Fast and stable computation of highly oscillatory and/or exponentially decaying integrals using a Clenshaw-Curtis product-integration rule
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| Format: | Preprint |
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2025
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| _version_ | 1866910015837700096 |
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| author | Dominguez, Victor |
| author_facet | Dominguez, Victor |
| contents | We propose, analyze, and implement a quadrature method for evaluating integrals of the form $\int_0^2 f(s)\exp(zs)\, {\rm d}s$, where $z$ is a complex number with a possibly large negative real part. The integrand may exhibit exponential decay, highly oscillatory behavior, or both simultaneously, making standard quadrature rules computationally expensive. Our approach is based on a Clenshaw-Curtis product-integration rule: the smooth part of the integrand is interpolated using a polynomial at Chebyshev nodes, and the resulting integral is computed exactly. We analyze the convergence of the method with respect to both the number of nodes and the parameter $z$. Additionally, we provide a stable and efficient implementation whose computational cost is essentially independent of $z$ and scales linearly with $L$. Notably, our approach avoids the use of special functions, enhancing its numerical robustness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast and stable computation of highly oscillatory and/or exponentially decaying integrals using a Clenshaw-Curtis product-integration rule Dominguez, Victor Numerical Analysis 65D30, 65D32, 65M70 We propose, analyze, and implement a quadrature method for evaluating integrals of the form $\int_0^2 f(s)\exp(zs)\, {\rm d}s$, where $z$ is a complex number with a possibly large negative real part. The integrand may exhibit exponential decay, highly oscillatory behavior, or both simultaneously, making standard quadrature rules computationally expensive. Our approach is based on a Clenshaw-Curtis product-integration rule: the smooth part of the integrand is interpolated using a polynomial at Chebyshev nodes, and the resulting integral is computed exactly. We analyze the convergence of the method with respect to both the number of nodes and the parameter $z$. Additionally, we provide a stable and efficient implementation whose computational cost is essentially independent of $z$ and scales linearly with $L$. Notably, our approach avoids the use of special functions, enhancing its numerical robustness. |
| title | Fast and stable computation of highly oscillatory and/or exponentially decaying integrals using a Clenshaw-Curtis product-integration rule |
| topic | Numerical Analysis 65D30, 65D32, 65M70 |
| url | https://arxiv.org/abs/2503.08169 |