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Understanding the ultraspherical spectral method

Lu Cheng, Kuan Xu

TL;DR

This paper analyzes the numerical stability of the ultraspherical spectral method for high-accuracy ODE solutions by identifying three sources of error in floating-point implementations, deriving an upper bound for forward error, and introducing an effective condition number that depends only on the top-left blocks of the system. Through the Airy equation example, it demonstrates that the apparent conditioning growth can be misleading and that the Cauchy error may decay due to numerical artifacts rather than true convergence. A Skeel-type, block-based condition number is shown to predict forward error more accurately than the classical condition number, with the bound remaining constant as the problem size grows. The work also clarifies the limitations of the Cauchy error as a convergence indicator and discusses extensions to other spectral methods and to PDEs, highlighting practical implications for stability and accuracy in spectral discretizations.

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.

Understanding the ultraspherical spectral method

TL;DR

This paper analyzes the numerical stability of the ultraspherical spectral method for high-accuracy ODE solutions by identifying three sources of error in floating-point implementations, deriving an upper bound for forward error, and introducing an effective condition number that depends only on the top-left blocks of the system. Through the Airy equation example, it demonstrates that the apparent conditioning growth can be misleading and that the Cauchy error may decay due to numerical artifacts rather than true convergence. A Skeel-type, block-based condition number is shown to predict forward error more accurately than the classical condition number, with the bound remaining constant as the problem size grows. The work also clarifies the limitations of the Cauchy error as a convergence indicator and discusses extensions to other spectral methods and to PDEs, highlighting practical implications for stability and accuracy in spectral discretizations.

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.
Paper Structure (6 sections, 3 theorems, 40 equations, 4 figures)

This paper contains 6 sections, 3 theorems, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Three sources of error
  3. The Airy equation revisited
  4. Conditioning of the ultraspherical spectral method
  5. The Cauchy error
  6. Closing remarks

Key Result

Proposition 4.1

\newlabelthm:vmu0 If $\epsilon \left \| |A^{-1}|E\right \| \leq c < 1$, where $c$ is a constant, $E$ is an $n\times n$$m$-Hessenberg matrix, and $\left \| \cdot \right \|$ is an absolute norm, it follows from ufb2 that where $B = A^{-1} = = $ and $E = $ with $B_{1}\in \mathbb{R}^{n\times k}$, $\tilde{B}_{1} \in \mathbb{R}^{n\times (k + m)}$, and $E_{11} \in \mathbb{R}^{(k + m)\times k}$.

Figures (4)

  • Figure 1: Solving \ref{['airy']} for $\mu=10^{-2}$ using the ultraspherical spectral method.
  • Figure 1: (a) Partition of $E$. The shade highlights the region where the entries are nonzero. (b) Estimates of the forward error in the computed solution to \ref{['airy']} for $\mu=10^{-2}$, compared with the actual rounding error $\varepsilon_S$.
  • Figure 1: The explanations of \ref{['lem:q']} and \ref{['lem:u2e0']}.
  • Figure 1: The total error $\varepsilon$ and the Cauchy error for solving \ref{['airy']} with $\mu=10^{-2}$ using the banded Galerkin spectral method (BG) and the integral reformulated tau method (IR tau).

Theorems & Definitions (6)

  • Proposition 4.1
  • Proof 1
  • Lemma 5.1
  • Proof 2
  • Theorem 5.2
  • Proof 3