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
University of Southwestern
and telephone, Email: email@example.com.
Condition numbers of functions (Study section 1.4 of the Neumaier
Forward and reverse mode of automatic differentiation, both univariate
and multivariate (Study section 1.1 of the Neumaier manuscript, and
also study the in-class notes, handouts, and electronically available notes
Interval arithmetic and its use in bounding error terms, such as
are found in Taylor series, the mean value theorem, and quadrature rules.
the material in the Neumaier manuscript, as well as in-class 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, in-class 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 in-class notes.)
The interval Gauss-Seidel 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.
how to discretize a linear or nonlinear PDE. (Rely mostly on in-class
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 Rayleigh-Ritz
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:
Know all three approaches, and be able to carry one out on a discretization
of a partial differential equation.
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."
The concept of order for integrating a system of ordinary differential
Postscript Copies of Previous Preliminary Examinations
Adobe Acrobat (PDF) Copies of Previous Preliminary Examinations
Note: The exam will
be closed book, and it will be fairly lengthy, so you will need
to prepare well.