Computing numerically with functions instead of numbers
Keynote address for Remembering Karl Menger 2008
Lloyd N. Trefethen FRS
Professor of Numerical Analysis and Fellow of Balliol College
Oxford University
For a long time there have been two kinds of mathematical computation: symbolic and numerical. Symbolic computing manipulates algebraic expressions exactly, but it is unworkable for many applications since the space and time requirements tend to grow combinatorially. Numerical computing avoids the combinatorial explosion by rounding to 16 digits at each step, but it works just with individual numbers, not algebraic expressions.
This talk will describe a new kind of computing that aims to combine the feel of symbo numerics. The idea is to represent functions by Chebyshev expansions whose length is determined adaptively to maintain an accuracy of close to machine precision. Our "chebfun" system is implemented in object-oriented Matlab, with familiar vector operations such as sum and diff being overloaded to analogues for functions such as integration and differentiation. The system is surprisingly effective, and a demonstration will be given together with a discussion of the underlying mathematics and of the prospects for the future. The chebfun system is a joint project with Zachary Battles, Ricardo Pachon, Rodrigo Platte, and Toby Driscoll.
14 April 2008, Location TBA, 4:30 p.m.
Nick Trefethen is Professor of Numerical Analysis and head of the Numerical Analysis Group at Oxford University. He was educated at Harvard and Stanford and held professorial positions at NYU, MIT, and Cornell before 1997. He is a Fellow of the Royal Society and a member of the US National Academy of Engineering.
As an author he is known for his books Numerical Linear Algebra (1997), Spectral Methods in MATLAB (2000), Schwarz-Christoffel Mapping (2002), and Spectra and Pseudospectra (2005). He is an ISI Highly Cited Researcher, with about 90 journal publications in numerical analysis and applied mathematics, and has served as editor for many of the leading numerical analysis journals. He has lectured in about 15 countries and 25 American states, including invited lectures at both ICM and ICIAM congresses.
Some of Trefethen's recent activities include the SIAM 100-Dollar, 100-Digit Challenge, the notion of Ten Digit Algorithms ("ten digits, five seconds, and just one page"), the chebfun system for numerical computation with functions instead of numbers, and a book in preparation called Neoclassical Numerics.
For more information on the Karl Menger festivities, please visit Remembering Karl Menger 2008.
