Explicit error bounds of the SE and DE formulas for integrals with logarithmic and algebraic singularity

TL;DR

This paper derives explicit, computable error bounds for the SE and DE quadrature formulas applied to integrals with logarithmic and algebraic singularities, including semi-infinite interval cases, improving accuracy guarantees.

Contribution

It introduces new, more accurate error bounds for the SE and DE formulas handling integrals with combined logarithmic and algebraic singularities, extending to semi-infinite intervals.

Findings

New error bounds reduce overestimation of divergence speed.

Bounds are applicable to integrals over finite and semi-infinite intervals.

Enhanced guarantees for numerical integration accuracy.

Abstract

The single exponential (SE) and double exponential (DE) formulas are widely recognized as efficient quadrature formulas for evaluating integrals with endpoint singularity. For integrals exhibiting algebraic singularity, explicit error bounds in a computable form have been provided, enabling computations with guaranteed accuracy. Such explicit error bounds have also been provided for integrals exhibiting logarithmic singularity. However, these error bounds have two points to be discussed. The first point is on overestimation of divergence speed of logarithmic singularity. The second point is on the case where there exist both logarithmic and algebraic singularity. To address these issues, this study provides new error bounds for integrals with logarithmic and algebraic singularity. Although existing and new error bounds described above pertain to integrals over the finite interval, the SE and DE formulas are also applicable to integrals over the semi-infinite interval. On the basis of the new results, this study provides new error bounds for integrals over the semi-infinite interval with logarithmic and algebraic singularity at the origin.