The Syllabus for the Qualifying Examination in Scientific
For the qualifying exam in scientific computing the candidate is
to prepare a syllabus by selecting topics from the list below.
The total amount of
material on the syllabus should be roughly equal to that covered
in a standard one semester graduate course. Once you have made your selections
discuss them with the professor who will examine you.
- 1) Hardware/Programming issues:
- Machine numbers, floating point arithmetic, accumulation of rounding
errors, memory hierarchy, arrays in C and FORTRAN, C++ scope,
C++ classes, organization of loops for computational efficiency.
- 2) Computational linear algebra:
- Basic linear algebra, solution of linear equations: direct and
iterative methods, convergence, matrix factorizations (LU, LL^T,
QR, SVD), linear equations and least squares, eigenvalues and
- 3) Iterative methods for nonlinear equations:
- Fixed point theorems, Convergence proofs, linear iteration methods,
Newton and secant methods for scalar equations, techniques for
enhancing global convergence, Newton and quasi-Newton methods for
- 4) Approximation theory and interpolation:
- Interpolating polynomials, Lagrange and Newton interpolation,
divided differences, piecewise polynomial approximation, least
squares polynomial approximation, orthogonal decompositions:
Fourier series/transforms and orthogonal polynomials.
- 5) Differentiation and integration:
- Divided differences, Richardson extrapolation, midpoint and
trapezoidal rules, the Euler-Maclaurin formula, Gaussian quadrature,
- 6) Initial value problems for ordinary differential equations:
- Finite difference methods: order of accuracy, stability analysis,
convergence results, Euler's explicit and implicit methods, local
truncation errors/rounding errors/accumulated errors, higher order
methods: Adams Bashforth and Adams Moulton methods, Runge-Kutta
methods, backward differentiation formulas, stiffness.
- 7) Boundary value problems for ordinary differential equations:
- Shooting methods, finite difference methods, finite element methods,
- An Introduction to Numerical Analysis (2nd ed), by K. E. Atkinson,
Wiley, New York, 1989.
- Analysis of Numerical Methods, by E. Isaacson and H. B. Keller,
Dover, New York, 1994.
- Numerical Analysis: Mathematics of Scientific Computing, by
David R. Kincaid, E. Ward Cheney, and Ward Cheney
- Introduction to Numerical Analysis,
by J. Stoer and R. Bulirsch
- Scientific Computing, by John Trangenstein (available online at http://www.math.duke.edu/faculty/johnt/math224/book.pdf)