The G. S. Watson Annual Lecture, 2004
Professor Nick Trefethen
Title: "Polynomial interpolation is much better than you think"
Presenter: Professor Nick Trefethen, Professor of Numerical Analysis, University of Oxford
Date: Tuesday, 25 May 2004, 5pm to 6pm, Room 2.30 - Business Building, La Trobe University, Bendigo
Abstract: Most mathematicians, and even most numerical analysts who should know better, think that polynomial interpolation is dangerous. This is a myth. In fact, polynomial interpolation is utterly robust and stable, provided that (i) you use Chebyshev or other clustered points and (ii) you evaluate the interpolant by the barycentric Lagrange formula. Interpolants through thousands or even millions of data points can be calculated fast and accurately. What's more, such interpolants provide extraordinarily accurate approximations to smooth functions and form the basis of some of the most powerful numerical methods known for solving ordinary and partial differential equations.
One aim of this talk is to dispel the myth and to say a few words on where it came from. The other is to show off a new "chebfun" software system, developed with Oxford DPhil student Zachary Battles, that is based entirely on Chebyshev polynomial interpolants. What we have done is take the usual commands of Matlab that act on vectors and matrices, like "sum" and "roots" and "svd", and overload them to commands that act instead on continuous functions and operators. To the user, it looks as if Matlab is continuous rather than discrete. Under the hood, everything is polynomial interpolants.
About the Presenter: Nick Trefethen is Professor of Numerical Analysis and Head of the Numerical Analysis Group in the Oxford University Computing Laboratory. Born in 1955, he grew up in Lexington, Massachusetts, attended Harvard and Stanford Universities, and has held professorial positions at New York University (Courant Institute), MIT, Cornell University and Oxford University. He is currently nearing the end of a 12-month visit to the Advanced Computational Modelling Centre at the University of Queensland. Nick Trefethen has published about 80 research papers and several books that span a number of areas within numerical analysis and applied mathematics, has won awards for graduate-level instruction at MIT and Cornell, and is a frequent invited speaker at international conferences. He is perhaps best known for developments associated with pseudospectra of non-normal matrices and operators, including theory, algorithms and applications in fluid mechanics, numerical solution of partial differential equations, numerical linear algebra, shuffling of cards, random matrices, differential equations, lasers and other fields.