Methods of numerical integration of oscillatory functions by the DE- formula with the Richardson extrapolation (Q1099924)

The numerical method \textit{H. Toda} and \textit{H. Ono} [Some remarks for efficient usage of double exponential formulas, Kokyuroku, RIMS, Kyoto Univ. 339, 74-109 (1978)] for evaluating infinite oscillatory integrals introduces a surplus decaying exponential factor into the integrand followed by a double exponential change of variable to produce an integral which is then evaluated speedly from a sequence of trapeziodal rules. Richardson extrapolation over differently decaying factors is used to estimate the integral's value in the absence of any surplus factors. The method has been successful in practice and the present study examines the several limiting processes to establish a theoretical basis. It is shown that good results would be expected for wide classes of analytic integrands of practical importance. The use of a different exponentially factor is also examined and numerical results indicate a much improved performance.