Computing numerically with functions instead of numbers

ABSTRACT

If we make a sequence of computations involving rational numbers, the numerators and denominators tend to grow exponentially in length; floating-point arithmetic contains this growth by pruning to 16 digits at each step. We present an analogous set of ideas, and the "chebfun" software system, for computing with piecewise-smooth functions on an interval. Functions are represented by Chebyshev series to approximately 15 digits of precision. Operations such as +, --, x, /, max, min, norm, and zerofinding are carried out fast and accurately within this context, always pruning the accuracy to 15 digits of precision at each step to contain the combinatorial explosion. This is joint work with Zachary Battles and Ricardo Pachón.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation
          Subjects: Nonlinear approximation

General Terms:
Algorithms, Theory

Keywords:
Chebyshev series, symbolic-numeric computing