Numerical Analysis

The topics of this exam are covered in undergraduate courses such as MATE 4061-4062 and in the graduate course MATE 6672.
Topics:

  • Floating point numbers: representation of real numbers in finite precision, arithmetic in finite precision, error bounds.
  • Numerical methods for solving linear systems: Gaussian elimination and its variants, LU factorization, Basic iterative methods: Richardson, Jacobi, Gauss-Seidel and SOR.
  • Numerical methods for solving non-linear systems: Fixed point iteration, Newton method, bisection, Regula-Falsi, Aitken’s method.
  • Basic theory of polynomial approximation: Weierstrass approximation theorem, Berstein polynomials, Lagrange interpolation polynomials, Legendre, Chebyshev, Hermite and Laguerre polynomials, least squares approximation.
  • Numerical integration: interpolatory and Gaussian quadratures, integration rules for singular and multiple integrals.
  • Numerical differentiation; approximation formulas of derivatives, error and order of approximation, introduction to the finite differences method for solving two-point boundary problems or partial differential equations, Von Neumann stability analysis.
  • Numerical methods for solving ordinary differential equations: Euler-Cauchy method, Runge-Kutta method, predictor-corrector, one step and multi-step methods, consistency, convergence and stability, shooting method for boundary value problems.


Bibliography:

  • An introduction to numerical Analysis, E. Süli and D. Mayers, Cambridge University Press, 2003.
  • Scientific Computation, Deuflard, Springer Verlag, 2000.
  • A first course in numerical analysis, A. Iserles, Cambridge University Press, 1999.
  • Numerical analysis: an introduction, W. Gautschi, Birkhäuser, 1997.
  • Numerical methods for Ordinary differential Equations, Hairer and Wanner, Springer Verlag 1999.
  • Finite difference Schemes and Partial Differential Equations, J.C. Strikwerda, 2nd Ed, SIAM 2004.
  • Numerical Analysis, K. Atkinson and W. Han. Springer Verlag, 2000.
  • Analysis of Numerical Methods, E. Isaacson and H. B. Keller, Dover Pub. Inc., 1994.

Previous exams:

Fall-2007

Spring-2007

Fall-2006