On the Dimensional Estimate of Rounding-Errors of a typical Computing Process

Abstract

The phenomenon of roundoff-error propagation is a well known problem in computations involving floating point arithmetic. Prominent works in the field of error analysis include (1) the error-analysis based on differential error-propagation model for computer algebra system (CAS), (2) the identification and reformulation of instability in a code generated by CAS, (3) estimating the bounds on errors in symbolic and numerical environments. The main concern in these attempts is to control error-propagation by using numerically stable code. Beside these attempts, only few efforts are made towards the theoretical understanding of the underlying process of error propagation. In this paper, we attempt to show that the roundoff-errors may propagate as a random-fractal process. We apply concepts of nonlinear time-series analysis on a series constituting successive roundoff-errors generated during the computation of Henon-map solutions. We estimate the correlation dimension, which is a measure of the fractal dimension, of the series as 5.5 ± 0.05. This low value of correlation dimension shows that the error series can be modeled by a low dimensional dynamical system.