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A practical error formula for multivariate rational interpolation and approximation

机译:多元有理插值和逼近的实用误差公式

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We consider exact and approximate multivariate interpolation of a function f(x 1 , . . . , x d ) by a rational function p n,m /q n,m (x 1 , . . . , x d ) and develop an error formula for the difference f − p n,m /q n,m . The similarity with a well-known univariate formula for the error in rational interpolation is striking. Exact interpolation is through point values for f and approximate interpolation is through intervals bounding f. The latter allows for some measurement error on the function values, which is controlled and limited by the nature of the interval data. To achieve this result we make use of an error formula obtained for multivariate polynomial interpolation, which we first present in a more general form. The practical usefulness of the error formula in multivariate rational interpolation is illustrated by means of a 4-dimensional example, which is only one of the several problems we tested it on.
机译:我们考虑通过有理函数p n,m 1 ,...,x d )的精确和近似多元插值> / q n,m (x 1 ,....,x d )并为差f − p <建立误差公式sub> n,m / q n,m 。与有理插值误差的众所周知的单变量公式的相似性令人惊讶。精确插值是通过f的点值进行的,而近似插值是通过以f为边界的区间的。后者允许对功能值进行一些测量误差,该误差由间隔数据的性质控制和限制。为了获得此结果,我们利用为多元多项式插值而获得的误差公式,我们首先以更通用的形式给出该误差公式。通过4维示例说明了误差公式在多元有理插值中的实际用途,这只是我们对其进行测试的几个问题之一。

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