Understanding the ultraspherical spectral method

TL;DR

This paper analyzes the ultraspherical spectral method, explaining its error sources, condition number, and why certain errors decay below machine epsilon, providing insights into its high accuracy and reliability.

Contribution

It identifies the sources of error and derives the effective condition number of the ultraspherical spectral method, clarifying its numerical stability and error behavior.

Findings

Effective condition number explains backward error stability.

Cauchy error can decay below machine epsilon, misleading convergence.

Analysis extends to other spectral methods and PDE solutions.

Abstract

The ultraspherical spectral method features high accuracy and fast solution. In this article, we determine the sources of error arising from the ultraspherical spectral method and derive its effective condition number, which explains why its backward error is consistent with a numerical method with bounded condition number. In addition, we show the cause for the Cauchy error to go below the machine epsilon and decay eventually to exact zero, revealing the fact that the Cauchy error can be misleading when used as an indicator of convergence and accuracy. The analysis in this work can be readily extended to other spectral methods, when applicable, and to the solution of PDEs.