Error autocorrection in rational approximation and interval estimates. [A survey of results.] |
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Authors: | Grigori L Litvinov |
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Institution: | (1) Independent University of Moscow B. Vlasievskii per., 11, Moscow, 121002 Moscow, Russia |
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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 (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. |
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Keywords: | error autocorrection rational approximation algorithms interval estimats |
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