Analysis of rounding errors in large systems of nonlinear equations

The behavior of rounding errors in computing complicated functions is analysed and the adequacy of the probabilistic model introduced to estimate the rounding errors is discussed, where the technique of Fast Automatic Differentiation plays an essential role and the rounding error of a function is regarded as an instance of the weighted sum of uniformly distributed independent random variables. A system of nonlinear equations with 108 variables, which represents a chemical plant, is investigated as an example. The estimated statistical parameters of the rounding errors given by the model are seen in good agreement with the observed ones. A one-parameter family of random variables defined by the weighted sum of an infinite number of uniformly distributed independent random variables with a geometrical progression as the weights is also introduced, which is shown to be a simple but plausible model for the distribution of the rounding errors in computing complicated functions. [In Japanese.] An English translation is available under the same title as Research Memorandum RMI 89-02, Department of Mathematical Engineering and Information Physics, Faculty of Engineering, University of Tokyo, Japan, 1989.