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\begin{document}

\title{Numerical Methods I}
\author{Review for Final Exam}
\date{Friday, December 12, 2003 \\
9:30-11:30, Research II Lecture Hall}
\maketitle


\begin{enumerate}

\item All the topics up to the Midterm Exam.

\item Polynomial interpolation: Know the Lagrange basis polynomials
and be able to apply them (for example for deriving Newton--Cotes
quadrature formulae, or Adams solvers for ODEs; problems like
Questions~6.3 and~6.4 from the homework); understand more generally
how interpolation with basis functions works (e.g.\ for Hermite or
trigonometric interpolation)---you don't need to remember formulae,
but should be able to use the ideas.

\item Spline interpolation: Basic ideas; be able to derive the system
of equations that determines a spline (with some hints given); what
type of system of linear equations do you get?  Can you derive some of
the minimal properties of splines?

\item Orthogonal polynomials: basic concepts.

\item Numerical Quadrature: Newton--Cotes formulae, Gauss quadrature;
understand the idea and use of the Euler--Maclaurin summation formula
(you do not need to remember its exact form); Romberg integration.
Questions~8.1 and~8.2 could be similar to exam questions.  (Do you
realize that you have derived a Gauss quadrature formula in Question
8.1?)

\item Basics of differential equations: Be able to explicitly solve
simple equations (e.g.\ the logistic equation, linear $2\times 2$
systems by diagonalization).  What is a critical point, how to check
its stability (Question~10.3)?

\item Matrix exponential: idea, compute the matrix exponential for
simple matrices, basic application to ODEs.

\item Methods for solving ODEs:
\begin{itemize}
\item Basic concepts: implicit vs.\ explicit, one-step vs.\ multistep.
\item Local truncation error (computing the local truncation error for
some method \emph{will} be on the exam!)
\item Convergence: consistent one-step methods always converge (why?),
multi-step methods converge if and only if they are $0$-stable.
\item Order of convergence
\item Absolute stability
\item Families of methods: Taylor, Runge--Kutta, Adams--Bashforth,
Adams--Moulton (ideas only, no need to remember formulae)
\item Think about root finding schemes for implicit methods.  When
does simple iteration work?
\end{itemize}




\end{enumerate}
\end{document}
