Chebfuns: A New Kind of Numerical Computing
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.
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.