Chapter 1 will more formally introduce and explain shadowing. Chapter 2 is the main body of the work, which introduces the optimizations to the shadowing algorithm of Quinlan and Tremaine , and talks about high-dimensional shadowing in general. Chapter 3 contains some preliminary results on high-dimensional shadowing that will be of interest both to practicing N-body astronomers, and to those interested in shadowing in general. Chapter 4 contains descriptions of the many open questions and further avenues of research in this area. The appendix contains a review of the basic N-body equations and the derivation of the Jacobian of an N-body system in which M particles move amongst N-M fixed ones.