An algorithm for calculating the spectral intensity of radiation due to the coherent addition of many particles with arbitrary trajectories is described. Direct numerical integration of the Lienard-Wiechert potentials, in the far field, for extremely high photon energies and many particles is made computationally feasible by a mixed analytic and numerical method. Exact integrals of spectral intensity are made between discretely sampled trajectories, by assuming the space-time four-vector is a quadratic function of proper time. The integral Fourier transform of the trajectory with respect to time, the modulus squared of which comprises the spectral intensity, can then be formed by piecewise summation of exact integrals between discrete points. Because of this, the calculation is not restricted by discrete sampling bandwidth theory and, hence, for smooth trajectories, time steps many orders larger than the inverse of the frequency of interest can be taken.