http://interval.louisiana.edu/courses/prelims/summer1998naprelim.html
Study Guide for the Numerical Analysis Prelim, Summer, 1998
Tuesday, August 18, 1997, 1:00 PM to 5:00 PM, MDD 206
Facullty consultant: R.
Baker Kearfott, Department
of Mathematics,
University of Southwestern
Louisiana
Office hours
and telephone, Email: rbk@louisiana.edu.

Condition numbers of functions (Study section 1.4 of the Neumaier
manuscript.)

Forward and reverse mode of automatic differentiation, both univariate
and multivariate (Study section 1.1 of the Neumaier manuscript, and
also study the inclass notes, handouts, and electronically available notes
carefully)

Interval arithmetic and its use in bounding error terms, such as
are found in Taylor series, the mean value theorem, and quadrature rules.
(Study
the material in the Neumaier manuscript, as well as inclass notes, old
exams, and electronically available notes.)

Quadrature formulas. Pay especially close attention to changes of
variables in both the formula and the error term. (Study section 9.1
of the Neumaier manuscript, inclass notes, etc.)

Least squares problems, with QR factorizations, the normal equations,
and the SVD. (Study 6.1, 6.2, and 6.6 of the Neumaier manuscript, as
well as your inclass notes.)

The interval GaussSeidel method with inverse midpoint preconditioner.
Know how to carry out iterations of this method. (Study your notes,
and Rigorous Global Search: Continuous Problems. If you need a copy of
this book, I can loan you one.)

The finite element method, piecewise linear basis functions, etc.
Know
how to discretize a linear or nonlinear PDE. (Rely mostly on inclass
notes. You may also want to consult "An Analysis of the Finite Element
Method" by Strang and Fix, as well as the section on the RayleighRitz
method in "Numerical Analysis" (fifth edition) by Burden and Faires. Strang
and Fix is in the library, as is Burden and Faires. See me if you need
a second copy of the latter.)

Eigenvalues and eigenvectors. Know how to compute them, and know
their significance in a stability analysis of methods for parabolic partial
differential equations. (Rely mostly on class notes.)

Stability analysis for systems of ODE. There are three approaches:

The eigenvalue approach, using a model problem, discussed in class.

The Fourier method discussed in Kincaid and Cheney.

Alternate approaches, such as discussed in Ames, "Numerical Methods for
Partial Differential Equations."
Know all three approaches, and be able to carry one out on a discretization
of a partial differential equation.

The concept of order for integrating a system of ordinary differential
equations.
Postscript Copies of Previous Preliminary Examinations
Summer,
1990
Winter,
1991
Summer,
1992
Winter,
1993
Summer,
1998
Adobe Acrobat (PDF) Copies of Previous Preliminary Examinations
Summer,
1990
Winter,
1991
Summer,
1992
Winter,
1993
Summer,
1998
Note: The exam will
be closed book, and it will be fairly lengthy, so you will need
to prepare well.