Understanding the ultraspherical spectral method

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.