[Paper Review] Roundoff errors in the problem of computing Cauchy principal value integrals

Paweł Keller, Iwona Wróbel|arXiv (Cornell University)|Nov 9, 2011

Mathematical functions and polynomials20 references4 citations

TL;DR

This paper presents a robust method for computing Cauchy principal value integrals using standard adaptive quadrature by transforming the singular integral into a sum of nonsingular integrals and controlling round-off errors through derived error estimates. The approach ensures high accuracy and reliability, especially when combined with error tolerance settings informed by theoretical bounds on numerical instability.

ABSTRACT

We investigate the possibility of fast, accurate and reliable computation of the Cauchy principal value integrals $\mathrm{P}\!\int_{a}^{b} f(x)(x-τ)^{-1} dx$ $(a < τ< b)$ using standard adaptive quadratures. In order to properly control the error tolerance for the adaptive quadrature and to obtain a~reliable estimation of the approximation error, we research the possible influence of round-off errors on the computed result. As the numerical experiments confirm, the proposed method can successfully compete with other algorithms for computing such type integrals. Moreover, the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine.

Motivation & Objective

• Address the lack of reliable, widely available algorithms for computing Cauchy principal value integrals in scientific computing systems.
• Investigate the impact of round-off errors on adaptive quadrature performance when computing singular integrals with a pole inside the integration interval.
• Develop error control mechanisms that allow standard adaptive quadrature routines to produce accurate and reliable results for Cauchy principal value integrals.
• Enable the use of existing, off-the-shelf adaptive quadrature subroutines for these challenging integrals without requiring specialized solvers.
• Provide a practical, easy-to-implement framework that ensures near-optimal accuracy and efficiency by dynamically adjusting error tolerance based on theoretical error bounds.

Proposed method

• Transform the Cauchy principal value integral into a sum of two nonsingular integrals using analytical identities, specifically splitting the integral around the singular point τ.
• Apply the identity $ I_{\tau}(f) = \int_{|x-\tau|\geq\delta} \frac{f(x)}{x-\tau} dx + \int_0^\delta \frac{f(\tau+x) - f(\tau-x)}{x} dx $, where $ \delta = \min\{1+\tau, 1-\tau\} $, to remove the singularity.
• Use adaptive quadrature on the transformed nonsingular integrals, leveraging existing reliable subroutines from systems like Maple, MATLAB, or custom implementations.
• Derive global error estimates that account for round-off error contributions, particularly in the symmetric difference term $ \frac{f(\tau+x) - f(\tau-x)}{x} $, to prevent misestimation of accuracy.
• Introduce a strategy to set the error tolerance below the threshold where round-off errors dominate, ensuring the computed result remains within a reliable error bound.
• Improve reliability by replacing the native error estimate from the quadrature with a composite estimate based on theoretical bounds, especially when the native estimator fails or overestimates.

Experimental results

Research questions

• RQ1 Can standard adaptive quadrature routines be effectively used to compute Cauchy principal value integrals despite their sensitivity to round-off errors?
• RQ2 What is the influence of round-off errors on the accuracy of adaptive quadrature when applied to singular integrals with a pole inside the interval?
• RQ3 How can round-off errors be reliably estimated and controlled to ensure both accuracy and efficiency in the computation of Cauchy principal value integrals?
• RQ4 Can a general-purpose method be developed that works with any adaptive quadrature subroutine while ensuring high reliability and near-optimal convergence?
• RQ5 What error tolerance setting ensures that the computed result is not compromised by numerical instability due to round-off, even when the tolerance is set very low?

Key findings

• The proposed method successfully computes Cauchy principal value integrals with high accuracy using standard adaptive quadrature, outperforming native implementations like qawc in terms of reliability.
• When the error tolerance was set too low (e.g., $ \sqrt{\varepsilon} $), the qawc algorithm from Quadpack failed or misestimated the error for up to 15,171 values of $ \tau $ in the case of $ f_5 $, demonstrating the fragility of default error estimation.
• By applying the derived error estimates and using $ \max\{T_{\text{qawc}}, E_{\text{qawc}}\} $ as the error estimate, the method achieved correct results for all tested values of $ \tau $, even in the most sensitive cases.
• The method significantly improves reliability: for $ f_9 $, the number of incorrect error estimates dropped from 1,440 to just 144, and failures were reduced from 1,694 to zero with proper tolerance and error estimation.
• The method is efficient and portable, requiring only access to a reliable adaptive quadrature routine and knowledge of its error estimation behavior.
• The use of global error bounds, particularly those derived from the symmetric difference form, enables accurate error control even when the adaptive quadrature fails to detect round-off effects locally.