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\title{Numerical Methods II}
\author{Review for Final Exam}
\date{Friday, May 21, 2004, 11:00--13:00 in the Research II Lecture
  Hall} 
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\begin{enumerate}

\item Gradient and Conjugate Gradient methods for nonlinear problems.
  Understand the following issues:
  \begin{itemize}
    \item Which method converges better for quadratic functions (i.e.\
    when you are effectively solving a system of linear equations)
    when the system is ill-conditioned?

  \item Why may you need to ``restart'' the CG method in the nonlinear
  case?
  \end{itemize}
  Look, in particular, at the results from Projects 4/5 on Homework~6.

\item Basic stochastic differential equations.  Be able to do problems
  like Question~2 on Homework~8, or compute $\mathbb{E}[\exp(\mu
  W(t))]$.

\item Stability of numerical methods for ordinary differential
  equations.  Think about idea of absolute stability in the context of
  stochastic differential equations.  (Where do the equations on
  p.~541 of Higham's article come from?)

\item Linear boundary value problems, in particular the computation of
  the local truncation error.

\item Matrix eigenvalue problems: Know the technical tools (plane
  rotation matrices, Householder reflections, Frobenius norm and its
  invariance under orthogonal transformations), Jacobi method, QR
  method, Gershgorin theorem, QR decomposition of a tridiagonal matrix
  by plane rotations, inverse iteration, elementary perturbation
  analysis.

  Revise, in particular, explicitly performing a Householder
  reflection, or one step of the QR algorithm.  There are many good
  examples in the book by S\"uli and Mayers.

\item Penalty method for optimization with constraints.  Questions
  like on Homework~7 are possible.
  
\end{enumerate}

Note: This list of topics is only a selection of what we covered this
semester.  Don't completely ignore topics that are not listed, in
particular if you are ambitious to solve the extra-credit questions.


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