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.
Efficient estimation of eigenvalue counts in an interval
Jülich Supercomputing Centre, Forschungszentrum Jülich（于利希超级计算中心，于利希研究中心）；Department of Electrical and Computer Engineering, University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校电气与计算机工程系）；Computer Science & Engineering, University of Minnesota, Twin Cities（明尼苏达大学双城分校计算机科学及工程学系）
Rational minimax approximation via adaptive barycentric representations
Univ Rennes, Inria, CNRS, IRISA（雷恩大学、Inria、CNRS、IRISA）；National Institute of Informatics（信息学国立研究所）；Mathematical Institute, University of Oxford（牛津大学数学研究所）；Laboratoire Paul Painleve UMR 8524, Dept. Mathématiques, Univ. Lille（保罗·潘勒瓦实验室UMR 8524，数学系，里尔大学）
