The Runge Example for Interpolation and Wilkinson's Examples for Rootfinding

TL;DR

This paper analyzes classical numerical analysis examples, the Runge interpolation problem and Wilkinson's rootfinding examples, using modern backward error analysis to explain their numerical behaviors and educational implications.

Contribution

It applies modern backward error analysis to classical examples, providing clearer explanations of their numerical phenomena and emphasizing the role of computer algebra in education.

Findings

Backward error analysis clarifies Runge and Wilkinson examples

Computer algebra enhances understanding of numerical methods

Provides educational insights into polynomial interpolation and rootfinding

Abstract

We look at two classical examples in the theory of numerical analysis, namely the Runge example for interpolation and Wilkinson's example (actually two examples) for rootfinding. We use the modern theory of backward error analysis and conditioning, as instigated and popularized by Wilkinson, but refined by Farouki and Rajan. By this means, we arrive at a satisfactory explanation of the puzzling phenomena encountered by students when they try to fit polynomials to numerical data, or when they try to use numerical rootfinding to find polynomial zeros. Computer algebra, with its controlled, arbitrary precision, plays an important didactic role.