CHEBFUN AND CHEBOP

Lloyd N. Trefethen, 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 symbolics with the speed
of numerics.  The idea is to represent functions by
Chebyshev expansions whose length is determined adaptively
to maintain an accuracy of close to machine precision.
The 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.  A further
overload gives the chebop system for solving differential
equations.  All this is joint work with Zachary Battles,
Folkmar Bornemann, Ricardo Pachon, Rodrigo Platte,
and Toby Driscoll.