Syllabus for the Comprehensive Exam in Numerical Analysis

  1. Basic Material: Fixed point iteration; bisection; Newton's method; the secant method; polynomial interpolation; numerical differentiation; numerical Integration.
  2. Properties of Numerical Methods:
    * From ODE and PDE methods: Consistency; zero-stability; accuracy; absolute stability; convergence
    * From PDE methods: Fourier analysis of error; stability; maximum principle; convergence; the CFL condition; the method of characteristics; domain of dependence
  3. Numerical Methods: Implicit and explicit schemes; initial-value ODE methods: one-step methods including Runge-Kutta methods, multi-step methods, predictor-corrector method; boundary-value problems: shooting, finite difference methods; for elliptic, parabolic, and hyperbolic PDE problems: Crank-Nicolson, Lax-Wendroff, Lax-Friedrich, leap-frog, alternating direction implicit methods, implicit and explicit methods, the finite element method, and conservative schemes
  4. Numerical Linear Algebra: Solving systems of linear equations; perturbation theory; Gaussian elimination; error analysis; pivoting and stability; sparse systems; the linear least squares problem; eigenvalue problems; iterative methods; QR factorization; singular value decomposition


Suggested textbooks: Numerical AnalysisNumerical Methods for Ordinary Differential Systems: The Initial Value Problem by Lambert; Finite Difference Schemes and Partial Differential Equations by Strikwerda; Numerical Methods for Conservation Laws by LeVeque; Matrix Computations by Golub and Van Loan
Suggested courses: 6643 and 6640
Other relevant courses: 4640 and 4641