Foundations of Interval Computation

This paper reports a study of numerical computation problems from a theoretical viewpoint. It shows that computer systems are not capable of computing of real numbers correctly due to the differences between the algebraic structures of real numbers and model numbers. These two classes of numbers are not isomorphic. From this study, we have learned that there are no machine errors or computation errors. If fact, one can view it is a human mistake by putting real valued problem onto a model number platform for computation. This paper proposes use of the concept of computer model numbers to approximate rough numbers for computation. Moreover, we revise an arbitrary initial compact interval to a shortest initial closed-open model interval for ordinary interval computation. This way, we can assure that the final resulting interval will be the shortest interval and that the computation will result in the greatest precision.