The purpose of this thesis is to study the factors involved in determining a most efficient method for the numerical integration of the differential equation x' = f(t,x) . By "a most efficient method" we mean a method requiring a minimum of computation to obtain a solution within prescribed error bounds. We outline two computational procedures and derive estimates for the propagated error of a general multi-step method when based on either procedure. These estimates, lead us to conclude that a stable single-iterate procedure, involving one evaluation of f at each step, will determine a solution most efficiently. In particular, this procedure based on Adams formulas is recommended. Experimental results support our conclusion in all stable cases. However, these results also indicate that the role of stability in the choice of a most efficient procedure is in need of further investigation. / Science, Faculty of / Mathematics, Department of / Graduate