The spectral-iterative technique, a numerical method recently developed for scattering applications, is extended to open waveguide discontinuity problems. In this method, the fields in the region near the discontinuity are calculated numerically with the spectral-iterative technique. Fields in the regions removed from the discontinuity are treated analytically using waveguide modes with unknown complex amplitudes. Field matching along the boundaries between these regions gives an iterative solution for the reflection and transmission coefficients on the guide. Use of the spectral iterative technique provides numerical efficiency due to the use of the Fast-Fourier Transform. Results are presented for transverse electric waves on a truncated dielectric slab waveguide and also for step discontinuities. Several methods for improving the convergence of the technique are discussed.