A user-friendly method for computing indefinite integrals of oscillatory functions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:315

Article

Hasegawa, Takemitsu¹  Sugiura, Hiroshi² 

[1] Univ Fukui, Dept Informat Sci, Fukui 9108507, Japan

[2] Nanzan Univ, Dept Mechatron, Showa Ku, Nagoya, Aichi 4668673, Japan

关键词: Oscillatory integral;    Chebyshev interpolation;    Error analysis;    Uniform approximation;    Three-term recurrence relation;    Numerical stability;   

DOI  :  10.1016/j.cam.2016.10.034

来源: Elsevier

【 摘 要 】

For indefinite integrals Q(f; x, omega) = integral(x)(-1) f(t)e(i omega t) dt(x is an element of[-1, 1]) Torii and the first author (Hasegawa and Torii, 1987) developed a quadrature method of Clenshaw-Curtis (C-C) type. Its improvement was made and combined with Sidi's mW-transformation by Sidi and the first author (Hasegawa and Sidi, 1996) to compute infinite oscillatory integrals. The improved method per se, however, has not been elucidated in its attractive features, which here we reveal with new results and its detailed algorithm. A comparison with a method of C-C type for definite integrals Q(f; 1, omega) due to Dominguez et al. (2011) suggests that a smaller number of computations is required in our method. This is achieved by exploiting recurrence and normalization relations and their associated linear system. We show their convergence and stability properties and give a verified truncation error bound for a result computed from the linear system with finite dimension. For f (z) analytic on and inside an ellipse in the complex plane z the error of the approximation to Q(f; x, omega) of the improved method is shown to be bounded uniformly. Numerical examples illustrate the stability and performance of the method. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free