[en] Although the klystron was invented over 40 years ago, it still is quite a scientific challenge to analyze this seemingly simple device. Accurate calculation of space-charge density and particle motions are required, but there are dozens of parameters that must be optimized. Thus the analysis must be inexpensive to run, because over 100 runs typically are required to optimize a particular design. A self-consistent ring model of the klystron interaction is being developed for use as a design tool, with the initial goal of improving the LAMPF klystrons to conserve electrical energy. A mathematical model that includes large-signal effects, relativity, harmonic and fundamental cavities, extended cavity fields, and self-consistent calculations of the induced currents is being developed. The model is self-consistent because the gap fields are calculated interactively, because the currents that drive the cavities are not known until after the beam transverses the cavities. Several test cases have been run, and the basic theory and results to date are discussed