Methods of controlling round-off error in one-step methods in the numerical solution of ordinary differential equations are compared. A new Algorithm called theoretical cumulative rounding is formulated. Round-off error bounds are obtained for single precision, and theoretical cumulative rounding. Limits of these bounds are obtained as the step length approaches zero. It is shown that the limit of the bound on the round-off error is unbounded for single precision and double precision, is constant for theoretical partial double precision, and is zero for theoretical cumulative rounding. The limits of round-off bounds are not obtainable in actual practice. The round-off error increases for single precision, remains about constant for partial double precision and decreases for cumulative rounding as the step length decreases. Several examples are included. (34 pages)

For scientists, the accuracy of their results is a constant concern. As the programs they write to support their research grow in complexity, there is a greater need to understand what causes the inaccuracies in their outputs, and how they can be mitigated. This problem is difficult because the inaccuracies in the outputs come from a variety of sources in both the scientific and computing domains. Furthermore, as most programs lack a testing oracle, there is no simple way to validate the results. We define a model for the analysis of error propagation in software. Its novel combination of interval arithmetic and automatic differentiation allows for the error accumulated in an output to be measurable at runtime, attributable to individual inputs and functions, and identifiable as either input error, round-off error, or error from a different source. This allows for the identification of the subset of inputs and functions that are most responsible for the error seen in an output and how it can be best mitigated. We demonstrate the effectiveness of our model by analyzing a small case study from the field of nuclear engineering, where we are able to attribute the contribution of over 99% of the error to 3 functions out of 15, and identify the causes for the observed error. / Thesis (Master, Computing) -- Queen's University, 2012-09-24 14:12:25.659