Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The search for non-terminating paths within a program is a crucial part of software verification, as the detection of anfinite path is often the only manner of falsifying program termination - the failure of a termination prover to verify termination does not necessarily imply that a program is non-terminating. This document describes the development and implementation of two focussed techniques for investigating the non-termination of affine loops. The developed techniques depend on the known non-termination concepts of recurrent sets and Jordan matrix decomposition respectively, and imply the decidability of single-variable and cyclic affine loops. Furthermore, the techniques prove to be practically capable methods for both the location of non-terminating paths, as well as the generation of preconditions for non-termination. / AFRIKAANSE OPSOMMING: Sagtewareveri kasie vereis of die bewys van die beeindiging van 'n program, of die deteksie van oneindige uitvoerings. In hierdie tesis ontwikkel en implementeer ons twee tegnieke om oor die oneindige eienskap van a ene lusse te beslis. Die tegnieke wat ontwikkel word is gebaseer op konsepte soos Jordan matriksdekomposisie en herhaalde groepe wat al in die verlede gebruik is om die beeindiging van lusse te ondersoek. Die tegnieke kan gebruik word om die uitvoerbaarheid van beide een-veranderlike en sikliese a ene lusse te bepaal. Feitlik alle nie-eindige a ene lusse kan ge denti seer word en die toestande waaronder hierdie oneindige eienskap verskyn kan beskryf word.