Error autocorrection in rational approximation and interval estimates

Abstract

The error autocorrection effect means that in a calculation all the intermediate errors compensate each other, so the final result is much more accurate than the intermediate results. In this case standard interval estimates are too pessimistic. We shall discuss a very strong form of the effect which appears in rational approximations to functions, where very significant errors in the approximant coefficients do not affect the accuracy of this approximant. The reason is that the errors in the coefficients of the rational approximant are not distributed in an arbitrary way, but form a collection of coefficients for a new rational approximant to the same approximated function. Understanding this mechanism allows to decrease the approximation error by varying the approximation procedure depending on the form of the approximant. The effect of error autocorrection indicates that variations of an approximated function under some deformations of rather general type may have little effect on the corresponding rational approximant viewed as a function (whereas the coefficients of the approximant can have very significant changes). Accordingly, deforming a function for which good rational approximation is possible may lead to a rapid increase in the corresponding approximant's error. Thus the property of having a good rational approximation is not stable under small deformations of the approximated functions: this property is "individual", in the sense that it holds for specific functions. Results of computer experiments are presented.