Error autocorrection in rational approximation and interval estimates. [A survey of results.]
Error autocorrection in rational approximation and interval estimates. [A survey of results.]
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有理近似和区间估计中的误差自动校正。
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发表时间：
2002
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通讯作者：
G. Litvinov
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作者：
G. Litvinov
误差自纠效应是指在一次计算中所有中间误差相互补偿，因此最终结果比中间结果准确得多。在这种情况下，标准区间估计（在区间分析框架中，包括所谓的 Yu. Matijasevich 的后验区间分析）过于悲观。我们将讨论函数有理逼近中出现的一种非常强的效应形式。误差自动校正效应出现在所有有效的有理逼近方法中（例如，最佳逼近、Padé逼近、多点Padé逼近、线性和非线性Padé-Chebyshev逼近等），其中逼近系数中非常显着的误差不会影响该逼近的精度。原因是有理逼近的系数中的误差不是以任意方式分布的，而是形成同一逼近函数的新有理逼近的系数集合。对该机制的理解允许通过根据近似值的形式改变近似过程来减少近似误差。给出了计算机实验的结果。误差自动校正的效果表明，在相当一般类型的某些变形下，近似函数的变化可能对被视为函数的相应有理近似值几乎没有影响（而近似值的系数可能有非常显着的变化）。因此，当对可能有良好有理逼近的函数进行变形时，相应的逼近误差会迅速增加，因此在逼近函数的小变形下，具有良好有理逼近的性质并不稳定。这个属性是“个体的”，因为它适用于特定的功能。
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 (in the framework of interval analysis including the so-called a posteriori interval analysis of Yu. Matijasevich) are too pessimistic. We shall discuss a very strong form of the effect which appears in rational approximations to functions. The error autocorrection effect occurs in all efficient methods of rational approximation (e.g., best approxmations, Padé approximations, multipoint Padé approximations, linear and nonlinear Padé-Chebyshev approximations, etc.), 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. The understanding of this mechanism allows to decrease the approximation error by varying the approximation procedure depending on the form of the approximant. Results of computer experiments are presented. The effect of error autocorrection indicates that variations of an approximated function under some deformations of rather a 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, while deforming a function for which good rational approximation is possible, the corresponding approximant’s error can rapidly increase, so the property of having 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.