SIX ADVANCED TOPICS IN NUMERICAL ANALYSIS
CSCIGA.2945001 / MATHGA.2011001
Professor Nick Trefethen
This sixweek course will be structured in an unusual way. Each of our six
meetings will be independent. At each meeting, the first hour will be a lecture
aimed at anyone interested in numerical analysis at a high level, organized
around a wellknown topic and mixing historical perspectives, recent
developments, and always some new mathematics. The second hour will be for
enrolled students only, a handson work session making use of Chebfun.
 Lecture 1. Gaussian elimination
 Lecture 2. Random functions and random differential equations
 Lecture 3. Chebyshev series
 Lecture 4. Rational functions
 Lecture 5. Quadrature
 Lecture 6. ODEs
Some topics to be included in the lectures.

Gaussian elimination
 Orthogonal vs. nonorthogonal algorithms in linear algebra
 Stability paradoxes and probabilistic analysis
 GE as an iterative algorithm
 Continuous analogue of GE
 Lowrank approximations in higher dimensions

Random functions and random differential equations
 What's the continuous analogue of randn?
 Smooth random functions and smooth random walks
 Random functions in 2D and 3D
 The limit dt → 0
 Stochastic DEs, Ito vs. Stratonovich

Chebyshev series
 Fourier, Chebyshev, Laurent
 Faber's theorem and inverse Yogiisms
 Smoothness and convergence theorems
 Chopping a Chebyshev series
 Chebyshev vs Legendre and other expansions
 Series expansions on the sphere

Rational functions
 Polynomials vs. rationals and two famous problems
 p/q vs. barycentric representations
 The AAA algorithm
 Other adaptive barycentric algorithms
 Rational functions, quadrature, and hyperfunctions

Quadrature
 Gauss, ClenshawCurtis, and periodic trapezoidal
 Computation of Gauss nodes and weights
 Cauchy integrals and numerical linear algebra
 Perturbed nodes and the Kadec 1/4 theorem
 Nonuniform resolution of polynomials
 Cubature and multivariate polynomials

ODEs
 Marching vs. global methods
 Block operators and spectral discretizations
 Highlights from Exploring ODEs