3  Presentation Topics

• You can choose to do a presentation for 10% of your grade (see syllabus for details),
• Presentations will be 10 minutes plus 2 minutes for questions (I will have to be strict on timings to fit everyone in),
• You will present to the whole group in one of our normal class times,
• You may present however you think best illustrates the key ideas (you may use the whiteboards, slides, jupyter notebook, ….),
• Note that 10 minutes is not very long! Pick something interesting but be wary of the time constraints,
• It would be nice if your presentation included some numerical implementation. You may submit accompanying numerical experiments,
• If you have your own ideas please let me know and I can advise if they are suitable,
• Here is a list of potential presentation topics,

3.1 Chapter 1: Basics of numerical analysis and floating point numbers

1. Research a time when real world problems have occured when floating point numbers have not been implemented properly!
2. Horner’s method,

3.2 Chapter 2: Solving nonlinear equations in 1d

(Pick a topic that you didn’t already learn about in Assignment 5)

1. Halley’s method,
2. Osada’s method,
3. Steffensen’s method,
4. Traub’s method,
5. Aitken’s \Delta^2 Method,
6. Efficient methods for finding zeros of polynomials,
7. Solving nonlinear equations in d > 1 dimensions.

3.3 Chapter 3: Polynomial interpolation

1. Weierstrass approximation theorem and Bernstein polynomials,
2. Lebesgue constants,
3. Companion matrices,
4. Neville’s method,
5. Cubic spline interpolation,
6. Investigate convergence of Chebyshev interpolation of functions of different smoothness.

3.4 Chapter 4: Numerical integration

1. Richardson extrapolation.
2. Romberg integration,
3. Gaussian quadrature with respect to more general weights w(x),
4. How to deal with improper integrals?
5. Adaptive quadrature.

3.5 Chapter 4.5: Integrating inital value problems

1. Runge-Kutta methods

3.6 Chapter 5: Solving linear systems of equations

1. Conditioning of the Vandermonde matrix and its implications,
2. Gershgorin circle theorem,
3. Lanczos algorithm,
4. Using LU decomposition to solve a tridiagonal system of linear equations,
5. Linear regression,
6. Circulant matrices,
7. Toepliz matrices,
8. Rodrigues’ formula for Legendre or Chebyshev polynomials.

3.7 Others

1. How JPEG works (we looked at SVD for image compression and JPEG does something different!)