Quadratic phase integration using a Chebyshev expansion

An integration algorithm is described which is particularly effective in the numerical treatment of integrands having rapidly varying phase and slowly varying amplitude. The algorithm involves approximating the phase function by a quadratic polynomial and rewriting the integrand without approximation as a slowly varying function multiplied by this quadratic phase exponential. The slowly varying function is then approximated by Chebyshev expansion and the desired integral is thus expressed as a sum of constituent integrals with integrands containing a Chebyshev polynomial multiplied by the quadratic phase factor. These constituent integrals are computed by means of LU decomposition applied to a system of linear equations with a banded coefficients matrix. Example results are presented indicating that a substantial reduction in computation time may be realized by means of this approach.