This is the full text of a thesis written by James Dishaw which was presented to the Air Force Institute of Technology, Wright Patterson Air Force Base, Ohio in September 2007. The DI algorithm is an alternative to source iteration that, in our testing, does not require an accelerator. I developed a formal verification plan and executed it to verify the results produced by my code that implemented DI with the above features. A new, matrix albedo, boundary condition treatment was developed and implemented so that infinite-medium benchmarks could be included in the verification test suite. The DI algorithm was modified for parallel efficiency and the prior instability of the refinement sweep was corrected. The testing revealed that DI performed as well or faster than source iteration with DSA and that DI continued to work where DSA failed. Performance did degrade when the diamond-difference (without fixup) spatial quadrature was used. Because diamond-difference is a non-positive spatial quadrature, it can produce nonphysical negative fluxes, particularly in higher dimensions. I developed a new fixup scheme to accommodate the negative fluxes, but it did not improve performance in XYZ geometry when the scattering ratio was near unity.[Taken from Abstract]. This is in PDF format so Adobe Acrobat software is required in order to read it.