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.
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 Menger 2008.
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