A method is presented for calculating pointwise average cross sections in the unresolved region. This method can assess the effects of interference and overlap up to the fourth decimal place in cross-section calculations, having the capability of considering the effects of interference among sequences of resonances and of overlap between resonances in a sequence up to third order in each. Thus, the method, while more sophisticated than justified in view of current cross-section data uncertainties, can be used to evaluate the validity of various approximations currently made in the determination of cross sections for use in Doppler calculations. The effects of the level of approximation on cross-section calculations are given, Hwang's method for calculating the overlap in a single sequence is assessed, and the range of validity of Hwang's method and the approximations used in it are evaluated. Results of studies to determine the relative speed and accuracy of methods for calculating the J integral, performing the statistical average, and computing the complex probability integral, W, are also presented. The effects of interference among sequences are shown to be small for the calculation of cross sections and the Doppler change in cross sections over a wide range of composition. The effects of resonance overlap within one sequence, however, may be significant for the calculation of both cross sections and Doppler changes in cross sections in some critical assemblies.