and reverse interpolation.
	\item	Specify convergence tolerance for any function so that excessive
			(unecessary) iterations of a root-finder are not performed.
\end{enumerate}

% -------------------------
% \subsection{Working with \MATLAB}
\noindent To perform more advanced root-finding with \MATLAB\ you will need to
\begin{enumerate}
	\item	Describe the role of global variables in finding the roots of
			$f(x,a,b,\ldots) = 0$ where $a$, $b$, \ldots are parameters, and
			the method returns the value of $x$ that gives $f=0$ for fixed
			values of $a$, $b$, \ldots
	\item	Write m-files that use pass-through parameters $a$, $b$, \ldots,
			to evaluates $y=f(x,a,b,\ldots)$ for use with the \texttt{fzero} command
\end{enumerate}

% -------------------------
\section*{Developing Mastery}
Working toward mastery of root-finding you will need to
\begin{enumerate}
	\item	Analyze the convergence rate of bisection.
	\item	Identify the behavior of Newton's method for repeated roots.
	\item	Connect the problem of root-finding of a scalar equation to the
			the solution of nonlinear systems of equations.
\end{enumerate}


\end{document}
