Extension of a class of periodizing variable transformations for numerical Integration
Extension of a class of periodizing variable transformations for numerical Integration
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数值积分的一类周期变量变换的扩展
DOI：
10.1090/s0025-5718-05-01773-4
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发表时间：
2005
影响因子：
2
通讯作者：
A. Sidi
中科院分区：
文献类型：
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作者：
A. Sidi
本文引入并研究了有限域积分数值计算中的一类具有m个整数的S m变量变换[A.Sidi，一种新的用于数值积分的变量变换，数值积分IV，1993(H.Brass和G.Hammerlin编)，第359-373页]。这类变换的一个代表是用于多维积分的格规则的sin m变换。这些变换使被积函数周期化，使得梯形规则能够达到很高的精度，特别是在偶数m的情况下。在本工作中，我们将这些变换推广到任意值m，并对所得到的变换梯形规则近似给出了详细的分析。我们表明，在适当的m下，它们在不同的情况下可以非常有用。例如，我们证明了如果被积函数在积分加在端点处为零的区间上光滑，则取2m为奇数可得到精度特别高的结果。这种情况通常可以通过从被积函数中减去积分区间端点处的线性内插式来实现。通过扩展的Sin-m变换，我们用数值例子说明了一些结果。
Class S m variable transformations with integer m, for numerical computation of finite-range integrals, were introduced and studied by the author in the paper [A. Sidi, A new variable transformation for numerical integration, Numerical Integration IV, 1993 (H. Brass and G. Hammerlin, eds.), pp. 359-373.] A representative of this class is the sin m -transformation that has been used with lattice rules for multidimensional integration. These transformations periodize the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with even m. In the present work, we extend these transformations to arbitrary values of m, and give a detailed analysis of the resulting transformed trapezoidal rule approximations. We show that, with suitable m, they can be very useful in different situations. We prove, for example, that if the integrand function is smooth on the interval of integration add vanishes at the endpoints, then results of especially high accuracy are obtained by taking 2m to be an odd integer. Such a situation can be realized in general by subtracting from the integrand the linear interpolant at the endpoints of the interval of integration. We also illustrate some of the results with numerical examples via the extended sin m -transformation.