An accurate numerical method for the computation of a class of generalized cosine integrals
DOI:
https://doi.org/10.5540/03.2025.011.01.0339Keywords:
Generalized Cosine Integral, Numerical Quadratures, Asymptotic Approximations of Integrals, Double Exponential Quadrature, Numerical Methods in C++, Fractional LaplacianAbstract
We develop an efficient and accurate method to compute the quadrature of an oscillatory integral arising in the discretization of the fractional Laplacian operator. The complete mathematical development is presented. The implementation, performed in modern C++, is provided as open-source software and proves to produce results with accuracy up to a few ulps.
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References
IEEE Std 754-2008. IEEE Standard for Floating-Point Arithmetic. Standard. Institute of Electrical and Electronics Engineers, 2008, pp. 1–70. doi: 10.1109/IEEESTD.2008.4610935.
M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. 9th rev. ed. Dover Books on Mathematics. New York: Dover Publications, 1965. isbn: 9780486612720.
D. H. Bailey, K. Jeyabalan, and X. S. Li. “A comparison of three high-precision quadrature schemes”. In: Experimental Mathematics 14.3 (2005), pp. 317–329. Available at: http://eudml.org/doc/52168.
DLMF. NIST Digital Library of Mathematical Functions – 5. Gamma Function – Properties. Release 1.1.12 of 15/12/2023. Online. Visited on 26/03/2024. url: https://dlmf.nist.gov/5.9.
DLMF. NIST Digital Library of Mathematical Functions – 8. Incomplete Gamma and Related Functions – Related Functions. Release 1.1.12 of 15/12/2023. Online. Visited on 26/03/2024. url: https://dlmf.nist.gov/8.21.
D. Goldberg. “What every computer scientist should know about floating-point arithmetic”. In: ACM Computing Surveys 23.1 (1991), pp. 5–48. doi: 10.1145/103162.103163.
Y. Huang and A. Oberman. Finite difference methods for fractional Laplacians. Nov. 2016. arXiv: 1611.00164 [math.NA].
Boost org. Boost C++ Libraries 1.84.0 – Math Library – Special Functions – Hypergeometric pFq. Online. Visited on 26/03/2024. url: https://www.boost.org/doc/libs/1_84_0/libs/math/doc/html/math_toolkit/hypergeometric/hypergeometric_pfq.html.
Boost org. Boost C++ Libraries 1.84.0 – Math Toolkit 4.1.1. Online. Visited on 26/03/2024. url: https://www.boost.org/doc/libs/1_84_0/libs/math/doc/html/index.html.
E. Y. Remez. “Sur la détermination des polynômes d’approximation de degré donnée”. In: Communications of the Kharkov Mathematical Society 10.196 (1934), pp. 41–63.
H. Takahasi and M. Mori. “Double Exponential Formulas for Numerical Integration”. In: Publications of the Research Institute for Mathematical Sciences 9.3 (1973), 721–741. doi: 10.2977/prims/1195192451.
