%%  File:  rootsGuide.tex
%%
%%  Study Guide for Numerical Methods with MATLAB
%%  Root-Finding Chapter
%%
%%  Gerald Recktenwald,  gerry@me.pdx.edu
%%  August 2000


\documentclass{article}

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

\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 the secant method
    \item   Identify situations that cause Newton'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 situations that cause the secant 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.

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\end{enumerate}

\bigskip
\noindent To perform basic root-finding with \MATLAB\ you will need to
\begin{enumerate}
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    \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
            (unnecessary) iterations of a root-finder are not performed.
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\end{enumerate}

\bigskip
\noindent To perform more advanced root-finding with \MATLAB\ you will need to
\begin{enumerate}
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    \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}