The Syllabus for the Qualifying Examination in Scientific Computing

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 eigenvectors.

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 nonlinear systems.

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, singular integrals.

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, eigenvalue problems.