Stable Algorithm of Integrating Rapidly Oscillating Functions

The work describes the new regularized algorithm for computing integrals of rapidly oscillating functions allowing effectively and accurately determine the required value in the presence of stationary points. In the case where the phase function has stationary points (its derivative vanishes on the interval of integration), the calculation of the corresponding integral is still a sufficiently difficult task even for the Levin method due to the degeneracy of the resulting system of linear equations. The basic idea of regularization, described in the article, is the simultaneous modification of the amplitude and phase functions, which does not change the integrand, but eliminates the degeneracy of the phase function on the interval of integration. The regularized algorithm presented in the work is based on the Levin collocation method and describes the stable method of integration of rapidly oscillating functions at the presence of stationary points. Performance and high accuracy of the algorithms are illustrated by various examples.