Error autocorrection in rational approximation and interval estimates

TL;DR

The paper investigates error autocorrection, where intermediate perturbations in rational approximants cancel to yield highly accurate results despite ill-conditioning. It analyzes both linear and nonlinear Padé--Chebyshev methods, introduces the error approximant concept, and demonstrates through theory and extensive computer experiments how such autocorrection enables robust construction of rational approximants (via PADE and REDUCE) and challenges conventional interval estimates. By linking autocorrection to interval analysis, the work shows standard bounds can be overly pessimistic and proposes a framework for more realistic error assessment and potential convergence acceleration. Overall, the results provide practical guidance for designing and analyzing rational approximants that remain accurate under substantial internal perturbations.

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.