New methods for evaluating the validity of the results of mathematical computations

From a mathematical standpoint, numerical methods can be divided in two classes: (i) Direct methods, which give an exact result after some finite number of computations, and (ii) approximate methods, which only give an approximate result after any finite number of computations.

When these numerical methods are carried out on a computer, they are projected into discrete space and performed with a limited precision arithmetic. Consequently they yield approximate results. A result given by a direct method contains only a computing error while a result given by an approximate method contains both a computing error and a method error. It is absolutely necessary to evaluate these errors. In this paper, we present new methods for estimating these errors and for evaluating the exact number of significant decimal digits appearing in computed results. Also presented here is a general methodology for improving the precision of computed results.