Title: Computational advantage in evaluating oscillatory integral using quadratic spline

Authors: M.P. Ramachandran; M.P. Ramachandran

Addresses: ISRO Satellite Centre (Retired), Esteem Classic, Industrial Suburb, Bangalore, 560022, India ' ISRO Satellite Centre (Retired), Esteem Classic, Industrial Suburb, Bangalore, 560022, India

Abstract: The quadratic spline is used in the conventional Levin's method to evaluate the oscillatory integral. Generally, the Levin method requires O(n³) computations and can be unstable. Here, the quadratic spline interpolation method requires solving recurrence relations of the derivatives of the given function and needs only O(n²) computations, where (n) is the number of selected nodes. The recurrence relations for large (n) are shown to be not ill-conditioned. Linear piecewise and cubic interpolation do not offer such advantages. The bound on the solution is obtained in terms of frequency. Numerical examples, including an application to a scattering problem, adequately illustrate the performance of the proposed method. They exhibit stability when the nodes are adequately large, unlike the conventional Levin method.

Keywords: oscillatory integral; Levin method; quadratic spline; stability; recurrence; ill-conditioned; interpolation; convergence.

DOI: 10.1504/IJCSM.2024.136812

International Journal of Computing Science and Mathematics, 2024 Vol.19 No.1, pp.28 - 38

Accepted: 14 Mar 2023
Published online: 22 Feb 2024 *