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\begin{document}
\begin{flushleft}
        {\normalsize \it \bf The University of New Mexico} \\
        {\normalsize \it E.A. Coutsias} \\
        {\Large \bf 504 '09-Project}
        \end{flushleft}
{\bf "To build an efficient symmetric eigensolver"}\\
We want to write a fast eigensolver for moderate-sized
symmetric matrices. It should be able to give efficiently either
a selected few or all eigenvalues and, optionally, eigenvectors.
All tricks you know must be incorporated, with key objective to
perform the calculation in the most efficient way possible:
\begin{itemize}
\item Conversion to (symmetric = tridiagonal) Hessenberg form.
\item Efficient/stable solution of triangular system (Rayleigh 
quotient iteration step) using either sparse LU with pivoting
or Givens QR.
\item for the case when all eigenvalues are required, follow the 
steps given in problem 29.1 of Trefethen and Bau. For the case
where only one eigenpair is sought, use Rayleigh quotient iteration.
\item if the largest eigenvalue is sought, use Gershgorin estimates
to get a good upper bound for its value before starting the iteration.
You must ensure that you did not converge to a lesser eigenvalue.
\item Document the flops as well as the convergence rate of your algorithms.
Employ random symmetric matrices of size $2^k, k = 4,\ldots,12$ and give the number
of iterations required by RQI method to find the largest eigenvalue. Also give the number
of iterations required by QR to converge. Verify that the eigenvalue found by RQI method is indeed
the same as the largest eigenvalue found by QR method.
\item Do you encounter any problems with the convergence of RQI? Discuss
\item If you can use f95 or C you may do so, however if you can only use Matlab, then you
      may have to restrict the size of the problem to get convergence in finite time. 

\end{itemize} 

\end{document}
