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

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.