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

Abstract

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.