Numerical indefinite integration by double exponential sinc method

Abstract

We present a numerical method for approximating an indefinite integral by the double exponential sinc method. The approximation error of the proposed method with N N integrand function evaluations is \[ O ( exp ⁡ ( − c 1 N / log ⁡ ( c 2 N ) ) ) \mathrm {O}(\exp (-c_1 N/\log (c_2 N))) \] for a reasonably wide class of integrands, including those with endpoint singularities. The proposed method compares favorably with the existing formulas based on the ordinary sinc method. Computational results show the accordance of the actual convergence rates with the theoretical estimate.