Gaussian quadrature rules for composite highly oscillatory integrals
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- by Menghan Wu and Haiyong Wang
- Math. Comp. 93 (2024), 729-746
- DOI: https://doi.org/10.1090/mcom/3878
- Published electronically: July 21, 2023
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Uncorrected version: Original version posted July 21, 2023
Corrected version: This paper was updated to add a corresponding author designation.
Abstract:
Highly oscillatory integrals of composite type arise in electronic engineering and their calculations are a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory of orthogonal polynomials and its nodes and weights can be computed efficiently by using tools of numerical linear algebra. An interesting connection between the quadrature nodes and the Legendre points is proved and it is shown that the rate of convergence of this rule depends solely on the regularity of the non-oscillatory part of the integrand. The second one is constructed with respect to a sign-changing function and the classical theory of Gaussian quadrature cannot be used anymore. We explore theoretical properties of this Gaussian quadrature, including the trajectories of the quadrature nodes and the convergence rate of these nodes to the endpoints of the integration interval, and prove its asymptotic error estimate under suitable hypotheses. Numerical experiments are presented to demonstrate the performance of the proposed methods.References
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Bibliographic Information
- Menghan Wu
- Affiliation: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
- Email: menghanwu@hust.edu.cn
- Haiyong Wang
- Affiliation: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China; and Hubei Key Laboratory od Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
- ORCID: 0000-0002-0360-4489
- Email: haiyongwang@hust.edu.cn
- Received by editor(s): December 27, 2021
- Received by editor(s) in revised form: January 12, 2023, April 22, 2023, and June 7, 2023
- Published electronically: July 21, 2023
- Additional Notes: This work was supported by National Natural Science Foundation of China under Grant number 11671160.
The second author is the corresponding author. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 729-746
- MSC (2020): Primary 65D30
- DOI: https://doi.org/10.1090/mcom/3878
- MathSciNet review: 4678582