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{\bf Math/AMath 594: Iterative Methods for Solving Linear Systems} \\
(Autumn 1998) \\
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Lectures: & MW 3:30--4:50, room 102 Loew \\ [5pt]
Professor: & Anne Greenbaum,\hspace{.1in} C-434 Padelford, 543-1175 \\ [5pt]
Office Hours: & T, Th 1:30--3:00, or by appointment. \\ [5pt]
e-mail: & greenbau@math.washington.edu \\ [5pt]
Web Address: & http://www.math.washington.edu/\verb+~+greenbau \\ [5pt]
& Course materials: Click on ``Math/AMath 594". \\ [5pt]
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\noindent
Text: {\em Iterative Methods for Solving Linear Systems} by Anne Greenbaum,
SIAM, 1997. (available at University Bookstore)
\vspace{.1in}
\noindent
Reserve list: The following books are on reserve in the Mathematics
Research Library.
\begin{enumerate}
\item
{\em Matrix Computations} by Gene H. Golub and Charles F. Van Loan
(QA188.G65, 1996)
\item
{\em Iterative Methods for Solving Linear Systems} by Anne Greenbaum
(QA297.8.G74, 1997)
\item
{\em Numerical Linear Algebra} by Lloyd N. Trefethen and David Bau, III
(QA184.T74, 1997)
\end{enumerate}
\vspace{.3in}
\noindent
The course will follow the development in the text, with occasional
outside papers to read for discussion. Topics to be covered include:
\begin{itemize}
\item
Review of relevant linear algebra: Vector norms and inner products;
canonical forms and decompositions (e.g., Jordan form, Schur form,
QR decomposition, singular value decomposition); eigenvalues; the field
of values.
\item
The most popular iterative methods: Simple iteration; the conjugate
gradient method for Hermitian positive definite problems; MINRES for
Hermitian indefinite problems; GMRES, QMR, and BiCGSTAB for general
non-Hermitian problems.
\item
Derivation of error bounds for CG, MINRES, and GMRES: The open problem
of precisely describing the worst-case behavior of GMRES; the open
problem of deriving realistic error bounds for other non-Hermitian
matrix iterations such as BiCG and QMR.
\item
Effects of finite precision arithmetic on the convergence rate of
Lanczos-based iterative methods.
\item
Discussion of the open search for an optimal or near-optimal short
recurrence for non-Hermitian problems.
\item
Preconditioners: Regular splittings; optimal diagonal and block diagonal
preconditioners; incomplete decompositions.
\item
Multigrid and domain decomposition as preconditioning techniques.
\end{itemize}
\vspace{.2in}
\noindent
There will be weekly homework assignments (with some programming) (30 \%),
a course project (40 \%), and a final exam (30 \%).
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