Numerical treatment of a twisted tail using extrapolation methods

Abstract

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 2 -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, theD 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.

Bessel, sine and cosine functions and extrapolation methods for computing molecular multi-center integrals

“…The analytical formulae are very useful when only single integrals are to be computed, whereas from a numerical and computational point of view, recurrence relations are more efficient when evaluating a sequence of integrals as is the case in molecular structure calculation. As an example, in the calculation presented in Table 1 the use of the analytic expression (21) is faster than the use of the recurrence relations (20) by a factor 1.20, whereas in the calculation of the complete integral I n 2 ,l 2 ,m 2 n 1 ,l 1 ,m 1 presented in Table 4, the use of recurrence relations is faster by a factor 1.12 over the analytic expression. These factors may increase with an increase in the size of the calculation.…”

“…A few classes of transformations are well suited to handle these difficulties, notably the nonlinear D [16],D [17] and G [18] transformations. These transformations have proved extremely useful in numerical integration of oscillatory integrals even in the case of challenging and pathological integrals such as the Twisted Tail [19][20][21].…”

A Generalized Technique in Numerical Integration

“…Traditional quadrature rules and summation techniques have failed to provide accurate approximations to such integrals. Numerous methods and techniques were developed for improving convergence of these challenging integrals [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and extremely efficient methods were introduced such as numerical steepest descent, Filon-type and Levin-type methods. Unfortunately, their application to complicated integrals is extremely challenging.…”

The S and G transformations for computing three‐center nuclear attraction integrals

“…Recently 1, we presented highly efficient algorithms for an extremely difficult integral, called the Twisted Tail, proposed in The SIAM 100‐digit challenge, launched in 2002 by L. Trefethen, and which consists of a collection of 10 challenging projects in numerical computation. The algorithms that we developed for the Twisted Tail were based on extrapolation methods and nonlinear transformations, namely the $\bar{D}$ transformation 2, 3, the $\bar{D}$ transformation with the W algorithm 2, 4 and the G transformation which is introduced in 5, and is extended to G n in 6 and to G in 7.…”