Scientific Computing

Scientific Computing Syllabus

For the Oral Qualifying Exam

For the oral 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.

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.
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.
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.
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.
Differentiation and Integration
  • Divided differences, Richardson extrapolation, midpoint and trapezoidal rules, the Euler-Maclaurin formula, Gaussian quadrature, singular integrals.
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.
Boundary Value Problems for Ordinary Differential Equations
  • Shooting methods, finite difference methods, finite element methods, eigenvalue problems.

References

  • K.E. Atkinson, An Introduction to Numerical Analysis, 2nd ed. (Wiley, 1989)
  • Isaacson and Keller, Analysis of Numerical Methods (Dover, 1994)
  • Kincaid, Cheney and Cheney, Numerical Analysis: Mathematics of Scientific Computing
  • Stoer and Bulirsch, Introduction to Numerical Analysis
  • J. Trangenstein, Scientific Computing