Comparing the Performance of Four Sinc Methods for Numerical Indefinite Integration

Abstract

Aims/ Objectives: To compare the performance of four Sinc methods for the numerical approximation of indefinite integrals with algebraic or logarithmic end-point singularities.
 Methodology: The first two quadrature formulas were proposed by Haber based on the sinc method, the third is Stengers Single Exponential (SE) formula and Tanaka et al.s Double Exponential (DE) sinc method completes the number. Furthermore, an application of the four quadrature formulas on numerical examples, reveals convergence to the exact solution by Tanaka et al.s DE sinc method than by the other three formulae. In addition, we compared the CPU time of the four quadrature methods which was not done in an earlier work by the same author.
 Conclusion: Haber formula A is the fastest as revealed by the CPU time.

Yet another DE-Sinc indefinite integration formula

“…Haber [2] derived a numerical indefinite integration formula of the form f (x j )w j (x), x ∈ (−1, 1), (1) where the weight function w j depends on x, but the sampling node x j does not. Furthermore, the formula can attain a root-exponential convergence: O(exp(−c √ n)), even if the integrand f has an endpoint singularity such as f (s) = 1/ √ 1 − s 2 .…”

“…As a subsidiary contribution, we compared the six formulas (SE1), (SE2), (SE3), (DE1), (DE2) and (DE3). Akinola [1] performed a numerical comparison between (SE1), (SE2) 1 and (DE2), and reported that the formula (SE1) exhibited the lowest CPU time among those formulas. However, he compared CPU times with respect to n, which is not suitable for a comparison in terms of efficiency.…”