The Runge Example for Interpolation and Wilkinson's Examples for Rootfinding
For students and practitioners of numerical analysis, it clarifies classic counterintuitive phenomena using modern theory.
The paper uses backward error analysis and conditioning to explain the Runge interpolation and Wilkinson rootfinding examples, showing that the observed instability is due to ill-conditioning rather than algorithm flaws.
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.