For pt. II see Technical Report No. 99-423, Dept. of Comput. and Inf. Sci., Queen's University, Kingston, Ontario, May 1999. Parallel computers can do more than simply speed up sequential computations. They are capable of finding solutions that are far better in quality than those obtained by sequential computers. This fact is demonstrated by analyzing sequential and parallel solutions to numerical problems in a real-time paradigm. In this setting, numerical data required to solve a problem are received as input by a computer system, at regular intervals. The computer must process its inputs as soon as they arrive. It must also produce its outputs at regular intervals, as soon as they are available. We show that for some real-time numerical problems a parallel computer can deliver a solution that is significantly more accurate than that computed by a sequential computer. Similar results were derived recently in the areas of real-time optimization and real-time cryptography.