Work

Fast and stable computation of highly oscillatory and/or exponentially decaying integrals using a Clenshaw-Curtis product-integration rule

Year: 2026

Type: article

Abstract: We propose, analyze, and implement a quadrature method for evaluating integrals of the form $$\int _0^2 f(s)\exp (zs)\, \textrm{d}s$$ , 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 the number of Chebyshev nodes. Notably, our approach avoids the use of special functions, enhancing its numerical robustness.

Source: Numerical Algorithms

DOI: 10.1007/s11075-026-02333-8

Author: Víctor Domínguez

Institutions: Universidad Pública de Navarra (UPNA), Institute for Advanced Materials and Mathematics

Language: English

Topic: Mathematical functions and polynomials

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,290

Funder: Universidad Pública de Navarra