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\begin{document}

\title{Numerical Methods I}
\author{Review for Midterm Exam}
\date{Friday, October 17, 2003}
\maketitle


\begin{enumerate}

\item Computer Arithmetic: Understand how floating point numbers work
(you do not need to know implementation details, but should have an
understanding of effects as exemplified in Homework~1 Questions~2, 5).

\item Condition number: definition (for the evaluation of a
real-valued function, for finding roots, and also for matrices), what
does it tell you?  (See Homework~1 Questions~3, 4; Homework 3
Questions~2b, 5.)

\item Root finding: Bisection method, Newton method; have an idea how
the secant method can be derived from the Newton method.

\item Order of convergence: hands-on definition of order of
convergence; how can you motivate by Taylor expansion that Newton's
method is (usually) convergent of order 2?  When does Newton's method
fail to converge quadratically?

\item Contraction mapping theorem: Understand basic idea.  (Think
about connection to iterative solvers for linear systems!)

\item Matrix norms: Definition, connection with condition numbers (see
Homework~3, Questions~1, 2).

\item $LU$ and $QR$ factorizations of a matrix; how to use the
factored form to solve a system of linear equations; importance of
pivoting.  (No need to study implementation details, but be familiar
with matrix manipulations, especially multiplication of block
matrices, and the basic ideas behind the algorithms.)

\item Can you compute by hand the $LU$ and $QR$ decompositions of
simple matrices ($2\times2$ or at most $3 \times 3$)?

\item Least squares: How to use $QR$ factorization to solve least
squares problems.

\item Iterative solvers for linear systems: Jacobi and Gauss--Seidel
method, convergence criterion for general iterative methods, basic
idea behind gradient and conjugate gradient method.

\end{enumerate}
\end{document}
