Numerical treatment of a twisted tail using extrapolation methods

Highly oscillatory integral, called a twisted tail, is proposed as a challenge in The SIAM 100-digit challenge. A Study in High-Accuracy Numerical Computing, where Drik Laurie developed numerical algorithms based on the use of Aitken’s Δ²-method, complex integration and transformation to a Fourier integral. Another algorithm is developed by Walter Gautschi based on Longman’s method; Newton’s method for solving a nonlinear equation; Gaussian quadrature; and the epsilon algorithm of Wynn for accelerating the convergence of infinite series. In the present work, nonlinear transformations for improving the convergence of oscillatory integrals are applied to the integration of this wildly oscillating function. Specifically, the transformation and its companion the W algorithm, and the G transformation are all used in the analysis of the integral. A Fortran program is developed employing each of the methods, and accuracies of up to 15 correct digits are reached in double precision.