http://interval.louisiana.edu/courses/prelims/summer-1998-na-prelim.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

1. Condition numbers of functions (Study section 1.4 of the Neumaier manuscript.)
2. 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 carefully)
3. 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 in-class notes, old exams, and electronically available notes.)
4. 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.)
5. 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.)
6. 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.)
7. The finite element method, piecewise linear basis functions, etc. Know 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.)
8. 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.)
9. Stability analysis for systems of ODE. There are three approaches:
1. The eigenvalue approach, using a model problem, discussed in class.
2. The Fourier method discussed in Kincaid and Cheney.
3. 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.
10. The concept of order for integrating a system of ordinary differential equations.

• Summer, 1990
• Winter, 1991
• Summer, 1992
• Winter, 1993
• Summer, 1998

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### 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.