Chopping a Chebyshev Series

TL;DR

This paper discusses the design and implementation of a chopping algorithm for Chebyshev series in the Chebfun software, which is crucial for accurate and efficient numerical function computations.

Contribution

It introduces the chopping algorithm used in Chebfun Version 5.3, detailing the considerations and design choices after years of development.

Findings

The algorithm effectively determines where to truncate Chebyshev series for optimal accuracy.

It addresses the complexity of balancing precision and computational efficiency.

The design has influenced subsequent numerical computing methods for functions.

Abstract

Chebfun and related software projects for numerical computing with functions are based on the idea that at each step of a computation, a function $f(x)$ defined on an interval $[a,b]$ is "rounded" to a prescribed precision by constructing a Chebyshev series and chopping it at an appropriate point. Designing a chopping algorithm with the right properties proves to be a surprisingly complex and interesting problem. We describe the chopping algorithm introduced in Chebfun Version 5.3 in 2015 after many years of discussion and the considerations that led to this design.