%% File: linSysGuide.tex
%%
%% Study Guide for Numerical Methods with MATLAB
%% Chapter 8, "Solving Systems of Equations"
%%
%% Gerald Recktenwald, gerry@me.pdx.edu
%% August 2000
\documentclass{article}
\usepackage{studyGuide}
\renewcommand{\guideName}{Study Guide for Solving Systems of Equations}
\title{Study Guide for\\
Solving Linear Systems of Equations}
\author{Gerald Recktenwald}
\newcounter{enumSave}
\begin{document}
%\maketitle
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\section*{Bare Essentials}
At the end of this chapter you should be able to
\begin{enumerate}
\item Transform equations written in the natural variables of an
applied problem to the canonical $Ax=b$ of linear algebra.
\item Explain the condition of consistency in terms of linear combinations
of column vectors.
\item Explain the condition of singularity of an \matdim{n}{n} matrix
in terms of linear independence.
\item Express matrix rank as a measure of linear independence.
\item Relate rank of the coefficient matrix to the consistency of a \matdim{n}{n}
system of equations.
\item Write the formal solution to $Ax=b$.
\item Explain why it is not a good idea to use the formal solution as a
computational procedure for solving $Ax=b$.
\item Describe the most efficient procedures for solving $Lx=b$ or $Ux=b$
when $L$ is lower triangular and $U$ is upper triangular.
\item Name the solution algorithm most commonly used for solving $Ax=b$.
\item Write the equation that defines the residual vector.
\item Describe the significance of $\kappa(A)$ on the reliability of
the numerical solution to $Ax=b$.
\item Describe the significance of \norm{r} for a well-conditioned $A$.
\item Describe the significance of \norm{r} for a ill-conditioned $A$.
\item Describe the reason for pivoting. Is pivoting a remedy for
ill-conditioned systems?
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\end{enumerate}
\bigskip
\noindent To perform basic solutions of linear systems with \MATLAB\ you will need to
\begin{enumerate}
\setcounter{enumi}{\value{enumSave}}
\item Assign the elements of matrix \texttt{A}, and vector \texttt{b},
for a system of equations.
\item Write a compact (one line) statement that uses the recommended method for
solving $Ax=b$, given that $A$ and $b$ are already assigned to
\MATLAB\ variables.
\item Compute \norm{r} of a system given that $A$, $x$, and $b$
are already assigned to \MATLAB\ variables.
\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 Describe the qualitative relationship between the magnitude of $\kappa(A)$
and the singularity of $A$.
\item Estimate the number of correct significant digits in the numerical
solution to $Ax=b$ given values of \epsm\ and $\kappa(A)$.
\item State conditions required for a successful LU factorization of $A$.
% $A$ is square, nonsingular
\item Write (describe) a procedure for solving $Ax=b$ given an LU factorization of $A$.
\item State conditions required for a successful Cholesky factorization of $A$.
% $A$ is symmetric and positive definite
\item Write (describe) a procedure for solving $Ax=b$ given a Cholesky factorization of $A$.
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\end{enumerate}
\bigskip
\noindent To perform more advanced solutions of linear systems with \MATLAB\ you will need to
\begin{enumerate}
\setcounter{enumi}{\value{enumSave}}
\item Write the preferred expression for solving $Lx=b$ or $Ux=b$
when $L$ is lower triangular and $U$ is upper triangular.
What algorithm does \MATLAB\ select to implement the solution
for these systems?
\item Use \MATLAB\ and the LU factorization of $A$ to solve several systems
of equations that have the same $A$ and a sequence of different $b$.
\item Use \MATLAB\ and a Cholesky factorization of $A$ to solve several systems
of equations that have the same $A$ and a sequence of different $b$.
\item Implement solutions of nonlinear systems of equations with
successive substitution.
\item Implement solutions of nonlinear systems of equations with
Newton's method.
\end{enumerate}
\clearpage
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\section*{Developing Mastery}
Working toward mastery of solving systems of equations you will need to
\begin{enumerate}
\item Given a variety of \matdim{m}{n} system of equations, where $m$ is not necessarily
equal to $n$, describe the method used by the \verb|\| operator to solve
$Ax=b$.
\item Given $L$, $U$, and permutation matrix $P$ from an LU factorization of $A$, apply
these to solve $Ax=b$. Specifically, use the $P$ appropriately.
\item Explain how \MATLAB\ uses the $L$ and $U$ factors returned from the \texttt{lu}
command to solve $Ax=b$ \emph{without} explicitly requiring $P$.
\item List the order of magnitude work estimates for Gaussian elimination with back
substitution, LU factorization, and Cholesky factorization.
\end{enumerate}
\end{document}