%%  File:  linSysGuide.tex
%%
%%  Study Guide for Numerical Methods with MATLAB
%%  Chapter 8, "Solving Systems of Equations"
%%
%%  Gerald Recktenwald,  gerry@me.pdx.edu
%%  August 2000


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\title{Study Guide for\\
       Solving Linear Systems of Equations}
\author{Gerald Recktenwald}

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\begin{document}

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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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\bigskip
\noindent To perform basic solutions of linear systems with \MATLAB\ you will need to
\begin{enumerate}
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    \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}
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    \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

% -------------------------
\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}


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