The recent detection of gravitational wave (GW) signal GW150914 by the Advanced LIGO experiment has inaugurated the long-anticipated era of GW astronomy. This event saw the merger of two black holes, having roughly 36 and 29 solar masses, as well as the ringdown of the resulting 62 solar mass black hole. The energy emitted in gravitational radiation was equivalent to about three solar masses. The detection underscored the importance of theoretical models for not only isolating signal from noise, but especially for the accurate estimation of source parameters. The two-body problem in Einstein's general theory has no exact solution, and so the development of these models is highly nontrivial.;We present in this thesis a set of original results on the dynamics of the inspiral for a class of binary systems known as extreme-mass-ratio inspirals (EMRIs), comprised of a small compact object (generically a stellar mass black hole) in orbit about a supermassive black hole. Our work also has potential application to intermediate-mass-ratio inspirals (IMRIs). IMRIs are thought to be a potentially strong source for ground-based GW experiments such as Advanced LIGO/VIRGO. Though not generally a good source for the LIGO network, EMRIs on the other hand are well-suited for detection by proposed space-based detectors, e.g. eLISA. Our work particularly constitutes a program of developing computational tools, methods, and results for eccentric E/IMRIs, which are thought to be astrophysically important but are much more challenging to model theoretically compared with circular orbits.;We begin with a brief review of relevant parts of general relativity (GR) theory, followed by overviews of two prevailing approximation formalisms in GR, black hole perturbation (BHP) theory and post-Newtonian (PN) theory. Our first original result is a high-precision computation of the first-order gravitational metric perturbation using a Lorenz gauge frequency domain procedure. Next, we present a fast spectral method for efficiently evaluating source integrals in eccentric (and/or inclined) BHP problems. We then apply this method to compute, to very-high-precision (as many as 200 significant digits), the energy and angular momentum fluxes for an array of binaries of varying orbital separation and eccentricity. The results are subsequently fit to PN expansions, and we are thereby able to determine a host of previously unknown, high-order PN parameters. We conclude with a short discussion of possible future extensions of this work.
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