A new conjugate gradient method (the LS method) that takes into
account the effect of inexact line searches is introduced and its implementations
discussed. The connections between existing conjugate gradient methods and
the LS method are used to provide a new explanation of the difference in
efficiency between conjugate gradient methods. [Continues.]
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy at Loughborough University.